UNIVERSITY OF CALIFORNIA GIFT OF Col. Glen F. PE.PRRTrQE.1MT DOCUMENT NO. ZO35 THEORY AND DESIGN OF RECOIL SYSTEMS AND GUN CARRIAGES Prepared in the Office of the Chief of Ordnance SEPTEMBER, REPRODUCTION PUANT WA5H\NQTON BARRRCKS O.C. -rinf Lilrary U.F ORDNANCE DEPARTMENT Document No. 2035 Off ic of the Chief of Ordnance WAR DEPARTMENT, Washington, October 1921. The following publication entitled "Theory and Design of Recoil Systems and Gun Carriages" is pub- lished for the information and guidance of all students of the Ordnance training schools, and other similar educational organizations. The contents should not be republished without Authority. By order of the Secretary of C. C. WILLIAMS, MAJOR GENERAL, CHIEF OF ORDNANCE. Jjlraiy FOREWORD This edition is published in its present form with lioeral margins and spacing so that corrections or additions may be freely made. In a document of this kind it is almost inevitable that ambiguities, errors and misstatenents will appear, and it is only in extended and repeated use that these are fully exposed. It will therefore be appreciated if all those to whom this volume comes and who use it critically will forward criticisms, corrections or necessary additions to the Artillery Division, Ordnance Office, Washington, D. C., so that these nay be incorporated in the master volume. After such changes have been received for a suitable period, it is expected to have the text printed in usual book form. PREFACE Although strictly artillery design may be considered a highly specialized branch of machine design, there are so many features that differentiate this work from ordinary machine design, it has been felt that a volume covering the specialized points is of fundamental importance in order that our designing engineers may have in a readily accessible from reference data covering the general subject and in particular those features of modern development not now covered in published works. Such is the purpose of this volume. Artillery design may be subdivided into the design of cannon, and the design of gun carriages and recoil systems. During the late war the ex- tensive introduction of self propelled gun mounts, such as caterpillar vehicles, has introduced automotive problems in the design of these types of gun mounts in addition to the ordinary consideration affecting design of gun mounts. Further in the design of artillery we have three important aspects, - (l) the technical and theoretical considerations of a design, (2) the fabrication, standardization and production features, and (3) the service and field require- ments to be fulfilled. All three aspects are equally important and a successful design results only from a balanced consideration of the three. This discussion has been written under the auspices of Colonel G. F. Jenks, Chief of the Artillery Division, Ordnance Department, U. S. A. and of Colonel J. B. Rose, Chief of the Mobile Gun Carriage Section of that Division. Effort has been made to arrange systematically in a form for reference tne great quantity of engineering data in the files of the office. In order to develop and analyze this data, it has been necessary to introduce a considerable number of original discussions and deductions. The work is an attempt to cover only the * technical aspect of the design of gun carriages and recoil systems. The fabrication and field service phases, though of course inherently coordinate in. a design are subjects of such com- plexity and broadness that they require for their full appreciation a separate treatment. These aspects have therefore necessarily been entirely onitted, except in so far as they are directly connected with the technical features involved. Acknowledgement and thanks are especially due to Colonel J. B. Rose, who has proof read the complete work in the view of bringing the data into conformity with the practice and standards of the Ordnance Department. It should be stated, however, that this has been done only to the degree which was found possible without destroying original con- clusions and discussions or without alteration of the system of nomenclature used. The latter is in partial but not complete agreement with the most general practice. Further acknowledgement and thanks for suggestions on the various parts of the work are due to: - Mr. D. A. Gurney, Ordnance Engineer, Mobile Gun Carriage Section, Artillery Division. Prof. B. V. Huntington, Professor of Mathematics and Mechanics, Harvard University Professor C. E. Fuller, Professor of Ap- plied Theoretical Mechanics, Massachusetts Institute of Technology. Professor G. Lanza, Professor Emeritus in charge of Mechanical Engineering Department, Massachusetts Institute of Technology. Acknowledgement of assistance on the Computation work is due to Mr. E. V. B. Thomas, Mr. Kasargian and Mr. McVey of the Artillery Division, also to Messrs. Murray H. Resni Coff and 0. L. Garver for preparing this data for publication. RUPBN BKSERGIAN, Formerly Captain, Ordnance Dept.U.S.A, Chapter I Chapter II Chapter III Chapter IV Chapter V Chapter VI Chapter VII Chapter VIII Chapter IX Introduction and Elements - Types of Cannon and Carriages - Classifi- cation of Carriages and Recoil Systems. Dynamics of Interior Ballistics as Af- fecting Recoil Design - The Para- bolic Trajectory. External Reaction on Carriage during Recoil - Stability - Jump. Internal Reactions throughout a Carriage during Recoil and Counter Recoil. Gun Hydraulic Principles as Applied to Various Systems of Recoil and Counter Recoil - General Theories on Orifices and Flow of Oil. The Dynamics of Recoil and Counter Re- coil - Differential Equations of Resistance, Braking, etc. and of Velocity - General "Formulas for Recoil and Counter Recoil, Classification of Recoil Systems - Derivation of General Formulas for Design and Computation - General Linitations, etc. Hydro-pneumatic Recoil Systems. Hydro-pneumatic Recoil Systems (Continued) 8 Chapter X Railway Gun Carriages. Chapter XI Gun Lift Carriages. Chapter XII Double Recoil Systems. Chapter XIII Miscellaneous Problems Discussions of Various Types of Carriages. CHAPTER I TYPES AND PRINCIPAL ELEMENTS OF CANNON AND CARRIAGES. The fundamental principles of gun carriage design are entirely the same as those of engine and machine design, and it is the object of this volume merely to bring out the specific ap- plication of these principles to the design of gun carriages . A gun carriage is a machine exercising primarily the following functions: (1) To provide a fixed firing platform which dissipates the energy given to the recoiling parts in reaction to the energy imparted to the projectile and powder gases. (2) To return the recoiling parts to their initial position for further firing. (3) To provide the mechanism for elevat- ing the gun for different ranges and angles of site, and for traversing the gun for changes in the direction of fire. The effect of allowing the gun and a part of the carriage to recoil is to reduce many times the stresses in the carriage and to maintain its equilibrium. A properly designed recoil system will give reactions consistent with the strength and stability of the carriage, and a smoothness of action which is essent- ial for long service and accuracy. The success of one design over another is due to perfection of many de- tails, which insures smooth action and long service and to a judicious compromise between many opposing conditions and requirements. To approach the study of carriage design, it is necessary to know the elements of interior ballistics and the characteristics of guns for meeting different 9 10 ballistic conditions in so far as these affect the form of carriage and determine the forces acting upon it. These subjects will, therefore, be briefly considered, but a complete discussion must be obtained from works treating them specifically. From one view point a cannon may be considered as a tube of proper thickness for strength, having a chamber in the rear of somewhat larg- er diameter which contains the powder charge. The powder charge is inserted by opening a breech block in the rear end of the cannon. This breech block necessarily must withstand the maximum powder pressure over its cross section and a power- ful locking device is therefore needed. The details of this mechanism are complicated, but need not be considered in carriage design, except in special cases where the breech mechanism is operated during counter recoil. The design of the rifling grooves and capacity of powder chamber will be considered later. The elements of a gun are shown in figure (1). "A" is the powder chamber, "B" the rifled portion of the bore, "C" the breech block, "D" the gun lug for the attachment of piston rods, which restrain the gun in recoil. * V- il \ . -L fV ^ The caliber of a gun is the diameter of the bore and is expressed usually in millimeters or inches. Speaking very roughly, small guns range from 37 m/m 11 to 75 m/m, and are suitable for mounting on aeroplanes or for use with infantry. Light field guns range from 75 m/m to 105 m/m. Ordinary medium artillery ranges from 105 m/m to 8 inches* Heavy artillery ranges from 8 inches upward. The above classification refers to mobile field materiel only. PQWTT7FRS AND flUNS Carriages are designed for either howitzers or guns. Howitzers are for high angle fire, the striking angle being generally above 25 de- grees. They have a medium or low muzzle velocity. A gun is designed for range and, there- fore, has a high muzzle velocity. The angle of elevation of howitzers is usually between 20"* and 70 a nd the muzzle velocities from 400 to 1800 feet per second. The angle of elevation of a gun is usually from minus 5 degrees to plus 45 degrees with muzzle velocities ranging from 1700 to 3000 feet per second. The angular velocity of the projectile is also considerably higher for a gun than for a howitzer. In modern practice the line of demarcation between guns, howitzers and mortars has become somewhat less distinct, and we may consider all of them as cannon which decrease in power in the order named and generally for use at elevations which increase ia the order named. Against aircraft, firing is at elevations from to 80 degrees, hut the muzzle velocity is high; hence, the pieces used in such work are properly classified as "guns". The traversing limitations of a gun and howitzer may be the same or different but do not enter in the differentiation between a gun and howitzer. The muzzle velocity of howitzers being lower than that of guns, it is possible with the same total weight of materiel to fire a much heavier projectile. 12 BECOIU BG_PABT-g The recoiling parts consist of the gun together with the various parts attached to it and recoiling with it. We have two methods of arrangement of recoiling parts: (1) the piston rods with their pistons attached to the gun lug and recoiling with the gua. (2) the pistons and their rods held stationary. So far as the recoil mechanism is concerned we are only concerned with the relative motion between the rods and pistons and their cylinders. The greater part of our guns in the service translate in recoil directly along the axis of the bore, others as on certain Barbette mounts and double recoil systems have a translation in addition to that along the axis of the bore. Guns on Dis- appearing carriages and special other types have rotation in addition to translation. In ordinary recoil systems the center of gravity of the recoiling parts is usually located slightly below the axis of the bore. This insures a positive jump (muzzle up) during the powder pressure period. If the center of gravity of the recoiling parts is great- ly below the axis of the bore considerable stresses are brought upon the elevating rack and pinion, due to the fact that the powder pressure causes an ex- cessive turning effect about the trunnions the amount depending also upon the location of these. For this reason when the cylinders recoil with the gun, extra weight is very often introduced on the top of the gun. This, of course, raises the center of gravity of the recoiling parts nearer the axis of the bore. The recoiling parts are constrained to recoil parallel with the axis of the bore by gun clips en- gaging in guides in a fixed cradle or by the gun itself sliding in a fixed cylindrical sleeve. Due to the fact that the braking forces developed in the 13 cylinders are usually considerably below the axis of the bore during recoil, considerable pinching action takes place at the front and rear clip contact with the guides. This causes somewhat greater friction than would be obtained by mere sliding friction. The clips attached to the recoiling parts, or rather to the gun itself, which in turn engage in the guides of the cradle, are usually either continuous or three to four in number. In order to maintain a con- stant friction throughout the recoil, clips should be evenly spaced along the gun and the front clip should engage in the guides before the rear clip leaves the guides. When the gun recoils in a sleeve or cy- linder which is a part of the cradle, it is conetiroes possible to distribute the various pistons and cy- linders symmetrically about the axis of the bore. As we shall see, this decreases the friction during the recoil and counter recoil. Figure (2) shows the recoiling mass where the pistons and their rods recoil with the gun. Below in figure (3) is shown a recoiling mass consisting of the cylinders grouped together in a single forging in a so-called slide or sleigh, and rigidly attached to the gun. 14 . 3 nt_CRADLE_ The cradle serves as a constrain- ing member for the sliding of tlie gun to the rear in recoil and as a support for elevating the gun. The cradle and the recoiling parts together are known as the tipping parts and turn about horizontal trun- nions fixed to the cradle and resting in bearings in the top carriage. To elevate the tipping parts, an elevating arc bolted to the cradle engages in a pinion fixed to the top carriage, or vice versa. The cradle and therefore the tipping parts are supported in the top carriage at two points: (1) at the trunnions and (2) at the tooth contact of the elevating arc and pinion. See figure (4). IS ng. 4- When the cylinders do not recoil they are in turn an integral part of the cradle, and therefore, the recuperator forgings and the cradle are one and the same. A sleigh may or may not he interposed between the gun and cradle. With guns, where the cylinders recoil with gun, the cradle merely serves the purpose of a constraining guide for the recoiling parts and rigid- ly attached to it are the piston rods and their pis- tons . TJPPIflG_FABT The term"tipping parts" applies to those parts of a carriage which move in the process of elevating the gun. In order to rapidly elevate the gun, it is considered very important that the tipping parts are nicely balanced about the trunnions. Thus the center of gravity of the tipping parts must be located at the trunnions. As the height of the trunnions and axis of the bore are governed by stability at horizontal elevation, clearance in traveling and accessibility for leading, the length of recoil at maximum elevation becomes limited. If a minimum elevation of about 20 degrees is allowed for a howitzer, we might raise the trunnions, thereby in- 16 crease the length of recoil, and thus maintain stability. When, however, a gun must fire at high ele- vation as in antiaircraft materiel, or when a carriage serves the double purpose of supporting a gun or a howitzer at high elevations, the maximum possible re- coil at maximum elevation becomes greatly limited. The recoil displacement at maximum elevation may be increased most satisfactorily by placing the trunnions to the rear and introducing a balancing gear for balancing tbe tipping parts about the trunnions. The balancing gear usually consists of an os- cillating spring or pneumatic cylinder, the trunnions of which rest in bearings in the top carriage, the end of the piston rod being attached to the cradle. Since it is difficult to obtain perfect balance by this method throughout the elevation, the maximum unbalanced moment in the process of elevation should be considered in the design of the elevating gear mechanism. A method by which exact balance can be maintained throughout the elevation is obtained by use of a cam and chain connecting tbe cradle with the spring or pneumatic cylinder. In this case the cam is fixed to the cradle and the spring cylinder to the top carriage. However, due to the variation in trunnion friction and other similar factors the former method is probably better since a very close approximation in balance throughout the elevation can be obtained. The reaction on the elevating arc and the trunnion reaction are modified by the introduction of tbe balancing gear, though ordinarily where the weight of tipping parts is relatively small as com- pared with the recoil reaction the effect of the balancing gear on the reactions may be neglected. When it is desired to use an independent line of sight, a rocker is introduced between the elevating pinion and cradle. The rocker, when moving, is a part of the tipping parts. In the process of 17 elevating the gun an elevating pinion rotates the rocker about the trunnions until the proper line of sight is obtained; the cradle is then brought into its proper position by gearing con- necting the rocker and cradle. TOP CARRIAGE The top carriage serves as an. intermediary piece connecting the tipping parts with the bottom car- riage, or in semi-fixed mounts, with the bottom platform. The top car- riage is supported at its bottom by a vertical pintle block and circular traversing clips. At the top it supports the tipping parts on its trun- nion bearings and elevating pinion bearing. The top carriage -together with the tipping parts are known as the traversing parts. To traverse the gun, the top carriage with the tipping parts are rotated in a horizontal plane about the pintle block by a circular traversing reck and pinion or worm gear. In certain types of field artillery the top carriage is an integral part of the trail, in which case traversing is obtained with respect to the wheels and axle by moving the trail along the axle and about the spade point as a pivot. Traverse by this method is naturally very limited as com- pared to traverse with a rotating top carriage. All stationary mounts or field platform mounts have a separate top carriage which serves this specific function of traversing about the vertical pintle support. In very large carriages the top carriage is supported by a circular ring of horizontal rollers, the pintle bearing merely serving as a constraining pivot. In certain types where the bottom carriage itself is traversed, the top carriage is used for translation only. It is then supported on rollers moving along an inclined or horizontal plane and the braking is affected by a recoil cylinder in the top carriage which 18 connects the top carriage with the bottom carriage through the piston rods. Fig. 5 Top carriages way be roughly classified into: (1) the ordinary type of side frames connected at the front or rear by cross beams or transoms which contain the pivot bearing. (2) pivot yoke type used on small mobile mounts and (3) trail carriages. The ordinary side frame type of top carriage is extensively used on the stationary mounts on mobile platform mounts and even on trail supported carriages. The pivot yoke type is especially useful when split trails are introduced, since it supports the equalizer bar for balancing the distribution of the load between the two trails. 19 TRAIL AND SPADE With mobile field artillery it is customary to use a trail and spade for the double purpose of preventing a backward motion of the carriage on firing of tbe gun, and of giving sufficient stability to the carriage in order that the wheels may not leave the ground. We have two classifications of trails, - (1) the single or box trail, (2) the split trail. With a single trail it is necessary to have a large U- shaped aperture or fork arrangement at the forward end in order to elevate, load and traverse the gun without interference. When split trails are used we have two separate single trails which may turn at the wheel ends about the axle. It is customary with the split trails to introduce an equalizing mechanism which connects the two trails and distributes the load between the trails on firing. The spade and float support tbe trail and are designed to take up the horizontal and vertical re- actions at the rear end. In tbe design of the spade and floats it is important that tbe unit bear- ing pressure be held to a low value. This should not be more than about 30 Ib. per sq. in. for the float and 40 Ib. per sq. in. for the spade. For wide traverse of the gun it is necessary to lift the spade from the ground and turn the carriage to the desired line of fire. For this reason the static load on the spades should not ex- ceed about 100 Ibs. for light carriages. Thus in a preliminary lay-out of tbe carriage, it is necessary to locate the center of gravity of tbe total system in battery very close to the axle in order that the static load under the float does not exceed the desired amount. This inherently makes the counter recoil stability in battery very small especially at horizontal recoil and requires con- siderable care in the design of a counter recoil 20 system. At horizontal elevation the carriage is usually designed with a very small margin of stability. Therefore, in firing the vertical load on the float practically equals the weight of the total system. The bending moment in the trail gradually increases from the spade toward the wheel axle. We have max- imum bending moment at the attachment of the trail to the top carriage or wheel axle. PLATFORM MOUNTS. With fixed mounts and heavier types of field artillery it is customary to support the travers- ing parts on a platform, that is, the top carriage rests upon a platform which serves as a bottom carriage. When a platform bottom carriage is used, it must be either bolted to a concrete foundation as in fixed mounts or else it must have a vertical projection similar to a spade on a field carriage to take up the horizontal reaction in firing. Further, the bearing surfaces of this platform must be suf- ficient to prevent overturning of the carriage firing at low angles of elevation or change in level in firing at any elevation. That is, the center of pressure of the reaction of the earth must be within the middle third of the length or diameter of the platform in the line of fire. Since platform mounts vary considerably in con- struction of detail no attempt will be made to catalogue the various types used. With fixed mounts the bottom carriage or plat- form is usually secured to a concrete foundation by a distribution of bolts along a circular flange; and since with fixed mounts all round traverse is possible, each bolt should be designed for maximum tension. 21 DC 22 CATERPILLAR MOUNTS. To increase mobility during the World War, cater- pillar mounts were developed extensively. A caterpillar mount consists of an ordinary gun mount including the tipping parts and top car- riage mounted on a bottom carriage which fits with- in the frame of the caterpillar. The caterpillar is propelled by its own engine, and traverse can be readily made by keeping one of the caterpillar tracks stationary and moving the other. For more delicate traversing the top carriage is provided with limited traverse about the bottom carriage. The essential features of the caterpillar proper are: (1) The frame which supports the bottom carriage and the principal bearings for the driving mechanism. The caterpillar frame in turn is generally supported on a series of roller trucks which travel on the caterpillar tracks. Between the roller trucks and caterpillar frame, spring supports are usually provided, and the roller trucks are built to have more or less up and down movement at their ends to conform with the contour of the ground. The frame may be either a casting or built up of structural steel. The structural steel frame is perhaps lighter but more subject to objectional de- flections . ., The reactions on the frame consist of the various spring supports from the supporting roller trucks, the reactions of the bearings of the running gear and the reactions of the gun mount transmitted by the bottom carriage to the frame on firing. 23 The frame of a caterpillar is subjected to a complicated system of stresses. Due to various possible loading conditions during traveling such as the entire weight of the caterpillar being carried in the center or else at the ends, we have different types of loading reactions. Further a wrenching action with corresponding large transverse, stresses are induced by the supporting reactions being on either side at the further extremities of the track. This requires considerable lateral bracing. In fact outside of fabrication and con- struction considerations, the design of the cater- pillar frame should be based on a careful analysis of the various types of supporting reaction com- binations that may take place in the traveling of the caterpillar. It will be usually found that the traveling stresses are somewhat greater than the firing stresses and are often of an opposite character. The driving mechanism of the caterpillar con- sists of two tracks each consisting of a continuous track or belt of linked shoes. The caterpillar track is driven, by sprockets usually at the rear end. The drive shaft contains at one end the track sprocket, and at the other end the drive sprocket gear, which meshes by a suitable gearing to a clutch, the system of gearing &nd clutch being symmetrically the same for either track. The clutches are driven by bevel gears or other forms of reduction gearing through a gear box, and sometimes a master clutch, to the engine crank shaft. The traction gearing is straight forward and is very similar to other types of drive gear transmission. Mechanical steering is obtained by operating either the right or left track, holding it stationary or sometimes reversing the motion and running the track backward. 24 Electric drive caterpillar mounts are in two (2) units and possess certain advantages'; first, the transmission can be greatly reduced in either unit by the use of compact motors a_nd gearing'; second, the units can be made similar and the mobility thereby increased; third, a better design cf gun mount is possible due to less limitations on clearance and other corresponding factors. The electric drive consists of the gun mount unit and the power plant unit. The power plant unit sup- plies power for driving itself, as well as the gun mount; fourth, the caterpillar is braked in traveling by suitable band brakes in the trans- mission. When, however, the gun is fired, it is necessary to brake the caterpillar from running back. The braking and torque being usually in an opposite direction and necessarily of a large value as compared with the traveling braking 1 ; it is usually customary to introduce a band brake on the final drive shaft and thus eliminate the stresses in the transmission during firing. The braking should be designed to produce a traction reaction equal to approximately 80 percent of the total caterpillar. Fifth, in a design of caterpillar mounts, stability is of prime im- portance due to the limited wheel base and necessity of maintaining as light a mount as possible. Stability may be increased by the use of outriggers attached to the caterpillar body. To decrease the overturning reaction of the recoil on firing and thus increase the stability, double recoil systems have been successfully introduced on larger caterpillar guns. A double recoil system consists of an ordinary recoil system between the gun and cradle of the top carriage and a lower recoil system between the top carriage and frame. The top carriage is designed to roll up an inclined plane of sufficient elevation to bring the recoiling masses into battery and the cate-rpillar lies in a r k tf 26 horizontal plane. This elevation is usually at from 6 to 7 degrees. By the use of double re- coil systems fhe stability is greatly enhanced, since the inertia resistance of the top carriage creates a stabilizing moment which is added to the inertia resistance of the upper recoiling parts. In the design of the double recoil system cater- pillar mount, it is highly desirable that the top carriage recoil as far as possible up the inclined plane. Due to less limitations and clearance, an electric drive of the two supporting units offers a very suitable gun mount and a long recoil of the lower recoil system is usually possible. Fig. 6 RAILWAY MOUNTS Railway mounts developed during the late war consist of three (3) systems: (1) these where the car mounted on suitable tracks, rolls back on firing;(2). those sliding back on a special track the tracks being disengaged, (3) platform or stationary railway mounts with suitable outriggers, the 27 trucks being entirely disengaged. In types (1) and (2) a very limited traverse is possible, whereas, in type (3) considerable amount of traverse is possible. Railway mounts of type (1), rest upon suitable girders, supported by the trucks at either end. The girder must be designed to carry the maximum firing load stresses at maximum elevation, as well as stresses due to the dead load weights. The trucks take the supporting reactions from the girders of the dead weight load as well as fhe firing load at maximum elevation. Great care is needed in dis- tributing the loading from the various axles by properly formed truck equalizers. In type (2) a special built-up track is necessary, the trucks being disengaged merely carrying the dead weight of the mount. The mount is designed to have a considerable bearing surface, and thereby the bearing pressures are greatly reduced In sliding railway types, recoil systems have in certain types been completely eliminated, the recoil being merely resisted by the friction of the track. Due, however, to the enormous stresses due to high caliber guns at maximum elevation, recoil systems should always be introduced. With stationary or platform mounts the question of stabilizers of corresponding outriggers become a fundamental feature in this type of design. Platform railway mounts have similar characteristics as ordinary field platform mounts in mobile artillery. 28 rig. 9 Fig. 10 CHAPTER II DYNAMICS OF INTERIOR BALLISTICS AS AFFECTING RECOIL DESIGN. The object of interior ballistics is partly to derive expressions for the acceleration and velocity of the projectile during the travel in the bore, and the corresponding pressures on the base of the shell and breech in terms of tne powder loading, the form of powder grain, the initial volume of powder chamber in the gun, and other variables upon which the velocity and pressure depend. In the design of the recoil mechanism as well as the carriage for its maximum stresses, it is very important to know the accelerations, velocities, and pressures in the gun to a considerable degree of accuracy throughout the time the powder gases act. In the study of interior ballistics, it is con- venient to divide the powder pressure interval into two periods: (1) The interior period while the shot travels up the bore to the muzzle. (2) The after effect period while the powder gases expand after' the shot has left the muzzle. During the interior period, we have considerable combustion of the charge and corresponding gas evolved in the powder chamber before the shot has left its initial position in the breech end of the bore, the temperature rising and the pressure reaching a value sufficient to force the projectile into the rifling groove and to overcome initial frictions, usually a considerable fraction of the powder pres- sure obtained. The projectile then moves up the bore followed by further combustion and expansion of the gases evolved from the combustion of the powder. The combustion exceeds the expansion up 29 30 to the tine of maximum powder pressure which is reached after a travel up the bore roughly from 1/6 to 1/3 the length of the bore depending greatly on the type of cannon, charge, etc. The energy of combustion is expended: (a) In Kinetic Energy of translation ? of the projectile. (b) In Kinetic Energy of translation of the recoiling mass (assuming the recoiling mass free). (c) In the Kinetic Energy of the charge itself. (d) In the work on the rifling and in friction. (e) In the angular energy given to the projectile. (f) In dissipated heat. The last three are very small as compared with (a), (b) and (c). Further (b) and (c) are small as compared with (a). "Ingalls" states that about 83* of the total energy of the work of expansion goes into the Kinetic Energy of translation of the shot, the re- mainder 17* going into the forms b, c, d, e and f. The rate of combustion depends upon the forn and size of the grain, it being an observed fact that powder burns in layers always parallel to the initial surface. Further the rate of combustion is a function of the actual pressure generated, vary- ing as some power of the pressure. The value used for this exponent is one of the most tentative features in the whole subject of interior ballistics. DYNAMIC RELATION- Let IT = the mass of the pro- jectile. * = the weight of the SHIPS IN INTERIOR BALLISTICS. projecti le . m = the mass of the charge 31 it = the weight of the charge. m r = the mass of the recoiling parts. w r = the weight of the recoiling parts. u = the travel up the bore. x = the absolute displacement of the shot in the bore . X = the corresponding displacement of the recoiling parts . v = the absolute velocity of the shot in the bore. v o = the muzzle velocity of* the shot. V = the free velocity of the recoiling parts (absolute ). Pxj= the total pressure on the breech. P = the total pressure on the base of the shot. Pk= the intensity of pressure on the breech (Ibs. per sq . in. ) . p = the intensity of pressure on. the base of the shot (sq. in.), f = the component of the rifling reaction parallel to the axis of the bore. Then, dv P - f = m - , for the motion of the d t, projectile dV and P^ - f = m r , for the motion of the recoiling mass in free recoil (2) and further assuming the charge to expand in parallel laminae with the successive laminae having velocities as a linear function of the end velocities, we have, p. _ p = 5 /dv dV . D 9 V - ) f n\ 2 dt dt (3 > where dv dV dt dt = the mean acceleration 32 of the powder. Combining (l), (2) and (3) I , dV ffi x dv <*< * > at = (n + T ? at (4) Integrating successively, ( m * JL) V < + 4-)v (5) (m r * -1-) x - ( + ~)x (6) c The absolute displacement of the shot in the bore is connected with the travel (u) up the bore by the following relation: x = u - X since the positive value of X is assumed opposite to x. Substituting in (6), we have, i . (m + - ) u X - 2_ (7 ) m r + m * m which gives the relation of free recoil to the travel of the shot up the bore. Obviously (5), (6) and (7) may be written immediately from the principle of "linear momentum" (that is, the total momentum of the system remains constant unless acted on by external forces) and the principle that tha center of gravity remains fixed unless acted upon by external forces. In free recoil the exterior forces are nil. The pressure on the breech exceeds that on the shot by the inertia resistance offered by the of the powder gases, 33 >n r dV mdv Neglecting ^ as small compared with m r , 2 p b - f (m P - f hence Since the rifling reaction expecially during the movement of the shot up the bore is roughly 2 per cent or less of the value of p, we may entire ly neglect the term f . ' in the above expression, which simplifies to ft m + P b = - 2 P (8) From a series of experiments conducted by the United States Navy the value E m + - - 2m - = 1.12 a constant, approx. hence p b - = 1.12 F approx. (9) It is to be noted that the acceleration of the powder is very likely somewhat different from the assumption upon which (8) was de- rived, but nevertheless equations (8) and (9) give a good approximation of the increase of breech pressure over that at the base of the projectile. During the "forcing in of the rifling" before the commencement of motion of the shot, obviously Pb = P- 34 According to the previous assumptions the pres sure varies progressively, decreasing from its maximum value at the breech block to a slightly smaller value at the base of the projectile. Therefore, if we let p^ be the average or mean instantaneous pressure or rather the pressure in the -powder chamber and bore, we have, p + p P = - * 2 In terms of the total pressure at the base of the projectile, m dV dv m p , . 7T . , 21 -3T, dv m * 2^ dt but dv hence -_ = p =p(i + ^_) = p (1+ _L_) (10) 4 in Aw or in terras of the total breech pressure dV " ra r dV m r dt + ^_ ~tt Pm 5 ~ (1 m " * 2 (11) ID * 1 2 35 EQUIVALENT The riflil1 g grooves in MASS OF the gun come in contact with PROJECTILE the copper rifling band on the projectile and angular motion is transmitted to the projectile in addition to the translatory motion. The object of the angular motion is to give the projectile a gyroscopic effect maintaining, with a combination of the air reaction, the axis of the projectile parallel to the tangent of the trajectory and further making an oblong projectile possible with greater ballistic efficiency. Let P = the reaction of the powder on the base of the shell. m = the mass of the projectile, f = the total rifling reaction normal to the rifling groove, uf = the friction component of the rifling reaction tangent to the rifling groove. 6 = the angle of pitch of the rifling, (i. e. the angle the rifling makes with the axis of the bore ). p = the pitch of the rifling d = the diameter of the bore k = the radius of gyration of the projectile.. x = the displacement of the projectile up the bore from its initial position. 5 = the corresponding angular displacement twist of the projectile. Then we have, P - f(sin e 4 u cos 9 ) = m 4!*- dt \i>) f(cos e - u sin 9 ) | = mk' f ( ^ } 36 Further since, the number of complete turns or revolutions of the projectile in its linear displacement x or its angular displacement ft, is JL or JL p 2* ie have d a # = 2* d a x (14) ar*" ~p~ "31"*"" In terms of the angle of pitch of the rifling, - 16 = x tan e or n tan e = "" hence 2 d*x T tan 6 dF~ (is) Substituting (14) or (15) in equation (13) we have * d a x d*x mk * 4 "P" dT mkMtan --- -------- = - - (cos 9 - u sin e)d (cos 9 - u sin e)d a (16 ) which shows the reaction f is always proportional to the linear acceleration of the projectile. Therefore, the friction uf , is also proportional to the linear acceleration. Substituting (16) in (12), we have or in terms of the rifling angle, .sin e * u cos e \ 4k*tan@. d*x P - I 1 ^cos 6 - u 51B b > ~li - J ffi ip 37 which shows that the powder reaction P is also directly proportional to the linear acceleration of the projectile. Evidently the equivalent mass of the projectile, is .sin e i i cos 6 < cos e - i i sin 6 j .sin Q + i i cos 6 > ^cos e - ' j sin e ; dp 4 tan e ^ a -, \ " Hence the rifling reaction and friction due to rifling are directly proportional to the powder re- action, that is the pressure on the rifling grooves always varies at any instant directly with the powder reaction. fbus we have the relationship that rifling. frlc.tj.on behaves exactly like, an additional mass: that jLa f it has an inertia^ef f ect since it is_prft- portional to the aeeeleratipn. The true equivalent mass due to the linear and angular inertia of the projectile alone, can be ob- tained by assuming the rifling friction zero, ( i.e., putting u = o) 4 n 8 k*v n' (1 - - ) m (i " ::" ) . (20) The true equivalent mass may be readily checked by a consideration of the total energy of the projectile,. that is, \ m'v 2 i mv a + y In* 2nv 2v tan and w - - 38 and I = mk a where k = radius of gyration about its longitudinal axis. hence . . * 4n k . ,, 4k* tan* 9 N B 1 - ( 1 + = ) m = (1 + ) m D* d* EQUIVALENT MASS For a differential layer of OF the powder charge at the base of POWDER CHARGE the projectile, its velocity evidently is equal to that of the projectile while for a dif- ferential layer at the breech, the velocity is equal to that of the gun. For intermediate layers, we must assume some law of variation of velocities, between the two end limits. For simplicity and probably a fairly close approximation, we will assume for the various laminae, a linear variation of velocity be- tween the end limits. Further since the velocity of the gun is small as compared with that of the projectile, in virtue of the approximation of the whole analysis, we are entirely justified in assum- ing the recoil velocity entirely negligible. If, Velocity of projectile * v (ft. sec.) Distance between breech and base of projectile = x (ft.) Velocity of any inter - mediate lamina = v 1 (ft/sec.) Distance from breech to lamina = x 1 (ft.) Then v 1 = v (ft. sec.) u If we assume the density of the powder is uniform through the distance x, so that the weight of the lanina is x dx , then the kinetic 39 energy of the lanina is W v ' W V 2 o V 1 I " * I * TT o dx or . -7T * *X 2 g x 2 g and the Kinetic energy for the total charge becomes, K. E. of w - f. _Ii r x x ' 2 dx 1 x 2g ^o ' J (i. ) T- (21) That is, the equivalent mass, when dealing with the energy equation, is 1/3 the mass of the total charge. It is important to note that when dealing with momentum, the momentum for the total charge becomes, on the same assumption w u x 1 w JL / JL vdx '=.JL v (22) gx o x 2g that is the equivalent mass from the moment or aspect is 1/2 the mass of the total charge. EQUIVALENT MASS It is convenient in deriv- OF ing the energy equation, THE RECOILING PARTS to express the Kinetic Energy of the recoiling parts in terms of the velocity of the projectile. Neglecting m ds small as compared with m r 2 and, neglecting the recoil brake reaction as small, we have, by the principle of linear momentum, f \ ( \ m r V = (m + - ) v ( approx.) 40 hence (m * ) Therecoil Energy, becomes, ? r v * = i I v a (23) and therefore the equivalent mass of the recoil ing parts, in terms of the velocity of the projectile, becomes, (m + JL ) 2 ENERGY EQUATION The mechanical work expended by the gases of the powder charge in the bore is equal to the external work ex- erted on the projectile and gun, plus the Kinetic Energy given to the gases themselves, plus the heat energy lost in radiation through the walls of the gun. If W = the Potential Energy of the Gases at any instant. P|,j= the total reaction exerted on the breech of the gun. P = the total reaction exerted on the base of the projectile. X = the displacement of the gun measured in the direction of its movement. x = the displacement of the projectile measured in the direction of its motion. E = the Kinetic Energy of the powder charge. Q = the loss of heat due to radiation. 41 J the mechanical equivalent of heat = 778 *** lb * B. T . U , Then for the energy equation of the powder gases, ire have, -Pb dX -P dx * d(E + W) * JdO (1) hence -dW P b dX + P dx + dE + JdQ. (2) that is the loss of the potential energy of the gases, due to a differential expansion goes into mechanical work (PjjdX + P dx + dE) and radiation JdO. Further by (19), (23) and (21), P dx = d[y(m"v a )) p b ax AS = tlf (5 v )] so that Further, in terms, of a hypothetical mean pressure P m (over the cross section of the bore) equation (3) may be expressed in terms of the travel up the bore u, (i.e. the relative displace- ment between the gun and projectile). where (6) (6) jec the bore of the gun, approximately since dv where v -- = the acceleration of the projectile up du 42 dv dx dv dv l~ dx v dV = df dx + dx = d? l~TdT> and dX is pared with dx, and m" = the equivalent mass of the projectile which takes care of its angular acceleration as well as the rifling friction, see equation (19). EXPANSION It will be assumed, that the OF expansion of the gases due to POWDER GASES the combustion of the powder charge obeys the law of a perfect gas. Hence, we have, PV = RwT where p = the Intensity of Pressure exerted by the gas Ibs/sq. ft. V = the volume of the gas. (cu. ft.) w = the weight of gas (Ib.) R = a coefficient (ft. Ibs. per Ib . gas.) T = absolute temperature reached. Further, with a perfect gas, the internal energy of the molecules of the gas is entirely in a Kinetic or Vibratory form, and therefore, is directly proportional to the temperature. Hence, we have, dQ dQ = cwd T and c = where dQ = the heat required to raise the gas for a change of temperature dT. c = Specific heat or the heat required to raise one Ib . of gas one degree of temperature at the temperature considered. We are concerned especially with the expansion of a gas at constant pressure or at constant volume or a combination of the two. 43 pdV Hence, dQ. = wC p dT = C p =-- at constant pressure C v dQ wC dT = R at constant volume. If the volume and pressure vary together, then, we have the sum of the partial variations, above, that is, dQ - (C p dV + C v V dp) Rw " but since, dT = (p dV + V dp) R we have, dQ = C y wdT + P ' v p dV and Q _ a = wC v /dT + p R v /p dv This relation can be interpreted, physically immediately, since the internal energy being entirely of a kinetic or vibratory form, must be proportional to the change in temperature at constant volume other- wise additional heat must be added for the external work. Hence wC y /dT measures the molecular kinetic energy. Considering an expansion at constant pressure, the total heat required is, Q = V 4-pdV I where U * the internal energy = wC y dT. Since the heat is added at constant pressure, we a Iso have, pdV = wR dT But the heat added at constant pressure is, Q wC p dT hence, substituting in the total heat equation, wC p dT = wC v dT w R dT ^ Dd R _R _ Cp - C v ' T- or J - P - <-v 44 If now the specific heats C v and C p are assumed constant for the range of temperatures during the ex- pansion of the powder gases, we have, C p - C v Q = *C V (T - TJ + -^ W (7) where If is the external work performed. T t is the initial temperature. Neglecting the loss of heat by radiation as small, we have practically an adiabatic expansion in the bore of a gun; that is, - C, wRT Since p = -=-, dividing by T, we have C v " + Without the vanes, the reaction on the gun breech be- comes,- R v {I ?* ; K - I ) and with the vanes the reaction on the breech is pro- bably different and modified to, / T Pfc dt = * ( V' - J) t o since some expansion probably takes place within the vanes, themselves. Now as to the actual reaction obtained, the ideal brake differs from actual conditions, essentially in the following points:- (1) Only a part of the total charge can be deflected through the vanes. (2) The entrance velocity can only be a component of the actual muzzle velocity of the gases. (3) Only a partial expansion of the gases can take place before entrance into the vanes . (4) The exit velocity can not, for practical considerations, be entirely to the rear, 30* from the rear, being like- ly the maximum angle that the gases can be deflected. (5) Only a very small expansion can take place through the vanes themselves; of the gases passing through the vanes the total expansion is small. In consideration of (1) (unless the vanes are ex- tended a considerable way out) the higher the muzzle 55 velocity the less the total charge passing through the vanes. It has been found experimentally that it is useless to add more than a given column of vanes, further addition of vanes having very little effect on the reaction. Further the first one or two vanes nearest the muzzle, are subjected to an intensity of pressure practically equal to that of the gases at the muzzle. Further development of the muzzle brake should be directed in obtaining greater expansion to the rear by a suitable combination of vanes, curvatures of same, etc. LEDUC'S FORMULA The empirical formula estab- lished by Leduc is especially service- able and sufficiently accurate for a predetermination of the reaction of the powder, during the powder period and its effect on the recoil. Leduc's formula, assumes that the velocity curve of the projectile during its travel up the bore follows that of an equilateral hyperbola, with parameters a and b, that is, If v = the velocity of the projectile at any point in the bore (ft/sec) u = the corresponding travel up the bore (ft) a and b being parameters of the hyperbola, then v = r-r: (ft/sec) b + u where a and b must be determined by the elemantary principles of Interior Ballistics. Determination of the parameters a and b:- When u is made infinite, that is u = a and v - a - a then a 56 a is therefore determined by considering the expansion in a gun of an infinite length. c If n the ratio of the heat capacities (*- *1.4l) and for an adiabatic expansion pV = k, v Then the work of an expansion from, initial Volume V t to final Volume V , becomes, i = / p d V, but p = * where k and n are con- Now when V f becomes infinite W * * t (ft. Its.) n - 1 V t Since. 1 Ib. of water * 27.68 cu. in. for unit density, k 1 i - 1 27.68" Expansion at Unit Work for an Infinite Expansio Density. Weight of given volume of powder gas Weight of same volume of water and if 7 C the given volume of the chamber (cu. in.) V t the volume of 1 Ib . of gas (cu. in.) then ^ ** 27.68 A * - per Ib. of powder gas. *a 27.68 hence the specific volume of the gas, becomes, 27.68 V *a ' "I" 57 Therefore, the work of expansion of 1 Ibs. of the gas to oc becomes, \f A n ~ l W 1 27.68 Since the gas evolved is proportional to the weight of the charge w, and a = v for an infinite expansion in the bore, we have wa f w E A * ^ or a com pi e te expansion of w 2g Ibs. of powder gas, hence a /2gE (-*)* A 2 Now S has a value = 653 ft. tons roughly, and by experiment "_lJ. s 1/12 ( appr ox.) Taking into account the various losses, it has been further found ex- perimentally that /2gE = 6823 for ordinary good powder. Therefore, the parameter "a" becomes, a * 6823 ( )t ATI ttf W 27.68 Now A a -n but with a powder chamber V c , loaded with w Ibs. of powder, the specific volume of 1 Ibs. of powder evidently becomes, t ^ '' t a * w assuming complete combustion of the charge, (2) that is the density of loading may be defined as the ratio of the weight of the charge to the weight of a 58 volume of water sufficient to fill the powder chamber, Hence the parameter a becomes, i __i a = 6823 (--y ,27.68 w.ta H f \ y / v c To evaluate the parameter b, we must consider the ac- celeration of the projectile, and the reaction of the ponder gases on its base, during its travel up the bore. The acceleration up the bore, becomes, jiv (b * u )a- av K a 2 bu V " = (b + u )' (b * u) 3 (3) since 3. V v = fr4. u from Leduc's formula, hence the pressure against the projectile, for a displacement v, becomes, w a a bu P = g (b+u) 2 (Ibs) Further the maximum pressure occurs, when T i.e. when, ] .4 .3 = -3u(b + u) + (b+u) (b+u) 4 and u = - (ft) that is the maximum pressure in the bore occurs at a displacement equal to one half the parameter b or the parameter b - twice the displacement of the maximum powder reaction in the bore. We have, therefore, substituting v = in (3) 2 59 P ' - (Ibs) (4) The mean powder reaction on the base of the projectile during its travel up the bore, becomes, where VQ = the muzzle velocity ft/sec. U Q = the total travel up the bore (ft) The pressure against the projectile when the shot is about to leave the muzzle, becomes, a bu Hence to determine the parameter "b" we have the following equations:- 4 w a . m 27 g D f -* \ where P m V Q and U Q are * a *^_ ( known. 2 wv and P Q P e , a and b are au v : b+tl ( unknown. ) Hence a solution is possible:- If A^ = Area of bore and P m = a given property of the powder used P m = 30,000 to 33,000 Ibs/sq.in. usually. P m = p^ A for the max. powder reaction. Substituting atl e 2g (b+u ) 2 o 2 (Ibs) hence a = 60 P . 4 (its) bence a" 27 g b 4 w Equating, we have, 2 P e (b * u o) _ 27 b P m hence 27 b + 2 b u * u* = o 27 t e + 12 - 3- ) v b + u 2 = 8 P. Solving, we have, (9 2? m 1 , S<9 " (2 " 8" Pi > "o ! /(2 ' g- F* ' ' o - (ft.) (7) which determines the parameter b, in terms of the travel up the bore, the given maximum powder reaction and the mean powder reaction, being determined from the muzzle velocity and travel up the bore. To completely determine the velocity, powder force, and time against the travel up the "bore, we have aa- (ft/sec) b + u 61 P = g (b+u) 3 and the corresponding time of travel, becomes, , du (b+u) * - / y- ' (au) du b , 1 = a l#e u * a u * Constant Now when u = 0, t = and log e u = - a, and the constant cannot be evaluated without making some as- sumption. Since the initial powder reaction required to force the projectile into the rifling grooves is large and the displacement u , to Max. powder pressure is small, we can reasonably assume the powder reaction constant and equal to the maximum- powder re- action during the initial travel u = ^ . Hence as- 2 suming the maximum powder reaction to be reached at the beginning of the travel of the shot up the bore, and then and substituting P raax> from to remain constant up to u = hence t. = v-~ (-) (8) Sunstituting in the previous time equation, we have, '27" b b b b (-) = log - - * constant a a 2 2a 62 and Constant = ((/27 - 1 ) -2 log e ^ ] a \ (2.098 log e - ) 2 "a therefore 2u u = - (2.3 log + - +2) (approx.) (9) a b b The powder reaction on the breech during the travel up the bore is somewhat greater than at the base of the projectrle due to the inertia resistance of the powder gases and charge. It has been shown previously that the breech pressure is augmented over that at the base of the projectile by either of the two following formulae:- v, + | Pfe = p (Ibs) w or P b = 1.12 P (Ibs) The former is based on a theoretical assumption, and gives an idea as to the change in the pressure drop from the breech to the projectile with different ratios of powder charge to weight of projectile. The latter is entirely empirical and it appears that the ratio of the weight of the charge to that of the pro- jectile has no effect on changing the ratio of the breech pressure to that at the base of the projectile. Unfortunately the latter empirical value is somewhat limited especially for extreme ratio of the projectile weights but is, however, reasonably accurate for ordinary calculations. The former is more or less in error due to the assumptions made, but it gives the 63 characteristics for extreme ratios. Therefore, for extreme ratios of charge to projectile weights, the former formula should be used, while with ordinary ratios, the latter should be used. R e c ap i t ul ait ion of the Various Formulae Originating from LIDUC'S Formula - Let v = Velocity of projectile up "bore (ft/sec) u = Travel up bore (ft) v = Muzzle velocity (ft/sec) U Q = Total travel up bore (ft) t = Time of travel up bore (sec) t o = Tine of total travel up bore (sec) W = Weight of powder charge (Ibs) w = Weight of projectile (Ibs) V c = Volume of powder chamber (cu.in.) A = Density of loading P = Powder reaction on base of projectile (Ibs) P^ = Powder reaction on base of breech (Ibs) PJH = Max. Powder reaction on projectile (Ibs) P e = Mean Powder reaction on projectile (Ibs) AVf = ( * \ )v and (w r = - )Xf = ( w + - )x 2 Now the absolute displacement of the shot in the bore is related to the travel up the bore u, by the equation x = u - X Hence, we have, and w (w + - )u X f = Since w and w are small as compared with w r , we have for a sufficient approximation w w + - Vf = 2- v (ft. sec) w_ X -iLlJl Af - ' *' U ft. The equation of velocity displacement and time of free recoil during the travel up the bore, becomes, 66 V f = (w r ) b + u (ft/sec) Xf-C- (ft) t = - (2.3 log + e * a b b With constrained recoil, assuming a recoil reaction X we have, dt Ubs) hence 1 P b d <- Kt = v * p b <' f -r: v but / - v f therefore, Kt V f - -- = V (ft/sec) m r Kt : 2m, X (ft) 2u (2.3 log ^ + _ a b b + 2 (sec) In the several equations, it will be noted, that the common parameter is the time of travel up the bore in the gun. Hence if for various values of u, we ob- tain correspondingly values of time, the free velocity and displacement is obtained for the given time and the corresponding effect of the recoil brake during this time is deducted from the velocity and displace- ment respectively. Further it has been tacidly as- sumed that the powder reaction with constrained recoil is the same as with free recoil at the same time in- terval. This, however, is not strictly true since the 67 powder reaction is somewhat modified due to the slightly different motion of the gun with constrained and free recoil respectively. The effect, however, is entirely negligible as compared with the magnitude of the reaction and other factors involved, even with the most refined measurements and analysis. EXPANSION OF THE GASES AFTER The manner of THE SHOT HAS LEFT THE BORE the expansion of the AND ITS EFFECT ON THE RECOIL powder gases after the projectile has left the bore is very difficult to calculate, and various assumptions based on empirical data have beeri formulated, for calculations during this period. The following theory though imperfect gives an idea as to the manner of the expansion of the powder gases in the "After effect Period". (1) The momentum imparted to the gun during this period evidently equals the momentum iwparted to the powder gases: n>r< V f * V fo> ' S < v w - I ) where Vf = maximum free velocity of the recoiling parts. (ft/sec) Vf = free velocity of recoil when the shot leaves the bore (ft/sec) v w = mean velocity of the powder gases attained (ft/sec) i = mass of powder charge (Ibs) v = muzzle velocity of projectile (ft/sec) Since m r Vj =(n + - )v we have m r Vf = mv + m v w In other words, the maximum free momentum obtained by 68 the gun, equals the sum of the total momentum of the projectile and the total momentum of the powder charge. It is important to note that the momentum relations are very nearly true provided we are able to calculate v w the mean velocity of the powder gases and can neglect the small effect of the air pressures exerted on the gases. (2) We have the following energy relations due to the expansion of the gases : (a) Initially the gases have a Kinetic Energy = * (-)v* 2 3 (b) The work of expansion of the gases in expanding from the pres- sure in the bore when the shot leaves the gun (i.e. the muzzle pressure) to the atmospheric pressure, becomes v a We = / pdV a. fee y, r o where V o = volume of powder chamber + volume of the bore of the gun. V a = volume of gases at atmospheric pressure . (c) The final Kinetic Energy of the gases may be approximately assumed equal to: 1 Sv^. 2 It is to be noted that the final Kinetic Energy of the gases is difficult to calculate due to the di- vergence or cone effect produced when the gases expand into the atmosphere The total Kinetic Energy equals 69 the sum of the Kinetic Energy of the center of gravity of the gases plus the relative Kinetic Energy of the gases relative to the center of gravity. From a series of experimental tests conducted by the Navy on the velocity of free recoil with guns of various caliber it has been ascertained that the momentum effect of the powder gases is equivalent to the weight of the charge times, a constant velocity of 4700 ft/sec. Assuming the divergence of the spreading of the gases to be similar at all muzzle velocities , it is possible to estimate the divergence factor and then in guns of very high muzzle velocities we may calculate the maximum free velocity by multiplying the work of expansion by the divergence constant and the solving for the mean velocity of the gases. The pressure of the gases rapidly falls to the atmospheric value or approximately this value, before the divergence of spread of the gases is appreciable, hence the maximum Kinetic Energy of the gases will be attained at approximately atmospheric pressure. The change in Kinetic Energy of the powder cases therefore, becomes, jSv w ---v = change in Kinetic Energy, and the work done on the gases, equals the work done by the external pressures p o and p a and the work of ex- pansion pdV. Hence, v d p V - P a v a * J P dv ~ tota i work done. v o To allow for the relative Kinetic Energy due to the spreading of the gases, we may multiply the work done on the gases by a constant, and then equate this value to the changes of the translatory Kinetic Energy of the guns. 70 v -n v + f a P dV ) = -mv 2 - im .,2 o v p p a v a J 2 w * q v V v o where K = the divergence constant to allow for the spreading of the gases at the muzzle. Now the work of expansion, becomes, , V a Po v o ~ Pa V a W e = / T> d V = : 'o where the expansion exponent k = 1.3 approx. Hence the total work done on the gases, becomes, Po V o - Pa V a k . Po v o - Pa v + ; = iT^I (p o v o ~Pa v a) K *~" X further, since p rt V ^ _ n yk o p a v a , we have, P " rrr-fc" v " Pa Va) = rri p v o [1 - (-^ ) ^ i PO Hence the energy expression reduces to the convenient form, p ^L_I_1_ K[r r p o o \l - ' / k >J = __ v a PO 2 3 from which knowing p o V Q , p a m and v enables us to immediately calculate v w , the mean free velocity of the powder gases. To evaluate the dispersion constant, to take care of the relative Kinetic Energy of the gases after expansion, the ballistic data of the 155 m/m Filloux gun has been chosen, since assuming a mean velocity of the gases 4700 ft/sec., calculated and experimental results were found to check very close- ly. 71 Weight of powder charge w = 26 (Ibs) Volume of powder chamber S = 1334 (cu.in.) Total length of bore u = 186 (in.) Muzzle velocity v = 2410 (ft/sec) Area of bore Aj, = 29.2 (sq.in.) Weight of projectile = 96.1 (Ibs) Max. powder pressure p m = 35300 (Ibs/sq.in.) Mean Powder pressure = wyg _ _ 19200 (Ibs/sq.in.) " "e 644 uAv Twice Abscissa of maximum pressure .27 "m *i\ j. -i/^ i \ 2 IT = *^7 ^fl Muzzle pressure when shot leaves muzzle _27 2 _u 27_ 2 185.68 x 35300 P = ~4~ S (e + u) 3 P7n " T~ 57.38 X (57.38 + 185. 68) 3 10140 Ibs/sq.in. we have then, 0.3 K 32.16 [ 10140 x U4 VIl - 0.3 V 10140 = i 2 i 26 2 - x 26 x x x 2 4700 2 3 2410 I Solving, we have, 156 x 1Q K V = (287 - 25) 10 6 = 262 x 1Q 72 Hence K = = 0.430 - 3.915 cu. Hence the energy of translation is but 43* of the total Kinetic Energy of the gases after complete expansion. Therefore with guns of numeral ballistic relations, we may estimate the mean translatory velocity of the gases after complete expansion, by the formula: o where v = muzzle velocity (ft. see) w = weight of powder charge (Ibs) b = 1.3 approx. . p a = atmospheric pressure = 2116 (Ibs/sq.ft.) p o = muzzle pressure of powder gases (Ibs/sq.ft.) VALLIERS The hypothesis of Vallier assumes, HYPOTHESIS that during the "after effect Period" in the powder period of the recoil, that the powder reaction on the gun falls off proportional to the time. That is, If P o b = the total breech reaction of the powder gases, when the projectile leaves the muzzle (Ibs) t o ~ time of travel of the projectile to the muz- zle (sec) t t = total powder period (sec) P^ = powder reaction on "breech (Its) t = corresponding time (sec) then ?b = p ob - c free ve i oc ity of recoil at end of * powder period. Vf o = Free velocity of recoil when the shot leaves the muzzle. / l [P ob f _ t (t - t ))dt = r (V f , -V fQ ) 1 Q t O Integrating, we have, P b ( t-*t.) -i . r (V I V Q ) 2m r (V f , -V fo ) hence t, _ t ,, __L_J - H_ (sec) and p ob C = 2(V f . -V fo )m r Therefore the powder reaction during the after effect period, becomes, o P b = p ob ^ ; (Ibs) 2(V f . - V fo )m r RECAPITULATION Of PRIKCIPLE FORMULAS OP INTERIOR BALLISTICS PERTAINING TO RECOIL DESIGN. The velocity and displacement of the recoiling parts during the travel of the projectile up the "bore have the following relations with the velocity of the 74 projectile up the bore and the relative displacement of the projectile in the bore. (Weight in Ibs. ) If m = massof projectile v 00 ., oc . la m = mass of powder charge m r = mass of recoiling parts" v = velocity of projectile (ft/sec) u = displacement of projectile in the bore from its breech position V = velocity of recoiling parts (ft/sec) X = free displacement of recoiling parts (ft) then =: ra (m + - )v m + ; V = 2 = _ v Approx. (ft/sec) ra m m,. + = m r , ffl (m + - )u m + - X - = u Approx. (ft) The pressure on the breech, in terms of the pressure on the base of the projectile, becomes If P b = breech pressure (total) (Ibs) P = pressure at base of projectile (total) (Ibs) m m * - P., = P = 1.12 P approx. (Ibs) m The mean pressure in the bore, becomes, I m + - 4 P m = p b ^ 1 ) (Ibs) 75 For building up the energy equation, we are concerned with the various equivalent masses of the moving ele- ments that the powder reacts on in terms of the major mass of the projectile. The equivalent mass of the projectile, becomes, if k = radius of gyration about its longitudinal axis (ft) p = pitch of the rifling " 9 = pitch angle of the rifling . 4k 2 tan 2 9 , lh _ m = (1 + - )m = (1 + - ) a |it P 2 d ' If we include the effect of the friction of the rifling we have, sin 9 + u cos 6 4 tan 9 k 2 , Ibs . m " =[1 + ( - 5 - : - ^) - - ] m ( - ) cos 9 - u sin d 2 The equivalent mass of the powder charge, for the energy equation = 35 3 ! A* " w (3) P m = P m Ad (Ibs) 2 W V (4) P e = g = 32.16 ft/sec-2 . (6) v = ^L- (ft/sec) b+u w a 2 bu (7) P = (ibs) g (b + u) 3 w a a bu (9) g (b+u Q ) 3 (Ibs) vr'iC ;O **a f- - (10) t = - (2.3 log + +2) (sec) a b b b 2U U t o = ~ (2.3 log - + + 2) (sec) a b b (ID t - H 2 VQ approx. The equations of velocity, displacement and time of free recoil during the travel up the bore, becomes, 79 w w + - p a*j V f = ( ^)( ) (ft/sec) w f b + u *5 X f = ( -) u fft) w r :*jp-~ee .- 20 u , = 1 (2.3 lo? r- + 5- + 2) (sec) a With constrained recoil, assuming a recoil reaction K Kt V = V* - (ft/sec) X = Xf - (ft) 2m r t = (2.3 log + * 2) (sec) a bo Theexpansion of the gases after the projectile leaves the hore causes an additional recoil effect. The hypothesis of Tallier assumes the powder reaction to fall off proportionally with the time. On this assumption: If Vf ~ the velocity of free recoil at the end of the powder period. Vf o = the velocity of free recoil when the shot leaves the muzzle. p t t and t the corresponding terms --- (sec) ob .' " - P ' 01> 2(V fl _ V )B r (IT,.) 80 >y THE PARABOLIC TRAJECTORY The nucleus of exterior ballistics is the differential equations of the parabolic path of a shot projected in a vacuum. These equations then nay be modified for air resistance and gyroscopic deflections due to the angular momentum of the pro- jectile and air reaction: Let x and y be the horizontal and vertical coordinates of the trajectory. n * the mass of the projectile. V o = the muzzle velocity. t the time of flight. 0' = the angle of elevation from the horizontal of the axis of the bore. = angle of elevation of the departure of the projectile from the muzzle. e * the increment angle or "jump" to the elastic deformation of the carriage and the move- ment of the gun in a direction not along the axis of the bore. 1 - r = angle of sight. O a a line of sight. L * range to given target. L Q = horizontal range corresponding. = striking angle from horizontal. m 1 = angle of fall from line of sight. The differential equations of motion give: dx dy Integrating successively, we have, ~ V cos ~ = - gt + V sin dt and . \\ 82 ^^^^^ x = V cos t y = - - + V Q sin t id * Pe ~ w r L r ~ ff a L a + v b L = hence (3) In like manner we have after the powder pres- sure ceases (4) Now considering the external reactions on the recoiling parts alone*, during the powder pressure period, we have figure (2) 86 Fig. 3. hence P - and when P figure (3) d'x k dt d'x = * B + R - If- sin j dt (5) ,t B + R - W r sin = M r ~ r dt (5 1 ) 87 Substituting (5) and (5 1 ) in (3) and (4) respectively, we have Kd + P e - W r L r - W a L a + VfcL - (6) K r d - V r L r - W a L a + V b L (7) Thus the external effect during the powder pressure period is always at every instant equal to the total resistance to recoil, that is, the sum of the total braking and guide friction, minus the weight component and a powder pressure couple Fe dependent upon the actual total powder force. In general, e is very small and usually for a first approximation the powder pressure couple can "be neglected. Further, for constant resistance to recoil K r - K a - K - B + R - VT r sin (8) which is the average external effect during recoil on the total system. As shown in Chapter VI on the "Dynamics of Re- coil" i K -* E b - E + V f T (9) where Vf * the maximum free velocity of the recoiling parts, that is _ W4700 + W Q w r (10) W * Weight of powder charge, W * Weight of shot and W r Weight of recoiling parts v * Muzzle velocity of shot u Travel up the bore in inches b * Length of recoil in feet d a Diameter of bore in inches E Unconstrained displacement of recoiling parts during powder pressure period. 88 T = Total time of powder pressure period. In equation (9) note that E = K t Vf T and K o 2 v Substituting these values in (9) and solving for a wide range of artillery material and thus evaluating the variables as a function of the diameter of bore, muzzle velocity and travel up bore, Mr. C. Bethel has given the very valuable and serviceable formulae, and accurate to one percent. M r Vf 1 = = - - - b + (.096+. 0003 d) uv f v o >*|H This formula holds only for constant resistance to recoil. It is important to note that the "total braking" sometimes called "the total pull" is not in general equal to the resistance to recoil, but is the total resistance to recoil plus the weight component, that is B+R=2P a +ZP n +2R s +2fig = K+W r sin (12) where 2P a = Total recuperator reaction ZPj, = " hydraulic reaction 2Rs = " stuffing box friction 2R ,< = Guide friction K = T M. V, i r v f (b - E + V f T) To obtain the external reactions on the carriage mount, it is convenient to know d in the previous moment formulae about A, in terms of the height of the trunnions and the distance between the trunnions and a line through the center of gravity of the re- coiling parts arjd parallel to the axis of the bore. 89 Let H+ = height of trunnions above the ground distance from trunnion axis to line through center of gravity of recoiling parts and parallel to l>ore. moment arm of K about A nor. horizontal distance bet-ween reactions A and B. from A to center line of trunnions. As the gun elevates, we have two cases: (1) When the line of action K passes above A, see figure (4) (2) When the line of action K passes be- low A, see fig. (5) t s = d = 1 = c = -*- K Fig. 4 90 r Fig. 5 Fig. 5' 91 For case (1), we note that h' - (d sin + c ) tan = d cos but s *' = H t + ^I~0 and a E + _f d sln _ c tan = d cos cos cos h t cos + s - d sin 2 0- c sin = d cos hence d = h t cos + s - c sin (13) For case (2), we note that h 1 + d cos = (c - d sin 0) tan but n ' = h t * ^ 1 cos cos cos 2 2 h t cos + s + d cos = c sin - d sin hence d = c sin - h t cos - s (14) If W = weight of the total system (gun, recoiling parts and carriage ), we have for moments about A *s ^s = v r L r + w a L a In battery or IB terms of the tipping parts = W t and the top carriage alone (not including the stationary parts of the tipping parts = W a W s L s = W t L t + V a L a In battery where L s = distance to center of gravity of' total sys- tem in battery from A If b = length of recoil, and the angle of elevation., and Lg - distance to center of gravity of system out of battery, we have W s 14 = W r (L r - "b cos 0)+ W a L a = r Ly + W a L a - r b cos hence TT S L^ = W s L s - ^ r b cos Out of battery 92 Hence the external reactions at A and B on the carriage mount become in terns of the resistance to recoil, powder pressure, height of trunnions and distance between trunnions and line through center of gravity of recoiling parts parallel to axis of bore, For low angles of elevation, Taking moments about A, we have, V b L + Kd + Pe - W s L s + tf r (x cos 0)= hence Wg L s - W r ( x cos 0)- Kd -Pe Pe disappearing for a finite value of x or in other words, when pe is used W r x cos may be neglected. And since V a r W s + K sin - V^ or directly from moments about B, noting that moment arm of K becomes d'= d + L sin = h t cos + (L-c)sin l * S we have, W a (L-L s ) + W P x cos + K(d+L sin 0) - Pe V 3 Va L and as before Pe disappearing unless x is very small. Obviously H a = K cos and is in no way directly ef- fected by the powder force. For high angles of elevation, the moment arm Kd reverses, and d and d 1 become respectively, d * c sin - h t cos - S and d 1 L sin _ d See(fig.5) = (L-c) sin + b t cos + S Now taking moments about A and B respectively W g L s - W r x cos IS + Kd - Pe b L and W 8 (L-L g ) + W r x cos + K(L sin - d) * Pe 93 and H a - K cos 0. For design use, the external reaction formulae be conveniently grouped. IN BATTERY: for low angles of elevation: j r Nvn* cos + S - c sin 0) - Pe ) g Ul a v ) V " "T~ ) W S (L-L S ) + K(h t cos +(L-c) sin +S)+Pe ( ) H a K cos ( for high angles of elevation: W g L g + K(c sin - h t cos - S ) - Pe ( L ) > W s (L-L g )+K[(L-c) sin +h t cos 0+S] +Pe ( ( V a ) (16) ) H a a K cos ( OUT OF BATTERY: for low angles of elevation: ( y a^s ~ w r k cos ^~ ^^ h t cos +S- c sin ) ) ) W S (L-L S ) + W r b cos 0+K(h t cos j0+(L-c)sin 0+S) ) v ~ T"" (17) c ) ) H a = K cos ( 94 for high angles of elevation: b cos + K(c sin 0- h t cos 0- s W S (L-L S )+ W r b COS0 + K(h t cos 0* (L-c )sin#+S L ( H a = K cos ) These formulae are immediately applicable to platform mounts traversing about a pintle bearing as well as field carriages. In platform mounts, the horizontal reaction of the platform on the mount is usually taken at the pintle bearing which is usually located in the front or muzzle end of the mount. Hence in place of H a we have HO = K cos 0. The reactions V^ and V a remain the same, V a now being the reaction of the platform on the traversing rollers of the mount. Very often V)j is divided into two equal vertical oomponents at the two ends of the traversing arc of the mount, and in such a case L is the horizontal distance in the projection of a vertical plane containing the axis of the bore from the pintle reaction to the traversing reaction, that is, if L is the actual distance from the pintle to the other end of the traversing arc, and 9 is the spread of the arc, then 6 L = L' cos - 2 In a field carriage, for a first approximation we may assume the horizontal and vertical reaction to be at the contact of spade and ground. These reactions are obviously H a and V a of the previous formulae and Vfc is the vertical reaction of the ground on the wheels, and L the distance from the wheel contact to the spade contact with the ground. For split trails, 95 V a and H a are obviously equally divided and if the gun is traversed, a horizontal reaction normal to the plane of H a an ^ V a is introduced; however, this re- action will not be considered until later, that is, the gun will be assumed at zero traverse. .A closer approximation to actual conditions in a field carriage is to regard H as acting at a vertical distance g from the ground line, usually when from 1/2 to 2/3 the vertical depth of the spade in the ground. The equations then will have an additional moment: H a g = K cos g, which is substracted from the moments of the numerator in the expression for V^ and added to the mpments in the expression for V a - The general equations for field carriages are then, for low angles of elevation: W S L S - W r x cos - K.(d + g cos 0)-Pe v b = . W 3 (L-L S ) + H r x cos + K(L sin - d + g cos#)+pe V a = H a = K cos d - h t cos + 3 - c sin and for high angles of elevation: Y^L, _ W r x cos + K(d-g cos 0) - Pe _. *S . ^ _____ _______ ___ L W S (L-L S ) + W r x cos + K(L sin0-d+g cos0)+ Pe H a = K cos d = c sin0 - h t cos0 - S where Pe disappears if W r x cos is used or vice versa. 96 Another class of mounts in which the previous formulae are not applicable, are known as pedestal or pivot mounts used on Barbette Coast mountings and for naval guns, as well. These mounts are attached to the foundation by bolts on a circular base usualljr equally spaced around the circumference. With such mounts the question of stability is of no consideration. The reaction between the foundation and mount and the distribution of the tension in the bolts, may be obtained approximately by considering the base of the mount as absolutely rigid. Then on firing, the front bolts become the most extended, the deflect- ions and corresponding stress being proportional to the distances measured from the back end of the base along the trace of the intersection of vertical plane, con- taining the axis of the bore with a horizontal plane, to the perpendicular chord connecting any two front bolts Thus if L , L, etc. are the lengths from the base end to the perpendicular chord connecting a set of two bolts, and if j o , j, etc. are the deflections of the bolts, we have j : j t : j t : L : L t : L, Now if the bolts are of equal strength, the ten- sions are proportional to the deflections, that is T O : T I : T f * j o : j t : j 8 = L : L t : L 8 ~ that is T Q = C L Q , T t * C L t , Q C T C : Hence the moment about the back end holding the pedestal down, becomes, C ll + 2 C L 2 + 2 C L* + C Lj = SM Considering now the gun and mount together we have, K d - W g L s - W r x cos = M hence Kd -(W S L S - W r x cos 0.) C = L* + 2L 2 + 2L 2 L 2 Oil n and the maximum tensiqn to which the bolt at the farther end is subjected, becomes, 97 [K d - (W S L S - W r x cos 0)]L L* + 2 ij + 21; L; If the gun traverses 360 every bolt should be designed for the maximum tension, T . The same method may be applied to various other combinations for holding a gun down on its foundation. BENDING Itf THE TRAIL In considering the strength AND CARRIAGE of a carriage body, the reactions at the trail, V a and H a , subject the total carriage to a bending stress. This is of special con- sideration in field carriages of the trail type. The reaction V a causes bending while H a decreases the bend- ing. Hence for maximum bending we should examine the conditions for maximum V a and minimum H a . Now, W S (L-L S ) + K[(L-c)sin + h^ccs + si +Pe \f s -^ ^_^__________^____^________________ a T : H a = K cos where L horizontal distance between wheel contact and spade contact with ground (in) c = horizontal distance from spade to vertical plane through trunnions (in) h t = height of trunnions from ground, (in) s = distance from trunnion to line parallel to axis of bore and through center of gravity of recoiling parts (in) L g = horizontal distance to center of gravity of total system, recoiling parts in battery (in) P S powder pressure couple (in/lbs) K = total resistance to recoil (Ibs) With a field carriage, since the trunnion position is very close to the wheel contact with the ground, 98 (L-c)sin is always very small compared with h t cos 0, hence, we have approx. W s (l-l s ) + K(h t cos + s) + P e V a = If L x = distance from trial contact with ground to any section in the carriage body or trail h v = the height of the section from the ground y we have, for the bending moment at section xy, "xy = V a^x - H h y Substituting the value for V a and neglecting s being small, we have L M xy = [W S (L-L^) + K'h t cos + P e^"r~ ~ K cos h y = W S L X (1- ~) + K cos (JL h t - h y ) + P s ^i BENDINd \N TRWL 8r Fig. 6 Now from fig. (6) it is evident y- h t is always greater than h y , hence for maximum bending moment we must have cos 0=1, that is = 0. Hence the maximum bending moment occurs at horizontal elevation, 99 At horizontal elevation, h t cos + s = h henca W-(L-U) + K h + P. V, = a L but we also have critical stability at horizontal elevation, that is K h + P e - W S L S = (approx.) therefore, V a =- W s (approx.) that is in virtue of the mount being just stable at horizontal elevation, or in practice approximately so, the vertical reaction at the spade equals the weight of the entire system, gun and carriage to- gether, yy L _ p Further H a = K = S (ibs) and the bending moment at section xy in the trail, becomes, h lui =WT ^ W T - P ^ * - ( -j n 1 K c ^ neglecting P e as usually small compared with W $ L S , ws have, M xv = W_(L, -L. ) (in Ibs) A jr o A. K For the maximum bending moment in the trail, we consider the section at the attachment of the trail to the carriage, then, L x = L s approx. and therefore, the maximum B. M. becomes, h - h Li Iff f f .^^^^^^^^^^^^ l ill ro-v \T ~ ** c He* \ ) \-Li a most useful formula in a prelinary carriage layout It is important to note that if the recoil varies the above formula and analysis do not hold. When, however, the recoil varies on elevation the maximum bending moment in the trail is obtained at the minimum elevation where the short recoil COTD- mences, that is, when cos is a maximum for the minimum recoil. If b_ = the short recoil at maximum elevation, then, we have, K S - maximum total resistance to recoil, then, 100 W.L^Cl - ~) * K cos (^1 h t - h y ) + P e JL where L x = distance from trail contact with ground to any distance in the carriage body or trail. h> * the height of the section from the ground. P e 3 maximum powder pressure couple. EXTERNAL REACTIONS DOSING Counter Recoil may COUNTER RECOIL be divided into two periods, the accelerating and the retardation period so far as the external effects on the mount are concerned. During the accelerating period, the external re- actions on the recoiling parts alone, are the elastic reaction of the recuperator in the direction of motion, the guide and stuffing box frictions and a hydraulic resistance during the whole or part of the accelerat- ing period, together with the component of the weight of the recoiling parts parallel to the guides, oppos- ing the notion of counter recoil. Hence, if x = the displacement from beginning of counter recoil of the recoiling parts with respect bo guides. Ff a = the resultant accelerating force of counter recoil. K r a the resultant retarding force of counter re- coil. F x * the recuperator reaction for displacement x trom beginning of counter recoil. R = the total friction. H x 3 the hydraulic resistance, if any, of throttling through recoil orifices or counter recoil buffer. Then, during the accelerating period 101 d x i i m j^j- = F x - R - H x - W r sin = K a and for the subsequent retardation . t - m = R + H x + W_ sin - F = K' dt* Considering now the external forces on the total system (recoiling parts together with mount) the braking resistance for the recoiling parts then be- come internal reactions, and considering inertia as an equilibrating force, we have, as before the fol- lowing external forces, d*x K a = m- - The inertia resistance during ac- celeration which is opposite to C'recoil. d'x * r m d~T* ^ ne i nert i a resistance during retardation which is in the direction of C'recoil. W r * Wt. of recoiling parts W a * Wt. of carriage proper H a and V a Horizontal and vertical reactions of spade and float y^ * Front Pintle reaction - horizontal com- ponent assumed zero as before. During the accelerating period, obviously, K a < K r that is, F X -R-HX - W r sin < F x + R + H x - W r sin hence, so far as stability and the balancing re- actions exerted i>y the ground or platforu on the carriage mount are concerned, the external effect during the acceleration period of counter recoil 102 need not be considered. If now, the inertia resistance is considered as an equilibrating force, we have Kr (d+L sin 0)-W r [ (L-L r )+b-x cos 0]- V a (L-L a )+V a L= Let d = d + L sin = h t cos + (L- c) sin + S Hence the limitation for counter recoil stability, noting that W r (L-L r ) + W a (L-L a ) = W s (L-L a ) becomes K r d' = W s (L-L S ; + W r (b-x) cos For a constant marginal counter recoil stability . moment O 1 this equation becomes K r d' =[G'+W S (L-L S ) + W r b cos 0] - W r cos x and the stability slope for a constant marginal counter .ecoil stability is evidently , W r cos that is decreasing as the recoiling masses move into battery. Minimum stability is evidently in battery position and 0=0, that is Wf r r \ s \L> -L> s I h where h = d for 0=0 In ordinary field- carriages, the weight of the system in battery is very close to the wheel axle or contact of ground and wheel, consequently (L-L S ) is very small . Therefore counter recoil stability is the primary limitation in the design of a counter recoil system. STABILITY The question of stability for field carriages is of fundament- al importance, it being a primary limitation imposed on the design of a recoil system. If a gun carriage is to be stable, then Kd - W a L a - * r (l r - x cos #) = 103 If we have a constant marginal moment G, that is an excess stability, we have Kd - W a L a - W r (l r - x cos 0) = G K = -G + VT S L S - x v r cos = A - m x A = where G + W I W cos \* L ! w W Jv Thus the resistance to recoil to conform with a constant margin of stability decreases in the recoil proportionally to the distance recoiled from battery. In battery, the resistance to recoil, -G * W S L S Kb = A = and put of battery, the resistance to recoil becomes, where b = total length of recoil -G + W S L S W r b cos d d consequently, fot a constant margin of stability, W r b cos K b d From this we obtain the equations of resistance to recoil for constant stability against displacement, W r cos IS -G + W s Lg K x = K b x, where G b = d In our Ordnance Department, K X = K Q = a constant during the powder pressure period. Thus if B represents the corresponding length of recoil, then for a constant stability moment G, -G + W S L S E W r cos K /-\ "~ v ~ , ' A ~~ 1 104 * r cos and K x - K (x - E) Obviously the stability "slope" or space rate of change of resistance to recoil for constant margin of stability, is ^ cos ID = d where d = ht cos + s - c sin A3 the gun elevates, W r cos remains finite, while d decreases to zero at the elevation Si, where the line of action of the resistance to recoil passes through the spade point. Thus the stability slope "m" thereby increases to an infinite value at that same elevation. But it is important to note that the resistance to recoil out of battery is finite and increases con- siderably as "d" decreases so far as it is limited by stability. Obviously in design it is inconsistent to ake the slope of the space rate of change of resist- ance to recoil consistent with the stability slope as the gun elevates, since the stability becomes sufficiently increased to allow a large resistance to recoil bo be used. We may, therefore, cause the slope to vary arbitrar- ily as a linear function from a maximum value at an arbitrary low angle of elevation, say some value from to 6, to zero at the angle of elevation where the resistance to recoil passes through the spade. Thus if, = the initial angle or lower angle of elevation from which the slope is to decrease arbitrarily. J t * the angle of elevation corresponding to where the resistance to recoil passes through the spade. d o = moment arm of resistance to recoil about spade point for angle to 105 8 SOO .j* 8^8 * *J " Z g* L, tf m = stability slope for any angle of elevation W r cos j0 m = - = stability slope at lower angle o d o of elevation. m = m - k (0 - ) then, m = m At angle, of elevation t , o = hence o * k ^t - *o> r k = TO . 0-0 Q hence m = m o - (-zr\ TT ) (# - ) or substituting for i " o W r cos - W r cos t - Thus the variation of the space rate of change of resistance to recoil may be divided into two periods, (1) from to 0Q w ~ C^ c i W r cos OB = which is parallel to the stability slope (2) from to W r cos 8 0-0 , where the slope ln W ~ & f\ is arbitrary. A graph of the variation of the space rate of change of the resistance to recoil against elevation conforming' to the assumption (1) and (2). If there is always to be an excess stability couple G we have from the previous discussion, fixed limitations for the resistance to recoil in and out of battery. Thus, from to o 106 - G + W S L S - G + W S L 3 W r b cos ft Kv = - : k = : - where throughout recoil G is a constant marginal stability couple, and from O to t - -G + W S L S r b cos the length of re- coil being as be- fore shortened as the gun elevates but if the stability marginal move- ment is never to be decreased for any part of the recoil below G, since the stability slope and space rate of resistance to recoil increase and decrease respectively as increases from Q to it is ob- vious that the minimum stability is in the position of out of battery. Therefore the resistance to recoil in battery is the resistance to recoil out of battery with a marginal moment G of actual stability, augmented by m b . That is, K = -G + W S L S W r b cos W r b cos ji + r J - 1 ] d d do < J=*0 cos cos 0- f * 1 \ LENGTH OF RECOIL Obviously the overturning CONSISTENT WITH force, that is the resistance to STABILITY OF MOUNT recoil, is a function of the length of recoil varying roughly inversely as the length of re- v coil. Hence as the gun elevates the stability in- creases and the recoil may therefore be shortened. In a preliminary design it is desirable to know the length of recoil as limited by stability, from or the lowest elevation wherein stability is de- sired to the elevation where the stability slope 107 is made to change arbitrarily. Let C s = the constant of stability = Overturning moment = where the overturn- Stabilizing moment ing moment = K r d and the stabilizing moment = W C L_ - W_ b cos OO I We may consider the limiting recoil at various elevations (1) with a constant resistance to recoil as would occur in certain types of re- coil systems. (2) with a variable resistance to recoil using a stability slope as outlined in the previous paragraph- For a constant resistance to recoil : = K : The critical position of stability is obviously with the gun at the end of recoil out of battery. Then C S (W S L S - W r b cos 0) K = - - r - >" -sj :?f.Q -j^bwcq arfJ $.iii;fe 1 v a 2 m r V f K = Ses "DYNAMICS OF RECOIL". Chap. VI. b-E +V f T Where E = displacement during powder period in free recoil. T = total time of free recoil. Vf = Max. free velocity of recoil, hence C S (W S L S - W r b cos 5) b-B+V f T d The above equation reduces to the quadratic form Ab 2 + Bb + C = and its solution is, /* - B / B - 4AC b = - 2 A 108 Where A = W r cos B = W r cos (V f T - S) - W S L S C = W s L s (V f T - E) For rough estimates, especially where the length of recoil is comparatively long, we may assume, ^ m r Vf C S (W S L S - W r b cos 0) - B + /B 2 - 4 AC C s W r cos b . and A . - C = - m r V For Variable Resistance to Recoil: - - -- -- ----- The resistance to recoil is assumed constant during the powder pressure period and thence to de- crease uniformly with a stability slope as given in previous article. Therefore, from the end of the powder pressure period to the end of recoil, the stability factor remains constant from to O (i.e. to where 'the stability slope is made to change arbitrarily) . At the end of the powder period. (See Dynamics of Recoil): a Kd = G.( 3 L S - W p (E - --) cos 2m_ hence C S (W S L S - *r E cos 6) do I w -- COS Now the resistance to recoil out of battery at the end of recoil, becomes, 109 KT* K - m (b - E + - ) (See Dynamics of Recoil) 2m r hence by the equation of energy VT* KT 2 KT 9 [2K-m(b-E+ -)] = m (V f - ) 2ra r m r now K = k + m(b-E + ) 2m r K = 5 = the constant resistance to mT 1 _ recoil during the powder 2ra r period, and C S (W S L S - W r b cos 0) k = = the resistance to d recoil at the end of recoil. Substituting these values in the enery equation, we obtain a quadratic equation in "b!! A sufficient approximation and simplification can be made, by not- ing that 2 E - - 0.9 E approximately and 2m r KT Vf = 0.9 Vf approximately Therefore, (K +~k) (b- 0.9E) = 0.81 m r vj and K = k + m(b- 0.9E) C s (W s L g - W r b cos 0) = + m(b-0.9E) d substituting in the energy equation, we have, 2C a (W_L S - W r b cos J0) + m(b- 0.9E)(b- 0.9E) = 0.81 m_V f d z Reducing and simplifying, we have the quadratic sol- ution, , -B / B - 4AC b = 2A s where A = m - 7- W r cos 2C, III 2C f * - 9E w r cos 2 Q = 0.8l(mE - m r V f o from to W r cos t - m , ( ) from to For a close approximation and when the resist- ance to recoil is not constant during the powder period, if K = the resistance to recoil in battery k = the resistance to recoil out of battery, we have, Kl- fe m V ? r R 111 M V f ( >b = (approximately) but K = k + mb c s and k = 7- 0_L, - W_b cos 0) d as Substituting, we have 20 s [- (W S L S - W r b cos 0) + mb] b = m r V f and the value b, becomes, -B /V - 4AC b = 2A where pp 3 nr A = m - ~r~*'r cos .fee d 2C S B = -r W S L S c = - r v f 8 N W r cos ra = " or any arbitrary slope as desired. d The above formula is sufficiently exact for a preliminary layout with a variable recoil and resist- ance to recoil provided the margin of stability is chosen fairly large, that is when a low factor of stability is taken. JUMP OF A FIELD CARRIAGE When the overturning moment exceeds the stabiliz- ing moment, we have unstabil- ity and an induced angular rotation about the spade point. After the recoil period, the gun carriage is returned to the ground by the moment of the weights of the system. This phenomena is known as the jump of the carriage. For the condition of unstability, we have: K d ~ *s^s * *r cos > where as before, K = total resistance to recoil W g = weight of entire gun carriage including gun 1 3 = distance from spade contact with ground to center of gravity of total system in the battery position. w r = weight of the recoiling parts x = movement in the recoil of the gun. To analyse the motion of the system, consider (a) the recoil or accelerating period. (b) the retardation or return period. The recoil period may be subdivided into the 113 powder period and the pure recoil period. During the recoil period the gun and gun carriage are given an angular velocity which reaches its maximum at the end of recoil. During the retardation the angular velocity is gradually decreased to zero, but with increased angular displacement, the maximum angular displacement occuring when the angular velocity reaches its zero value. Further change in angular velocity results in a negative velocity and a corres- ponding angular return of the mount to its initial position. ttniliootn jo x-* Y "''^ The acceleration during the recoil period is not constant, even with constant resistance to recoil, due to the fact that the moment of inertia and the moment of the weights of the recoiling parts about the spade point varies in the relative recoil of the gun. Therefore, the angular acceleration is not constant during the accelerating period. Likewise during the return of the recoiling parts into battery. Further the effect of the relative counter recoil modifies the return angular motion. Consider the reaction and configuration of t"he recoiling parts and carriage mount respectively. See figure (7). 114 Let X and Y = the components of the reaction between the recoiling parts and carriage mount, parallel and normal to the guides res- pectively. M = the couple exerted between same. I a = the moment of inertia of carriage mount about the spade point. I r = moment of inertia about the center of gravity of the recoiling parts. d x = perpendicular distance from spade point to * i line of action of X. d x = perpendicular distance from X to center of gravity of recoiling parts. d = d x + d x = perpendicular distance to line parallel to guides and through center of gravity of recoiling parts from the spade constant with ground. Q = angle made by d with the vertical = angle of elevation of the gun (in battery) x = distance recoiled by gun from battery position x s distance from "d" to center of gravity of re- coiling parts in battery measured in direction of X axis of perpendicular to line d. r = distance from spade point to center of gravity 5 J of recoiling parts, e = angle r makes with vertical l r = horizontal distance to center of gravity of recoiling parts from spade contact with ground . W a = weight of carriage proper (not including recoiling weights) r a = distance from spade point to center of gravity of carriage proper, o = angle r a makes with horizontal l a = horizontal distance from spade point to center of gravity of carriage proper. Then l r = (x Q -x) cos 6 - d sin 6 115 l a = r a cos (9 + a - 0) where in battery 6 = and for any other angular position during the jump of the carriage, 6=0+6 B=a variable angle during the jump, For the angular motion about the spade point, For the carriage mount, without the recoiling parts, d 6 i " " u a ~* "** d + 2 Xd x - Y(x - x)+ m - w a l a = I a T - (i) and for the recoiling parts, adding (1) and (2), we have, d*6 (3) Xd - Y(x Q -x)- w a l a = (I a + I p ) - Since the recoiling parts are constrained to rotate with the carriage mount, they partake an angular acceleration about the spade point combined with a relative acceleration along the guides. The acceleration of tne recoiling parts is AfT Jtfc> 9& divided into: (1) The tangential acceleration of the recoiling parts about the spade point; due to the constraint in the guides, dt 2 a d is divided into components in the x and y direction d *e / j d2 e ) - cos (e + e) = d - f dt a dt 2 116 (2) The centripetal acceleration of the recoiling parts about the spade point due to the constraint in the guides, s ,de . 2 rw - r (. ) dt and divided into components in the x and y direction. ,d9,* , N ,d6v! r( ) sin (9 + e) = (x - x)( 4 r(- )* cos (e + e) = d at (3) The relative acceleration of the recoiling parts d'x <*v r - -T along the x axis (4) The relative complimentary centri- petal acceleration due to the combined angular and relative motion of the re- coiling parts: de (5) The angular acceleration of the recoiling parts which obviously equals the angular acceleration about the spade point, that is d 2 e dt ! For the motion of the recoiling parts along the x axis, we have dv r d% P b - m_ - a r d 4 w r sin e - m p (x Q - dt dt (4) 117 For the motion of the recoiling parts normal to the guides, x d*e de ,,de .a Y - ra_(x n -x) -- w r cos 9+2 n r v r - + m_d( ) = dt dt d t (5) Substituting (4-) and (5) in (3) we have, 1 2 2 2 ur 1 mi 1 + 9m v ( v v ^ - - aL . _ at _ "r^r W a 1 a 60B r v,Ax o x; *9 d 2 9 (I a +I r ) - = (6) dt dt where I r =(x ~x) cose - d sin ) ' functions of the , . variable angle l a =r a cos (e+a-6) ) e From equations (4) and (6), we have 8 as a function of t. An exact, solution of these dif- ferential equations is complicated and therefore an approximate solution must be resorted to. APPROXIMATE SOLUTION 09 THE JUMP Of A FIELD CARRIAGE. The static equation of recoil, that is the equation of motion of the recoiling parts upon the carriage is stationary, becomes, s > W_ sin e -X R * and the equation of motion of the recoiling parts along the guides when the carriage jumps, becomes, 9 - X - o, r d L - r (A 118 Now the term m r (x rt -x)( ) is small and may be neg- dt lected, but on the ,2 d 9 other hand the term m_d may be considerable. dt 2 Furthermore the braking X and X s may differ considerably as well. The term 2 P b -m r - -= K s in static recoil. whereas with the jump of a carriage ,2 P b - m r = cK s where c = 0.9 approx. dt a During the pure recoil on retardation period of the recoiling parts, we have m_- - = K, in static recoil. r dt 2 whereas when the carriage jumps, ,2 m_ - = cK, where c = 0.7 to 0.9 r dt 8 Considering the moment equation for the movement of the total mount about the spade point, we have, (p b -ra r * )d-[m r (d 2 +(x -x) 2 )+I r +I a ] +2m r v r (x o -x) dt 2 dt 2 d6 _ ** r l r _ I _ Q dt a a where l r =(x o -x) cos 6 - d sin 6 l a =r_ cos (0 + a -v ) de The term 2m r v r (x o -x) is always small and may be Q t neglected . If b = length of recoil, for the average during the recoil, let b v y Y - *o x 2 119 Then, we have, for an approximate solution, 2 cKd-[m r (d%(x - |) 2 + I r + I a ] i-i - w s l s = (7) where w s l s = a l a + r l r c = 0.8 to 0.9 and w s = w a + w r and approximately, if the jump is small, l r = (x - -) cos - d sin# _ . ~ _ . .. i ' t l a = r a cos a = angle of elevation of the gun b = length of recoil d = perpendicular distance of line parallel to guides and through center of gravity of recoiling parts from spade contact with ground. X Q = the perpendicular distance from d to the center of gravity of the recoiling parts r a = distance from spade point to center of gravity of carriage proper a = angle r a makes with horizontal Hence for the angular acceleration, d 2 9 _ eKd- w s l s _ rad ,2 x,2 D72 I I - dt m r (d +x o - -) +I r + I a sec* If we assume a constant acceleration, we have, for the angular velocity attained at the end of recoil, d0 (cKd - w,,l s )t rad ( \ At 2 1 m r (d(x - |)-I r -I a where t^ = the time of recoil we have approximately, cK s wv -H w 4700 where V = 0.9 120 w= weight of projectile (Ibs) w= weight of charge (Ibs) w s = weight of recoiling parts (Ibs) c s 0.9 approx. and t p = the total powder period ob- tained by the methods of interior ballistics. The angular displacement during the first period of the jump, becomes, t (cKd-w s l 3 )t* 6 = - B radius. - During the second period of the jump we have, the angular velocity decreasing but the angular displacement still increasing: then rad ~ Integrating, we have t r (d(x - |) 2 ) + Ir +I a and for the angular displacement, S I *,!< (- - O x* L)^m it - w s ! 3 t de - *) 2 ) +1, +1, + ~ fc " 9l To determine the time of jump required to attain the maximum angular displacement, we have the angular velocity reduced to zero, whence, ae t w s i s t a j i. ' ~ , from which we may determine t s> Therefore, the 121 maximum angular displacement, becomes, _I *r de m r ^d 2 +(x - ' 2 t f The effect of counter recoil is to increase t 2 and decrease the negative moment (- w l ). S S RECAPITULATION OF FORMULAS: EXTERNAL EFFECTS AND STABILITY, Resistance to recoil: J Mf K = (Ibs) constant resistance through- b-E+V f r out recoil. m p = mass of recoiling parts = w r (16s) I g = 32.16 ft/sec. a b = length of recoil (ft) E = free recoil displacement during powder period (ft) T = time of free recoil (sec) Vf= Max. velocity of free recoil ft/sec. BETHEL'S FORMULA . m r v f 1 (Ibs) Constant lD + (.096 +. 0003d ) M V f resistance. V o M = travel up bore (inches) V Q = muzzle velocity (ft/sec) d = diam. of bore (inches) Assuming a gun carriage to be supported by a hinge joint at the rear (A) and a vertical support 122 in the front (B) we have the following equations for the reactions of the supports: Let H a -= horizontal component at rear hinge support or spade of carriage. (Ibs) V a = vertical component at rear hinge support (Ibs) Vjj= reaction of front support assumed vertical (Ibs) L = horizontal distance between carriage supports (in) ht= height of trunnion above support (in) s = perpendicular distance from center of gravity to recoiling parts to line of action of the resistance to recoil (in) c = horizontal distance from rear support to trunnion (in) K = total resistance to recoil (Ibs) <& = angle of elevation of gun g = vertical distance from ground to horizontal .component of resultant spade reaction. IN BATTERY: For low angles of elevation: w < .L-K(h+ cos + s-c sin 0)-Pe ' : v - ' - K[h t + cos 0+(L-c)sin ) H a = K cos 123 For high angles of elevation w s L, +K(c sin -h t cos - s)- Pe ) ) V b = ( w s (L-L s ) +K[(L-c) sin + h + cos 0+s]+Pe < (Ibs) ( v a - - r ~) ( H a = K cos ) OUT OF BATTERY: For low angles of elevation: w" 3 L s - W r b cos 0-K(h t cos + s -c sin ( W S (L-L S ) +W_b cos +K(b+ cos 0+(L-c)sin0+a> ) V a = - * - ( ) ( ( H a = K cos ) For high angles of elevation: ( ) ) w g L s - r b cos 0+K(c sin -h t cos 0-s) ( ( V b - L - ) ) ( ( ? g (L-L s ) + r b cos +K(h t cos 0(L-c)sin0+s) ) H a K cos ( With a field carriage where the spade is in- serted in the ground, the center of pressure lies a distance "g" inches vertically down. The general equations for the support of a field carriage, therefore become, 124 For low angles of elevation: w s L s - w r x cos 0-K(d+g cos 0)- Fe ( b L ) w s (L-L s )+w r x cos 0+K(L sin 0-d+g cos0)+Fe ( Va = ) ( H a = K cos ) ( d = h t cos + s - c sin ) For high angles of elevation: \ - *s Ls w r x cos *K(d-g cos 0)-Fe ; V^_ ( V a =w(L-L s )+w r x coS0+K(L sin0 -d+^ cos 0)+Fe ) ) ( ( H a = K cos 0; d=c sin 0-h t cos - s ) In certain types of Barbette mounts, we have the bottom carriage held down by tension bolts to a circular base plate. If we draw a series of parallel chords through the bolts on either side of the axis of the gun, and if we let the distance from those several chords measured from the rear bolt, be L Q , L - - - I n . we have, for the maximum tension induced in a tension bolt given by the ex- [Kd-(w s L 3 -W r x cos 0)JL TO = : : BEHDINa III THB TRAIL AND CARRIAflB. Considering the section at the attachment of the trail to the carriage, for a constant length of 125 recoil the maximum bending in the trail occurs at horizontal elevation and is given by the following expression: h _ h B.K. at the attachment of trail to carriage. h = the "height of the center of gravity of the recoiling parts (axis of bore practically above the ground when the gun is in its horizontal position. h vs = the height of the neutral axis of the section above the ground. w s - weight of entire mount including the gun. L s = horizontal distance from the spade to the center of gravity of the weight of the entire mount. When the recoil varies on elevation, the maximum "bending moment in the trail is obtained at the minimum elevation where the short recoil commences, we have, ^s k x L x M xv =w s L x (l )+K s cos S ( h t -h v )+Pe - L L !- where K s = maximum total resistance to recoil corres- ponding to short recoil "b s . S = minimum angle of short recoil. L s = distance from trail contact with ground to any distance in the carriage body of trail. h y = the height of the neutral axis of the sect- ion from the ground. Pe = maximum powder pressure couple. STABILITY OP COUNTER RECOIL. In the design of a field carriage counter re- coil stability is a basic limitation. We have for counter recoil stability that. 126 The equation stability, gives, for variable resistance to recoil, for low angles of elevation consistent with the stability slope, -8+ /B* -4AC b = 2 A where ff_cos o ^o A = = C s -* (from to O elevation) B SSI - 2K-2mE m r and C S (W S L S - w r E cos ) (Ibs) w T 2 d - C 3 E- cos J25 2ra r After an arbitrary elevation (approx.5 ) the stability of the mount greatly increases with elevations and therefore the stability slope is made to arbitrarily decrease with the elevation arriving at constant resistance to recoil at the elevation corresponding to where the line of action of the resistance to recoil passes through the spade point. To estimate the minimum recoil allowable for the various angles of elevation in this range, we have -B /B 2 -4AC b = 2A . jsstaas jjaiJ BC J0.-0 from < .0 to 3 * ^ A = D - 5f_ COS d 2C a B = (w a L s * P.9E w r cos 0)-1.8 mE 0.81 127 < (L-t.) R r " "~"h" where K r = the total resistance of counter recoil at horizontal elevation. w s = weight of entire mount including gun. L 3 = horizontal distance from spade to center of gravity of w s . L = horizontal distance from spade to wheel contact with ground. Further 2 d x (4 f- may be obtained from the velocity curve of counter recoil towards the battery position). and K r = H x +R+w r sin - F x where H x = hydraulic or buffer brak- ing at end of counter recoil. (Ibs) R = total friction resistance w r sin =0 weight compound equals zero at horizontal elevation. F x = recuperator reaction. (Ibs) RECOIL STABILITY The stability limitation of the resistance to recoil varies in the recoil due to the movement of the recoiling weights. The slope or rate of the variation in the recoil of the equivalent force applied through the center of gravity of the re- coiling parts and parallel to the guides that will just overturn the mount, is given by the following expression: w r cos ra= from to 128 where ra = the stability slope 3 angle of elevation d = perpendicular distance from spade to line through center of gravity recoiling parts parallel to the guides. w r = weight of recoiling parts = the initial angle or lower angle of elevation from which the slope is to decrease arbitrarily. If from O the slope is made to decrease ar- bitrarily with the elevation, to the elevation 0, the angle of elevation corresponding to where the line through the center of gravity of the recoiling parts parallel to the guides passes through the spade point, we have for the stability slope w r cos t - m = ( ~r ) where the slope is d ~0 arbitrary. LBMQTH Of RECOIL COM3ISTBNT WITH STABILITY OP MOUNTS. The equation of stability, gives, for constant resistance to recoil, -m r Vf _C s (w g L s -w r b cos The solution of this quadratic equation for b, gives: -B /B -4AC b = 2A where A= i r cos ) where all units 8= w r cos 0(V fT -)-w s L s ( are in feet ..2 s ) and pounds. _ i"ri ci ) CHAPTER IV -soo INTERNAL REACTIONS. <: - ' ' In the design of the various parts of a gun carriage it is of fundamental importance that we have a coraplste knowledge of the stresses to which each member is subjected, and the variations of such throughout recoil and the position of elevation and traverse . We have already considered the external reactions on the whole system, and such reactions are useful in computing the stresses in the supporting structure for a gun mount as the strength of concrete emplace- ments for barbette mounts, or the strength of a rail- way car or caterpillar frame. The primary internal reactions within a gun and its mount may be classified as follows: (a) The mutual reactions between the recoil- ing parts and the carriage proper or gun mount. (b ) The mutual reaction, between the tipping parts or cradle and the top carriage. (c) The mutual reaction between the top carriage and bottom carriage. The mutual reaction (a) is between the moveable and statipnary part of th3 total system during" the recoil; that of (b) between the moveable and station- ary parts during elevation of the gun; and that of (c) between the moveable and stationary parts in traversing the gun. The mutual reaction (a) may be subdivided into individual or component reactions as follows: (.1) The reactions of the constraints due to the guides or clip reactions at 129 130 the two ends of the clips in contact with the guides, which may be subdiv- ided into friction and normal com- ponents . (2) The mutual reaction of the elastic medium connecting the recoiling parts to the carriage proper, that is, the hydraulic brake and recuperator re- action, together with the joint frictions. This will be known as the elastic reaction between the recoil- ing parts and carriage proper. The mutual reaction (b) may be subdivided into: (1) The trunnion reaction between the tipping parts and top carriage. (2) The elevating arc reaction between the elevating arc of the tipping parts, and the pinion of the top carriage. The mutual reaction (c) may be subdivided into: ,(1) The pintle or pivot reaction between the pintle bearing on the bottom car- riage or platform mount and the pintle of the top carriage fitting within this bearing. (2) The traversing arc reaction, that is, the reaction between the traversing arc of the top and bottom carriage. These are usually roller* reactions for platform or pedestal mounts, the rol- lers being either a part of the top or bottom carriage or else clip reactions field carriage and may be more or less distributed about the arc of contact. Let X and Y - the coordinates of the center of gravity of the recoiling parts along end perpendicular to the guides with origin at center of gravity of re- coiling parts. 131 x t and y t = coordinates of front clip reaction measured from the center of gravity of the recoiling parts. & t = Normal component to guides of front clip reaction. uQ t = Frictional tangentional component of front clip reaction. x and y = coordinates of rear clip reactioa 2 2 ^^^ measured from the center of gravity of the recoiling parts. Q = Rear clip reaction normal component. u& 2 = Rear clip reaction frictional component. B = nQ + nQ = total guide friction. 1 2 B = elastic reaction (hydraulic breaking and re- cuperator reaction including friction of joints) assumed parallel to the guides. F = the total powder pressure on the breech of the gun which necessarily lies along the axis of the bore. e = the perpendicular distance from center of gravity of recoiling parts to line of action F, that is to axis of bore. Assuming as in Chapter III, the mount to be hinged at the rear or breech end to its support and resting on a smooth surface at the front end, and if d = perpendicular distance from hinge to line through center of gravity of recoiling parts, parallel to guides. djj= perpendicular distance from hinge to line of action of B. l r = horizontal distance from hinge to center of gravity of recoiling parts in battery. l a = horizontal distance from hinge to center of gravity of stationary parts of system (includes stationary parts of tipping parts). From fig.(l) considering the reactions on the re- coiling parts alone, we have from the equations of motion: 132 Fig. 1 133 for notion along the x axis, 2 .F" B uQ - uQ + w* r sin = m r (1) and since there is no motion along the .y axis, .-; -ie* adi oj ; .- 'r^ ifljft?j' ; ^ i'-'i c '~ Q f - Q i = w r cos ^ (2) tdJ *c and taking movements about the center of gravity since there is no angular acceleration B(d-d h )-u(Q y -Q y )+Fe-Q x - Q x =0 (3) u 22 1122 Now in fig.(l) considering the gun carriage or mount including the stationary parts of the tipping parts, we have for the moments of the reaction of the recoiling parts on the gun mount about the hinge A, l r -x cos 1 - x cos Q t (x t * + d tan 0)-Q 2 ( +d tan 0-x ) oo: C 3 WxiEWUAiU^aaaal cos * ?bi?ib +uQ (d+y )+uQ (d-y )+Bd h = 2M_ Q (4) 1 " 1 2 2 D ~ a but uQ i y i -uQ z y a =B(d-d b )+Pe-Q i x t -a 8 x 2 and uQ <-uQ =R 1 2 Substituting these values in (4), we have, l r -x cos (Q t -Q 2 )( + d tan 0)Bd+Rd+Ae=SM ra cos that is, -W r (l r -x cos j0)-W r sin d+Bd+Fe+Rd=2M r or simplifying and combining, we have, (B+R-l r sin 0)d+Fe-W r (l r -x cos 0)=2M r (5) at maximum powder pressure, x is usually negligible and the equation reduces to: (B+R-W r sin 0)d+F9-W r l r = SM ra (5 1 ) From this we observe that the reaction between the recoiling parts and the mount is equivalent in effect to a force (B+R-W r sin 0), the line of action of which is parallel to the axis of the bore or guides and 134 passes through the center of gravity of the recoiling parts, and a couple of magnitude Fe, due to the powder pressure, together with a component equal to the weight of the recoiling parts and in its line of action assumed concentrated . Thus the reaction on the gun mount of the recoil- ing parts, therefore, is equivalent to a single con- centrated force, the resultant of (3+R-W r sin 0), equal to the total resistance to recoil and a force equal to the weight of the system together with a couple Fe. Since a couple and a single force in the same plane are equivalent in effect to a single force, parallel to the former, and displaced from it equal to the couple divided by the force, the resultant reaction on the mount of the recoiling parts reduces to a single force; the resultant of B+R-W sin and W r , which be- comes, since B+E-W r sin = K, equal to J= /K 2 +yf* -2KW r sin and the line of action of J makes an angle -l(W r cos 0) -KW-cos 0) & =Tan *- =Tan * (K+W r sin 0) (B+R) with the axis of the bore and is displaced a dis- tance Fe. frora the center of gravity of the recoiling J parts. It is however, more convenient in computation to resolve this resultant into its components, K and W r together with Fe. If now we consider the equilibrium of the gun carriage mount, we have for moments about the hinge point, 2M ra - W a L a +V b 1=0 that is, (B+R-Vf r sin 0)d+Fe-W r l r -W a L a +V b 1 = C (6) and since W a l a + W r i r = W S 1 8 135 136 Equation (6) reduces to (8+R-W r sin 0)d+Fe-W s L s +V b 1 =0 (6') which of course is the same as the equation obtained in Chapter III. It is, however, very often more convenient to regard the mutual reaction between the recoiling parts and carriage as divided into component .re- actions along and normal to the axis of the bore to- gether with a couple. See fig. (3). Let x and y be the coordinate axes along and normal to the axis of the bore or guides. X r = the sum of the component reactions along the x axis. Y r = the sum of the component reactions along the y axis. Now by introducing a couple Mr between the re- coiling parts and carriage, it is entirely immaterial where we assume the line of action of X r and Y r . Let r = the perpendicular distance from X r to the center of gravity of the recoiling mass. z = the perpendicular distance from Y r to the center of gravity of the recoiling mass. Considering the equations of motion of the re- coiling mass, we have, ,2 P*w r sin /HC r =M r ~j- Y r W r cos ) (7) Pe +X r r -Y r z - But X r -W r sin = P-M r 777 =K a is the resistance to recoil during the accelerating period, and 2 X r -W r sin = - M r -jpr = K r ia the res i s tance __ to recoil during the retardation period. In general K a and K r are 137 Fig. 3 138 different in value. Hence, let K = X r ~W r sin for any given displacement x of the recoil. If we now consider the reaction of the recoiling parts on the carriage proper, the moments about the binge A of this reaction, becomes, l p - x cos X r (d-r)-Y r ( + d tan 0-z)+M r =ZM ra cos Inserting values for Y r and M r from the equations of notion of the recoiling parts, we have: X r d-W r l r +W r x cos 0-W r sin 0d+Pe = 2M ra hence, (X r W r sin 0)d-W r (l r - x cos 0)+Pe= 2M ra ) or ( Kd-W r (l r -X cos 0)+Pe= SM ra ) which is the same expression as obtained before. We may regard the line of action of X r and Y r to pass through the center of gravity of the re- coiling mass, together with a couple M=Pe, the powder pressure couple about the center of gravity of the recoiling parts. See fig. (4) Taking moments about A we have directly 1 - x cos X r d-Y r ( ) + d tan 0)+Pe = 2U ra (8) cos and since Y r = N r cos 0. we have as before (X r -W r sin0)d+Pe-W r (l r -x cos#)= 2M ra (9) where JC r =B+R The object of this analysis has been to show that so far as the external effect of the reaction of the recoiling parts on the carriage mount is concerned the exact location of rod pulls or the lins of action of the guide frictions, is entirely immaterial, though as we shall see immediately, the value of R, the sum of the guide frictions, does depend upon the line of actions of these pulls together with the friction line 139 140 of action of the guide friction itself and thus in- directly the external effect on the carriage mount is affected slightly. Further the location of the center of gravity of the recoiling parts may considerably change the guide frictions during the accelerating or powder pressure period. BRAKING PULLS The total resistance to recoil if assumed constant throughout the recoil is readily evaluated from the following relations: If K = total resistance to recoil (assumed constant) (Ibs) b = length of recoil (ft) Vf= velocity of free recoil at end of powder period (ft/sec) B = displacement of free recoil at end of powder period (ft) T= time of powder period (sec) Then from the energy equation for the movement of the recoiling parts after the powder period, we have, (ft . lbs) Simplifying, K = mr f (Ibs) b-E+V f T With a variable recoil consistent with a stability slope "m", and assuming a constant resistance during the powder period, we have, If K= the resistance to recoil in battery k= the resistance to recoil out of battery instability slope = "r h (h=height of axis of bore above ground) 141 2 then :^l{b-(E- -)]= m r (V f -~)* (ft. Ibs) m and k =K-m[b-(E- - )] (Ibs) Combining and simplifying, we have, K = ~ L ^~ ~* (lbs) 2[b-E+V f T- - (b-E)] in battery ffl p at ettikjj si :!!.<> ri V"^^*^ 09 KT 2 k =K-m(b-(E- - )] out of battery II ) , r B-the total braking pull (Ibs) R=the total guide friction (Ibs) Pk=tne total oil pressure on the hydraulic pis- ton (Ibs) p ' r j ) =the hydraulic reaction plus the joint frictions (stuffing box + piston) (Ibs) P a =the total elastic reaction(due to compressed air or springs) (Ibs) p' r a =the total elastic reaction plus the joint frictions (stuffing box + piston) (Ibs) ft i =the normal front guide reaction (Ibs) ^=tbe normal rear guide reaction (Ibs) u=coef f icisnt of guide friction (0.15 to 0.25) Then K=B+R-W r sin (Ibs) Total resistance to recoil where B=P' + P' (Ibs) Total braking h a R= u(Q t +Q 8 ) (Ibs) Total guide friction The stuffing box friction is usually assumed at from 100 to 150 Ibs. per inch of diameter of rod, and if d u and d a are the stuffing box diameters of the hydraulic and air cylinders respectively, we have 142 (Ibs) P a =P a +100d a (Ibs) GUIDE OR CLIP REACTIONS The recoiling mass is constrained to translation V < C 1 1 parallel to the axis of the jc. 3 (i r "bore by the recoiling masses engaging in suitable guides in the cradle of the top carriage. In general the re- coiling mass may recoil in a sleeve, a part of the cradle, or along guides considerable below the axis of the bore and the center of gravity of the recoiling parts . For the former case, considering the external re- actions on the recoiling mass, fig. (5) a a -a t =W r cos Q x l + & 8 x 2 + u (Q t y t -Q- 2 y 2 )-Fe-B b=0 (moments about the center of gravity of re- coiling parts) where e b = d-d b , then & t x t +(a t +ll r cos 0)+u (Q^-Q^y.,- cos 0)-"Fe-Bb = cos J0(x 2 ~uy 2 )- Fe-Bb=0 Hence Fe+Be h -W_ cos 0(x -uy ) Q t = * (10) Further Fe+Be h -W_ cos x +W_cos uy +Vl r cos x +W_cos x + U* 2* 2' 1* 2 W r cos uy -Vf.cos uy i 'II "2 143 0(x +uv ) Hence Q 2 = - - - (11) When the guide reactions are below the axis of the bore as in Fig. (4) y 2 remains the same in the above formulae whereas y t reverses in sign. Hence for case (2) Fe+Be b -W_cos 0(x -uy ) Q= - - - - - 2_ (12) and Fe+Be h +HLcos 0(x -ay ) ft= - loti - * - L_ (13) The total guide friction becomes, R=(Q 1 + Q 8 ) hence 2(Fe+Be b )-W r cos x 2 +W r cos ny 2 + r cos x t + x 2 + u(y t - y a ) 2(Fe+Be h )+W r cos 0f (x -x J + ii(y +y )] (14) x +x +.u(y -y ) 1 2 w 1 ' S Now if M=x 1 -t-x,+ u(y 1 -y, ! ) for case I or M=x 1 +x s -u(y i +y a ) for case II and if N=(x t ~X 2 )+ u(y t +y e ) for case I N=(x i -x 2 )+u(y 2 ~y 1 ) for case II we have therefore, in general that 2(Te+Be b )+W r cos N R= - - - - - u (15) M which gives the total guide friction. The value of the coefficient of friction u ranges from 0.15 to 0.20. The total braking evidently becomes, 2(Pe+Be b )+W r cos N B+ - u-K+W r sin M or B(M+2uh)=(K+W r sin 0)M-(2Pe+W r cos N)u hance (K+W r sin 0)M-(2Pe+W r cos 0N)u BS . M+2Uh which gives the total recuperator reaction in terms of the total resistance to recoil. Denoting as before by Pjj= the hydraulic reaction plus the joint frictions (stuffing box and piston) P ~ the total elastic reaction plus the joint 3 frictions. 6^= distance from center of gravity of recoil- ing parts to line of action of hydraulic brake pull P n e a = distance from center of gravity of recoil- ing parts to line of action of the re- cuperator reaction P a The front and back clip reactions become, , a a r , g ( and cos (17) y ^ reversing in sign when the guide reactions are entirely below the axis of the bores. Combining as before and noting that F=u(Q +Q ) we have t 2 2Fe+22p' e a +2ZPu e h +W r cos N R- h h r (18) M where M and K are the constants referred before. Now K= P a +2P h +R-W r sin (19) 145 and combining (18) and (19) we get KM=(2P a +2P h -W r sin 0)M+ n(2Fe+22Pae a +22Phe h +W r co5 N) Simplifying, KM=((2P a -W r sin0)M+ZP,! | x M+ u(2Fe+22P a e a +W r cos or further simplifying, a +W r cos N) Hence, M ( K _ sp ' +Win 0)- u (2Fe+2XP'e a +N Wnos 0) I at a M+2U6J, which gives the gross hydraulic pull in terms of the total resistance to recoil, the gross air or spring reaction and the maximum powder force. APPROXIMATE FORMULAE Assuming the reaction be- GUIDE FRICTION tween the recoiling parts to be equivalent to a normal force passing through the center of gravity N, a couple M, and the braking and guide friction forces B and R having moment arms about the center of gravity of the recoiling mass equal to dv and r respectively where r is the mean distance to the guids frictions, we have, for moments about the center of gravity of the recoiling parts, Be^+Rr=M neglecting the powder effect which is usually very small and N=W r cos for the total reaction. Obviously the actual normal guide reactions, becomes, x and . W_ cos x. fl s -M+ i. * 1 1 where 1 = x +x also R=u(N +N ) 12 12 146 2M+lf r cos 0(x -x ) hence R=u ( ) Substituting the value of M, we obtain, u W r cos 0(x t -x 2 ) (21) l-2ur Very often x l ~x i is small and in a preliminary design x t nay be assumed equal to x e Hence 2ufie b : l-2Ur (22) which gives an approximate value of the guide friction, useful in a preliminary design - u may be assumed from 0.13 to 0.25. Very often as in symmetrical barbette mounts, the value of Be b may be small due to a small value of e b and a certain limitation arises as to the use of the friction formula previously derived. When lf cos x * 1 1 that is, W r cos x f = Be +Rr = 8e b approx. we have continuous contact along the guides, the distributed guide reaction oalancing the weight comoonent normal bo the guides. For such a condition the guide friction, becomes, R=0.2 to 0.3W r cos (23) INCREASE OF GUIDE FRICTION If we assume the total DURING POWDER PRESSURE oraking B to be constant PERIOD. during the powder pres- sure period, the guide frict- ion R is augmented by the powder pressure couple together with the increased friction couple. Let B = the constant braking force f a the varying powder force 147 N t = the normal reaction of the stationary part on the recoiling mass. R = the guide friction during the powder pressure period. M = the reacting couple of the stationary part upon the recoiling mass. R S and M 2 are the corresponding values during the retardation period. Then, during the powder pressure period, we have N-W r cos 0=0 (24) and during the retardation period, we have, -W r sin 0=M r d'x dt N-W r cos 0=0 (25) Further, let AM=M 1 -M 3 and AR=R t ~R 2 , then subtracting (25) from (24) we have AM=Fe+AR r (26) Now during the accelerating period the normal guide reactions, become, , M t W r cos x f (27) U ' - *- K r COS Xi and during the subsequent retardation 14 2 W r cos x ? W * = T 1 (28 M 2 W r cos x i Na= T~ "T~ 148 Adding the two equations in (27) and (28) respect- ively and subtracting (28) from (2?) and multiplying by the coefficient of friction n, we obtain obvious expression: 2AM *" T U (28) Substituting (29) in (26), vie have AM=Fe+ ur (29) Fe Fel and AJ4- - = - 1_ iHE 1 - 2 ur (30) 1 and substituting in (29), ire have for the change of friction during the powder period, (31) Thus the guide friction is continuously augmented always proportional to the total powder pressure, providing the braking is assumed constant. We also note an additional cause of first class importance for the reduction of "e", that is, the importance of locating the center of gravity of the recoiling mass along the axis of the bore. Another cause for a change in guide friction during the powder period is due to the torque reaction of the rifling, T r though the total guide friction remains the same. The normal reaction on the left guide, becomes, m M_ w r coa 0* a T r Nt l 3 2, " 2dg (32) 149 and the same for the right guide, becomes, U W-COS 0X., T r N s * E 1 + -_ (33) *r 2 X 2 l 2dg W r cos 0x t T r 2 l ' 2dg" where dg is the distance between guides. In the gun recoiling in a sleeve this torque must be balanced by the reaction of the key way. Noting that, R-w(N tl *N tl + N +H tr )u(N t + N t ) the total friction remains the same. It is important to note that the friction on the left guide over that on the right due to this rifling introduces a couple in the plane of t"he guides which tends to cause rotation about an axis normal to the plane of the guides. Therefore, it is always essential that small side grooves or flanges on the clips be introduced. The additional friction on the flanges is entirely naglsgible, but the normal reaction to the flange in extreme cases may be considerable. During the recoil the guide friction is seldom constant since the distance between clip reactions pro- gressively decreases in the recoil, that is the front clip approaches the rear part of the guide in recoil. When the recoil is long it is desirable on field carriages to have an additional clip near the muzzle which engages in the guide sometime later in the recoil. Due to this cause the guide friction continually increases until the engagement of the outer clip and then we have a sudden drop in the magnitude of the friction. When the braking pull remains constant and the powder pressure coupls is small and n o outer clips are introduced during the recoil, the clip reactions should always be designed for the condition of out of battery. To recapitulate in the limitations in design so far as guide is concerned, we note, 150 (1) The bearing pressures and consequent friction of the guide are reduced by increas- ing the distance between the clip reactions nearly directly, consequently for a given guide reaction and friction, we have a minimum distance between clip contacts on the guides. (2) The guide reactions are reduced by bringing the resultant of the rod pulls through the center of gravity of the recoiling parts. (3) The moment effect and consequent guide reactions are further reduced by bringing the resultant guide friction line through the center of gravity of the recoiling mass. (4) It is highly desirable to center the center of gravity of the recoiling mass midway between the guide reactions. This condition is usually impossible to attain especially out of battery, but may be compensated by increasing "1" the distance between the clips, by an additional front clip near the muzzle. (5) Proper functioning of the recoil may be srrtirely destroyed by having the center of gravity of the recoiling mass too far below the axis of the bore, thus introducing a powder pressure couple with excessive guide friction during the powder pressure period. This powder pressure couple may cause a "springing" of the guides and considerable heating as well. The center of gravity of the recoiling mass should never exceed 1.5" from the axis of the bore unless a friction disk for rotation during recoil about the trunnions is introduced. 151 COMPUTATION OP BRAKING We have seen from the POLLS previous discussions that the guide friction is not independent of the braking pulls due to the hydraulic and recuperator reactions. These pulls tend to cause rotation and thus augment the guide friction over that due to the weight component. The total resistance to recoil is given by:- (1) when constant during recoil, (lbs) b-E+V f T where Vf= maximum free recoil velocity (ft/sec) T = total powder period (sec) E=free recoil displacement during powder period (ft) b= length of recoil (ft) r = recoiling mass (2) when variable consistent with the stability slope "m", m r Vc+m(b-E) 2 K= - ; - (Ibs) 2[b-E+V f T- ^ (b-E)] 2 m r where K=B + Rg-W r sin0=P n +P a + u (Q 1 --Q 2 )-Hf r sin0 (Ibs) and 8=P n +P a = total braking Rg=u(Q l +Q 9 )= mean guide friction assumed constant 2uBe h We have seen - Ibs.approx. where nf=0.15 to 0.2 e)j= distance from center of gravity of recoiling parts to line of action of B. (in) 162 B 3 total hydraulic and recuperator pull (Ibs) I 3 total distance between clip reactions (in) r= distance from center of gravity of recoiling parts to mean friction line (in) then, 2uBe b ' - " (Ibs) hence, (K+_sin 0)(l-2ur) B T - - l+2u(e b -r) Further since, (Ibs) 8in we have on simplifying ~~J"" (K+W r sin j0)( (Ibs) l-2u(e b -r) Very approximately, Rg=0.3 W r cos and B=K+W r sin 0-0.3 W r cos (Ibs) P n K+W r sin 0-F a -0.3W r cos (Ibs) INCREASE OF RESISTANCE TO During the powder RECOIL DURING POWDER PERIOD period, the powder pres- sure couple may be suf- ficient to cause a large increase in the guide friction, whereas the braking pulls due to the hy- draulic resistance and recuperator reaction are not 153 affected. Prom the previous discussion on guide friction, the increment in guide friction equals, 2Feu (lba) or more exactly 2Feu 2 M where M=x l +x 2 +u(y t -y s ) Guides above and below axis of bore =x t + x 2 -u(y t +y 2 ) Guides entirely below axis of bore. Hence the total resistance to recoil becomes during the powder period, K*=K + AR g (Ibs) and this value should be used in the computation of the trunnion and elevating gsar reactions. Strictly speaking, the value of K is slightly high, since the augmented friction due to a large powder pressure couple, will diminish the maximum velocity of restrained recoil and thus the resistance to recoil for a given displacement b. A more exact value of the resistance to recoil can be estimated as follows: Thus, (K+AR tf )T* i (K+AR ff )T K[b-(E )]= - m r [V f * ] 2m r 2 m r Simplifying, we have, * r f *' J 2m_ R. ? (Ibs) 154 bore 2uP a e 2nF m e R tf - - (approx)(lbs) "+x+u(-y) l-2u r (lbs) Fro interior ballistics, we have, ' w aVu P* 1.12 - (b'+u ) 8 w = weight of projectile (Ibs) u o - total travel up bore (ft) v = muzzle velocity (ft/sec) b'-u [(|l-*-l) /(l-ll-) a -l] <"> ~ (ft/sec) u where P,= total maximum powder force (Ibs) Unless the powder pressure couple is excessive, that is the center of gravity of the recoiling parts is considerably below the axis of the bore the above refinement in calculation is unnecessary. When e exceeds 1.5 to 2 inches the above effect becomes of consideration. INTERNAL STRSSS IN THE It is very important to RECOILING PARTS observe that the braking force 8 when treating of the external forces on the re- coiling masses as in the previous discussions refers always to the reaction of the oil in the hydraulic brake and the spring or com- 155 pressed air reaction of the recuperator. During the accelerating period the reaction on the gun lug nay differ considerably from the braking force B due to the acceleration of the piston and rods where these recoil with the gun or to the acceleration of the re- cuperator sleigh or slide when the sleigh recoils and the rods are fixed to the carriage. If now we consider a recoiling mass consisting of a gun together with a single cylinder recoiling with the gun, figure (4) and if we let, B 3 the total braking force along the axis of the cylinder. B =the normal reaction of cylinder on the gun lug. j =the tangential or shear reaction on the gun lug M -the bending moment reaction on the gun lug. Neglecting the guide friction, let, Q I and Q S be the normal guide reactions x t and x ? the coordinates along the axis of the bore of the clip reactions with origin at the center of gravity of the gun. x and x the coordinates parallel to the axis of the bore with origin at the center of gravity of the recoiling parts. x c and y c =the coordinates of the center of gravity of the recoiling cylindr with respect to the center of gravity of the gun as origin. e^- distance from the center line of the recoil cylinder to a line through the center of gravity of recoiling parts parallel to the axis of the bore. .M r and W r = mass and weight of the recoiling parts. M c and tf c * mass a n d weight of the recoiling cy- linder. j = the distance from the shear reaction on the gun lug to the center of gravity of the gun 156 itself. F the maximum powder force along the axis of the bore. If the mass of the lug is negligible as compared with the mass of the gun, the coordinates of the center of gravity of the recoiling mass with respect to that of the gun becomes Vc Vc x r -r and y r = e= "r "r Further * t - - - and y y - -^j " " .- s * and *. 3 ^ Now considering the gun with its lug alone, the reaction of the recoil cylinder on the lug, consists of the pull B a bending moment W c cos (x c + j ) and a shear reaction W r cos 0. Taking moments about the center of gravity of the gun, we have fi'(b+e) * \ c cos A j-W c cos 0(x c +j ) aQ 8 x ^* B'e b *B'e-W c co. x c =Q t (x t + -~^)*(a r w r w r Simplifying, we have, B'e b +B'e = O t (x t +x t ) + W r cos x a Considering the recoil cylinder alone we note that during the accelerating period, d'x B -B+W c sin0 M C - df 157 but from the recoiling mass, we have, d*x F-B+W r sin dt W r hence B' = B-Vf c sin + r-(F-B+W r sin 0) and substituting in the previous equation, we have, V (B-W c sin + --(F-B+W r sin 0)(e b +e)=Q t "r "r 5 but y c = 8t>+ and y r = e hence (ejj+e)= - 'r W r e "c Therefore, substituting in the above equation, we have, 8(eb+e)-W r e sin 0+Fe-Be+W r c sin = Q 1 (x t +x t )+W r cos ) ; hence Beh+Pe =Q (x +x )+W_ cos x. Obviously if we consider the recoiling mass, fig. (4) we have, taking moments about the center of gravity X Qx tut Q-Q s W cos A Hence Be^ + Fe - Q t (x t + x f ) + W r cos x a the same equation as obtained above as of course we should ex- pect. The above discussion shows the importance of considering either the mass of the gun with its proper external reactions or the mass of the recoiling parts with its proper external reactions and not confusing the mass, of the gun and recoiling parts, and the co- ordinates of their center of gravities. The maximum stress in a section m - m, see fig. (4) of the gun lug obviously occurs when the bending moment due to the weight of the recoil cylinder is a minimum and the braking force B a maximum that is at maximum elevation, In the above discussion the normal reaction between the 158 piston surface and cylinder was assumed zero. This reaction obviously depends upon the weight and relative deflections of the rods and cylinders. If these weights were equal and at the same distance from the point of support, and with equal elasticity, this reaction becomes zero and we have the bending moment assumed; but since the rods are relatively very elastic as compared with the cylinder in general the moment W c cos (x c + j) if anything is augmented. If I mn is the moment of inertia of the section " - m", AJJ-U its area of cross section, and y is the distance to the edge from the neutral axis of the section and "g" the distance from 6 to the neutral axis, we have for the maximum fibre stress cos 0(x c tJ)]y f c cos + - (34) c where B =B-W c sin + (F-B+W r sin 0) w r Since the weight components are small as com- pared with the powder pressure force and braking for a first approximation, we have, w c [B+ -(P-B)lg W r _ a-n> = (35) l m-m which is a useful formulae for practical design. TIPPING PARTS The tipping parts consist of all the parts that move in elevation with the gun. The two principje parts of the tipping parts are the recoiling parts and cradle, the one moving in recoil and the other remaining stationary. The cradle supports by its guides the 159 recoiling parts on recoil, it takes the reaction of the braking exerted on the recoiling mass and is sup- ported by trunnions resting in bearings in the top carriage and is further prevented from rotating about these trunnions during the recoil by the reaction between the elevating pinions of the top carriage and the elevating arc of the cradle. When a rocker is in- troduced between the elevating pinion and cradle for an independent line of sight it should, properly speak- ing, be included in the tipping parts. It is of fundamental importance to always balance the center of gravity of the tipping parts about the trunnion axis since with massive parts the elevating process must be done quickly and with the minimum re- action on the elevating pinion of the top carriage. Let x and y = the coordinates parallel and nor- mal to the axis of the bore. X and Y = the x and y components of the trun- nion reactions. F = the total powder pressure force. E = the reaction between the pinion and elevating arc. j 3 the radius of the elevating arc. 6 e a the angle between the "y" axis and the radius to the elevating pinion contact with the elevating arc. The mutual reaction between the tipping parts and 'top carriage may be divided into the component reactions X and Y of the trunnions and the elevating arc reaction E. By D'Alerabert's principle, considering the inertia of the recoiling mass as an equilibriating force, we have during the powder pressure period assuming the gun practically in battery, for equilibrium of the tipping parts, that, fig. (5) ,t F-M r ~ -2X +W t sin +E cos G e * (1) d u for motion along the "x" axis, 160 9 ~. s ?.} no tsdiexe fco -;*oq for motion along the "y" axis, and ^ M t, w d*x bnr p ( + s)-M r J^T 8-E j = For moments about the trunnions, the weight of the tipping parts having no moment since the center of gravity is at the trunnions in battery. 1 i d*x But F-M r T~T =K a fcns V otal resistance to recoi|., T *pj ?. 1 -,..-, . dur in g th8 accelerating period. Hence equation (3) reduces to: Pa + K.s-E j = and DO ^ the reaction on the elevating arc in battery becomes, a ( 4 V ' J fU lc eixs 5dJ p4 and the trunnion reactions in battery, becomes, (Fe+K a s) 2X=K a *W t sin * cose e ) ( fcn* nointq edJ ne>ev?ift.7^nT~-- ;- fc{1 .* fl * '5 5 j_/ ,13JJuOf Sri-* / / dd^ 1C 8S4*M!ibT003 "^" bft* "x" &sntiL'to , * . J.-- aQnsi& .! .- *di to ' ^ * (17) (Ej-W r h r ) < 2Y r -E sin 9 -W r cos - cos B ) K which shows the rocker reactions depend only on the elevating arc pressure. If now we consider the equilibrium of the tipping parts not including the rocker, we have, fig. (7) 2J('=K r +(W t -W r ) sin 0+M sin B ) ( 2Y'(W t -W r ) cos -ti cos B ) (18) and K r s+W r x cos 0-V' h r M= (19) k since the moment of (Wt~ w r ^ about the trunnions= -W r h r but now from equation (12), we have 2X=lfX'-2X r ) 2Y=2Y'-2Y r ( hence substituting the values obtained in (17) and (18), we have, * To account for tooth contact, more strictly, re- olaoe j to j cos 0.; oos 0. to oos.(0 +20) and sin. to 168 w -t bnsqsb eueJ:49Ai won . -. o ft. r i fit . "I . i * tn & ( Ot 169 ( 2X=K r +(W t -tf r ) sin 0+14 sin: B- -* - iiu^nof; gnt-rqe thus checking the formulas derived for the rocker reactions. oa be;.. boil\o s/iiiqe 8.Hj ni SfiiTqs tsriT TRUNNIONS LOCATED AT In guns shooting at -hijjjh J THE REAKr BALANCING^ r . ,elevAtiV< p o + ' H t h t cos rn ~ C[d h -(r +r t r n _,)A0] d h~ d o = r o+ r t~" r n-i ) A0 Strictly speaking the angle A0 in the above pro- cedure should be augmented by the angle r " n ~ r radians- Where D = the perpendicular distance from the trunnions to the extremity of the equivalent cam radius. From the equivalent radius thus obtained the cam contour may be drawn by drawing in a curve always tangent to the 172 perpendiculars to these radius, drawn from their f. extremities. With a balancing gear or equilibrator, the trunnion reactions are modified and now become ___ - r it T = the tension in the chain. a = the angle T makes with axis "X", (takea *) j along the axis of the bore) Ks*W r Xcos 0+Fe -a iui)deos e *T cos *>3)e-c onj- \n& ;,,^ bns > )sin6 e *T sin a iT( J I v,d Jni:oq")B and soihsT IsiJ/ni: ftri' J Y ) j -f"' ei^ns-Jnsaaeioni ne i!8 Usually d = 0. and thus the Y component of the trunnion reaction is unaffected. DIRECT ACTING BALANCING Another form of GEAR balancing gear, for nedj balancing the tipping '3+ 'O~ i,P arts at. all angles of <, ' elevation about the trunnions which are located to the rear, consists of a spring or pneumatic oscillating cylinder and its rod directly connected between the tipping parts and the top carriage. In the position of the tipping parts at zero elevation, gives maximum moment and therefore re- quires the axinua balancing reaction. $rtj moil ocf.flJaib IB i jdW ~ To .ccount for tooth contact of .levatl.g ohaniSB, replace j to j COS 20, oos S to oos (O t 20) rd .In d, to .in( e+ 20). n - sri.t OJ 173 M ..-? .noiievele ed.4 &ijtor> ae eseeio W t = wei S nt f th8 tipping parts n t = horizontal distance from the trunnions tcbijr;nu} the center of gravity of the tipping parts -11^0 %nii el j(g un- i^ b *Vt &ry.<\ r, o x n n 01 4 eri J fil,f.! .u^jfc *od y t - coordinates alon^ and normal to bore ; -oneisd orjfiiiiuffroift- trunjiion to center of gravity of tipping parts (gun in battery:)i.2.rq 5jni - angle of elevation., Juov.*! VJnifil9Tq s io1 m = maximum elevation>.. erfj Ic sixs oornnot^ arii gni r = radius from the trunnion to the crank pin b ft! j, llE: f*hich connects the tipping parts to the pi*-; vfciJfcon rod of the oscillating cylinder, (in.) iBi^cni R = reaction exerted by the balancing gear along the piston rod of the oscillating cylinder;. s) 3 iscfnos iiori TO Isi^iai siU : d t = moment arm of R about trunnion. (in) d = deflection of spring at horizontal elevation. d o = deflection of spring at maximum elevation. (in) ,ooi**veJs uwsilxsa icl seasooed C = spring constant. RJ = initial, balancing reaction (0 elev.) R t = final balancing reaction (0* elev.)(lbs) S = stroke of piston in oscillating cylinder. (in) p t - final ,air piressure j.^ pneumatic balancing cylinder (Ibs/sq.in.) ,| = initial air pressure in pneumatic balancing cylinder (lbs/sq.in.1 ^,f, effective area of balaacing pistoa ..(sjj.> c inj._) j V = initial air volume (cu.in.) *\L ,'J3bniI^o Jniiq* 6 ri; :? At any angle of elevation 0, we must have, R d t E t (x t cos J0-y- t sin 0) io? In general the center of gravity of the tipping parts lies approximately along the axis of the trunnions, and therefore, R d = W x cos 0. 174 If d t remains constant, the reaction R should de- crease as a cosinef unction in the elevation. Since it is usually impossible to decrease R according to a cosine function, we may so locate the trunnion of the oscillating cylinder so that the product R d t = Wt x t cos j0. By properly locating the trunnion axis of the oscillating cylinder a very close balancing is possible throughout the elevation either with a spring or pneumatic balanc- ing piston. For a preliminary layout, we may start by locat- ing the trunnion axis of the oscillating cylinder somewhere depending upon clearance considerations, along a line parallel to the chord joining the assumed initial and final positions of the crank and midway betwean the chord and middle of arc. The crank turns an angle equal to the total elevation & m* Then the initial or horizontal balancing reaction, becomes, r 0m R^ - (1 + cos r)=W t x t and the final balancing 2 A-, hence the moment equation about the trunnion reduces to , m r - . t - t x t cos + t y t sn + r x cos 0-Ej= QTf but due to the balancing gear, we have, "t x t c 3 ~yi s ^ n and further K=m r - that is, the inertia resistance, d t equals the total resistance to 180 recoil. Hence K s + W r x cos For motion along the x axis, we have, ( 2X=K+R sin e r +E cos e e +W t sin ( ) and for motion along the y axis, ( 2Y=f t cos 0+E sin e+g sin9 s -R cos e r ) I where as before ; 2W t x t cos R= roughly. ) o> ( r( 1+cos - ) f ) From the above analysis, we see, therefore, that the elevating arc reaction remains the same with or without a balancing gear, while the trunnion reactions may be increased or decreased according to the location of the line of action of the balancing gear. INTERNAL REACTIONS OF The rocker reactions depend TIPPING PARTS WITH solely on the elevating gear BALANCING GEAR - reaction, and with a balanc- ROCKER INTRODUCED. ing gear, the elevating gear reaction is independent of the eccentricity of the center of gravity of the tipping parts from the trunnions. Therefore, the rocker re- action on the trunnion is entirely independent of the reaction exerted by the balancing gear or counter- poise. In brief, the rocker reactions remain the To account for tooth contact of elevating echanisB, substitute j cos 2O for ,i E oos(O.-20) for B oos 6 8 and B Bin (O - 20) for B sin O. 181 same with or without a counterpoise or balancing gear. The reactions exerted by the top carriage on the trunnion do however depend on the mag- nitude and direction of the balancing gear. Therefore, the shear and bending at the section of the trunnion adjoining the cradle must also depend on the balancing gear or counterpoise re- action. . An analytical proof of the reactions is given as follows:- Let X and Y = trunnion components of the reaction of top carriage. X r and Y r = trunnion components of the re- action of the rocker. X and Y = shear components of the trunnion pins on the cradle. W r = weight of rocker. E = elevating gear reaction on rooker M = cradle reaction on rocker. j = radius to elevating gear arc. k = the perpendicular distance from the trunnions to the line of action of M which is the line through the axis of the elevating screw, when used, or the normal to the contact surface of rocker and cradle when an elevating screw is not used. B = the angle between the line of action of M and the "y" axis. x ffl and y m = the "x" and "y" coordinates of the cradle hinge joint of rocker elevat- ing screw or the center of contact of rocker on cradle. x r and y r = the "x" and "y" coordinate of the center of gravity of the rocker and is measured from the trunnions towards the breech and downward. 182 Evidently for the shear at the cradle section of the trunnion, x'= x + x r ) ( Y' = Y + Y r ) For the angular equilibrium of the rocker, Ej-Hk-lf r (x r cos - y r sin 0) = if we let x r cos 0-y r sin = h r , then Bj-W r h r Ha 1 , where k = x ffl cos B + y m sin B, that is, the cradle rocker reaction depends solely on the elevating gear reaction. For the translatory equilibrium of the rocker, 2X r = M sin B-E cos e e -W r sin 2Y r =E sin (J e ~W r cos -M. cos B which shows the rocker reaction at the trunnion de- pends only on the elevating arc pressure and there- fore is independent of the counterpoise reaction. Considering the equilibrium of the tipping parts not including the rocker, we have, 2x' = K r -i-(W fc -1f r ) sin +M sin B*R sin 6 r 2Y*=(W t -W r ) cos 0-M cos B-R cos e r Further if measured from the trunnion axis, l t =x t cos -y t sin 0= the horizontal distance to center of gravity of tipping parts(re- coiling parts in battery) l t = the horizontal distance to center of gravity of tipping par.ts (recoiling parts out of battery) l r = the horizontal distance to center of gravity of recoiling Parts in battery. 183 l c = the horizontal distance to center of gravity of cradle . -h r = the horizontal distance to center of gravity of rocker measured in a negative direction from the i's. d^ = moment arm of the counterpoise reaction R about the trunnions. Then since the moment of the tiooing parts minus rocker about the trunnions is equal to the moment of the weight of the tipping parts minus the moment of the weight of the rocker, we have, W~ h=?1 ~ x cos Now If r l r +W c l c = T t l t +W r hr hence W t l^+W r h r =W t l t +W r h r ~W r x cos =W t (x t cos 0-y t sin J) + W r h r ~W r x cos therefore, K r s+Rd t -W t (x t cos 0-y t sin 0)-l? r h r +W r x cos -Wk= but for equilibrium of the tipping parts in battery Rd t =W t (x t cos 0-y t sin 0) hence K r s+W r x cos 0-1 M = Since, however, 2X=2X'-2X r ) 2Y=2Y-2Y r We have in substituting the previous values for 2X* and ZXp^j, 2X=K r +W t sin +R sin 9 r +E cos 6 e 2Y=lf t cos 0-E sin 9 e ~R cos 6 r 184 K r s +W r x cos and E = J In the preceeding analysis it is important to note, that the center of gravity of the rocker is assumed to the rear of the trunnions, and the elevating gear reaction is considered positive when the radius to the pinion contact of the elevating rack is measured counter-clockwise with respect to the "y" axis through the trunnions. Evidently when Q Q is negative (i.e. clockwise from "y" axis, E cos 6 e remains the same but E sin 6 e becomes negative in the above equations. ol' EFFECT OF RIFLING Cue to the rifling, the TORQUE ON TRUNNION torque exerted on the gun by REACTION the shell aust be balanced in considering the equilibrium of the tipping parts by an equal and opposite moment exerted by the top car- riage on the trunnions(assuming due to the ouch greater flexibility of the elevating arc and pinion that the elevating arc reaction is entirely unaf- fected). If the rifling is right handed, then in the di- rection of the muzzle, the Y component of the left trunnion is increased and the Y comoonent of the right trunnion is decreased by the amount equal to the torque of rifling divided by the distance between the trunnion bearings on the top carriage. Usually this affect is quite negligible as compared with the ther forces exerted. STRENGTH OF THfe TRUNNIONS. The critical section of the trunnions is usually where the trunnion joins the cradle. Let "n" represent this section on the trunnion, see fig. (9). 185 xr.-s no n 186 a = the distance from "mn" to the center of the top carriage oearing. b = the distance from "mn" to the center of the rocker bearing . M X = the bending moment at "urn" in the plane of the X component reactions. Hy = the bending moment at "mn" in the plane of the Y component reactions. M = the resultant B. M. on the section, "mn" . f = maximum fibre stress. D = diameter of the trunnions at section "mn" I = the moment of inertia of the section about diameter. Then M x = Xa+X r b M y = Ya+Y r b and M ,/^J~" further nD 4 MD I = and f = 64 21 32M 10.18M hence f = = r nD D Usually the fibre strass is limited from to of the elastic limit of the material used, 2 3 and the minimum diarae ter of the trunnions becomes /10.18M D = 7 The shear stress is usually negligible as compared with the bending stress. LIMITATIONS ON THE EXTERNAL In considering the REACTIONS OP THE TIPPING external reactions on PARTS. the tipping parts, we have to consider the limitation imposed on the elevating arc reaction and the reaction on the trunnions by the top carriage. 187 ELEVATIHG ARC REACTIOH: (1) The elevating arc reaction is reduced by reducing the perpendicular distance between the line of action of the re- sistance to recoil, which passes through the center of gravity of the recoiling parts parallel to the axis of the bore, and the trunnion axis. When the line of action of the resistance to recoil passes through the trunnion axis, the reaction on the elevating arc in battery is zero if we neglect the effect of the powder couple and is equal to the moment effect of the recoiling weight when the gun is out of battery. The elevating arc reaction is re- duced proportionally to the increase of the radius of the elevating arc. The elevating arc reaction should al- ways be considered in the limiting con- ditions of in and out of battery, that is, with the maximum powder pressure couple acting and out of battery when the maximum moment effect of the re- coiling weight about the trunnions exists. (4) When the resistance to recoil does not pass through the trunnions the elevating arc reaction due to the short- ening of recoil is a maxim at max. ele- vation. TRUNNION REACTIONS: (1) The "X" component of the trunnion reaction (i.e. the component parallel to the bore), depends upon the total resistance to recoil and the component of the elevating arc reaction parallel 188 to the bore as well as the weight component of the tipping parts when the gun elevates. (2) The "Y" component depends upon the weight of the tipping parts and the component of the elevating arc pres- sure parallel to the "y" axis. (3) As the gun elevates the component of the elevating arc pressure parallel to the "x" axis decreases but the weight component increases and due to the shortening of recoil on elevating the resistance to recoil increases. (4) The component parallel to the "y" axis of the elevating arc reaction increases but in a negative direction, thus tending to decrease the Y component of the trunnion reaction. On high elevation because of the large re- sistance to recoil for a short recoil, the elevating arc pressure parallel to the "y" axis more than compensates the decreased weight component of the tiooin^ parts thus causing a reversal of direction of the Y component re- action of the trunnions. (5) Thus in general the X component in- creases while the Y component decreases, verv often reversing on elevating the gun and thus the trunnion bearing con- tact may shift 90 or over. STRESSES IN CRADLE OR The reactions on the RECUPERATOR FORGING. cradle or recuperator are: * (1) the trunnion reaction of the top carriage on the cradle: (2) the reaction of the elevating arc which is equivalent to a single force in the direction of the elevating pinion re- 189 action on the elevating arc together with an addition al moment: (3) the reaction of the recoiling mass on the guides: (4) a result and reaction parallel to the longitudinal axis of the cradle or the guides due to the various "pulls" exerted on the recoiling mass and (5) a distributed load which is uniform if the cross sections remain the same due to the weight of the cradle. In an accurate computation of the stresses in a cradle it is necessary from a preliminary layout of the cradle to locate roughly the neutral axis of each section and connect these points for a longitudinal neutral axis line. "We may then treat the cradle as a simple beam, talcing into account the bending moments caused by eccentric loads such as pull reactions off the neutral axis, guide frictions, etc. The trunnions usually are located considerably above the neutral axis and the X component of the trunnion reaction causes a large eccentric load with a consequent large abrupt change in the bending moment diagram. This is usually a characteristic in the bending moment diagram for all cradles or recuperators using guides. Let us now consider the various diagrams showing the characteristics for bending moment, direct stress and shear for the "Filloux" cradle as well as for the "240 m/m Schneider Howitzer" cradle representing typical cradles with guides (figures 10 and 11). Neglecting the weight of cradle as relatively small, and letting M t = max. bending moment at the trunnions. M c = max. bending moment at the point of contact of the elevating arc with cradle. Q t and Q ? = the front and rear normal clip re- actions. x' and x' = the "x" coordinates of these re- 1 2 actions with respect to the trunnions, d and d = the distance of the friction components 1 2 of Q. and Q from the neutral axis. 190 -- SENDING STRESS 155 Mr^n GUN CRRRIRGE: MODEL OP 1916 FILLOUX CRRDLE SHE1RR Fig. 10 191 B = the resultant of the "braking "pulls" reacting on the cradle. djj = the distance from the neutral axis to "B" d x = the distance from the neutral axis to the trunnions . X and Y = the trunnion reactions on the cradle. Xg and Yg = the elevating arc reaction on the cradle . Me = the moment exerted by the elevating arc on the cradle. = the distanc , _^^ the distance from the neutral axis to X e x e = the "x" coordinate of the elevating arc contact with respect to the trunnions. The bending moment changes at the trunnions by the amount 2X d x . Now for guns with "braking pull" reactions on the cradle in the rear as caused by the compression of the oil and air in the recuperator as in the 155 m/m Filloux, we have for the bending moments at the trunnions, M t = Q t (x t + ud t ) just to left of trunnion. M t = Q t (x t + ud t ) - 2X d x (just to right of trunnion) and at the elevating arc contact, M C = V 1 - x e * Ud 8 ) - Bd b As a check, we also have, M c = Q t (x^ + x e + ud t ) + 2Yx e - 2Xd x - Me The bending moment M C is usually distributed caus- ing a parabolic curve as shown in the B. M. diagram of the 155 m/m Filloux.. For guns with "braking pull" reactions on the cradle in the front, due to the tension in the stationary hydraulic piston and recuperator rods, as in the 240 m/m Schneider Howitzer, we have for the bending moment at the trunnions 192 /"ft "1 ' P J \ CRADLE 240 MM HOWITZER SCHNEIDER 1916 BENDlNCa MOMENT DIRECT STRESS SHERR rig. ii 193 1 ) -Bdj ) ( just to left of trunnions) and M t =Q t (x| + ud t )-Bdjj-2X d x ( just to right of trunnion) Further M c = Q t (x|*x c +ud t )+2Yx e -2Xd x -Bd b or as a check M c =G z (x^-x c + ud,) In order to compute the maximum fibre stress at a critical section, we must include the direct stress caused by the component of the reactions parallel to the X axis in addition to the fibre stress caused by bending. Therefore a direct stress diagram has been drawn for typical gun carriage cradles. For the case where the braking reaction it in the rear from the front clip to the trunnion, we have, A compression = uQ t which is small and nay be neglected as compared with the bending. From the trunnion to the rear clip, we have at the trunnions (just to right of trunnionslsee diagram] a tension = X-uQ ) at the elevating arc contact section a tension. = 2X-uQ 1 -X c Where the braking reaction is in front, we have from the front clip to the "braking yoke" on the cradle or ac a section through the point of application of the tensions of the rods on the cradle, we have a compression =uQ From the braking yoke section the compression increases to ud t + B at the trunnions A compression =uQ t +B(just to right of section) A tension =2X-uQ t -B 194 At the elevating arc section.: A ^fltftf j .,to.< ri ' 3 -ax-ua -B-X- rfodffo s e-.- x c latUio'? Shear diagrams for the 155 ra/m Filloux and 240 m/m Schneider Howitzer show the variation of the shear in these typical cradles. To recapitulate if y t and y c = the distance to the extreme fibres from the neutral axis at the trunnion and elevating arc section, aod if A t and A c are tbej3 2yeo areas of these respective sections, I t and I c cor- responding moments of inertia, we have for the ex- n n9 uj,. $ ._ ^ ^ |_! ~t ~t -~-f-^- I" t 1 * * 4 4 1 I i > r 1 c > -* * fl p ^| 4 ^ 6 5f! -. U U E^i i bn* O\RE1CT STRESS 4s ^c / ^^ aoiloee add 0^ I*ii?a r. ?> ol rose aai^Bib it- ^^^^ SHERR _ r^ ^j ! &n i 9 d _ \NCH RRU-W/AY HOV/\TZE1R MODE.U OF CRADLE Fig. N3 199 being no eccentric pull on the recoiling mass we have a distributed load due to the weight of the recoiling mass as shown in the figure In the 8. M. diagram we find the B. M. at the trunnions to be relatively small it being merely equal to that due to a distributed load equal to the weight of the recoiling mass, together with ths friction caused by this loading. w Thus, the intensity of loading = x +x *'Y3 "50 HTCH Hence the B. N. at the trunnions, becomes, t . jt t W r x W r + u d !R* ( AT o - and (P^-Q^ud+x'+XhVzYx^jy X-P h -uQ t ^t = " - i at right of section. 200 ?or barbette sleeves with pulls symmetrically balanced about the axis of the bore: Section at trunnion: STRENGTH OP CYLINDERS AND The strength of a RECUPERATOR PORGI>K3S. recuperator forging is a matter of vital im- portance since in modern artillery the tendency is to use higher and higher pressures con- sistent with the various cylinder packings used, and to stress the forging higher with smaller factors of safety. Hand in hand with this goes the metallurgical side where improvements in the quality of the steel with higher ultimate strength are con- stantly being made. High stresses in the recuperator forgings as with high ultimate strength and low factors of safety reduce the weight of the carriage and its cost considerably . Weight of course is of fundamental consideration for mobility. Hence it is of importance to calculate the stresses in the cylinder walls to a considerable degree of accuracy. The maxim stress in a recuperator forging is a combination of the following: (1) A bending fibre stress normal to a plane section perpendicular to the longitudinal axis of the forging which is caused by the external reactions exerted on the forging during firing. (2) A radial compression stress along a radius of the cylinder or normal to a cylindrical surface which is equal to the pressure in the cylinder for the inner surface. 201 (3) A tangential hoop tension, which is normal to a plane passing through the longitudinal axis of the cylinder. Obviously these stresses are principle stresses and with the aid of Poisson's ratio we may arrive at the resultant intensity of stress. In a first ap- proximation however it is sufficient to consider the tangential hoop tension alone, the effect and magnitude of the other stresses small. Consider now a single cylinder 1" long, and sub- ject to an internal pressure p, and external pres- sure p t and of inside radius R and outside radius R t respectively. Further let r = the inside radius to any differential lamina of the cylinder wall (in inches) d r = the radial thickness of the lamina p r = the radial compression at radius r (Ibs/sq.in) p t the tangential or hoop tension (Ibs/sq.in) the modulus of elasticity. Cj = the longitudinal strain. e r 3 the radial strain e t = the tangential or hoop strain. Then, for the equilibrium of a differential lamina d r , of length "1" along the axis of the cylinder and a peripheral length equal to the circumference, we have, 2p r lr-2(p r +dp r )l(r+dr)=2p t ldr hence d Pr d(rp, -pp-* 1 - E- d, Pt a ~Pr- r - - -T-^ (D V Let us further assume plane transverse sections to remain plane under pressure. This assumption is reasonably close to actual conditions for plane trans- verse sections some distance from closed ends, and in the case of a recuperator forging where the intensity 202 of longitudinal stress, i. e. the bending stress on transverse sections, is relatively small, except for extreme fibres from the neutral axis of the transverse section. We are, therefore, not greatly in error in as- suming the longitudinal strain to remain constant over the entire cross section, hence, vswcd c.: ej = - (pi ) = constant^ 1 q< . ii ; edJ lo where p^ = intensity of longitudinal stress, ev nt oi Jost hence p t -p r =k (2) ^ MU( d Pr d P Pt"Pr = ~ 2 Pr" r cT^ or k + 2p r = -r ^7- therefore dpr ^ k+2 Pr " ~ Integrating, log(k+2p r )= - leg r* + c or k+2p r = - r c k hence p r = - - - (3) ^ r* & ^c_ k * qoc 2r* 2 , Substituting (3) in (2) where p r =p o r =R Q inside conditions r = R t outside conditions c k hence P Q - po 2 . c k P = ' - - oJ 1 2R 2 2 i_ p _p _ _ f K \ -39 88T9V r o r i vo^ ,T a to 203 Now eliminating c and k, respectively, we find t c - -- - - and k = r i riiwi fi^.f*f bearing which is usually in the front and the V com- ponent exerted by the traversing circular guides in the rear. Sufficient horizontal play is allowed so that the reaction of the horizontal traversing guides is only vertical, the H component being taken up entirely at the pivot bearing. As a typical class (1) top carriage we may illus- trate by the Vickers 8", Mark VII, British Howitzer. ; siU a-oil --SJ.TBCJ ni EJieq 211 Further let 1 = distance between supporting reactions measured horizontally in the direction of the axis of the bore at traverse. A = the front pivot point. r B = the resultant of the distributed vertical re- actions of the horizontal traversing arc guide. 1 k = the horizontal distance to trunnions from B in the direction of the axis of the bore o at traverse. h t = height of trunnions above the traversing guides. S = the perpendicular distance from the trunnion center to the line of action of the resistance to recoil which necessarily passes through e center of gravity of the recoiling mass. = height of horizontal component of pivot re- action above the horizontal traversing guides. = weight of top carriage proper. = moment arm of W about B. Considering fig. (15) we have for the horizontal component of the pivot reaction, H a =K cos /5 and taking moments about fl, the center of pressure of the traversing guides, we have, Kd-W^- t *W r x'cos 0-W tc l tc +V a l-H a h a = ( . W t l t +W tc l tc -W r x cos 0-K(d-h a cos 0) ) hence V a = ' ' *-h a cos 0)+W t (l-l t )+lf tc (l-l tc )--H' r x cos ( and V b = where for low angles of elevation, d=h t cos 0+S-lt sin d' = h t cos ^--(l-l t )sin 0+S 212 Fig. 15 213 and for high angles of elevation, d*l t sin 0-b t cosl 0-S d' = (l-l t ) sin 0+h t cos 0+S y mn = the horizontal distance along the axle from the center of the wheel bearing pressure. Considering, max. traverse, right handed, at max. elevation, the reactions on the axle to the left of the section, become, (1) The components of the trail con- necting arm reaction on the axle:- X,Y and Z together with a couple M xy in the horizontal plane. (2) The vertical reaction exerted by the left wheel, S a . Therefore at section "mn", we have, (1) Bending in the vertical plane: (2) Bending in the hor- izontal plane: < in lbs -> (3) Shear in the vertical plane : X+S a (Ibs) (4) Shear in the hor- izontal plane: X (Ibs) (5) Torsion about the y axis, or in the x z plane: T=X Z Bn (6) A direct thrust: Y (Ibs) Thus, we have bending in two planes combined with torsion, and a direct thrust as well. Then for a 214 round section, as would^ Dually be the case, we h*,jon6 f 3 - - + - _ Max% normal fibre - 78J thrust on outer layer oil fx* etu c,ns.t8ifa 1*5 (Ibs/sq.in) gniised Jaedw &dJ lo -iscfnso sr)J , .- O j elxfi eri^ no iis-U od^ io -ralxs ori^ no noj ";s iniJoso The m-axi/num fibre stress, therefore, becomes _ ._antq Ifi-tnoxiiori erid ni f =-f + /**" Jo* en ]80lj18v ftbs/sq.in) 3 4 a 2 * j 2 which should not exceed of. the elastic limit of the material to be used. As a typical class o'f pivot yoke type, consider the reactions on a Barbette or Pedestal mount, figure (16) and a pivot yo.ke top carriage used on a trail carriage, fig.ure (16A). In the first type, the lower bearing sustains both horizontal and vertical com- ponent reactions, whereas the upper is merely a floating bearing and therefore sustaining only a hor- izontal component and designed to prevent bending in the lower pivot. In the second type, the middle bearing has a tapered fit within the axle, and therefore sustains both horizontal and vertical components, but suffers no bending moment since the axle is free to rotate. To prevent the top carriage and mount from rotating about the axle a lower cylindrical vertical pivot fits within an equalizer bar below the. axle, the equalizer bar being attached to the trails. (See Theory of Split Trail - next section). In this type of mount it is more convenient to compute the supporting reactions in terms of the hor- 6 10^ nsriT .Ilsw 8 Jsind.* joetifa b.~. 215 foul as o l3JUitr bn* isJ-nosx aoioiq SaiTOKE. TOP cerfT 1 J II ^4^ IS 8OO ~n ale {- aie :n -ZH 216 izorual and vertical components of the trunnion. If n e = angle made by radius to elevating pinion contact on elevating rack with the ver- tical. j = radius of elevating rack Then for the horizontal and vertical components of the trunnion reaction, we have, Fe+Ks >K cos 0+ ( ) cos n e ) In battery j ( Fe+Ks ) = B z x o- B x^o^> therefors (A-B ) (Z+g) 'Z "Z X, but lience A z =~ 2x ~~^~ (6) B z =W> + K s sin e) m -(S a +S b ) - A z (7) Let X, Y and Z = components of the reaction of the trail arm on the axle, M XV = moment reaction of trail arm on axle, in the X Y plane. Considering moments on the left trail and trail arm together about the axle, we have, A z x Q -A x (Z +g)-Q d e = hence, the horizontal shear reaction of the equalizer becomes, *. Next consider the various reactions on the trail arm, and we have, 224 = along the x axis, -Y+A = along the y axis. -Z+A Z = along the z axis. and further, -M xy +Ay(x o -x p ) '0 In the x y plane Therefore, the reactions of trail arm on the axle, becomes, A_x o -A x (Z +*) X=A+A z o * o + A., (Ibs) (9) d o Y = A y (Ibs) (10) Z A, (Ibs) (11) M xy A y (x -x p ) (in. Ibs) (12) Of AXLS MAXIMUM TRAVBR8B, MAXIMUM ELEVATION: This critical section of an axle is at a section near the center where the axle becomes enlarged for holding the vertical pivot of the top carriage. If the axle is made straight, we have no torsion on the section but mersly bending in a vertical and horizontal plane. If, however, the axle is underhung for clearance and lowering the top carriage, in addition to the bending, we have torsion as well, the nagnitude of the torsion depending upon the depth of the underhang. Let mn be the section under consideration near the center of the axle. x nn ^mn an ^ z mn = ^ 9 component distances from the center of contact of the trail connecting arm and axle. REACTION BETWEEN RECOILING During the counter re- PARTS AND MOUNT IN COUNTER coil, we may distinguish RECOIL. between the accelerating and retardation period so far as the. reactions between 225 the recoiling parts and mount are concerned. The re- actions during the acceleration however are of exactly the same character as during the recoil only of less magnitude. Therefore, from either the point of view of the internal reactions or stability of the mount, we are not concerned with the acceleration period of counter recoil. Therefore let us consider the various recoiling parts and mount coming into play during the retardation period of counter recoil, - Let (see figure 18) x t and v t = coordinates, along and normal to bore, of front clip reaction with respect to center of gravity of recoiling parts. x and y = coordinates, along and normal to bore, of rear clip reaction with. respect to center of gravity of recoiling parts. Q t s front clip reaction. Q 2 = rear clip reaction. W r = weight of recoiling parts. = unbalanced retarding force exclusive of f rict ion. ^0 - distance from center of gravity of re- coiling parts to line of action of 0. n = coefficient of friction = 0.15 usually. d 1 = distance from front wheel ground contact to line parallel to tore through center of gravity of recoiling parts. l r = horizontal distance from line of action of W r to front wheel ground contact. x = displacement along bore or guides from out of battery position. M a = moment of reaction of the recoiling parts on mount about front wheel contact and ground. Then, for the motion of the recoiling parts, we hava. 226 REACTION BETWEEN RECOILING PRRT5 RND MOUNT IN COUNTER RECOIL RECOILING PRRT5 227 d x 0+n(Q t +Q 2 )+W r sin 0=-m r - (1) d t Q t -Q 2 =W r cos (2) and d0-ft l x 1 -Q f x 8 +n Q t y t -n Q 8 y 2 = (3) Next, considering the reactions on the mount and taking moments, about the point of contact of the front wheels with ground A , we have, (4) - - d'tan 0+x J=M r A ' Substituting Eq. (3) and (2) in Eq. 4, we have immediately d"x rQ )d +W r sin 0.d -W_l = * 2 or (0+n(Q t +Q 2 )+W r sin 0)d '-W r l r =MA Further from equation (1) (-m r ^ )d'- W r l r = My Q t Hence, the reaction on the mount during counter recoil is equivalent to the total resistance to recoil acting in a line parallel to the axis of the bore and through the center of gravity of the recoiling parts, together with a component in line and equal to the weight of the recoiling parts. If further, we let, F y = the recuperator reaction. RQ = total guide friction R S+P 3 total packing friction. B x = total counter recoil buffer reaction. Then = B '+.. n - F,, a a ) = n W r C os (approx.) and the overturning force, passing through the center of gravity of the recoiling parts and parallel to the 228 bore, becomes, i dv - [F v -W r (sin +n cos 0)-R s +p-B x )= -ra r v - (Ibs) GRAPHICAL CONSTRUCTION AND Very often it is more EVALUATION OF THE REACTIONS convenient to evaluate IN A GUN CARRIAGE. the various reactions by graphical methods. Graphical constructions are of special value since they give a vivid picture of the relative magnitude of the various reactions. Further the method is comparatively simple and the closing of force and space polygons combined with overall methods gives an admirable check. The ac- curacy of the method even -with rough layouts is suf- ficient for the computation of the various reactions required. If we consider the kinetic equilibrium of any piece of the carriage, we have, by introducing the . kinetic reactions or inertia forces with the actual reactions exerted on the piece, a dynamic problem re- duced to a problem of statics. For equilibrium of the piece, we have, 2X = ) ZY = ( for a coplanor set of forces. ZM = ) Now the considerations ZX = 0, ZY = are met by the vector diagram of reactions having a zero re- sultant, that is the vector polygon of the piece closing. The condition ZM = 0, requires a consideration of the lines of action of the forces in a space diagram in addition. Since the moments may be taken about any point, there can be no resultant moment exist- ing. The condition 2X = 0, ZY = implies the result- 229 ant force to be zero, but does not imply the existence of a couple. Condition ZM 3 0. indies that a resultant couple cannot exist. A graphical method, therefore, always consists of two sets of diagrams, (1) a space diagram of forces and (2) a vector diagram of forces. The space diagram requires a layout proportional to the actual piece under consideration and the placing on this diagram the lines of action of the forces. The force diagram requires a layout proportional to the direction and magnitude of the various reactions exerted on the piece. The two diagrams must be carried on simultaneously since the direction of a resultant required in a sp^ace diagram, is obtained by the vector addition of the forces which are the com- ponents of the resultant. Since Vector addition is commutative, the order of Vector addition is immaterial. REACTIONS ON THE RBCOILIMG PARTS The known reactions consist: (1) The powder force along the axis of the bore Pfc . (IJbs) (2) The inertia force along an axis parallel to the bore and through the center of d gravity of the recoiling parts - - m r x ' s Zt r ^ (3) The weight of the recoiling parts acting vertically through the -center of gravity of the recoiling parts - - W r . The unknown reactions consist: A a; ** ^ ** * u " * (1) The resultant braking force B Ibs. (2) The front and rear clip reactions Q, and Q 2 Ibs. 230 The lines of -actions of these forces however are known or can be readily determined. Procedure: Layout a space drawing proportional to the dimensions of the recoiling parts, showing the assumed lines of actions of the various forces. See fig. (19;. 2 Since P^ - r = K the total resistance to recoil which is assumed as known, we have the effect of P K and m ^ * equivalent to, b G "r dt 2 (1) a couple Pfc % (2) a force K through the center of gravity of the recoiling parts parallel to the axis of the bore. Since a couple and a single force may always be reduced to an equivalent single force, we have (1) and (2) combined into a single force K acting at a distance above the axis through the center of gravity of the recoiling parts equal to P. e h p 9 (in) ( where CK is in inches) K The reactions and Q due to the friction in 12 i the cradle sleeve make an angle u = tan -*f with the normal to the guides, where f = coefficient of friction Q = 0.15 usually. Hence u = 8.5 approximately. Referring now to the force polygon or diagram, lay off K in the direction and equal to the magnitude of the total resistance to recoil. Lay off K = a b From b lay off b c = W R , the weight of the re- coiling, in magnitude and direction. Draw K + Hf = a c 231 232 Referring now .to tha space diagram lay off K at a perpendicular distance b 9 b K above the center of gravity of the recoiling parts and parallel to the axis of the bore. At the intersect- ion of K and Wp , draw J k parallel to a c until it intersects the line of action of the motion of the reaction Q 2 at k. From c of the force polygon, lay off c d and fn the direction of the rear clip reaction Q Z . Draw k c from k to the intersection of Q t and 8 in the space diagram. Draw a d parallel to k c in the force diagram until it intersects a d at d. This limits and de- termines the magnitude of in the force diagram. From d, draw d e parallel to B and a e parallel to Q . The intersection of a e and d e determines B and Q t respectively. Thus from a combination of the space and force diagram we obtain Q 2 B and Q t respective- iy- REACTIONS ON THE CRADLB. Referring to figure (20): The known reactions consist:- (1) The rear clip reaction Q 2 (Ibs) (2) The front clip reaction Q t (Ibs) (3) The weight of the cradle W c (Ibs) (4) The braking force B (Ibs) The unknown reactions consist:- (1) The trunnion reaction T (Ibs) (2) The elevating gear reaction E (Ibs) The direction of the latter being' known. Referring now to the force diagram lay off a b = in the direction and proportional to the magnitude of fl f . From fa draw c parallel and equal to B the brak- ing force. Draw a c. Referring now to the space diagram J k from the 233 in the force polygon, to the intersection of Q In the force diagram, draw c d = Q and parallel to Q. t . draw ad. In the space diagram draw J L parallel to a d to the intersection of W c . In the force polygon draw lf c equal and parallel to W c the weight of the cradle. Draw a c. In the space diagram 1 m parallel to a e to the intersection with E at m. From m draw m n to the trunnion axis, which gives the line of action of the trunnion reaction T. In the force polygon draw e f in the direction of E and a F in the direction of T. The intersection at f determines the magnitude of E the elevating gear reaction and T tha trunnion reaction. t yf "X REACTIONS OH THE TIPPIHG PASTS. Locate the trunnions along the resultant of the "battery position of W r and W c --- See upper diagram. Without balancing gear:- Considering the external forces on tipping parts, we have, the known reactions, (1) The total resistance to recoil K (l"bs) (2) The weight of the tipping parts W t (Its) The unknown reactions, (1) The elevating gear reaction E (l"bs) (2) The trunnion reaction T (l"bs) the direction of E "being known. In the space diagram lay off X parallel to the bore and at a perpendicular distance from the center of gravity of the recoiling farts = p b e ( . . In the force diagram, lay off ab = K and be = W t . Draw ac. In the space diagram, lay off J k from the inter- section of K and W t parallel to ac of the force diagram 234 235 I \ 1 Z o ^^-"' I r>- 238 and t o the intersection of E. Draw k 1 to the trunnion axis in the space diagram. The line of action of T is then along h 1 produced. In the force polygon, draw cd parallel to E and a d parallel to k 1 of the space diagram. Their inter- section at d determines the magnitude of and T respectively. With balancing gear:- Determi nation of the balancing gear reaction R. On the space diagram lay off Wj. the weight of the tipping parts in its battery position as well as the line of section of R. From the intersection of R and W t draw o m. This must be the direction of the result- ant of W t and F since the condition is that we have no moment about the trunnions when W^ is in its battery position. In the diagram below, lay off W. and R and draw o.m parallel to o m in the space diagram. This determines the magnitude of the balancing gear reaction R. Referring to the force diagram, lay off a b equal and parallel to K the total resistance to recoil, and be = W t the weight of the tipping parts. Draw ac. In the space diagram K is at a perpendicular dis- D p tance _fc_- f ro m the center of gravity of the recoil- ing parts and V^ at a distance W r x cos * from its battery position, where x is the displacement in the recoil At the intersection of K and W t draw j k parallel to ac to the intersection of R. In the force polygon draw cd parallel and equal to F. Draw a d. In the space diagram draw 1 k parallel to a d of the force polygon to the intersection of the line of action of E ac 1. Draw 1 m to the trunnion axis, thus determining the line of action of the trunnion re- action T. In the force polygon draw d e parallel to m E and 239 a e parallel to 1 m, thus determining the magnitude of E and T respectively, R3ACTION3 ON THB TOP CARBIAGB Without balancing gear:- The known reactions consist: (1) The weight of the top carriage H tc (Its) (2) The trunnion reaction T (Ibs) (3) The elevating gear reaction E (Ibs) The unknown reactions consist: (1) The horizontal component of the pintle reaction - H (Ibs) (2) The vertical component of the pintle reaction N (Ibs) (3) The front vertical clip reaction M (Ibs) The lines of actions of these forces are given from the construction of the piece. Referring to the force polygon fig. (25), draw ab =T equal to the magnitude and in the direction of T the trunnion reaction. Draw be parallel and equal to E the elevating gear reaction. Draw ac. In the space diagram draw j k parallel to T. At the intersection of j k and E produced draw k 1 parallel to a c in the space diagram to the intersection of W tc . In the force polygon draw c d equal and parallel to Wt c - Draw a d Prom 1 in the space diagram 1 IE parallel to a d to the intersection of N produced. From m draw m n to the intersection of H M. Draw a e in the force polygon parallel to mn in the space diagram. We thns have d e in the force polygon = N and ef = M and ja= H. 240 Thus the pintle reactions H and N and the clip reaction are determined in magnitude and direction. With balancing gear:- The "known reactions consist: (1) The weight of the top carriage W tc (2) The trunnion reaction T (3) The elevating gear reaction E (4) The balancing gear reaction B The unknown reactions consist: (1) The horizontal component of the pintle reaction H (2) The vertical component of the pintle reaction N (3) The front vertical clip reaction M The lines of actions of these forces are given from the construction. Referring now to the force polygon fig. (24) Lay off ab = T and be = E. Draw ac. In the space diagram from the intersection of T the trunnion reaction and E elevating reaction pro- duced at K. Draw k 1 parallel to ac of the force polygon. Con- tinue in the force polygon c d = R the balancing gear reaction. Draw ad. In the space diagram draw in parallel to ad and the intersect! on of W^ at m. In the force polygon draw de. Draw ae . In the space diagram draw mn parallel to ae to the intersection of N. Fron N draw n o to the intersection of M and H. In the force polygon draw a f parallel to o n. From E in the force polygon draw e f parallel to N to the intersection of e f . Draw f g and & a as shown. Thus we determine the reactions M, N, and H respectively. 241 REACTIONS ON THE ASSEMBLED CARRIAGE GUN ASP CARRIAGE TOGETHER. Location of the weight of the total mount:- Assuming a static reaction of 200 Ibs. under the spade, we lay off o'm = 200 Ibs. Then o N = W g = 200 under the wheel contact. The resultant of o'm and o n W 3 obtained by the additional construction lines o'q and op. Hence we determine from the triangle of forces the line of action of *L. The external reactions on the as- 9 sembled carriage consists of :- The known reactions - (1) K = the total resistance to re- coil. (2) W s = the weight of the total mount. The unknown reactions - (1) The horizontal spade reaction H a . (2) The vertical spade reaction V a . (3) The normal reaction under the wheels The direction of these forces are known. Referring to the force polygon lay off ab = K the total resistance to racoii and be = weight of total system W 3 . Draw ac. In the space polygon from the intersection of < and W g draw j k to the intersection of the reaction V a - Prom k draw k 1 to the intersection of H a , V b at 1. Referring to the force diagram draw ad parallel to 1 k of the space diagram to t"hs intersection of c e produced. We thus determine c d - V a , d e = V b and e a - H a< 242 Thus the reactions H a , B a and V^ are determined in magnitude. PROCEDURE IN THE CALCULATIONS FOR THE PRINCIPLE RE- A.CTIONS IN A GUN CARRIAGE MOUNT. (Illustrated by calculations on 240 n/m Hewitmer) REOUIRBD DATA. Type of Gun Howitzer Diameter of bore d (in) 9.45 Total Weight of recoiling parts W r (lba) 15790 Weight of Powder Charge W (Ibs) 40 Muzzle Velocity v (ft/sec,) 1700 Travel of Shot in Bore u (in) 160 maximum 60 Angle of Elevation ninimum 10 short 3.74 Length of Recoil b (ft) long 3> g Intensity of Powder Pressure p^dbs/sq.in) 32000 Initial Air Volume of Recuperator V a i 2970 (cu.in) Initial Air Pressure of Recuperator P a i 576 (Ibs/sq . in ) 243 INTKRTOR BALLISTICS. Maximum Powder Pressure on Breech 2,245,000 F P b = 0.7854 d 2 p m (Ibs) Maximum Powder Pressure on Base of 2,005,000 Projectile p m (Ibs) P = A: (Us) Mean Constant Powder Pressure jQ * * 7 Q 5.36 x 160 5 * 36U 1,350,000 1 = twice abscissa of Max. Pressure - ~ )* - 1 3.996 P OD = Muzzle Pressure on base of breech 622,000 Ibs. Vsl. of free recoil: 7f wV m + 4700 W = 50.25 ft/sec W r Vel. of free recoil - Shot leaving Muzzle 0.5W V m w r 40.50 ft/sec, Time of Shot to Muzzle t s i- 0.01175 sec. 1 2 12V,, 244 Time of Expansion of Free Gases - ob 32.2 0.01538 sec, Free Movement of Gun while shot travels to Muzzle _ u"(w+0.5W) l ~ 12(W r +w + w) 0.31 ft. Free Movement of gun during Pow der Expansion P v t* 0.7179 ft. Total free Movement of gun; Pow- der Pressure Period: I Z 4 + X, 1.0279 ft. Time of Powder Pressure Period r - t t * t f 0.02713 sec, BRAKING PULLS AMD STRESSES IN CYLINDERS. x axis taken along bore: v axis taken normal to bore. Mass of Recoiling parts r "r '' 32.16 15790 32.16 491 Constant of Stability C 0.85 to 0.9 Calculations only for max. elov. Height of center of gravity of recoiling parts above ground h (ft) Calculations only for max. elev. Stability Slope elf. Calculations only for max. elev. 245 Total Resistance to Max.Elev. Recoil 491 * 50.75 Hor.Elev. ,2 K = 2(b-+V f T) (Ibs) 2 (3. 74-1. 0279+50. 75*. 02713) 152,000 Variable Resistance to re- Calculations only for max. coil in battery (at elev. horizontal elev.) jnV K-- rf 2[b-E+V f T- - - 2 M, (Ibs) Variable Resistance to Re- Calculations only for max. coil out of battery (at elev. horizontal elev. ) k = K-m(b-E+ - ) 2m., Initial Recuperator Re- action, P a i = approx. 1.3W r (sin m +0.15cos0 m ) Ibs. (unless given) 1.3 x 15790 (gin 60+0. 15cos60) = 19300 used 18800 Ibs. Total Initial Recuper- ator Pull, P ai = P^i 100 d. (Ibs) 18800+100x2.938 19094 d a = diam. of recuperator rod. (in) 0. Effective Area of Recuper- 35.756 ator Piston - A a (sq.in) Initial Air Pressure ai (Ibs/sq.in) 18800 32.6 576 246 Initial Air Volume V ai (cu.in) 2970 Final Air Volume V af (cu.in) v af - V ai -12 Ab a Final Air Pressure P., "".'" Final Recuperator reaction 2970 - 12 x 32.6 x 3.74 = 1510 S76 1214 1214 x 32.6 = 39600 af = p af a air P af = approx. 2P ai (lbs)metallic J_Distance from axis of bore to 3.038+ 3.850 mean guide contact r(in) 3 3.4444 Distance between clips 1 (in) 86.25 J. Distance from axis of bore to 16.365 center line of hydraulic pis- ton e^ (in) J_Distance from axis of bore to line of action, of recuperator 15.656 reaction e a (in) Assumed coefficient of guide friction u = 0.15 to 0.25 Guide friction constant 2u A f l-2ur 0.15 0.15 86.25 -2x0.15x3.44 .00352 247 Total hydraulic Pull 152000+157908in60-18800( 1+.0635) (max. elcv.) 1+.0663 137500 UA f e h (Ibs) Total hydraulic Pres- 2 hydraulic cylinders: ^-100 d n ^ ' diam. of brake rod P h P^-100 d n ^ 68750-4.72x100= 68280 (in) Effective Area of Hy- 31.2 draulic Piston Max. Pressure in Hy- 68280 - = 2200 draulic Cylindsr 31.2 ' (Ibs/sq.in; Inside Diam. of brake 7.874 cylinder d ih - 1, (in) (dn* diam. recuperator rod) Outside diam. of brake 9.450 cylinder Hoop tension in brake cylinder wall 2 2 0(== ) = 12150 9-45' -7.875' Ibs/sq.in. 248 Max. pressure in recuperator 1214 cylinder v Paf=?ai Ubs/sq.in) Inside Diam. of recuperator 7.087 cylinder d ia = 1.13/A +0.785d| (in) Outside Diara. of recuperator 8.267 cylinder d oa (in) Hoop Tension in Recuperator Cylinder Wall , d a +d !a 1 d 2 -d * ' u oa u ia . in) .8.267+7.087 12140==1) =8020 .267-7.087 Inside Diam. of compressed air storage tank d (in) 84-66 Outside diam. of compressed Air Storage tank d oc ^ in ^ Hoop Tension in compressed air storage tank Paf ( 9.45 a __2 1214 C^ * ' ) 9.45-8.466' 11000 d 2 -d 2 . oc ic (ibs/sq.in) Width of Wall between ad- jacent cylinders^ (in) Hoop tension between adjacent - - - - - cylinders p = p h d ih*Paf d ia 1.8 w .i n) 249 GUIDE, ELEVATING GEAR AND TRUNNION REACTIONS: '. x axis taken along bore: v axis taken normal to bore. Coordinates from center of X s 37.843 gravity of recoiling parts to front guide reaction y t *-3.038 x t and y t (in) Coordinates from center of x g 48,4O7 gravity of recoiling parts to rear guide reaction y a = 3.86 x, and y 2 (in) J_ distance from center of 16,365 gravity of recoiling parts to brake piston rod axis e^ J_ distance from center of 15,656 gravity of recoiling parts to recuperator piston rod axis e a (in) U^esa-r - : -* U ( $* o Max. powder reaction P^T (Its) 2,245,000 (See Interior Ballistics) J_ distance from center of 6.13 gravity of recoiling parts to axis of bore e (in) Front guide reaction: gun re- coiling in sleeve: Fe+P n e h +P a -W r cos0(x 2 -uy 2 ) Q 5S I - 0.15 to 0.2 (Ibs) 250 Rear guide reaction: gun recoil- ing in sleeve Q = Fe+Pe n +P a e a +W r cos (Ibs) Front guide reaction: gun recoil- 2,245,000x5.13+137500 ing in guide below axis of bore 37. 84+48. 41-0. 15><6. 91 Fe+Pe n +P a e a -W r cos0(x 8 -uy 8 ) t + x -u(y t +y Rear guide reaction: gun recoil- ing in guides below axis of bore Q xl6. 365+19094x15. 66 -7895x47.63 2,245,000x5.13+137500 37.84+48.41-0-15x6.91 x!6. 365+19094x15. 66 +7895x37.38 162600 Max. guide friction Kg u(0 t +a 8 )- (Ibs) u = 0.15 (approx. ) Weight of Tipping Parts W t (lbs) Max. Resistance to recoil (dur- ing powder period) 2Peu Bg = 0.15(154800+ 162600)= 47,620 21,021 =137500 + 19094+47, 620- 13670=191000 =152000+ 5.13x0.15 2x2,245,OQQx 85.21 =192000 251 I distance from trunnion axis 3.73 to line parallel to axis of bore through center of gravity of recoiling parts s (in) Radius to pitch circle of 35.57 elevating arc. j (in) Angle between "y" axis and the 60 radius to elevating pinion con- tact with elevating arc 9 e * + ne Elevating gear reaction (in battery) E - F e ]K'a. (Ibs) Angle of E with horizontal Top carriage trunnion reaction (in battery with balancing gear) 2X=K+W r 3in0+Bcos9 9 +Rsin9 r (Ibs) 2Y=H t cos 0+Esine e -Rcos9 r (Ibs) (E is sans with or without balancing gear) Top Carriage Trunnion reaction Not used. (out of battery with balancing gear) 2X=K+Rsin9 r + B'cos9 e + W t sin (Ibs) 2Y=W t cos0+E'sin9 e -Rcos9ii (Ibs) (E 1 is same with or without balancing gear) 11,513,884*191,000x3.73 35.57 344,000 Not used. Estimated Weight of Bocker W- (Ibs) Neglected as small 252 Horizontal Distance from Trunnion to center of gravity of rocker h r (in) "measured to rear" Not used. Angle between line of action of rocker reaction on cradle and "y" axis. B I distance from trunnion to elevating sere* or normal to rocker cradle contact, k = x^os B+y^sin B (in) "x m and y m coordinates of rocker contact with cradle from trunnion to rear and down" . + 30 29. 43x. 866+15. 71 *0.5 - 33.35 Rocker Reaction on Cradle M= -^ (Ibs) 344000^35.57 33 . 35 367000 Elevating Bear Reaction (out of battery) E 1 = Ks+W r bcos (Ibs) Calculations max, elev. in battery, Top carriage trunnion reaction (in batterv)(X and Y components) 2X=K'+lf t 3in 0+E cos 6 e (Ibs) 0-E sin 6 2X=197000+18200+ 17200=381200 2Y=10510-198,000= -287,500 253 Top carriage Trunnion Reaction (out of battery) (X and Y com- ponents ) 2X=K+W r sin 0+ E cos 9 e (Ibs) cos 0-Ein (Ibs) Calculation at max. elevation in battery, Vith balancing Gear: Distance from trunnion to center of gravity of tipping parts (in battery) along x axis: x t (in) Not -used. Radius of bell crank (balancing gear) r,, (in) Not used. Balancing Gear Reaction: 2W t x t cos r a (l+cos (Ibs) Not used. (very approx.) or calculated from layout 0m = max. elevation. Angle made by balancing gear: reaction with "y" axis 9, (See layout) Not used. Rocker Trunnion reaction (X and Y components) 2X r =M sin B-E cos 8 e ~V* sin (Ibs) 2Y r = E sin 9 g -* cos B (Ibs) 183500 - 172000 11500 = 2X r 297000 - 318000 - 21000 = 2Y r 254 Total shear reaction of trunnion 190600 + 5750 on cradle, - X'=X+X r (Ibs) 196350 = X 1 Y'=Y+Y r (Ibs) -143750-10500' 154250 = Y 1 Total spring reaction of Top Carriage on trunnion sin (Ibs) cos (Ibs) 10000 * .866 8660 = X 3 10000 * o.S = 5000 - Y e Total rigid bearing reaction of top carriage trunnion X b = X -X s (Ibs) Y b = Y -Y s (Ibs) 190600 - 8660 181940 = Xfc -143750-5000= -149750= Y, Bending moment at cradle section 8660 x 5.5 + of trunnion 181940 * 2.9 + 5750 M x = X s a + X b b + X r c (Ibs) *.0.9 ~ 580780 M y = Y s a + Y b b + Y r c (Ibs; 5000 x 5.5 - 149750 *2.9 - 10500 x 0.9 = -416,950 Resultant B. W. at cradle section of trunnion (in Ibs) /580,780 2 + 416,960' 716,000 Max. fibre stress due to bend- ing a 10.18 M (Ibs/sq.in) 10.18 x 716000 8* 355 n rJ evi 7.5 3.6 TfcVJNN\QN PIN ~ n -. 256 SHEAR REACTION OF CRADLE. ON TRUMNtON PW : RERCT\ON OF ROCKER ON TRUNNION P\N : RERCT\ON OF TOP CRRR\RG. ON TRUNNION \9O6OO RERCT\ON ON P\N \N X PURNE.. 257 258 CALCULATIONS FOR STRENGTH OP CARRIAQ1 AXLE Proposed 75 m/n St.Chamond 50 Elevation and 22- traverse: Maximum Peak Resistance to Recoil - - assumed at 20,000 Ibs. The resistance to recoil may then be divided into a horizontal and vertical component in the vertical plane of traverse. T"hen, the horizontal component in the vertical traversed plane, nay "be divided into a component along the horizontal axis of the mount and a transverse component at right angles to the longitudinal axis of the mount. The components in the vertical traversed plane are:- Horizontal comp. = 20,000 * cos 50 = 12820 Ibs. Vertical comp. = 20,000 * siii 50* = 15320 Ibs. The longitudinal and transverse "horizontal com- ponents are:- Horizontal comp. = 12820 * cos 22.5 = 11800 Its Transverse comp. = 12820 * sin 22.5 = 4900 Its. 859 Then, 15320 + 4000 = 19320 (Total Downward Force) S x 130 = 4000 x 120.25 + 15320 * 128.2 - 11820 * 47.2 4000 x 120.25 = 481000 15320 x 128.2 = 1970000 11820 x 47.2 = 19320 14550 4770 2451000 558000 1893000 S = 14,550 4,770 A, + B 2 = 4770 nef 260 12800 x cos *2 = 11800 12800 x S in22- * 4900 2M A 8 X x 142.4 -11800 x 71.2 + 4900 x 128.2 11800 x 71.2 = 840000 4900 x 128.2- 629000 211000 H800 ... B 2110 1481 ) 1481 L42 - 4 ( A, = 10319 M about vertical pin for loft trail A y 126.38 * 10319 x 55.2 4900 .'. A , 4500 4500 400 B y - 400 IM axle - A, 130 -10319 x 32 * B a 130 - 1481 130C/1, -8 Z ) = 10319 x 32 - 1481 x 32 =283000 .'. A Z -B Z = 2180 B z = 4770 2A Z - 6950 .*. A z = 3475 ) 8 Z = 1296 j M about left wheel base in Z Y plane: -4900 x 41.2 + 15300 x 30 + 4000 x 30 - 4900 x 6 + 3475 x 41.2 - 12.95 x 101.2 - S p x 60 - 4900 41.2 * - 202000 49000 x 6 - 29400 -1295 x 101.2 = - 131100 - 362500 261 15300 x 30 * 459000 722000 4000 x 30 = 120000 362500 3475 x 41.2=143000 359500 722000 S g = 5980 ) S A = 8570 ) Reactions on Trail Axle. X and Y reaction on vertical pin of left trail: E x = 10319 t E y = 4500 B. . in XY plane on axle: E y x 10 = 4500 x 10 = 45000 " # XY plane: Thrust along X axis = 10319 Shear reaction of equal- izing bar. Thrust along Y axis = 4500 Thrust along Z axis = 3475 Shear reaction of Equalizer bar = 452000 - 331000 7.75 Thrust along K axis 15600 10319 15600 25919 EXTERNAL FORCE.S ON RXUE FOR SECTION -(m-n & -45OOO n 262 Section n-n 5" x 5" Torsion = 25919 x 2.2 = 57000 (" *) (B.M. zy ) = 3475 12.2 + 8570 x 2'6 42300 22300Q B. M. jjTy = 265300 " * 265300 (B.M. zy ) = 25919 x 12.2 - 45000 316000 45000 B. M. = 271000 (" *) ~27100(T f I \ * 30032- = Ci > :'0---. * 01 * ^1 265300 x 2. . 12700 5 x 25 271000 x 6 13000 f y ' 125 " 25700 n 5 4 625 n 32 32 . 46500 1.4 12850 + S\ x 25700 2 + 4650 2 12850 + 13620 = 26,470 BICAPITDLATION Qf gQ R HUT. AE OH THR TNT^BKAT. BBACTIOHS THROliaHnilT A GHH C*RBTGB. F = Powder reaction (Ibs) B = Total braking force not including guide friction (Ibs) 263 "b = distance from center of gravity of recoiling parts to line of action of 8. (in) R = total guids friction (Its) r = mean distance from center of gravity of recoiling parts to guide friction (in) e - distance from center of gravity of recoiling parts to line of "bore. (in) P n = total oil pressure on the "hydraulic piston. (Ibs) P'= the hydraulic reaction plus the joint frictions ^ (stuffing box at pistons) (Ibs) P a = the total elastic reaction (due to compressed air or springs) (Ibs) P a * the total elastic reaction plus the joint frictions (Ibs) Cj, 3 distance from center of gravity of recoiling parts to line of action of P n . (in) e a = distance from center of gravity of recoiling parts to line of action of P a . (in) d}, = stuffing box or rod diam. of hydraulic cylinder. "V 5 (in) d a - stuffing box or rod diam. of air cylinder. (in) Q = normal front guide reaction (Ibs) = normal rear guide reaction. (l"bs) x t and y t = coordinates from center of gravity of re- coiling ]!>arts to front guide reaction, (in) 1 = distance between line of action of Q t and Q f (in) x and y f = coordinates from center of gravity of re- coiling parts to rear guide reaction, (in) !f r = weight of recoiling parts. (Ibs) = angle elevation. u = coefficient of friction. X. and Y = component trunnion reactions (Ibs) X r and Y r = component roc"ker reactions at the trunnion (Ibs) & = elevating gear reaction J = radius to pitch circle of elevating arc. (in) 9 a = angle between "y" axis and the radius to elevating pinion contact with the elevating arc. 264 K * total resistance to recoil. (Ibs) s distance from center of gravity of recoiling parts to trunnion axis measured along the "y" axis, (in) Total resistance to recoil on recoiling vass. becomes. K * B + R - W r sin (Ibs) but B = P n + P a where P h = P h + 100 d n ) assu>in g 100 1T)8 . p Cr and i i (in diaro. for frictions P. P.+ 100 d. ) in stuffing box. hence K * P * ? + R - Vi sin QUIDK OR CLI? KACTIQ8 TO QUIDS FRICTIOM. Gun recoiling in sleeve, front guide reaction, Fe+Bb-W_ cos 0(x. -uy. ) Q t - - - - ' - * (Ibs) x t +x f +u(y t -y t ) and rear guide reaction. Fe--BbCW r cos (x * uy,. ) /1V . \ (Ibs) Gun recoiling in guides below the axis of the bore. front guide reaction, Pe+Bb-W.cos (x -uy ) Q = - - - - - 2_ (Ibs) and rear guide reaction, Fe+Bb+W r cos 0(x -uy ) Q^ = - - - - - V * (Ibs) ,+ x 4 -u(y f +y a ) If R * x t +x t +u(y t -y a ) for sleeve guides M = x t + x a -u(y i +y 2 ) for guide below axis of bore and 265 H * x -x +H (y t +y t ) for sleeve guides * * x t -x t +u(y t -y t ) for guides "below axis of bore. then the total guide friction equals, 2(Fe+Bb)+W cos . N R = - - - u ,(lbt) and for the total braking force B 4 (K+W_ sin 0)M-(2Fe+W_ cos fS N) u B = - 1 - 1 - (It.) X +2 u b In terns of tbe pulls, we bave for the clip re- actions, Fe + P a + 2Pe ~ lf cos ^( x ~ u ) Q , - (Ibs) * t ** t * u <*t-y> Fe+IP a e a + 2Pv h +W P cos 0(x +uy ) " - i - J- (Ibs) x t + and tbe guide friction becomes, 2Fe+22P^e h + 22P a e a + W r cos K R = - (Ibs) M and tbe hydraulic pull in terns of the total re- sistance to recoil and recuperator reaction, becomes, M(K-Pa~W r sin 0)-u(2Fe+22P'e fl + N *f r cos 0) , A r - A. a . ., r_ , _< (Ibs) For approximate calculations, the guide friction equals, 2u8d r R " ~ From tbe foregoing analysis we observe, that tbe guide friction and bearing pressures are reduced: (1) By increasing ths distance between the clips. 266 (2) By balancing the pulls about the center of gravity of recoiling parts or bringing the resultant pull closer to the center of gravity of the recoiling parts. (3) By "bringing the resultant friction line of the guides closer to the center of gravity of the recoiling parts . (4) By reducing the powder pressure couple Fe, that is by reducing the distance from the center of gravity of the recoiling parts to the center line of bore. The distance from center of gravity of the recoiling mass to the center line of bore should never exceed 1.5 inches unless a friction disk is introduced with angular notion about the trunnion. Stress QB Let W c = weight of piston and rod or the weight of recoiling cylinder. (Ibs) d- = distance from center of recoil pull to section "mn" adjacent gun of the gun lug. (in) I mn = moment .of inertia of section. (in)' y - distance to extreme fibre from -neutral axis. Cin) f nn 3 nax. fibre stress (Ibs/sq.in) then, W' [B+ -2. (F-B)]dg y -n (Ibs/sq.in) Trunnion and Elevating gear reaction: When the gun is in battery the tipping parts are balanced about the trunnion axis. This condition 267 implies that with the gun in battery, the center of gravity of the tipping parts passes through the trunnion axis. When the recoil is limited to a short movement under the breech when the gun is fired at high elevations the center of gravity of the tipping parts is placed forward if the trunnion axis and the balancing gear or counterpoise is introduced, balancing the weight of the tipping parts about the trunnion. The trunnion reactions are modified by the introduction of a balancing gear. Trunnion and elevating gear reactions when no balancing gear is used: (a) During the' acceleration period, ,Fe+Ks, 2X=K+W t sin * ( J ) cos 9 (ibs) (i bs ) (b) During the retardation period, Ks+W_x cos 2X=K+W t sin 0+( )cos 9 e (Ibs) Ks+W x cos 2Y=W t cos 0-( ) sin 9 P (ibs) J Ks+W_ X cos E * (Ibs) J where x = the recoil displacement out of battery. Rocker Reactions: T^s reactions on the rocker are primarily three: (1) The reaction of the trunnion upon the rocker, X r and Y r . (2) The reaction of the elevating gear, E. 268 (3) The reaction of the cradle , M, and the weight of the rocker, 1f r . If k = the perpendicular distance fro the trunnions to line of action of M. B = the angle between the line of action of M and the "y" axis. h' r = the horizontal distance to the center of gravity of the rocker from the trunnion. J = the perpendicular distance from the trunnion axis to the line of action if the elevating gear reaction (i. e. equals the radius of the circular elevating rack on the rocker). Then, the cradle reaction on rocker, becomes, Ej-W r h r Fe+Ks-W r h r M = = (in battery) (Ibs) k k Ks-W r x cos 0-W r h r = (out of battery)(lbs) k K approximately M = k The rocker trunnion reactions become, 2X r = M sin B -W^ sin tf-E cos 6 e (Ibs) 2Y r =E sin 9 e -W r cos 0- M cos B (Ibs) Layout of Balancing Gear: Two types of balancing gear have been used ex- tensively in gun carriage construction: (1) A cam with chain type for small field mounts . (2) A direct acting balancing gear. For type (1), let W t = weight of tipping parts. (Ibs) hj. = horizontal distance from the trunnions to the center of gravity of the tipping parts 269 (gun in battery) (in) r o = equivalent radius of can at horizontal elevation (in) r n = final equivalent radius of the cam where the cam arc has turned through the maximum angle of elevation = (in) R = niean radius of can. (in) d n = deflectioa of spring at zero elevation (in) d Q = deflection of spring at maxim-urn elevation (in) c = spring constant. 0= angle of elevation expressed in radius. If d s - deflection of spring at solid height, take d n (J to j)d solid ) ( d = (^ t i) d solid ) then _ *t h t " r o d n r n d o and d n -d =( To layout the radii of cam, we have divided into n parts, then, t h t Wh cos r t n t r = c(d n -r A0) Wh cos cos With a balancing gear of this type, the trunnion reactions are modified and now become, 270 if T = the tension in the chain d = the angle T oalces with the axis X(taken along the axis of the bore) Ks+W r x cos 0+Fs 2X=K+"Vf c sin + ( - - ) cos 8 Q - T cos d (Ibs) J Ks+_x cos 0+Fs = 2Y = t cos0-( jsin 6 e + T sin d (Ibs) J The elevating gear reaction obviously remains as before that is, Ks+\f r x cos E = - - - - (Ibs) for type (2), 1st Tf t = weight of tipping parts (Ibs) h t = horizontal distance from the trunnions to the center of gravity of the tipping parts (gun in battery) (in) x^ and y t = coordinates along and normal to bore from trunnion to canter of gravity of tipping parts (gun in battery) - angle of elevation. 5 m = max. elevation r = radius from tbe trunnion to the crank pin which connects the tipping parts to the piston rod of the oscillating cylinder, (in) R = reaction exerted by the balancing gear along the piston rod of the oscillating cylinder. (Ibs) d t = moment am of H about trunnion (in) d^ = deflection of spring at horizontal elevation d]j= deflection of spring at maximum elevation (in) c = spring constant Hj = initial balancing gear reaction (0 elev.) R t = final balancing gear reaction (0 elev) 371 S = stroke of piston in oscillating cylinder (in) p t = final air pressure in pneumatic balancing cylinder (Ibs/sq.in) p^ = initial air pressure in pneumatic balancing cylinder, (Ibs/sq. in) A = effective area of balancing piston (sq.in) V o = initial air volume (cu.in) With a metallic balancing gear, the dimension of the spring" may be approximated by the solution of the following equations: = cos 0_ ) from nrhich we may obiain d , dv, Q , ( s and c of the spring. a S r(l+ cos ) ^ 2 ) S = 2 r sin - With a pneumatic balancing gear, we have, for a pre- liminary approximation, Pf 2W t x t S ( Pl ^ ( ) ( _ r ( 1 +c< lDJi \ ) (.CM . in / ) 3S 2 1 ( Pi ) S = 2r sin * f . . Pf (in; = cos 0. ( 2 Pi (approx) ) With a direct acting balancing gear, the trunnion re actions are modified and become, Z72 if R = balancing gear reaction (Ibs) q r - angle between R and y axis d t = moment arm of R about the trunnion at any elevation (in) when the recoiling parts are in battery: 2XK+W t sin 0+E cos 8 e + R sin 9 r (Ibs) 2Y=Hf t cos J0+E sin 9 e -R cos 6 r (Ibs) W t x t cos 21 *t x t cos ^ R : = a (Ibs) Ks + P^e when the recoiling parts are out of battery :- 2XK+R sin 6 r +E cos 6 e +W t sin (Ibs) ) ( 2YW t ces U + E sin 6 ft -R cos e r (Ibs) ) ( 2W t x fc cos t R = . (roughly) (Ibs) ( r(l+cos -) ) 2 ( Ks+W_ x cos I s (Ibs) ( J ) It is evident that th elevating gear reaction remains the same with or without a "balancing gear while the trunnion ractions are modified both by the position and Magnitude ef the balancing reaction. 273 Strength of the trunnions The critical section ef the truT>',is is usually where the trunnion joins the cradle. Lt, "n* represent this section. [See fig. (9)]. a = distance fro "mn 11 to center of top carriage bearing . b * distance from "mn" to center of rocker "bearing MX = the bending moment at "mn" in the plane of the X component reactions. My= the bending moment at "mn" in the plane ef the Y, component reactions. M the resultant tending moment on section "an". f = aax. fibre stress (Ibe/sq . i'ff) D = distance ef fhe trumnien at section "mn" then M x = X.a+X r b (in Ibs) and M = M y = Y*+Y r b (in. Ibs) hence /10.18 M D = / (in) Stresses in cradle or recuperator forging: Let Q l and Q Z = the front and rear normal clip re- actions . x t and x g ~ the x" coordinates of these re- actions with respect to the trunnions. d x and d a = the distance of the friction co- ponents ef Q and Q Z from the neutral axis . B = the resultant of the braking pulls re- acting on the cradle. d-= the distance from the neutral axis to "B". 274 I t = moment of inertia of a cross section at the trunnions. y t = distance of extreme fibre from neutral axis at trunnion section. f t = fibre stress due to bending and direct pull or thrust at the trunnion section. I c = nonent of inertia of a cross section at the point of contact of the elevating arc with cradle. A C = area of cross section, at the point of con- tact of elevating arc with cradle, y = distance to extreme fibre from neutral axis of elevating arc section. f c = fibre stress due to bending and direct pull or thrust a t the elevating arc section. A^ = area of a cross section at the trunnion, then )yt U Q- + for the braking reaction in the rear, f t = - i + i for the braking ^ reaction in the front . *x, * x, "" ~^*~ " w j^ UV A for the brak- ing reaction in the rear. U 2 f = " ' " = ^^ + i- for the bralc- T A A c c ing reaction in the front. - -Jfc'^SJ APPENDIX----- APPENDIX CHAPTER IV- INTERNAL REACTIONS. BKACTIOHS AMD STRESSES IKDPCSD IK ELEVATING AMD TRVERS- IH8 MECHANISMS: STRESSES DUE TO The reaction exerted on the FIRING. elevating mechanism due to firing equals, In Battery, Out of Battery Fe + Xs ?%v Ks*W r x cos J0 J cos 20 J cos 20 where F = max. powder force K = Total resistance to recoil YT = weight of recoiling parts, r x = displacement out of battery. J = radius to pitch line of elevating arc fron center of trunnions, e = J_ distance fron axleof core to center of gravity of recoiling parts. S a J_ distance from line parallel to axis of gun through center of gravity of recoil- ing parts to center of trunnions. It is highly desirable to reduce the reaction E, since it stresses the teeth of the elevating mechanism. To reduce this, we may, (1) decrease "e" by so distributing the mass of the recoiling parts as to bring its masses ss near coincident with the axis of the bore as possible. (2) decrease "S" by bringing the trunnion axis along a line through the center of gravity of the recoiling parts and parallel to the axis of the bore. (3) increase "J" whenever feasible in a construction layout. 275 276 In certain types of oounts as those contain- ing a recoiling cylinder, the piston and rods "being fixed to cradle, the center of gravity of the recoiling parts is necessarily considerahly lowered froa the axis of the bore and therefore "e " is in- herently large. With large mounts, counterweights or bob weights are sonetines introduced to decrease "e". In this type of mount without a counterweight or "bob eight a friction clutch or hand brake are often in- troduced on the elevating gear shaft or adjacent gear shaft. Then E becomes limited to that required to overcome the friction of the clutch or brake and a large reaction on the elevating mechanism is thus re- duced. With a cone clutch, we have, uPr E = : - , where P= total spring load. r = mean radius of clutch r e = pitch radius of gear or pinion, n - coefficient of friction = 0.15 approx. 2 = cone angle With a dislc clutch, we have, ), where P = total spring load. r 2 = outer radius: r t = inner radius of dislc. k = total no. of friction surfaces . n = coefficient of friction - 0.15 approx. FRICTION OF TRUNNIONS Tn elevating, or traversing AND TRAVERSING PIVOTS. a gun, a large amount of the energy needed is that required to overcone the friction of the pivot about which the gun is traversed. 277 Trunnion friction: During the elevating process the load on trunnions equals the weight of the tipping parts, when the trunnion is sufficiently free from binding, the con- tact is along a narrow strip. Then u t nR sin 0+R cos 9 = -r ) u= coefficient of where tan * n lO friction. Wt ) R = or*l prs- . * .R (sin tan J+ cos 0)=~r ( SUP ) r = rsdius of trwa- ( nion. and the friction moment M t = R tan .r = r sin m Since is small, tan = sin t approx. hence W W t M+ = n r = 0.15 r approx. $o l 2 2 In starting n nay be as great as 0.25 an& proper allowance should be nade. Since the load brought on the trunnions during firing is greatly in excess of that on elevating the gun, the bearing contact may be divided, one part to carry the major of the firing load and the other to carry merely the weight of the tipping parts. This is ac- complished constructively by allowing play in the bearing which sustains the firing load, and holding the tipping parts for elevating or transportation merely on a spring cushion, the reaction of the spring, for a deflection just sufficient to lift the tipping parts just clear from the firing bearing, being equal to the weight of the tipping parts. Thus it is possible to reduce the friction by using a 278 smaller trunnion diameter, in tliat part of the bearing that is spring borne since the bearing surface for a nominal bearing pressure can be greatly reduced. Pivot friction in traversing: This friction will vary considerably according to the type of bearing used. We will consider three types of pivots, 1* flat circular pivot, 2 flat hollow circular pivot, and 3 conical pivot. To esti- mate the load brought on the pivot, let, V a = pivot reaction or load (vertical ) V^ = normal load of traversing guides (vertical) Wt = weight of tipping parts. 1 = horiiontal distance between V a and Vv d U l + = horizontal distance from W + to Vv v u U W c = weight of top carriage. l c = horizontal distance from t to V^ then ff 1 +W 1 V a = - load on pivot during traversing. If K t = the friction couple exerted at the pivot during the process of traversing we have for the various types of bearings, 1 for flat circular pivot: The friction on an elementary zone = 2 re r dr The moment of this friction about the center = j2nr*dr L 27 a r o 2 2V a nr o The total friction = V If the cone makes an angle 2, and p n equals the in- tensity of normal pressure, then, rd6dr rd9dr the normal pressure on area = p_ . since B fJin t the vertical component of this pressure = rdedr rd 6dr sina but the pressure on the projected area rd6dr = p rd9dr hence p = P n = ?. 2nrdr the friction on a differential zone = n 2 the total friction moment, therefore becomes, If then we let n = 0.15 280 VELOCITY RATIOS OF ELEVATING Elevating and travers- AKD TRAVERSING MECHANISMS ing mechanism consists usually of a train of spur, bevel, helical screw and worn gears. l)-Velocity Ratio of spur gear: Since * t r t = v t r a ) * = angular velocity ^ r = radius to pitch line. we nave - = L - _L ( n = no. of teeth. 2)-Velocity Ratio of Bevel gears: Again w r = w r where r and r are the outside 1 t Z 2 1 2 radii of the gears: The angle of coning for the first gear, equals, r. " * tan 6 t ( 6 t = 1 angle of cone, ) or the second gear tan 6 2 = (6 2 = - angle of cone) hence w 1 2 1 = tan 9. and - = tan w Therefore we may take any two common radii in ob- taining tlie velocity ratios, again 3)-Helical screw gears: Tfe have for the velocity of the common normal, w r cos r 2 cos 9, r cos 9 but also, Pn = then = pcos cos 9. Pn cos 9. 281 ) t = angle be- l *(F* tween axis ) of gear fl ( and perpend- icular to ( common normal. . ) 9 8 = angle between ( axis of gear ) #2 and per- ( pendicular to ) common normal. ( p n = common normal ) pitch. ( p t = circuraferent- ) ial pitch gear ( #1. ) n = circumferent- tial pitch gear #2. n = no. of teeth gear fl. n 2 = no. of teeth gear f2, Hence = = r cos9 r cos9 i 1 If = the total angle between the axis of the gears in mesh, then since p = p cos Q = p cos 9 cos 9 t = cos e = " si " ~ f P 2 ~ 2 P t P 2 cos therefore 282 Further the axial pitches, become, TB X = p t cot 6 t and m 2 = p a cot 9 f 4)"Velocitv Ratio Worm gears: Though a worm gear is a specified type of nelical screw gear when Si = 90. it is convenient to consider this type as a separate classification. When = 90, cos 6. * = sin 9 therefore the axial pitch of one equals the cir- cumferential pitch of the other. The worm of a worm gear has one to two or three threads while the gear has many threads. Now, for a single thread worm, r w sin 9 r., cos9 - tan 9 Directly, we have. % P ^T but 2n = tan9.wr w l w ; *g= ang. velocity ( of gear wheel . ) w w - ang. vel. of ( worm wheel. ) rg - pitch radius ( gear. ) r w = pitch radius ( of worm ) p = axial pitch of ( worm ) 9 = angle of helix. = tan 9 Thus the ratio of angular velocities depends upon the angle of the helix of the worm. With a "n " threaded worm, : n wP n g r and = n -^ w w r g tan 6 283 p In terms of the number of teeth, since = tan 9 w p n w n... ian 9 - 2nr n g and for a single threaded worm, since n w = 1 * - Velocity ratio in gear trains: Combining the previous equations from one pair of elements to the adjacent pair, we finally arrive at the velocity ratios of the first and last wheels of the trains in terms of the number of teeth or radii of pitch circles: In this combination, it is always preferable to set the general equation up in terms of the number of teeth rather than the radii of pitch circles, for then the relations are independent of the type of gearing and velocity ratios between a meshing pair are inversely as the number of teeth or threads. Thus assume worm #1 to drive worm gear #2, while bevel gear #3 on same shaft as gear #2, drives bevel gear #4, then helical screw gear #5 on same shaft as gear #4, drives helical screw gear #6 and finally gear #7 on gear shaft #6, drives pinion #8. Since 2 and 3, 4 and 5, and 6 and 7 are on same shafts, we have, then, wwwwnn nn 13 5794 88 hence * * * = * * * 284 w t n t n 4 n n therefore x x x w n n n n 8 1 3 5 7 If T t = torque on worm shaft #1 and T the torque on pinion shaft and e the efficiency of the total gearing, then T w a = e T^ hence -p n xn xn Mn T t * ( ) where T t = required power torque and T $ s load torque at end of train. REACTION BETWEEN GEAR PAIRS:- The efficiency of EFFICIENCY. spur and bevel gears is hifh compared with helical screw gearing, especially of the worn gear type. The very large force and velocity ratio attainable by the latter makes this type preferable. 1 Spur Gears: For approximate calculations, the normal reaction between 'the teeth will be taken at an angle of 20* with the tangent to the pitch circles. The effect of friction between the teeth is to cause the resultant reaction to make an angle of 25 with the tangent to the pitch circles. Therefore if T is the torque to be transmitted, the reaction between the teeth R, becomes, T x 12 where T is measured in (Ib.ft) r cos 25 , . . r is neasured in (in.) Ifhen smoother running is required with high velocity ratios helical spur gears have been extensive- ly introduced. If B = the angle between the normal to a tooth surface and the tangent to the circumference (i. e. normal to axis of rotation), then 285 T * 12 r cos 25 cos B If b = tooth rim breadth, the mean pressure is dis- tributed along a linear element - b sec 6 and therefore the pressure on an element becomes per linear inch, proportional to T * 12 r cos 25. b the same as in ordinary spar gearing. 2* Bevel Gears: The reaction between bevel gears takes place at the intersection of the common pitch circles of the cone elements of the gears, and this intersection is in the plane of the axis of the gearing. The neutral reaction between the teeth makes an angle approximately equal to 20 with the normal to this plane due to the contour of the tooth. The tangential component pro- duces no axial thrust. The component parallel to the plane = P tan 20, where P is the tangential component. This component is also perpendicular to the common intersecting line of the two cones. If the cone angle of gear #1 equals 28 then the cone angle of gear #2 = The axial thrust for gear fl becomes, P tan 20 sin6 The axial thrust for gear *2 becomes, P tan 20 cos6 Further the radial reaction "between the teeth and there- fore the radial bearing loads for gear #1 and gear #2, becomes, R ' = /p 2 4. (p tan 20 cos8) 2 = P /l + (tan 20 cose)' Where T x 12 P = ; and 28 - the cone angle of gear #1 r 71 2(- - 8)= the cone angle of gear #2. 286 3 Helical Screw Gears: Assuming the axis of the gears to make an oblique angle J t the angle 6 between the contact line of the teeth and axis of gear #1 is given by the expression P 2 sin cos 9 t = while the angle 9 between the contact line of the teeth and axis of gear #2, is given by the expression p sin cos 9. = ,. . * .. where p t and p z are the respective circumferential pitches of the two gears. The reaction between the teeth makes a resultant angle i with the normal to the contact line, where tan i = n the coefficient of friction Then, if T t is the external torque exerted on gear #1, we have T = R cos (9 t - i).r while if T g is the torque on gear #2, T a =R cos(9 2 +i).r. Work expended = T w w ' r cose Work delivered = T 8 w t Then the efficiency E becomes, T w cos(9 +i)cos9 E = -2-2- ? - tt cos(0 t -i)cos9 2 The reaction on the teeth is given by T t r.cos(9 -i) T sin(6 -i) R sin(e t -i)= ^ 287 and the thrust along gear shaft f2, is t R sin(8 + i ) = r cos(9 t -i) The total radial bearing load for shaft of gear #1 balances, T cos(6 -i) T R cos(9 -i)= -^ _ i- = -1 r t cos@ f -i; r x and the total bearing load of gear shaft #2 balances, ^ R. cos(9 +i)= - 4 Worm Gear: Though, worm gearing is a special case of 3, a separate analysis will be made due to the greater use of this type of gearing as compared with helical gearing when the shafts are not a t right angles. Let xx and yy* be the coordinate axis along and perpendicular to the axis of the worn in the plane perpendicular to the radius of the pitch line of the worm through the common pitch point as origin. Let S = the angle that the contour of the tooth makes with the normal to the xy plane at the pitch point, and 6 = the angle of helix. Let R = normal component between worm and gear tooth, nR friction component between worm and gear tooth, then the axial thrust along worjn wheel is X - R cos S cos 9 - nR sin 6 and the turning component on the worm is Y = R cos S* sin 6 + nR cos and the thrust tending to separate the teeth is Z = R sin S . It is to be noted that tan S = tan S cos 6 288 If T_ 3 torque applied to worm gear Tg= torque on gear wheel then, T w =Yr w and Tg=Xr^ r w = radius of worm gear r g= radius of gear wheel To determine the efficiency, .vs hava but = tan then ! cos S cos0-n sin8 . e = tan 6 cos S'sin0+n cos6 n tan0 cosS 1 -) tan n cosS 1 tan e n e= ; ~~ where k = tan" 1 - tan(e+k) cos S' and tanS'-tanS cos 6 COMBINING THE REACTIONS In gear transmission ha we A FROM ONE PAIR TO ANOTHER, between two elements, #1 and 2, f m w = angular velocity hence ;r- = 8 t Likewise between gear elements T i w t #3 and #4, -p w Then if gear #2 is on same shaft as gear |3, we have T = T and w = w hence 23 ? 3 T T w w 42 's- 1 - x = 6 E T 3 T l * 4 ^4 = e e ia 34 289 w Now the velocity ratio may be obtained as outlined w in previous discussipn on velocity ratios. In the proceeding discussion the inertia effect of the gear elements has been neglected in comparison with the friction developed between the gears. TORQUE AND POWER REQUIREMENTS In elevating ertravers- FOR ELEVATING AND TRAVERSING ing a gun, we nave three MECHANISMS. important periods :-(a) accelerating period, (b) the period of uniform motion and (c) the retardation period. The maximum torque obviously occurs during the acceleration and power is continued through period (b), while the friction of the mechanism brings the system to rest during period (c). Let 1^ = moment of inertia about the trunnions of the tipping parts. 1^ = moment of inertia about the vertical traversing pivot of the tipping parts and top carriage. E = elevating gear (tangential reaction.) J - radius of elevating arc. r = radius of traversing arc. Mt~ friction moment of trunnions M^= friction moment of traversing pivot. Then during the acceleration, J t t dt for e i eva ting the gun E . r -M'=l' ill fc * dt for traversing the gun Now MI and M^ are constant depending approximately on the weight on the bearing, while on the other hand E and E' depends on the elevating or traversing motor characteristics. 290 Neglecting the inertia of the gear elements, we have, the torque transmitted varying directly as the number of teeth, that is between any two gear elements, T i for gear pair 1-2 3 = -- - for gear pair 3-4 Ten 4 34 for gear pair 7-8 If gears 2 and 3, 4 and 5, 6 and 7 are assuned on sane respective shafts, T, = T 3 , T 4 = T s , T 6 = T 7 then Ii . L . L. , !i . L = A i. , L. ^i,J_ . ^_ , T . T , T . T . T . 2 "4 ". ", Si '.4 e t 5 7t Now TV = E r e and = * * * e e e e e t t 34 S 8 78 then g r n. n n n T = - ( - x -i x -5. x 1) n a n 4 n e " hence T n n n d 2 e ( x i x i x S)J-M 4. = !+ f r elevating r e n i n a n , n r dt the gun. 291 l 2 4 s II e ( * * * )r-M* = I t - for traversing ' * z the gun. dt z and for the sngular velocity ratios, we have, J and w = w t ; - for spur or bevel gears: (elevating; r i * = *t : ~ ^ or 3 P ur or tevel gears: (traversing) r e = -2 w t : for worm gear in contact with e np elevatin arc (elevating) 2nr . = -TP w t : for worn gear in contact with n p traversing arc (traversing; CHAPTER V. RECOIL HYDRODYNAMICS. OBJECT. The modern recoil system is essentially a hydropneunatic device for dissipating the energy of recoil by so called hydraulic throttling losses, and returning by means of the potential energy stored up in. the compression of air, the recoiling mass into battery. The potential energy at the end of recoil required to return the piece into battery is relatively small compared vritb the energy dissipated by the hydraulic braking. Further the potential energy of counter recoil is in greater part dissipated by the hydraulic counter re- coil buffer in the return of the recoiling mass into battery. In the design of the braking system misunder- standing may result due to incomplete comprehension of the fundamental principles underlying the hydraulic throttling and the various hydraulic reactions. Hence, in this chapter a resume of the essential principles underlying the hydraulic phase of recoil design will be attempted. ELEMENTARY HYDRAULIC Consider an ordinary tension BRAKE brake (fig.l) the oil being throttled through apertures in the brake cylinder from the front or rod side of the piston to its rear. Let a x = area of the variable apertures or orifice . A n = effective area of piston on rod side. A = total area of cylinder. a r = area of rod. P n * total hydraulic pull. 293 294 u Dp L J CvJ 00 L 295 V x =velocity of recoil at displacement x. v x = velocity of oil through apertures. D = weight of fluid per unit volume^ p = p^= intensity of hydraulic pressure. C = contraction coefficient of orifice. K = reciprocal of contraction coefficient. For a displacement dx, the mass of liquid moved by the displacement of the piston, becomes, D A h dx and due to the contraction of the liquid g in the throttling aperture or orifice, its effective area is reduced to C a x , therefore, the mass is accelerated to a velocity A v \r A w H h v x H h Y x 1 v x = = , since K = -5 now the energy SLy L of the jet, * D A h dx . v x becomes, dissipated by a loss due to sudden expansion in fhe rear part of tlie cylinder, where we find a void equal to: (A-Aj 1 )x= a r x . By the principle of virtual work, evidently x A, v - D A h dx *h v x p * d - -J^- ( T^-> hence 1 D Au V* g c a x that is in terms of the liquid pressure : D K A> V Consider again a brake where the throttling occurs between the hydraulic cylinder A and a re- cuperator cylinder B containing a floating piston which is contact with the oil on one side and the air on the other. See fig. (2). Let p = pressure intensity against hydraulic or recoil piston. 296 Aj, = effective area of hydraulic piston. a x = throttling area between the two cylinders which we may assume is controlled by a spring. v x = velocity through orifice. V x = velocity of recoil. V a = velocity of floating piston. A a = area of floating piston. p a = pressure intensity against floating pia- ton. x = displacement of floating piston. Then by the law of continuity, A^ dx = A a dx Due to the contraction and sudden expansion of the liquid from the throttling apertures, the loss due to eddy currents becomes, D A dx h "x By the principle of virtual work, we have, r D A h dx A h V x , ,1 2 i) / U A \l Ph Ah dx -p a A a dx = - ( - ) g C a x Neglecting the slight change in the total kinetic energy of the liquid in its virtual displacement. Simplifying, we obtain, p K ' v g a x which gives the drop in pressure through the orifice, or the so called throttling drop, Obviously, P n = Ph*h as before, i - D K*AV (4) PRINCIPLES OF (1) Though in the analysis HYDRODYNAMICS. of recoil brakes, liquid viscosity is an item of importance, the viscosity effect in modifying pressures is, with a few exceptions, small, and therefore, for a first 297 approximation we will consider an ideal fluid, that is a liquid with no viscosity. (2) It may be shown by simple analysis in the consideration of a small tetrahedron or triangular prism that the pressure intensity on all planes at a given point within a fluid is the same, the bodily forces such as gravity, inertia resistance etc. in limit being eliminated since they are functions of high- er order (three dimensions) than the surface pressures (two dimensions). (3) By higher analysis it may be shown that fluids flow in so called stream lines and therefore the variation of pressure with velocity at various points along the stream line as well as the change in such due to the acceleration of the fluid as a whole, may be determined by a consideration of the pressures on continuous differential elements. Due to the mutual action between differential elements, we nay, by simple integration along a stream line determine the pressures at the extremities of a stream line tube, that is the end pressures as well as the terminal velocities. Consider a differential element A 8 C D along a stream line, of cross section w of length ds and a circumferential perimeter c. Let, the intensity of pressure on A D be p, the weight per unit volume be G, then for the pressures on the surface A B - C D and the wall of the tube, we have dp pw-(p+ ,ds)(w-dw)-pcds sin <*-D " ds sin J0 = ds but cds sin < 6) g dt ds Integrating from (1) to (2) along a stream line, since the mutual reactions between contiguous particles can- eel out, we have, ; i ' d * 4? 1 1 ,dv v " v Obviously, / * ds sin =Z -Z hence dv The term / ds is of special interest and when it dt occurs the motion is not steady. This tern is theoretical, always existing in a recoil brake, since the fluid in addition to a space variation of velocity due to changes of sections, is on the whole ac- celerated as well. . , dv dv To evaluate / ds it is necessary to express - dt dt as a function of s. If now we assume the same stream lines to exist whether accelerated or under uniform steady motion, we have, by the equation, of continuity, w i V i = *2 V 2 = *3 V 3 and dv t dv^ dv a 1 dt dt ' dt hence knowing the acceleration at one section, dv n w ! dv - = r for any point "n", hence if w is a con- dt w n u t tinuous function of s, we have dv 1 dv t, 1 w t - hence the line integral dt w dt dt of the acceleration along a stream lines, becomes, 299 dv d_v ds 3 dt " " l dt f(s) The line integral of the acceleration may be obtained to a sufficient degree of exactness by dividing stream lines into a linear group of columns of various sections, obtaining the proper acceleration. To form (8) and multiplying by the length of the res- pective columns and then adding these columns together. 1 dv The term - / ds ( ) is found usually to be relatively g dt small compared with the pressure drops due to throttling and the changes of pressure due to changes of section. Hence (7) reduces to the energy equation for uniform or steady flow, known as Beraoulliis theorem, that is, t t P T D 2 D z 2g p v* The term * + Z t + x is known as the total head at section (1), composed res- pectively of a pressure head, gravitational head and a velocity head, (4) When friction, viscosity or turbulent motion occurs Bernoulliis theorem is modified by a friction head hf. Considering a tube of a stream, we have for steady motion Dw ds P t * t d^ t !ftd,+D" t -lu 7 dv dv (P t -P. Integrating, us have ' + C dv when = 0, C t = (p.-pX Integrating again, v = - - - * C. when v * 0, 2ul ( Pl -P,)H f and C g = - - Hence the distribution of velocity across a section is given by the equation, * ^ v = ^ul - (^ "" 7"^ (ft/sec) as measured from the center. For a dif- ferential flow, we have P -P, 2 H* dQ = vbdh = ~ - (h -- ) bdh and for the 2ul total flow, summing up oo both sides of center line, we have, Q = - bH Therefore the drop of pres- 12 ul sure between flat surfaces in a rectangular channel becomes. 12ul A ,., N p t -p 2 = - ft (Ibs) bH For the particular case of a square section, 12ulQ Pi-Pa = "TT" (Ibs) n 305 2 Plow through a circular section: p, 1 p 1 1 1 1 1 2- I rh 1 - X-Y = 0, which is immediately obtained since there is no change in the total momentum of the fluid, as we should expect from first principles, since the fluid acts as a medium for the transmission of the reaction between the re- coil cylinder and the recoil piston. Hence pA - Y = X. which gives the actual reaction exerted on the "brake piston. Since C vr v = A V, by the law of con- tinuity, then 2 AV Cwv , v D A v , . v - and V = - and X = pA - - (ibs) cw Dv* now p = , from Bernoulli's theorem, 314 Hence X = (A - 2 cv) 2g = \ I (A-2 cw) (Ibs) 2gc w D v* DA* Y 2 but p = - = - hence X = p(A-2 cw) 2g 2gc a w a That, is the reaction on the piston equals the product of the pressure in the recoil cylinder and the effective area of the recoil piston, where the effective area of the recoil piston equals the an- nular area betneen the recoil cylinder and piston rod decreased by twice the contracted area of the orifice. A physical explanation i$ that due to the pressure of the orifice, we have the pressure lowered around the orifice. Hence we must not only subtract the area of the orifice, but also an additional equivalent area which is to account for the lowered pressure about the orifice. Since c = 0.6 approx., then 2 cw = w approx., and therefore for practical calculations, the an- nular area of the recoil piston is merely decreased by the total throttling area through the piston. Ifhen the rod is assumed to extend through both ends of the recoil cylinder, we have a continuous rod in the cylinder and therefore no void is pro- duced during the recoil. Assuming the same symbols as before, we have, since the total change of momentum of the fluid is nil, pA - X - Y = 0. Hence X = pA - Y and the fluid merely transmits the mutual reactions be- tween the recoil cylinder and recoil piston. Let p w = the pressure in the orifice. (Ibs/sq.ft) ) Xf = total reaction ( on front of re- ) coil piston. for the momentum of the fluid contained in the front 315 part of the cylinder to the orifice, and DAY Y-p w w-X r = - v ) X f = total reaction on ( rear of recoil ) piston for the momentum of the fluid contained from the orifice to the rear end of the cylinder. Now X = Xf-X r = total reaction on recoil piston. Due to the sudden expansion of the fluid after leaving the orifice, the pressure on the rear face of the piston, becomes, p w (A-w)=X r (assumption from ex- periment - sudden expansion), hence Y -p w A = D A V v and X. = X f -p w (A-w) DAY =(p-p w )A -- - v Dv* Applying Bernoullis' theorem, we have p-p w = - but by the law of continuity c w v = A V therefore Dv* = (A-2-cw) (Ibs) Since p w is negligible *f compared with p, we have Dv 2 P - P w - P = 2g hence, as before X = p(A-2cw) That is the total reaction on the recoil pis- ton equals the product of the pressure in the re- coil cylinder and the effective area of the re- coil piston, when the effective area of the recoil piston equals the annular area between the recoil cylinder and piston rod decreased by twice the contracted 316 area of the orifice. Since c = 0.6, 2cw = w approx., and therefore again for practical calculations, the annular area of the recoil piston is merely decreased by the total throttling area through the recoil piston. DERIVATION OF RECOIL We may consider the throttling THROTTLING FORMULAS, effected in either of the follow- ing manners: (1) throttling through grooves in the cylinder wall or through a variable orifice in the piston itself, -(2) throttling through a stationary orifice. (1) Throttling through a variable orifice in the piston or grooves in the cylinder walls. Let A = effective area of the piston, i. e. the cross section of the cylinder minus the cross section of the rod. (sq.ft) p = the intensity of pressure at the pressure end of the cylinder (Ibs/sq.ft) D = the density of the liquid (Ibs/cu.ft) V = the velocity of the recoil (ft/sec) w = the area of the orifice (sq.ft) v - the velocity of flow through the orifice. (ft/sec) X = the total fluid reaction against the piston (Ibs) Then, we have, (neglecting the small pressure in the orifice) D A v pA -X v - - - for the momentum generated in the jet, Dv z and p = _____ f or the energy of the flow in the jet. AV = cwv ----- from the law of continuity of the flow, then 317 DA 2cw 2 X = ~ (1- ), 3 2 DA V , 2cw, (1 r-) (Ibs) 2gc*w 2 A Since the reaction on the cylinder is the difference between the force pA at the pressure end and the reaction of the jet D A V - v flowing from the orifice we have the reaction on the cylinder also equal to D A V DA 3 V* 2cw as would be expected from the equality of action and reaction. Ulith a continuous piston rod through both ends of the cylinder we may neglect the pressure through the orifi*ce and since by experiment the pressure on the rear face of the piston is practically that through the orifice, the reaction on the piston remains the same. Here again the reaction on the cylinder is DAY .. D A V pA-p A = pA -- v, since p A - - v as would 5 3 be expected from tlie equality of action and reaction. The reaction X on the cylinder may be written Y PA^ 2 2cw X = 2gc*w a A ' Dv 2 Further since p = = , , we have also. Zg 2gc a w* X = p(A - 2 cw) = p(A - w) approximately. Thus, knowing the pressure in the pressure end of the recoil cylinder to obtain the reaction on the piston, we must multiply this pressure by the ef- fective area of the piston minus the area of the re- coil orifice. (2) Throttling through a stationary orifice. With a stationary orifice, the throttling 318 usually takes place between the recoil or brake and recuperator cylinders. The loss of head or pres- sure drop is mainly due to the sudden expansion of the flow from the orifice, though with a relatively long orifice the loss due to sudden contraction may become appreciable. If w = the area of the orifice (sq.ft) A = the effective area of the recoil piston (sg.ft) V = the velocity of recoil (ft/sec) v = the velocity through the orifice (ft/sec) c = contraction factor of the orifice. H = the area of the channel leading away from the orifice, (sq.ft) Then from Bernoulli's theorem, we have p-p a ) where p = the pressure in the ~~~ = ^t ( recoil cylinder. ) p a = the pressure in the ( recuperator. Mow ) hf = total head lost due to hf=hf c +h.f e ( throttling. ) ^fc = l ss f head due to ( contraction. ) n fe~ l ss f head due to ex- ( pans ion. T* Now h f ., = where 5 may be taken 0.35 to 0.5 and gf and * z v , cw.a v ... cw.* f . h fe = "^ (1 ~ "1L ) hence h f = ~~ [(1 > * * ] 2 2g W In recoil mechanisms W is usually made from 2.3 to 3.0 tines w. Then, we have, if c is taken approximately = 0.65 (1 --J.)* = 0.515 to 0.614 For flow from an orifice into a large reservoir 319 4 > and (1 - -)* < 1 n * Hence usually cw [(1 - -T- ) + &] = 1 approximately, D A*V -55 hence p-p. = - for the drop of 2gc a w z pressure through the orifice. The reaction on the recoil piston is, D X = pA = - . . + p.A ** In recoil design, it is customary to measure areas in sq. inches and pressures in Ibs/sq.in. Further the average specific gravity of the re- coil oils used in our service may be taken at 0.849 and therefore the density D becomes, D = 62.5 x 0.849 (Ibs/cu.ft). The recoil throttling formulas become, therefore (1) For throttling through a variable orifice in the piston or grooves in the cylinder vralls:- X = P = 6 K 2 AV 2 (Ibs) w = (Ibs/sq.in); KA*V /6~ (sq.in) (sq.in) 175 w 2 13.2 / x KAV 175 w* where K = = 1.6 to 1.3 approx. 6=1 : c =- 0.6 to 0.8 approx. (2) For throttling through stationary X = orif ices:- KAV 175 320 ~ P * * 175* (Ibs/sq.in) CW ft where K = 1.6 to 1.3 approx. 6 =(1 -- ) + E VARIATION OF THE THROTTLING ffe have seen the CONSTANT IN THE RECOIL total braking on the recoil piston may be expressed, when throttling through a variable orifice in the piston or through grooves in the cylinder, as K A V and when throttling through a stationary orifice, as XIL I IV A V f * * \ = a - + p.A (Ibs) 175w . 2cw .i cw.a _ where 6=1 -- - and o = (1 -- ) + Since w varies throughout the recoil, 6 and 6' must also necessarily vary in the recoil. Calculations with the omission of the term 6 or 6* have been found slightly in error and this error has been ascribed to variations in the contraction factor of the orifice. The contraction factor may also vary but it seems more probable that the error is due to the omission of the term 6 or 6 1 . With stationary orifices -^ and 5 can very- H often be neglected and therefore the variation in the throttling constant can be neglected. With throttling through the piston or by grooves in the cylinders -2filL i s small but not negligible^ hence with this type of throttling variations in the orifice are more marked. 321 For a preliminary design 6 and 6' may be assumed equal to unity; but on recoil analysis and careful tests 6 and its variation in the recoil should be taken into consideration. CHAPTER VI DYNAMICS OP RECOIL. ELEMENTARY PRINCIPLES. The object of the recoil is to reduce greatly the stresses induced in the car- riage. Without recoil, the reactions brought on the various parts of the carriage are direct functions of the maximum powder force, which would require a very massive carriage for guns of large caliber. The mutual reactions created by the powder gases between the gun and the projectile is of very short duration compared with the time of recoil and for a rough approximation nay be treated as an impulsive reaction. Neglecting the mass of the pow- der gases, we have /Pdt = mv and /Pdt = MV. Therefore mv = MV, where m = mass of the projectile M = mass of the recoiling parts v = velocity of projectile V = velocity of recoil /Pdt = impulsive reaction of the powder gases. The momentum generated by the action of the pow- der gases in the projectile and gun is the same, as is immediately obvious from the principle of con- servation of momentum. It is to "be further noted that finite forces, as the resistance to recoil, can be neglected in the consideration of impulsive actions, and since the generated velocity of recoil acts for a differential time, the recoil displacement during the impulsive action can also "be neglected. The kinetic energy of the recoiling parts, after the impulsive action, is A. ~ MV' Since V = , the recoil energy in terms of the 323 324 IB velocity of the projectile becomes, A = (- mv ). Hence the energy of recoil is but n - of the energy of the projectile. The total energy generated by the impulsive action of the powder gases, is, therefore i m - (i . 5 > Obviously the greater M, the smaller the energy of recoil. The reaction R between the gun and raount for a recoil displacement b, is - MV a R = ^ or in teras of the velocity of the projectile _ "* , * * \ ~H ( ; BV } The reaction is thereby reduced proportionally to the increase of ths recoiling mass M. Hence to reduce the recoil reaction we increase the recoiling mass 14 and the length of recoil h . The dynamical relations for an elementary recoil analysis in terns of the relative velocity of the projectile with respect to the gun vp can "be readilj obtained as follows:- Vp = v + V assuming V measured in the direction of recoil from the conservation of momentum m Vg MV * mv = m(v R - V): hence V = - M + m The energy of recoil is and the recoil reaction If the recoiling parts are hrought to rest hy friction alone, R = u Mg 325 1 V 2 hence b = - 3 2 ug DOUBLE RECOIL SYSTEM: When a gun is mounted on a movable mount as a car body or itself rolls along a plane, we have virtually a doubl.e recoil systen, the upper recoil being between the gun and mount, and the lower between the mount and plane. As a first approximation we will neglect the resistance between the mount and plane as small com- pared with the upper recoil resistance. Let MR = mass of upper recoiling parts MC = mass of lower recoiling parts ra = mass of the projectile v o = the muzzle velocity of the projectile V = the initial velocity of the recoiling parts v = the velocity of combined recoil Then, during the impulsive action, neglecting the mass of the projectile, we have, T for the projectile / Pdt = mv o (1) T T for upper recoiling parts / Pdt - / Rdt = MV (2) o o Where F is the vertical reaction between the upper and lower recoiling parts. T How R is a finite force, .*. / Rdt - 0, if t is o very small. Further the displacement of the upper and lower recoiling parts inappreciable, since T T / Vdt = and / Fdt = respectively o o Hence, nv o = MpV with no appreciable displace- ment of either the tipper or lower recoiling parts and no moraetitura imparted to the lower recoiling parts. During the recoil, after the impulsive action, we have 326 T for the upper recoiling parts / Rdt=M R (V-v) o T for the lower recoiling parts / Rdt=M c v o hence, the combined velocity of the system when the relative recoil between the upper and lower recoiling parts ceases, is M R V v = T. : If the mutual recoil reaction R between upper and lower recoiling parts is made constant, then v" c n V R = M c - or T = - where T is the time of the relative recoil. The relative displacement Z is, ,V+v v" V ? 99 t & 6 Substituting for T, we have M c M R V* Z 3 for the relative displacement M R + M c 2R The relative displacement can also be obtained from a consideration of the energy relations in the recoil. We have V 1 X _* T) = **R(V -v ) for the upper recoil- parts V i a -T~ T) = - M c v for the lower recoil parts Subtracting: RZ = J M R (V a -v 2 )- f M c v* that is the energy of recoil, j M R V = is dissipated in friction and throttling (RZ) and 327 remainder is the kinetic energy of the combined masses. Now since, M v - R = M R + *c we have * i M R M c t M R +M C Therefore as before, the relative displacement becomes M R +M C 2R ELEMENTARY RELATIONS. During the travel of the projectile in the bore of the gun, neglecting for a rough approximation the mass of the powder gases, a mutual reaction is created "by the powder gases between the gun and projectile, which generates equal momentum in both projectile and gun provided no extraneous forces are exerted on the gun. The resistance of the recoil brake is very small compared with the powder force, therefore its momentum effect is negligible. After the projectile leaves the bore, further expansion of the gases take place and the reaction due to the momentum generated in these gases causes an addition* al increment in momentum of the gun. This additional momenta is commonly known as the after effect of the powder gases. Assuming free recoil of the gun, if m = mass of projectile M = mass of the gun or recoiling parts P = total powder reaction v = absolute velocity of projectile V = absolute velocity of gun in the recoil u = relative velocity of projectile in bore then during the travel up the bore / Pdt - mv = MV but u = v + V for the relative velocity of the pro- 328 jectilc, hence m(u-7)=MV and the velocity of recoil becomes . m . mv V = ( )u= m + M N Since m is snail compared with M, we are not great- ly in error in assuming u = v in approximate cal- culations. At the end of the travel of the projectile up the bore, we have mv and 7 ( After the projectile leaves the bore if P = the reaction exerted by the gases, then ** / it / Pdt = M(7f - V ) = nv where v = the mean tg velocity of the gases "m" after expansion. For a first approximation v will "be assumed a function of the muzzle velocity v and we will place BV =cv Q m Hence MVf=(m+cin)v o . For computations c will be taken equal to 2.3. The energy of free recoil becomes hence T. = i M How the recoil brake exerts a resistance R through a recoil displacement b, "hence Rb= -MV* roughly, and R = 2M.b The recoil reaction R is a measure of the stressed condition of the carriage and very often for a given carriage m, u, v o and b may one or all be changed. To compare the recoil reactions, we have for the sane gun, t t R t ( t *ci t ) v 0i b, ,,_ and for R =R =R, then = where c = \ (-.*,) v*, 2.3 approx., and for b t = b f , = "b, then R t (in^+cl^) 3 v r~ a ~, ., : where c = 2.3 approx. R, (,+co> t )* v, These equations are important in order to estimate with a given change in the ballistics of a gun, the necessary change in either the recoil or recoil brake reaction. The energy of recoil nay "be expressed as m +cu, t _. a . E = 1 r (m+cm) v } M r jM-f mv o) very roughly m = - (muzzle energy of the projectile) (approx.) M Therefore, to decrease the recoil energy M should "be made as large as possible. Since further The recoil reaction varies inversely as the recoil- ing mass, and therefore to decrease R, M s"hould "be made large. EFFECT OP POWDER GASES The effect of the pow- ON THE RECOIL. der gases on the recoil may be considered during two periods:- (1) while the pro- jectile travels up the bore, (2) after the projectile leaves the "bore and the ex- pansion of the gases ta"kes place. In either case an approximate assumption is- necessary in order to represent the phenomena with sufficient simplicity. During the travel of the shot up t"he bore it will be assumed that the gases expand in parallel lamina, and the motion of any differential lamina to be a linear function of the distance. from the "base of 330 the "bore to the lamina in question, that is i v + V v = c s + c where c = - V and c = u v = velocity of projectile V = velocity of recoil u 3 travel of projectile up the "bore hence with free recoil TO I mv + 2 - v = MnV during the travel up the bore u but i m , u i . m(v-V) 2-v =-/ vds= u u 2 The equation of momentum of the system during the travel up the bore becomes, therefore, - (Y ~ V) (m*0.5S)v v + m - = MV or V = M R +0.5m Further since the relative velocity of the projectile is ~ = v + V then, l+0.5D(?r- V) =(M+0.55)V at at du therefore (m+0.5m )T~ and for the displacement of recoil in terms of the relative displacement of the projectile, (m+0.5l) M+V+I If P = the reaction of the powder gases on the "base of the projectile Pfc 3 the reaction of the powder gases on the base of the "bore of the g"un then, for the powder gases, we have I d(v-V) 5 d_v _ ra d_V ( ^>" " 2 " dt Z dt ~ 2 dt for the motion of the recoiling parts in free recoil, 331 P b - *R ST ia4 **' ai ^* a ' 1 ^" and for the motion of the projectile If tbe gun moves backwards a displacement X, while the projectile moves forward an absolute displacement x, then X * / Vdt, x = / vdt (4) Prom (2) and (3) in (1), dV dv . i. dv _ dV MR dt " dt 2 dt ~ 2 dt hence (Mp+0.51) = (B+0.5 1)^ (5) Integrating, we have as before, (M R +0.5ii)V=(m+0.5i)v (6) and. (MR+O.SijX'dn+O.SSJx (7) For the relative displacement u = / l (V+v)dt or du = (v+V)dt o du (N R +0.65)d-u V+v . v . .... r / . . X - / Vdt = / ( _-) du o o Mp+tn+n hence m+0 . 5n X * .. as was obtained by direct sub- Mo +ra+m stitution of displacements. With a constant powder pressure during the travel up the bore, the time of travel becomes, 2u 2u (M R +0.5in) (2W R +m) U Q * = = i = -^^-^ v+7 (t Actually since the powder reaction varies during the travel up the bore, / U * u . f U d" o K Since m and if are always small compared with Mp, we have , u o du t = / 7 very closely o The relation between P^ and .P may be obtained as follows: m A ( v V ) at * p 1 . dv ' T~ approximately 2 dt hence or 0.5 Since however the linear motion of the powder gases is an assTaraption', we "have more accurately, dv P-JJ = (ID + Bi) - where for a first approximation B = 0.5 The rngaa powder pressure lies "between P^ and P hence P ffl = (1 + B - SL -) P where for a first approximation 8" = 0.3 ELEMENTARY ENERGY The Kinetic energy of the pow- RELATIONS. der gases may also be considered a summation of the elementary energies of the differential lamina. Assuming the gases to move up the bore in parallel lamina, with the velocity of any lamina a linear function of the end velocities and neglecting the velocity of the gun as relatively 333 small compared with that of the projectile, we have, for the kinetic energy of the powder gases, i = total mass of powder gas u = travel up "bore of projectile where i s hut v = - yS.O v = velocity of any given lamina s = distance from "base of Tbore to lamina in question 1 / m * * V ( 3 )V . 0+ : ' The Kinetic energy imparted to the recoiling parts IS 22 1 (m+Q.Sm) v ED = ""-" " 2 M ".' qi" , ' ** t ooien*Qxe i>Ai \ trevij ex ic Further if, W = the potential energy of the gases at any instant P^ = the total reaction exerted on the treech of the gun P = the total reaction exerted on the base of the projectile X = the displacement of the gun in the direction of its movement x = the displacement of the gun in the direction of its movement Q = heat lost in radiation J = the mechanical equivalent of heat then, the equation of energy of the powder gases he- comes - P b dX - Pdx = d(B p +W)+ JdQ that is the external worfc on the powder gas system goes into kinetic, potential or configuration energy and lost heat energy. The above equation may "be written -dW = P^dX + Pdx + dE p + JdQ Further since P b dX = d(J (m *' 5i) ' V> ) Pdx = d( mv* ) We have, - . 4 [ i ( ( "*- Sii> \ .*|>,'] JdQ * M 3 The work done on the system may "be represented by an equivalent force P m acting through a distance cor- responding to the travel of the projectile up the "bore, then - dW = P m du + JdQ and since du = dx, very closely, we have t r ((n+O.Sm) m , dv P m = t - * m + r 1 v M 3 du Thus the equivalent mass of the system gun, projectile and powder gases, referred to the displacement up the bore is given "by the expansion, (w+0.5i) i M = - + m + - R 3 RECOIL AND BALLISTIC The recoil reaction, say, when MEASUREMENTS. the gun is mounted on a ballistic pendulum and the reaction of Vhe projectile when fired into a ballistic pendulum, differ by fhe reaction caused by the ex- pansion and consequent acceleration of the powder. Obviously the snaller the charge the wore closely would the swings of these pendulums "be alike. BALLISTIC PENDULUM - QUM HOUHT8D OB PEHDULUM. (a) When the powder charge is very small, we have an equal impulsive action on the projectile and gun. If d = the distance from the axis of rotation to 335 the center line of the bore. M the mass of the pendulum and gun combined, k = radius of gyration about the axis of sus- pension. 9 - angle turned by the pendulum h distance from the center of gravity to the axis of suspension. Then in consequence of the mutual impulse during t~he fire, mv.d = Jfk*w and the initial angular velocity is. therefore. mv.d w = (rad/sec) Hk _, d e h The subsequent motion is given by, = - *j- sin 8 Integrating, .de.i 2gh W = ^ COS e + c de when 6 = 0, cos 9=1 and = w dt therefore t' 2gh c = " -v~ and ,d8 2gh 2 (-rH = ~ (cos 6-1) + w U U 1C Q w At the maximum swing ( ) * 0, and 6 = 9 Q , hence Q t , - cos e) This is immediately evident from the equation of energy, since ,, , Hk w = Mgh(l - cos e ) 2 o e The cliord of an arc radius "c" is 1 = 2c sin- 2 e Further since, 1 - cos 8 Q = 2 sin - "- So mv.d 336 M It 9 o hence v = -- 2 sin n d 2 M k 1 = v^gli mac whic"h means the velocity of the projectile approximate- ly. The radius of gyration may readily "be obtained experimentally by noting the time of swing. (b ) When the powder charge is com- para"ble with the weight of the pro- jectile, we have to consider the additional momentum generated by the powder gases. Assuming the center of gravity of the powder mass to have a mean -velocity equal to one-half the velocity of the projectile, we have (1) during the travel up the bore, (m+ ) v as the momentum m generated in the gun. (2) after the projectile leaves the lore we have an additional impulse p due to the expansion of the gases. Hence the equation for the motion of the "ballistic pendulum becomes, - 2 d[(m-- )v+p] = Mk w 2/pT e o 1 IvTO but w * sin -- * k 2 c i( 1 , Mkl hence (+ )v+p /gh 2 cd If now we repeat the experiment with the powder gases as done in the experiments on the Ballistic Pendulum "by Dr. Hutton, we "have 7 V * p * where obviously V is greater than v, 337 S Mk(l-I ) Subtracting, we have mv+-~-(v-v )= * /gh 2 cd i M (l-l)fc or v- -- (v_-v)= 2m in cd To account for the powder gases experimentally, Dr. Button proposed measuring with and without the pro- jectile as follows: Mkl rov+p 1 = T /gh with the projectile cd hence (i-i)1c o - - m cd The previous expression indicates this expression in error "by the amount _ which for small charges is relatively small tut for large charges may be appreciable and therefore can- not be neglected. As an approximation, however, in ordinary tests, the method of Dr. Button is suf- ficiently accurate, for the measurement of the velocity of the projectile. BALLISTIC PEMDOLUM - IMPDLSE OP PROJECTILE The "ballistic pendulum serves as a valuable mechanical means of measuring the velocity of the projectile though this method has been discarded in modern practice. The dynamics involved is worthy however of consideration in the general recoil pro- "blem. The time of penetration is sufficiently s~hort for no appreciable movement of the pendulum. Let d = the perpendicular distance from the axis to the line of penetration of the pro 338 jectile. J the distance from the axis to the position of the projectile when the penetration ceases. B the angle between "d" and "J" Then, the impulsive moment of the projectile Mp equals the change in its angular momentum, hence Mp * mv.d - mJ*w and the corresponding reaction on the pendulum "becomes M_ = Mk a w. Therefore mvd = (m"k*+-mJ a )w or mvJ cos B = (Mk*+mJ*)w. The initial energy of the system consisting of the pendulum and projectile is, therefore w and the worlc done by the weights in the movement to the maximu swing, "becomes, Mgh(l-cos 6)+mgJ[cos B- cos(9 - B)] hence, from the principle of energy, we have, j(Mk+mJ)w=Mgh(l-cos e)+mgj[cos B-cos(6 -B)] If B * 0, the equations reduce to mvJ =(Mk*+mJ*)w (Mk+mJ*)w*=2(Mgh-mgJ)(l-cos 8 Q ) Combining these equations and noting that e o 1 - cos 9=2 sin* , we have, for the initial 2 velocity of impact for the projectile, 2 + mJ)(Mh + mJ)g ] sin - J 2 GENERAL THEORY OF In the preceeding paragraphs the RECOIL. theory of recoil was greatly simplified by assuming the powder period to "be of such short duration as to be in the nature of an im- pulsive action, and therefore the momentum of recoil being generated practically instantaneously. In tha theory of impulsive forces, we may neglect finite forces such as the resistance to recoil since the time of action is negligible. Further the displacement in an impulsive action is entirely negligible. This method gives fairly accurate results for long recoil 339 but when fbe recoil is shortened the results "by this method of computation are only very approximate. Fortunately due to considerable progress made in interior "ballistics of late, the powder reaction can be determined as a function of time and displace- ment up the bore. It, therefore, "becomes a finite force and the recoil problem during the powder period can be treated with a considerable degree of accuracy. Let Pjj = the total powder reaction on the breech in Ibs. Its line of action is necessarily along t"he axis of the bore. B = the total braking due to the hydraulic and recuperator pulls. R - the total friction, (guide and packing frictions) in Ibs. K - the total resistance to recoil. H r = the mass of the recoiling parts r = the weight of the recoiling mass in Ibs. X = the displacement of the recoiling mass from battery in the direction of the glides. = the angle of elevation of the g"un a - the angle of the guides constraining the recoiling mass with respect to the horizontal . From the theory of energy, we have the fundamental principle: The work done on the system consisting of the recoiling part.g "hy t.hft pnariar gagfts must ptQiial tha work dons on fha system T">y t^ift t. otal ra gi gt. anr*.?*. tin recni 1 for t.hfi *nf.i rr rftnni 1 an nf./ t."ha enftrrfy r>f t.Vis ^ysteyn ar t,Tia bstfinning and nd rvf rf.r'.DJ] i g 7.rr>- Froro this theorem we may prove that with a re- sistance to recoil action throughout the powder period, the energy which the powder imparts to the recoiling mass whan free is always greater than the energy which must be developed by the brafce in the recoil. The greater the resistance to recoil during the powder period the greater this deviation. 340 In the following proof the time effect of the powder gases during free and constrained recoil is assumed the sane or, in other words, the powder reaction is regarded the same for any given time whether the recoiling mass is contrained or free. Theoretically of course due to the slightly different motions in the two cases, the notion of the powder gases themselves will be slightly different and therefore a slightly different reaction on the breech clock in the two cases. Since, however, the difference in motions is so small and the powder re- action so great, we may entirely neglect this fact and assume the powder force to be entirely a function of time and quite independent of the slightly different motion in constrained and free recoil. Supposing the gun to recoil along the axis of the bore as is usually the case, the total resistance to recoil evidently may be expressed as: K = B+R-W r sin 0. Therefore, the equation of motion for the re- coiling mass for constrained recoil, becomes, dV dv f Pv - K = ID -T and for free recoil, we have Ph=ffl r - dt Integrating for any given time, evidently, V < V^ The work done by the powder for contrained recoil is therefore less than with free recoil, since *i t Pt,V dt < / l F b Vf dt where t t = the total time 00 of the powder period. Kow the work done by the brake must equal the work done by the powder gases in constrained recoil, hence, b t / Kdx = / PV dt b t, / Kdx / P b V f dt 341 t b but / * P b V f dt = j ra r Vf therefore / Kdx < j m r V* { o o that is, the braking energy or rather the work done by the resistance to recoil provided the braking is effect- ive during the powder period, is always less than the free energy of recoil. When, however, no braking resistance acts during the powder period, the work done by the resistance to recoil or braking energy must equal the free energy of recoil. Therefore, for a given length of recoil, the recoil reaction is re- duced by maintaining a resistance during the powder period in a twofold way: (1) due to the fact that gun recoils over a greater distance, (i. e. the displacement during the retardation and in addition, the displacement during the powder period), (2) due to the fact that the braking energy is always less than the free energy of recoil. In the design of a recoil system it is there- fora, highly desirable to maintain a large resistance to recoil during the powder period and thus effective- ly to reduce the required braking and the consequent stresses set up in the carriage, as well as to give better stability to mobile mounts. GENERAL EQUATIONS (1) When the direction of OF RECOIL. recoil is not along the axis of the bore. Consider the re- coiling paris to be constrained along guides or an inclined plane making an angle "a" with the horizontal, and the axis of the bore to make an angle with the horizontal. Neglecting the reaction of the projectile normal to the bore, as small compared with the other reactions, we have for the equation of motion for the recoiling mass . 342 Pv, cos (0+a) - B - R - W p sin a * m, -2.JL (D dt hence Pt, cos ( 0+a)-B-R-W r sin a) dt m p dv and /P b cos (0+a)dt - /(B+R+W r sin a)dt = m r v but the powder force is measured by the rate of change of momentum imparted to the recoiling mass when free, that is dVf P " ' 'r IT hence P b cos (0+a)dt m r cos (0+a)d Vf Substituting in the above equation, we have m r Vf cos (0+a) = /(B+R+W r sin a)dt = m r V (2) When the resistance to recoil is constant, K = B + R + lf r sin a = a constant, and we have V f cos (0 + a) t = V (3) Integrating again, we have, Kt* /Vf cos (0 + a) dt - = X 2m r which gives the displacement from battery of the recoil during the powder period, but /Vf cos (0 + a) dt = E cos (0 + a) which is the component displacement for free recoil in the direction of recoil. The constrained recoil at the end of the powder period, becomea ? KT X = E' = E cos (0+a) - T (4) 2m r and the corresponding velocity at the end of the powder period, becomes, v r * V f max. c s ^ +a > ~ < 5 where T is the time of the powder period. Proa the energy equation in the motion from the end of the powder period to the end of recoil, we 343 have ~ m p v p = K(b-x t ) hence j m r [V f cos(0+a) - ] = Kb-K[E cos(0+a)- j^-] 6) Expanding and simplifying, we have K[b-E cos('0+a)+VfT cos(0+a)= jm r Vf cos*(j+a)] hence t ,,a x , ., . -m r Vf cos (0+a) K = b-(E-V f T)cos(0+a) (7) or in terms of .the component reactions, 1 2' 2* -,m r Vf cos (0+a) B+R+W. sin a = - (?') b-(E-V f T)cos(f+a) where WVQ+ 4700 ^ Vf = - from the principle of linear w r momentum. E = total free movement of gun during powdei period. T = total time of powder period, To deduce E and T we proceed as follows: (See Chapter II) Calculate rf Z1 Pm ti fa 27 ^N* -. i b 3 u o [( TS r~ ~ 1} i /(1 " T^ r~ ) ~ 1 ] 16 P e 16 Pe where p^ = max. powder pressure X area of bore and also, ~ b Pbm : then compute wv + 4700 w 2 V f = - - - j V= L *a J ... 344 where w - weight of projectile w weight of charge w r = weight of recoiling parts V Q = nuzzle velocity The time of the travel of the projectile up the "bore and the time during the expansion of the powder gases are respectively: b . 2u u 2(V fl -V fo ) w r *o - ; <*'3 log - * - * 8) t lo . ^ _ 3 u o * - -- approx. v o Therefore the powder period, "becomes T = t o +t lo The free recoil displacement during the travel up the bore, and during the expansion of the gases are respectively: u (*+0.5w) Therefore, the total free movement of gun during the powder period, becomes, E = X f o + MOTE: In the above and further formulae the units employed are : displacement in feet velocity in feet per second force in pounds mass in pound units With a void in the recoil cylinder during part of the powder period, equation (7) becomes slightly modified. Let S = length of void in recoil cylinder t g = time of free recoil to end of void Neglecting, R+W r sin a as small compared with B, ws find K =0, until distance S is reached in the recoil. 345 Therefore we "have = E cos (0 + a) - K(T-t s ) 2*. cos(0+d) - K (T-t s ) (8) (9) where T - time of total powder period. Substituting (8) and (9) in the energy equation, -, m r v r = K(b-x t ) and simplifying, we have 1 2 f 2' - m r VfCOS (0+a) K = b-CE-V f (T-t s )]cos(0+d) To evaluate t s , t"he time of recoil with void, we have t. = -(2.3 log ^- +~ + 2) a D D . where (w+--)cos(0+a) - D /l- ~) 16 = P (11) Chater II. (12) S "being the length of void. See III v o = muzzle velocity in feet u o = total displacement up "bore in feet p m = max. powder pressure, Ibs. per sq. in. 64 ' 4 u (15) mean powder pressure, Ibs.per sq. in. 346 A b area of bore of gun. If, ho*ever, the length of void corresponds to displacement greater than the recoil displacement for the projectile to travel up the bore of tbe gun, we have, b 2u a t. - - (2.3 log * * 2) + t a oo or approx. > (16) 3 u o A i 5 T. * * where t t is obtained froa the solution of the cubic equation, C-|j ( -ji - - ) + V fo } cos (0 * ) - x; - (17) n here r fo : V, wv + 4700 2(V f ,-V fo ) 32.2 and X^ a o cos (0 + a) (18) 27 a u also P b * 4" * ( b + u ) d- 12 P B V < 19 > Powder reaction on breech when shot leaves muzzle. COI8TBAIS1D VILOOItT Of BBOQILi (1) During powder pressure period. Knowing R from the previous formulae, tbe con- strained velocity of recoil nay be computed from the 347 free velocity curve as follows: From equation (3) we have, V * V* cos (0+a) - r and the corresponding displacement * Kt* X / ? f cos (D + a) dt - ~- o 4m r Kt* X* ees (0 + a) X QH sm r Thus we see the free velocity curve of recoil both against time and displacement of free recoil is re- quired in order to compute the constrained velocity curve. The free velocity curve during the powder period is divided into two periods, (1) the velocity of free recoil while the shot travels up the bore, and (2) the velocity of free recoil during the expansion of the powder gases after the shot has left the muzzle. Lednc's formula gives us a means of computing (!) while Vallier'a hypothesis serves for the computation of (2). From Lednc's formula, we have, during (1) of the powder period, v - r^ (20) b + u "b 2u u t - 5 (2.3 log -5- +-B- + 2) (21) where a * travel up the bore in feet Q O a travel up the bore to muzzle v * corresponding velocity of projectile in the bore of the gun (feet per sec) v o * muzzle velocity of projectile t * corresponding time of the travel in seconds. . t 27 Pm /I 27 Pm.a ~ . *-..t(r B --/a- I e-) -13 348 p m max. powder reaction on base of projectile B wv p e 3 - * mean reaction on base of pro- jectile during travel up bore. a 3 (b ^ o) 12 Farther from elementary dynamics, (see Chapter II) (w+f)v V f (22) w r ( *\ 2 X a i . r or approx. ^ 2 X, - (23) where w > weight of projectile in Ibs. v * weight of powder charge in Ibs. w r > weight of recoiling mass in Ibs. The procedure therefore, to compute the free Telocity carve against time and displacement during period (1) is as follows (a) Compute b and from it a, (b) For various displacement up the bore: compute v and t. (Equation 20 and 21). (c) Then from equations (22) and (23), compute V and X . Arrange the data in a table with corresponding values of V , X and t. 349 Prom these values the constrained velocity carve during (1) nay be computed from equations (3) and (4). Front Vallier's hypothesis, we have, during (2) of the pow- der period, for the total pressure on the breech Ft, - P ob - C(t -t ) (Valuers' hypothesis) where C t - t t t - t 2(V f , -V fo )m p hence Now, from elementary dynamics, the change of momentum along the axis of the bore, becomes, t / Pfc dt = m r (V f - V fo ) (25) *o Substituting (24) in (25) and integrating, we have obo 4m r (V f .-V fo ) r fying, we have, for the free velocity of recoil, V f - V fo - (t-t )(l - The corresponding displacement of free recoil, along the axis of the bore, t X f * X fo + / V f dt (27) *o w where w + - Xf - u u o = total travel up the r bore in feet. if if.*/* v {0 dt^A,-t ><. t - 4 .. P (v t .. Tfo) A*-* 350 Simplifying, the displacement of free recoil for tine t, becomes, The following initial values and constants are to be substituted in equations (27) and (28). fo " ~ *fo wv^ + 4700 w t - (2.3 log * 4 2) 3 u o - approximately. 3 v 27 B . , 27 P m aax. powder reaction on base of projectile. a wv o ^ * ft 7 ' J " * mean reaction on base of projectile daring travel op bore. 27 t a P b " 7- b . .^' 1.13 P ffl reaction on breech of gun when the shot leaves the muzzle. The procedure, therefore, to compute the free 351 velocity curve against time and displacement daring period (2) is as follows: (a) Compute P m , P e , and then b and a as before. (b) Compute P ob , t o , (V f i-V fo and Xfo. (c) Then from arbitrary time intervals between t t = T and to compute from equations (26) and (28) V f and Xf Arrange the data in a continued table as in (1) with corresponding values of V f , X f and t. Prom these values the constrained velocity curve during (2) may be computed from equations (3) and (4). MAXIMPM V1LOOITY OF COJ8TBAIIHD BIOOILt The condition of maximum velocity of constrained recoil is when the powder reaction exactly balances the resistance to recoil, since before this condition the recoiling mass is accelerated and immediately after it is retarded. Hence P b cos (0 + a) - K = t - t. 2. r (V -V to ) Hence the time at the maximum velocity of constrained recoil, is obtained from either of the following equations:- t ~ t Solving for t, we have (30) 352 2m(V f i-V fo )[P ob cos(0+a) - K] or t. * P ob t (30') P ob cos A * a Substituting t m in (26) and (28), we have, Vfm - v fo + -7 ^m-to) t 1 ~ 4mr ( Vft -v fo ) and (32) where Vf m and X fm are the free velocity and displacement corresponding to the maximum constrained velocity 'of recoil. BEOAPITDLATIQH Of FORMULAE FOB PRIBCIPLE PEBIQDS DURIKQ PQITDER PRESanHE P1BTQD. In the constrained velocity curve daring the powder period, we have the following important points: (a) Velocity and displacement of the recoil when the shot leaves the muzzle. (b) Maximum velocity and its corresponding displacement of recoil and time. (c) Velocity time and corresponding displacement at end of the powder period. Given data: r = wt. of recoiling parts. V Q = nuzzle velocity. w = weight of projectile. w * weight of powder charge. u * total travel of shot up bore. 353 P m = max. powder reaction on "base of shot. P^ = max. intensity of ponder pressure assumed X area of bore, b * length of recoil. INTERIOR BALLISTIC OOMSTAHT8 BgflPIBED FOB VELOCITY CUBVB; P e * mean average powder reaction on base of shot * 2 !I- 2gv B = twice abeissa of max. pressure, 27 P m ,, /7 27 re?--" i^-u * max. velocity of free recoil a velocity of free recoil - shot leaves muzzle w + 0.5 w p ob * total pressure on breech when shot leaves muzzle. * 4 (B+u ) t Q * time of recoil while shot travels to muzzle B , 2u v 3 u o = - (2.3 i g -- + - + 2) - - ~ approx. a v o t t * time during the expansion of gases after shot leaves muzzle. m a(v f .-v fo ) j^ p o b < 354 t * T time for total powder period """"'*. * , " " c " X fo free movement of gun while shot travels to muzzle u (w+0.5 I) w+0.5 w - = u approx. \f i o = free movement of gun during expansion of powder gases. Total free movement of gun during powder period B *fo * x f'o K resistance to recoil: t * angle of elevation: a * angle of plane of guides with horizontal, * f* \ - r Vfi cos (0+a) b-(E-V f ,T)cos(0+a) VKLOCITY AMP PT 8PL AGKMR MT8 AT PERIODS (a).(b) and fe). At Period (a): V and X the constrained velocity and dis- placement in recoil for period (a) when shot leaves the muzzle. Kt V V fo cos (0+a) Kt X X fo cos (0+a) - ^ 355 At Period (b): t m time at max. velocity of constrained re- coil. K(T-t ) P ob cos(0+a) m and Xf m = velocity and displacement of free recoil at the instant of maximum velocity of constrained recoil. p ob ^ob^m"*) x fm - x fn * tVfo * ( t m~ t o) ~ TTT- 10 2m r 6m r (Vf V- and X_ = maximum constrained velocity and cor- m in *i^ ^ ^ responding displacement of recoil. Kt m x^ \ "I V m = Vf m cos (0 + a) - x m = x fm cos At Period (c): V = V r = constrained velocity of recoil at end of powder period. X t - E P a corresponding displacement of constrained recoil at end of powder period. Kt V V_ = V f i cos (0+a) m r Kt 2 t X, E P = Xfi cos (0+a) oin_ f 356 UNITS TO BE EMPLOYED IN THE ABOVE AND FURTHER FORMULAE; BRITISH 8Y8TIM MITRIO SYSTEM METRIC SYSTEM QBAYITATIONAL 9R AV IT AT I 01 AL GRAVITATIONAL UNITS. OMITS. UNITS. Displacement in feet - ft. in meters*n in centimeters' cm Telocity in feet per in meters in centimeters see. -ft/sec. per sec.> per see * m/se c . em/se c . Force pounds - Ibs . Kilograms = Kilograms kg. kg. Pressure Intensity lb s . sq. in, Kg. per sq. cm. Kg. per c q. en. Pressure Area . inches Sq.cn. Sq. em. Mass Lbs/g (-52.2) Kgs/g Kgs/g = 981 Tie Seconds =Seo. Seeonds=8ec* Seconds - Sec. OOJ8TRAH1D VKLOCITY COBVK: (2) During Retardation Period of Recoil. After the ponder period the recoiling mass is brought to rest by the resistance to recoil. The recoiling mass then reaches the extreme out of battery position. At the beginning of the 2 period of recoil, the recoiling mass has an initial velocity V t * V r i and an initial displacement from battery X t * E r . 357 A V Prom the equation of motion, we have K - m_V dX Integrating, between the limits X, to any given displacement X, and between corresponding velocity V, to Vg we have X V x / K dX - - m r / V dV (33) X V. Hence, retardation period of the recoil. Hence K(X-X ) - which is the equation of 2 energy during the 2K(X-X. ) (34) ra r A simpler and more direct form for computing the constrained velocity during the 2* period of recoil is as follows: We have, as before K d x * - m r V d V Integrating between the limits X and b in the displacement and V x and o in the velocity, we have m.V x K(b-X) - -J-^- (35) /2K (bH Hence V x - / (b-X) (36) ro r showing that the velocity during the retardation period is a parabolic function of the displacement. It is to be especially noted that a characteristic of a constant resistance to recoil is a parabolic function of velocity against displacement. GENERAL EQUATIONS OP RECOIL In the previous formulae CONTINUED.- VARIABLE RESIST- the resistance to recoil ANCE TO RECOIL. was assumed constant throughout the recoil. It is however often de- sirable for stability to decrease the resistance to re- 358 coil in tbe out of battery position and thus partially compensate for tbe decreased stability due to the moment effect caused by tbe overhang of the recoiling mass in the out of battery position. With a variable resistance to recoil it is customary to maintain a constant resistance during tbe powder period and thence decrease tbe resistance proportional to tbe displacement to the out of battery position, with a given arbitrary slope "m". See Chapter III. Let K Q * the constant resistance during the pow- der period. V t and V r * tbe velocity of constrained recoil at the end of the powder period, b = total length of recoil. Then tbe equation of tbe resistance to recoil against displacement of recoil becomes, K constant, from to X x or B r (37) K * K o m(X-X ) from X^ to Further, g j V t V p V f cos(0+a) 2. (38) m r K T X * E r B cos (0+a) - (39) o rn Now fron tbe equation of motion of the recoiling parts during the retardation, we have, K dX - m r V dy Integrating between limits, X, and b: and V, and 0, K dX d V Hence ^ m y* [I - m(X-X t )J dX > -j-i Integrating, we have for the energy equation, 359 K (b-X t ) (b-X t ) m r V i (40) Substituting (38) and (39) in (40) and neglecting the terra j/* T 4 m ^o 1 - in the expansion as small, we have ra "o j 2 4 frVf. < - 2 cos*(0+a)+ ^ [-b- ] > , T b-B cos(0+a)+ V f cos(0+a) T - - [b-E cos(0+a)] o ffl Thus from the ballistic constants E and T, together with the length of recoil, maximum free velocity of recoil and any given arbitrary slope "", the re- sistance to recoil maintained constant during the pow- der period may be computed. Substituting K Q in place of R in the proceeding formulae during the powder pressure period enables as to compute the retarded velocity curve daring the powder pressure period. During the retardation or second period of recoil we have, K dx - m r V dV Integrating, from the displacement x to the end of recoil, we have b v / Kdx * m r / J Y dV therefore / (K o ~ (X-X t )]dX x Hence m P V* K(b-X) - X mX,X and simplifying, we bave 360 "rVl [K - - (b+X^X, ' 2 Hence j = :(K - - tb+X - 2X)](b-X) O (42) where as before, m the arbitrary slope of resistance to recoil. o m* X a E cos ()0 + a) - iv- *m r GENERAL EQUATIONS OP When the direction of recoil RECOIL - Cont. is along the axis of the bore, (a) Constant resistance to recoil throughout recoil, let K B + R - W r sin = total resistance to recoil B * total braking R = total friction E 3 displacement in free recoil during powder period. T = corresponding time for free recoil, then for the motion of the recoiling parts, dV T p b KT P b - ' m rdT ( ~ f dt * m7 3l V r but as before / ~ dt = Vf = max. free velocity of recoil, hence KT* and the corresponding displacement, X *E r E 0IHB After the powder period, from the equation of energy, iH-V.. K(b X.) KT * / .. Ri.a .. /. Rl v - m_(V f ) ' K(b - E * - ) m P c ffl j. 361 t Tf *"r*f and simplifying, we have K (43) This equation obviously is a special case of equation (7) since when (0 + a) * 0, cos(0 + a) = 1 and a = - 0, (b) Variable resistance to recoil. The resistance to recoil as before is assumed constant during the powder pressure period and thence to decrease uniformly consistent v/ith stability, that is with a stability slope as given in Chapter III on stability. At the end of the powder period, we have for the constrained velocity of recoil and corresponding dis- placement, At the end of recoil, the resistance to recoil be- comes K Q - mfb-S r ) where m * the stability slope (See Chapter III). The mean resistance from the end of the powder period to the end of recoil, becomes, 2K-m(b-E r ) - = K - ? < b -*r> 2 and from the equation of energy of the recoiling mass, we have K 0>-B r ) - 2 (b-E p )* - 7 m r V* (46) Substituting the values of E r and V r from (44) in m 2 (46) and neglecting the term _*_ m K T we have, mV * m(b-B) a 2 -r 362 This equation obviously is a special case of equation (41) since when (J + d) 0, cos(j0+a) * 1 and d - 0. .*.. (c) Dynamic equation of recoil during powder period. Since during the ponder pressure period, the re- sistance to recoil is assumed constant even with variable recoil, we have, therefore, the same dynamic equation with either variable or constant resistance to recoil during the powder period. Dividing the powder period into two intervals t Q and t x - t o while the shot travels up the bore and during the expansion of the powder gases after the shot has left the bore, respectively, we have (1) During the travel up the "bore, / ^ (48) Kt u - (49) 8 2u u and t - -(2.3 log + - + 2) (50) a B B Thus V, X and t are functions of the parameter u. The ballistic constants a and B have been determined previously in this Chapter as well as in Chapter III in "Interior Ballistics". 363 When the shot reaches the muzzle, / \ ("*") 2 Kt (B + u ) m p (61) w (52) o o and t - - (2.3 log + + 2 ) 3 u o * approx ^ o (53) (2) during the expansion of the powder gases, we have au u o where u = the total travel up the "bore, the dynamic equation of recoil during this period becomes, ir a dV - K ra r - dt (54) 2 where P ob B 2 -- - - 1>12 p m (See Chapter III) 4 \B + u - / Integrating, we find (55) l "O- 364 Hence V - V.+ - - - [1 -- - ] _ _ (t-t )(56) 2(t t -t ) The corresponding displacement is obtained by integrat- ing equation (55) 2 6(t-t Q ) V d(t-t) + m r V (t-t ) + Const. (57) Now m r / V d(t-t ) = m(X -X Q ). Hence where t = t Q , X X and const. * 0. Simplifying (57) we obtain for the recoil displacement during the second period of the powder period, X = X + V (t-t ) 2m (58) To obtain the maximum restrained recoil velocity and corresponding displacement, we must equal the total powder reaction to the total resistance to recoil, that is P b - K Pobd -- ) - K where P O ^ the pressure on \~t-o tnc Creech when the shot leaves the muzzle. t = total time to maximum restrained recoil velocity, hence solving for t m , we have t a t t - (t t - t o ). Substituting in equation ob (56) and (58) we have Ppb^nrV . t m" t o . K 2(t t -t Q ) m 365 (t -t )* P b(tm ' to) M- *"* * 3 2 ^ ** (62) (d) Dynamic equation of recoil during the retardation or the pure recoil period. (1) constant resistance to recoil: Since the total resistance to recoil is constant, the velocity must be a parabolic function of the displacement of the recoil, Prom the principle of energy, we have, M V* /~~~t \~ K(b-X) hence V /2 2 m r (2) Variable resistance to recoil The resistance to recoil out of battery, becomes, k * K - m (b-E + - ) where K = 2m r "~*~ The average resistance to recoil in the displacement b - X, becomes k * \ (b-X) From the energy equation, we have, 366 k(b-X)+ I (b-X)' - j-j r V* I 2 /Kb-XHTc* (b-X)] v COMPONENT REACTIONS OF Let K * total resistance THE RESISTANCE TO RECOIL. to recoil. (Ibs. or Kg) B = total braking. (Ibs. or Kg) R total friction to recoil. (Ibs. or Kg) P h = reaction of hydraulic brake. (Ibs. or Kg) P v * reaction of recuperator. (Ibs. or Kg) p x * hydraulic brake pressure. (Ibs/sq. in) or (Kg/I 2 A 3 effective area of hydraulic brake piston, (sq.in. or m ) p y recuperator pressure. (Ibs/sq.in) or (Kg/m ) Ay * effective area recuperator piston. (sq.in or m ) V o " initial volume of recuperator. (cu. ft. or m ) X * recoil displacement. (ft. or m) Sf * final spring reaction. (Ibs) or (Kgs) S o = initial spring reaction. (Ibs) or (Kgs) The total resistance to recoil then becomes along the bore along special guides K * B + R - W r in KB+R+W r sin 6 where angle of elevation, 9 * angle of guides. Now in systems where the hydraulic brake is independent of the recuperator system, B P n + P y In systens where the brake and recuperator are connected B Pj, For independent systems P V J(T T~\~] ^ or pneumatic recuperators "" 367 S f- s o v S + ( ) x for metallic recuperators o and P yi 1.3 W r (sin 0, + u cos m ) approx. hence KA S -7- where c = -7- w x "x V k * v o * cv V A* r o ) * ~r~ for pneumatic re- "v cuperators. s f- s o cv* B = S + ( ) x + 7 f or metallic re- x cuperators. For systems where the hydraulic brake and recuperator are directly connected, , KA' P-P V = where c = * Mf* M* w x "x p vi A * P y j * 1.3 K r (sin + u cos 0) approx. therefore P = o v pA ' i * c AT since and c A C hence B cv ~ which is an equation of exactly the sane form as for a system where the recuperator is independent of the hydraulic brake. 368 The general equation for tbe resistance to recoil "becomes, (a) when the recoil is along axis of tbe bore: a cv * ~ + R - W r sin t, for pneumatic V recuperators. K - S + ( - - )x + -yr + R - W r sin 0, for metallic recuperators, (b) when tbe recoil is along special guides: V * O CV K = p vi(VTT4 ) * ~T~ * fi * w r sin e for pneumatic V f^ AX W y recuperators. SfSo + -7- + B + W r sin 6, for metallic recuperators. a CV K -7- + R + ff p sin 9, for gravity mounts. GENERAL EQUATIONS OP The function of the recuperator COUNTER RECOIL. is to return tbe recoiling mass into "battery. The stability of a mount in counter recoil is greatest at the beginning of counter recoil and least at the end of counter recoil or when the gun enters tbe battery position. To prevent shock and unstableness as the gun arrives in battery it is necessary to introduce some form of counter recoil buffer towards tbe end of counter re- coil. Very often a buffer resistance of varying amount is introduced throughout the counter recoil. In addition we always have the resistance of tbe guides. Without a recuperator tbe recoiling mass must be 369 returned to battery by the gravity component due to the inclination of the guides with the horizontal. If this inclination is small, the gravity component does not greatly exceed the friction and thence a very elementary buffer may be used, the return velocity being always small. Let K V 3 total unbalanced force in counter recoil. F y = recuperator reaction. = variable orifice for counter recoil buffer. By = counter recoil buffer resistance. A y = effective area of recuperator piston p y = pressure intensity in the recuperator cylinder. p a = pressure intensity in the air reservoir. R = total friction of counter recoil. During the accelerating period of counter recoil, we have dv K w = HD v and during the retardation dx dv K v - m R v - dx During the acceleration K v is necessarily always smaller than the total resistance to recoil, "hence during the acceleration counter recoil stability is of no consequence. During the retardation, if d 1 = the distance from front hinge or wheel contact with ground in a field mount, to the line of action of the total resistance to recoil. L = horizontal distance between front and rear supports of mount. L s = horizontal distance from rear support to center of gravity of total system with recoil parts in battery. b = total length of recoil. 370 * 3 = weight of total mount. Then, for a gun recoiling along the axis of the bore during the retardation, K v d' ^ lt g (L-L s ) +H r (b-X)cos and the minimum stability occurs when the gun enters "battery, that is K y d ' ^ W S (L-L S ). The stability slope of counter recoil, becomes ^ cos ^ m 1 * d 1 To consider the components of the total resistance to counter recoil, we have three classifications: (1) recuperator systems independent of the hydraulic bralce and with no throttling between the air and recuperator cylinders. (2) recuperator systems independent of the hydraulic brake, with throttling between the air and recuperator cylinders. (3) recuperator cylinders connected directly with the brake cylinder. In all systems an independent buffer may be in- troduced in either the recuperator or brake cylinder front end. In certain types the buffer acts as a plunger brake within the piston rod of the recoil brake. Then, (1) for recuperators independent of the "bralce cylinder and with no throttling between ths air and recuperator cylinders, K v = F v - B' X - W r sin - R (1) when V o = initial volume 1.3 W r '(sin 0+0.3 co 0) approx. (2) for recuperators independent of the bralce cylinder, with throttling between the air and recuperator cylinders, 371 p v A y -BjJ - W r sin - R where 2 c v p v " p a 2 (W Q = constant orifice usually) w o V^ Fyj. = 1.3 W r (sin + u cos 0) hence 2 c v i KV = (Pa 2~' A v ~ B x " w r sin & ~ and since 2 "o w o then the equation reduces to same form as (1), that is K y = F v -(Bi + BJ ) - W r sin - R, (3) for recuperators directly connected with the recoil bralce cylinder, K y * p y A - B - W r sin - R where c v P\r = Pa W 2 (" x = variable orifice by buffer rod on a floating piston in recuperator or air cylinder.) r t ^V -A(b-x) J - F v F . = 1.3 W r (sin * u cos 0) hence, K tt = F - (B* + B") -W r sin - R, v v " 2 C V where B x = ~ A v A ,,7 V "x which is again an equation of same form as (1). 372 The general equation of counter recoil, therefore, becomes *V ~(&x + Bj) - W r sin J - fl mg v - where 9 t a i L/n Y * | y B x " c ~ x 2gc"w* o" DAjv* * V B x * " * 1 c . 2gC w CALCULATION OP RECOIL It is often convenient to CURVES. calculate the retarded velocity curve against displacement, especially when the resistance to recoil is not made constant. In all cases we have seen the resistanc-e to recoil is in general a function of both the displacement and velocity of recoil, that is the recuperator component of the recoil resistance is a function of the displace- ment, whereas the bralce component is a function of the velocity and the variation of the throttling orifice. Hence K = f(x t v) and the dynamic equation of recoil 18 dV Pb cos(0 + 0) - K WR - or when the recoil translates in the direction of the axis of the bore, K dV To measure Pj, we may consider the momentum im- parted by the powder gases in free recoil, then P b " R aT or J Pfcdt B R^f"^f t ) Therefore, for l i the same interval of time (t-t t ) we have 373 R (V f -V fi ) cos (9*0) - K (t-t t ) R (V-V t ) be nee V V t +(V f -V ft )cos(9+0) U-t t ) or when the re- "R coil translates in the direction of the axis of the bore, R * V = V +(V f -V f .) (-t.) Further since X = X +/ Vdt, Bo t we have t X = X t *V t (t-t t )+ / V f dt cos (9+0)-V ft (t-t t )cos(9+^) - K ( t . t )* now J V f dt = X f -Xf t hence 2 R 4 t X = X t + [ t -V f t (t-t ) or when the recoil translates in the direction of the axis of the "bore, x - x^O^-v^Mt-^MXf-x^)- Jj- (t f -t a )* Therefore the velocity and displacement, for any given interval (t t -t s ) (a) along guides not parallel to the "bore: v t v t +(v ft -v fl )cos(e+0) - (t t -t t ) R x t =x t +(? t -v ft cos(e+0)](t t -t t Mx ft - (t,-t t ) (b) alon^ guides parallel to the axis of the "bore: 374 After the powder period these formulas reduce to w -".-'t) R 2n R and obviously are independent of the direction of the guides with respect to the axis of the tore. K t +K 2 The value of K = - , which may he closely approximated by a repetition of the substitution in these equations, since from the first substitution we closely ap- proximate V 2 and thereby can determine K a =f(X a V ) for the second substitution. CALCULATION 0? ACCELERATION, TIME AND DISPLACEMENT PROM A GIVER VELOCITY CURVE: Recoil and counter recoil velocity curves are usually drawn experimentally as functions of the displacement though they may be drawn as well as functions of the time. The customary method of obtaining a velocity curve, is to set a tuning fork vibrating and allow the vertical oscillations to form a sinuous curve along a narrow soot covered strip recoiling with the gun. Then if f * the fre- quency of oscillations of the fork, we have for the time of one oscillation, T = -^- If n = the number of oscillations for an interval Ax, the velocity becomes, v 3 , where At = nT if x is measured in inches, At ' ' 13 ZT (n/sec) 375 To obtain the time as a function of the displace- ment, since vdt=dx t = / - dx o v 1 * 1 and if x is measured in inches, t = / - dx v o v Hence the area under the reciprocal of the velocity curve against displacement is the time of recoil. We may then draw the velocity curve as a direct function of the time of recoil. When the recoil velocity is measured as a function of the time, the acceleration is dv 7- = the slope of the velocity curve at When the recoil velocity is measured as a function of the displacement, the acceleration is, dv v ~ = the velocity * the slope of the velocity curve . = t~he sub-normal of the velocity curve. If dx is measured in inches, the acceleration is 12 v (ft/sec*) dx From the relations, v=f(x) and t=/ - dx =/ - - dx v f(x) we see that the velocity curve may "be readily expressed either as a function of the displacement or as a function of the time or both. CHARACTERISTICS OF RECOIL From Proof Firing Tsts, CURVES. recoil curves are obtained for both recoil and counter recoil. From these curves, it is possible to determine the variation of the reactions throughout recoil or counter recoil. 378 In the analysis of curves during the powder period, since the mutual relation connecting the variation of powder force and the retarded recoil is the common time, it is necessary to express the forces, velocities and displacements as functions of the time. In the analysis of curves during t~he retarded recoil and counter recoil it is possible to express the forces and velocities as direct functions of the displacements which considerably simplifies the work. (1) Powder Pressure Period: Recoil along axis of "bore. The equation of recoil is dV Pjj - K = Dp T where K = B+R-W r sin With a given velocity curve, the velocity and displace- ent should be tabulated as a function of the time; then for any interval ( t a ~ t t ) "e have (v ft -v fi >- < V v t )-L(t t -t t ) = o = o If K is assumed constant or found to be constant by brake measurements or if it is determinate as a function of the displacement, we nay evaluate Vf the free velocity of recoil. More often however, the free velocity and displacement curves can be evaluated as a function of the time, and knowing the retarded velocity and displacement curve as a function of the time we may calculate the resistance to recoil from the above expressions. Then u _y p b = "R / l a - *t and ,4? dV dV p b-R jf '- * here "R d"t s "R v al ' "R dV It is to be noted that Pv and - Dp r are the external d t ^^^ 377 recoil forces during the powder period. Further P^ acts along the axis of the bore and - n R *- acts through the center of gravity of the recoiling parts parallel to the axis of the bore or guides. If e = the distance from the center of gravity of the recoiling parts to the axis of the "bore, we have for the external reactions on the mount a couple Pue and a force parallel to dV the axis of the "bore, Pv - mo = K. The balancing dt forces are the weights and reactions of the supports. For stability the moment of the weights about the rear support must exceed the moment of Pve and K about the rear support. (2) Retardation Period: Recoil along axis of bore. During this period, we have simply dV dt applied through the center of gravity of the recoiling parts, R V = - K parallel to the axis of the bore, which together with the weights and balancing support reactions are the external forces on the mount. It is to "be further noted that since X * P h * P v * R ~ w r sin ^ we nave velocity curve, dV P h - o R V P v - R + W r sin f dx V k where P v * ?vi I ) f r pneumatic recuperators V - A v o *x R = 0.25 lf r cos + R_ approximately where R p = estimated packing friction. 378 (3) Counter recoil: C'Recoil along axis of bore. During the accelerating period of counter recoil, the inertia resistance is directed towards the breech the same as in recoil. Here dv K v = mp v to t"he rear dx and during the retardation the inertia resistance is directed forward and "here, *v - * R V which together with the weight of the system and balancing supporting reactions are the external forces on the mount. Since further, during the retardation, dv i - m R v = F v - B x - W r sin - R we have sin and v o F v * F v f [ ] fof pneumatic recuperators V ft -A(b-x) and R 3 0.15 W r cos + Rp approximately where Rp = estimated packing friction. Since critical counter recoil stability is at horizontal elevation, C'recoil curves are usually ob- tained at "horizontal elevation. Then, i dv B x ? v -R+pV for the buffer force where ^ x the overturning force is dv D v - along the axis of the bore forward, dx RECOIL BBAKII0 WITH A CONSTANT ORIFICE: As a first approximation we will assume tne re- cuperator reaction not to vary greatly in the recoil. 379 Then K = A * Bv where A = P v + R - W r sin t B = the hydraulic "brake throttling constant. (1) During the powder period, we have Pu - (A+BV*)m R dt V +V . m R Expanding, we have which is a quadratic equation of the form aV* * bV + c = 2 Z and - b A) -4ac V, where a . ^(t^J V B b 1 * -i- (t,-t t ) 2 nn c = ,- t - t - 4 R "R If the intervals are talcen very small, then A +BV* (t,-t t )*(v fl -v ft ) Then solving for V 2 we may repeat with the expression V +V V, = V -- - (t.-t t )*(V f ,-7 ft ) 380 for a closer approximation. The displacement is obtained from the expression, V +V (2) During the retardation, we have V - A - BV* hence dx R v a ' A+BV R A * Bv Integrating, we have X -X * log e 28 ' In particular if X t * Eg the constrained recoil dis- placeaent at the end of the powder period and v i = Vp tbe constrained recoil velocity at the end of the powder period, then, for any displacement x and recoil velocity V, we have, R , x " ER " 7* loge or with common logarithm, X - ED 1.15 log - B A+BV when V = the length of recoil "becomes, b - B R + 1.15 log (1+ - Vj) B A As first approximation, we may take Eg * E the displacement in free recoil during tbe powder period and V R * Vf the maximum velocity of recoil* then h - B * 1.15 -| log (1* j- CO CHAPTER VII CLASSIFICATION AND CHARACTERISTICS OP RECOIL AHD RE- CUPERATOR SYSTEMS. Recuperator systems nay be divided into: (1) Hydro pneumatic recuperator systems (2) Pneumatic recuperator systems (3) Spring return recuperator systems. (1) With hydro pneumatic systems, we have two fundamental arrangements:- (a) The hydraulic brake separate from the hydro pneumatic re- cuperator. This requires two or more rods, a brake rod and a re- cuperator rod. Further we have in general two or more cylinders, a brake cylinder, a recuperator cylinder which may have passage way or connection with an air cylinder. The recuperator and part of the air cylinder is filled with oil. The oil nay be in direct contact with the air in the air cylinder as in the Schneider and Vickers material or it may be separated from the air by means of a float- ing piston in the cylinder. (b) The hydraulic brake cylinder connecting directly with the recuper- ator cylinder. The oil must be throttled between the recoil and re- cuperator cylinder, and thus the oil at lower pressure reacts usually oa 381 382 a floating piston separating the oil and air in the recuperator cylinder. To augment the initial volume for the air in the re- cuperator an additional air cylinder connecting with the re- cuperator may be introduced. Thus with this arrangement we have from two to three cylinders. (2) With pneumatic recoil systems, we have usually, (a) One or more pneumatic cylinders, according to a satisfactory layout. The piston compresses the air directly, no oil or other liquid being used for transmitting the pres- sure. (3) With a spring return system, we may have various arrangements: (a) One or more spring cylinders separate from the recoil brake cylinder. (b) With small guns, the spring con- centric and around the recoil brake cylinder. The potential energy or the energy of compression of the recuperator during the recoil, becomes ?OR PNEUMATIC OR HYDRO PKBUMATIC 3Y3TIM8; If p a j - initial air pressure. (Ibs/sq.in) p a f = final air pressure (Ibs/sq.in) Paf = m = ratio of compression Pai V Q initial air volume Vf * final air volume K y * recuperator reaction 383 Un T\\\\\\V? \\\\\\\\ Z1C K\\\\\\V> oo ul VA'.W 384 Vf k f / Pa* - - Pai 5 / 1-k V, V Paf ? o k Now m - 3 <.r~) 3 the ratio of compression Pai v f Therefore, the work of compression becomes in terms of m, and the initial volume, I(p) - 1] ft. Ibs. (1) where p a s is in Ibs. per sq. ft. and V Q in cu. ft. [.(, - u f t . lba . (1 ., 12 k-1 k when V Q is in cu. inches and p a ^ in Ibs. per sq. in. The recuperator reaction, becomes for any displacement X in the recoil, Pa *a Pai V k - A . o fl a x where p a i is in Ibs. per sq. in. and A a in sq. in. Jf in inches and V Q in cu. inches. The initial volur becomes, v V ^7- (s) where b = length of recoil. With the oil in direct contact with the air, we will assume that the 385 temperature remains approximately constant through- out the recoil and k Mill be taken at 1.1 With a floating piston interposed between the oil and air, or with a pneumatic recoil system, we will assume a negligible radiation, that is the com- pression approaching an adiabatic condition. Hence k will be assumed = 1.3 FOR SPBIH3 RITUHH SYSTEMS: If S o * initial spring recuperator reaction Sf * final spring recuperator reaction Then the potential energy stored in the spring at the end of recoil, becomes b S f -S ft P,g, . / (* + _L_2. x)f (4) b" and if b is inches, we have P,E,*(S o +Sf )- mm The reaction exerted by the springs at any displace- ment of the recoil X, becomes s f -s K v * S o + r x (5 ) RECOIL BRAKES. In the broad classification of recoil brake systems, we have those: (a) where the brake system is independent of the recuperator system, (b) where the brake system is part of or connects with the recuperator system. (a) In consideration of independent brake systems, we nave a further 386 classification- CD Brakes with throttling orifice through the recoil piston, the vary- ing aperture during the recoil being produced by the difference in areas of the constant apertures in the piston and the area of the bar or rod of varying depth or diameter fixed to the recoil cylinder and moving through the aperture; or the throttling nay be around the piston by varying grooves in the cylinder walls along the cylinder. (2) Brakes with varying apertures through the recoil piston, the aperture being cut off during the recoil by a rotating disk about the axis of the piston, the disk being rotated during the recoil by a projecting "toe" engaging in a helicoidal groove in the cylinder wall. This form of brake is known as the Krupp valve and is extensively used not only by the Krupp but other types as well. (3) Brakes with the throttling taking place around the piston, [not through as in (1) and (2)], through a sleeve perforated with boles along the recoil. The piston cuts off the number of boles during the recoil thus decreasing the effective throttling area. (4) Brakes with the throttling taking place through a spring controlled valve. With independent brake systems the spring valve is contained in the piston. Where the brake is part of the recuperator the throttling takes place through a fixed orifice sonewhere between the two cylinders (5) Brakes with a constant orifice. On- 387 less the air pressure is fairly large, and the throttling relatively small constant orifice control should be avoided since it gives a large peak in the braking. In consideration of brake systems as a part of the recuperator, the throttling takes place between an orifice fixed somewhere between the two cylinders, and usually of the spring controlled type though sometimes with high air pressures a constant orifice may be used. In general it nay be stated when the recoil energy is large the throttling may be very satisfactorily con- trolled, as in brake systems of the type (1),(2) and (3). Where the energy of recoil is small as in small caliber guns, the throttling areas especially at the end of recoil must be small. This can not be satisfact- orily met with "bars" or "grooves" due to the inherent tolerance making very often the clearance greater than the required throttling areas towards the end of recoil. This difficulty has been repeatedly met in the design of small recoil systems. On the other hand spring controlled valves are admirably adopted for small recoil systems, since the throttling towards the end of recoil can be finely controlled. COUNTER RECOIL SYSTEMS OR HUNHIKJ 70BWARB BRAKES: In the classifications of counter recoil systems, we have two general types of systems: (1) Where the brake comes into action daring the latter part of the counter recoil. (2) Where the brake is effective throughout the counter recoil. With (1), the buffer action can only take place after a displacement of the void (the displacement 388 of tha recoil piston rod A r b), which with guns of large piston rods May be a considerable part of the counter recoil. With (2) the buffer must be filled daring the recoil, otherwise no buffer or braking action can take place. The brake with systems where the buffer action takes place towards the end of counter recoil, con- sists usually of a buffer chamber as an extension of the recoil cylinder in the front and spear buffer pro- jecting from the front side of the piston or with a buffer chamber within the piston rod itself the spear buffer rod being attached to the front bead of the cylinder. In the former type we must have a projectory chamber from the cylinder, while in the latter we must have a larger piston rod with consequent larger void to overcome during the counter recoil. With guns of high elevation in order to meet the demands of counter recoil at maximum elevation, we have a surplus potential recuperator energy in the recuperator and no means of checking or regulating the velocity during the greater part of horizontal recoil: therefore at the initial condition of counter recoil stability, we have unfortunately an inherent con- dition of a large buffer force, making the mount un- stable at the end of counter recoil. Therefore, this type of counter recoil brake, which is effective only during the latter part of counter recoil should only be used for guns of low ele- vation. Counter recoil brakes of type (2) which fill during the recoil end are effective throughout the counter recoil, are really the standard form of counter recoil regulator to meet the varying con- ditions required in modern artillery. Varying forms of this type of brake are used. Thus in the Filloux and Schneider reeoil system the buffer is at the end 389 of a counter recoil rod which serves also as a throttling bar through the recoil piston. The buffer head is enclosed and slides within a buffer chamber in the piston rod. Apertures near the piston in the piston rod adait the oil daring the recoil into the buffer chanber, the oil passing through a valve in the buffer bead which can open during the recoil and closes during the counter recoil as in the Schneider material. In the Filloux, though we have a filling in buffer in the recoil piston rod, the buff- ing action takes place only at the end of counter re- coil but a positive filling is ensured. The velocity of counter recoil is maintained low in this system by lowering the recuperator pressure during the greater part of counter recoil by throttling through a constant orifice in tbe air cylinder. Various forms of filling in buffers are shown in figs. (1), (2), (4). APPROXIMATE FORMULA FOR If tbe total resistance TOTAL RESISTANCE TO to recoil is assumed con- RECOIL. stant throughout the re- coil, we have when the recoil is along the axis of the bore, which usually occurs in practice, that t ..a 7 "r V f b-B+? f T where B * free reeoil displacement during powder period. T * tine of powder period. wv + 4700 w V * max. free velocity of recoil. "r (ft/sec) b * length of recoil, (ft) Now B * C t V f T and T * C t 390 where u o = travel up the bore and v o * muzzle velocity. Substituting, f v v u o * c ~~ u o This value of E may be obtained in another way, "T v, B = C( - )u - C u r hence " 1 K 7 m r v f< * > v f o b-C C u +C V f v o v o - 7 -r v f< u o v f b*(C,-C C ) - v o Mr. C. Bethel found from computation on a series of guns of various calibers that the value (C j -C 1 C ) eould be represented very closely to a linear function of the diameter of the bore, that is C f -C t C f where d = diameter of the bore, (in) If u o travel up the bore (in) ? o = nuzzle velocity (ft/sec) Vf velocity of free recoil (ft/sec) b * length of recoil (ft) then C f - C t C t .096 + . 0003 d 391 and we bare, K 2 u o v f [b+(.096+.0003 d) (Bethels Formula) The formula applies to a constant resistance to recoil and is especially useful, since the computation of E and T are not needed. GENERAL EQUATIONS OP The characteristics and RECOIL AND COUNTER functioning of the various re- RECOIL.- RECOIL coil systems may be shown in SYSTEMS. an unique way by a study of the equations of recoil and counter recoil. Let K = the total resistance to recoil assumed constant throughout the recoil, (in Ibs) P b = powder pressure on breech p * the pressure in the recoil brake cylinder. (Ibs/sq.in) A = the effective area of the brake piston. (sq. in) p y the recuperator pressure (Ibs/sq.in) A T the effective area of the recuperator pis- ton (sq.in) B = pA + P v *v s tne total braking, (in Ibs) R_ * the total packing frictions, (in Ibs) Rg * the total guide friction (in Ibs) R * R p +Rgthe total recoil friction (in Ibs) angle of elevation of the gun. X = displacement from battery along the recoil (in ft) b = total length of recoil (ft) Then during the recoil dv Pjj-K * m r during the acceleration - K - m_v T- during the retardation. * at 392 The external force on the mount during the acceleration is dv Pjj - r * K, as well as the weight of the dt recoiling parts, and a couple P^d^, where d^ = distance from the center of gravity of recoiling parts to the * *U V axis of the bore. During the retardation, the external force on the mount in the duration of recoil is, dv p v * K , together with the recoiling dx weight, (1) when the brake system is independent of the recuperator system, then K B * R - w r sin pA+p v A v +R-w r sin 18 Now the hydraulic pull becomes, C v* pA - 5 where v * the velocity of recoil at displacement x (ft/sec) N x - the throttling are at displacement x. Further, with pneumatic or hydropneumatic re- cuperators, k PV A V ' Pvi ) where k = 1.1 to 1.3 and with spring return recuperators P T A T - S - S * -~ - x Hence the total resistance to recoil, becomes, with pneumatic recuperators, V, 393 and with spring return recuperators * + Pa 2 Rf R-W r sin R-w r sin * A 395 STABILITY COH8IP1HATIQM Now if K h horizontal resistance to recoil h = height of center of gravity of recoiling parts above the ground. w g > weight of the total l s * horizontal distance from spade to center of gravity of W 3 with recoiling parts in battery. w c * weight of carriage proper (not including re- coiling parts) l c horizontal distance from spade to center of gravity of carriage proper. V r * weight of recoiling parts. l r * horizontal distance from spade to center of gravity of recoiling parts in battery. e constant of stability then since W s l g W r l P + * c l c for any displacement x, the stabilising moment becomes, W r (l r - x cos 0)+W c l c =W s l g - W r x . Therefore, with a given Margin of stability, we have, K D h * c(W s l s - W r x) and hence for a constant margin of stability throughout the recoil at horizontal elevation, e W 8 1, e * r b b That is, the resistance tp recoil at horizontal recoil, should decrease with the recoil consistent with this equation. When a constant resistance is maintained through- out recoil at horizontal elevation, K h should be limited consistent with stability in the out of battery position. Advantage of the total resistance to recoil following the stability slope: (1^ More energy is dissipated by the brake during the powder period, by fol- lowing the stability slope and thus gives a greater decrease of the recoil dis- placement during the powder period. 396 (2) The braking forces being bigber during the greater part of the recoil, the re- maining energy or energy of constrained recoil, is dissipated in a shorter re- coil displacement. Hence the total recoil displacement is decreased over that with a constant resistance to recoil. Farther, since the stability slope causes a smaller resistance to recoil in the out of battery position with a longer recoil, the total resistance to recoil if Maintained constant throughout recoil Bust be smaller, and the recoil displacement greater for a given energy than when the resistance to recoil fol- lows tba stability slope. The relation can be shown analytically as follows: Assuming a constant resistance to recoil maintained during the powder period and varying with the stability throughout the remaining part of the recoil, we have for a variable resistance to recoil throughout recoil, K T K T* V r ' V f ' V ; ' ' B ' T~ r "r where K Q = the resistance to recoil maintained constant during the powder period. Since c Bgl c the stability slope becomes, m - ( therefore, the resistance to recoil in the oat of battery position becomes, k * K O - m(b-S r ), we have therefore, r Substituting for V p and E r , we have solving for b, b, B * (1- f ) *o (1- f & in m 6 T* where A - m r ?j energy of free recoil 397 o 398 c. * r stability slope K With a constant resistance throughout the recoil, K(b-B r ) - ^ m r V (1) KT* KT where E_ E - - and V_ * V* - - 2m r m r c W.l_ c W and K r 8 * E b (2) a h Combining (1) and (2) we obtain the length of recoil for a constant resistance throughout recoil, and consistent with the out of battery stability, c a VfT-E i / b c * T -( T > r- /[(VfT-EJ-C.j'^BlA+C^B-vyr)] t m o o m where r w l r n 3 S v*TfB C- * ; m stability slope h h A * - n r Vf energy of free recoil. b v Length of Recoil for Variable Resistance The ratio r * b c Length of recoil for constant Resistance to Recoil to Recoil gives the percentage of recoil by following the stability slope to that of a constant recoil consistent with stability in the out of battery position. The relation can be shown graphically, fig.C ). The ordinates to the line A8 represent the maximum possible overturning force consistent with stability. The slope of 399 cWj, C Wglg AB and the ordinate oA , Main- fa h taining a constant resistance to recoil during the powder period consistent with stability we have ordinates to DE, in the powder displacement oH. The resistance to recoil decreases according to the EF con- sistent with a constant margin of the stability. The area OD, EF Go, represents roughly the energy of recoil A - m r Vf If now a constant resistance is to be maintained we have diagram o J P C where the constant resistance to recoil o J = C P, and CP = c 8 C, that is, is consistent with a given margin of stability in the out of battery position, and further the area oJPCA*-jm r Vf (the energy of free recoil). METHODS OP CALCULATING With independent recuperator THROTTLING ORIFICES. systems, the throttling is usually either by throttling grooves or bars or by a mechan- ically controlled orifice as in the Krupp valve mechanism previously described. Let us now consider the necessary throttling orifice variation along the recoil. Daring the powder period, we have two methods, (1) To maintain by a proper variation of the throttling grooves a constant resistance to recoil during this period. (2) To maintain a constant orifice during the powder period. In method (1), we have, r ( ) + R t - W r sin i K A con- V o~ AX stant during the powder period. Therefore C A- V = 13.2 / K - p a - F t + W r sin 400 where K for a constant resistance to reooil. b-E+7 f T K ^^ for a variable re- 2[b-B+V f T- 5- (b-B)J 'Stance to recoil. tr 2 B- f Vp .\t Pa * Pai A v^ ' p ai A v approx. during powder YQ AX. period unless the recoil is relatively short. R t > Rg+ R p total friction: guide friction * pack- ing friction, further from interior ballistics, av. total powder pressure w v * o Pe a 2g a w * weight of shell T O * u8le vel. (ft/sec.) u * total travel up bore (ft) 27 a Initial pressure on breech in gas expansion p o ^ = c 4 1.12 p M (Ibs) where p a total powder (b * u) " pres.ure.(lbs) and 27 P / 27 P e u( I D t /i - *L Ji). - ! 16 p e 16 p e 3 u wv +4700 w t " 7- approx. V f - - * ? o "r r fo ) W p (W*0.5 w)y Q * ~~^r T Vf "~r r T * t + t total ponder period (sec) 401 and g = Xf + f^ + total free recoil daring powder period, (ft) Three points are sufficient for the orifice curve during the powder period and the corresponding constrained velocity and displacement to sub-stitute in the orifice equation with its lay out are: K l o 7 f * V fo (ft/sec) "r lii 2m, (ft) when the shot leaves the muz- zle. Kt m = v -- (ft/sec) fm where 'f.*:* (ft) > the maximum restrained recoil velocity and corresponding orifice. 4. r (f r v go ) T + P ^ b d r K(T-t ) Pob 6 r (V -7 ) sec. Rt - (ft/sec) < ft > At the end of the powder period. 402 After the powder period, that is during the retardation period, we have for a constant resistance to recoil, simply, and therefore CA o- 13.2 /K-p a -R t +W r sin which gives up the required throttling with a con- stant resistance to recoil during the retardation period of recoil. When the resistance to recoil is variable, we have during the retardation period, that 1 m r vj -[K - 2 (X+b-2E r )](b-x) K- (b+X-2B r )](b-x) and there- / , fore . >^IK-- (b.X- 2Br )](b-x) CA 1 / ~ T w , === ^ = ^ == ^ =: _i. (sq.in) 13.2 /K-p a -R t +W r sin which gives the required throttling with a variable resistance to recoil during the retardation period. With a pneumatic or hydro pneumatic recuperator system, VQ k Pa " PaiM , A y> where k 1 - 1 to 1 ' 3 o v V - initial volume. 403 S t~ S 6 With a spring return recuperator, p a S o - X b b - length of recoil (ft) where S o initial compression of the springs (Ibs) S * final compression of the springs (Ibs) and 2 approx. s o In method (2), with a constant orifice during the powder period, we have cVv 1 dv * * iTTTf p -"**" ain * ' "r JT Since an integration of this equation is complicated an approximation is made by assuming the recoiling mass to recoil during the powder period "a" given mul- tiple distance of the free recoil displacement when the shot leaves the bore. Let E r * length of constrained recoil during powder period, and corresponding length of constant orifice (inches) u total travel of shot up bore (inches) I * constant from (2 to 2.5) use 2.24 w * weight of powder charge (Ibs) W s weight of projectile (Ibs) P n * total hydraulic pull (Ibs) w x = area of orifice (sq.in) at recoil displace- ment x (in) A effective arc of hydraulic recoil piston (sq.in) c * coefficient of contraction - - - 0.85 to 0.9 d - S. G. of fluid 0.849 a 2 u (1) Now the total drop in pressure through the recoil orifice, becomes, 404 7 (d 62.5)A* 7 X p (ibs per sq.ft)(See Hy- gc w x dro dynaaics) or 62.5 d A 7 P * (Ibs. per sq.in) 64.4 x 144 c'wj During the retardation period of the recoil, we have fro* the equation of energy, K(b-x) "r . 64.4 K(b-x) * - 7- hence 7- ,. ' 12 * 12 r therefore, .*,,/u \ 62.5 dA K(b-x) 12 x 144 c'wj W r and . w. -.0361 -^ - (2) d A 8 K(b-B r ) .0361 i*J- - - (3) which gives the orifice at any displacement x in terms of the total resistance to recoil, R and the total hydraulic pull P h . When the resistance to recoil is made to conform with the stability slope, we have t 64.4[K-0.5(b+X-2E r )](b-X) 12 w p 62.5 d A*[K-0.5B(b+X-2E r )](b-X) P 12 x 144 c* w* W r 405 hence w. * .0361 C* "r Ph and d A*[K-0.5(b+X-2B r >](b-E r ) .0361 C" W r Further P - K + W .in * - R - for pneumatic or hydro pneumatic recuperators, and ss P b - 8 + f r sin - 8 t - (S + ^ x ) for spring return recuperators. METHODS OP THROTTLING (1) The simplest net hod f throttling is by vary- ing an orifice through the piston by throttling bars fixed to the recoil cylinder and moving in the apertures through the , piston. Let w x * the throttling area (sq.ia) as previously calculated. S * width of throttling bar or whole in piston (inobes) b - depth of hole in piston fro* cylinder surface (in) d = depth of throttling bar (inches) d * initial of bar (inches) n number of bars (usually n = 2 Then the initial or maximum opening w nX(b-d )(sq.in) approx. and for any other point in the recoil, w x * nS(h-d) (sq.in) approx, With grooves in tbe cylinder wall. w x * n S d (sq.in) where d depth of rectangular groove (in) 406 (2) When the throttling takes place around a long buffer rod of varying diameter and passing through a circular hole in the piston, as in the Schneider material, we have, if diam. of bole in cylinder (sq.in) d x diam. of buffer rod passing through hole in cylinder (sq.in) then x - 0.7854(D*-d x ) which gives the variation of the diameter of the buffer rod along the recoil. la the Pilloux recoil mechanism, we find grooves of varying depth in the buffer rod, engaging with holes through the piston. The object of this arrangement is to pass from one set of grooves to another by turning the buffer rod on elevating the gun, thus making it possible to shorten the recoil on the elevating the gun. If n * number of grooves engaged during the recoil, s = width of groove (in) and d = depth of groove (in) then w x = n 3 d. (3) When a constant orifice is main- tained during the powder period we may use the so called Krupp valve, which has bad a wide application for artillery brakes. The initial orifice is closed uniformly by a disk on the piston rotated by a heliooidal groove in the cylinder wall of constant pitch. Let * initial angle moved by valve disk during powder period before engaging the throttling area in the piston. t * angle moved by valve during the retardation period. p * pitch of helieoidal groove in cylinder wall (inches) ( Linear displacement per complete 407 408 revolution of disk.) r o * radial of cylinder (inches) r * radios to bottom of tbrotting opening con- tour (inches) then the number of turns and the linear displacement x, becomes, 2n x - p 2x Hence with a constant pitch with the total recoil displacement b inches, we have 3xb hence 2n(b-B r ) * ' Further the throttling area becomes, dw x ( )d0 hence * . r w x - / * -2s d0 dx (sq.in) For computation and design it is more convenient however, to express the throttling area in terms of the displacement from the end of recoil, since the area is zero at this point and opens up gradually to its maximum near the battery position. We have then, w / (b " x) Mr e~' > d (fc- x ) wnepe r , o, where x b, In the forn of a summation which lends a simple practical method of laying out the contour of the aperture in the piston, we have, if the displacement of recoil is divided into "n" parts from E r to b, 409 Starting from the out of battery position, we bare, n^-r^) * * I A(b-X__ r ) (sq.in) 'n-i -X n .) (sq.in) a-g M " A(b-X_) (sq.in) and *g * ~ 2 o( r o~ r n-g> A ( b ~ x n-g> Orifice area at point g from the out of battery position. Thus from a step by ste{. process ire lay oat the contour of the aperture in the piston, and the total area of the orifiee at any displacement of the recoil, measured from the out of battery position, must equal the required throttling area at this point. (4) Another form of geometrical throttling, devised in order to effect variable recoil consists essentially of cutting off holes in a perforated sleeve by the piston, the throttling taking place through the sleeve in the front and rear of the piston. We have therefore two distinctive throttling drops, that in front of the piston, and to the rear of the piston through the recoil sleeve. If w x = the throttling area in front oi the piston at any point in the recoil (sq.in) w y the throttling area to the rear of the pis- ton at any two points in the recoil (sq.in) 410 p x the drop of pressure through the throttling areas w x , in the sleeve, (Ibs/sq.in) Py * the drop of pressure through the throttling areas w y , in the sleeve (Ibs/sq.in) We have for the total drop P P - P x + P, (assuming the throttling 175 w x 175 Wy constant C the same) henee F * ( + 175 w ; where * c is the equivalent throttling area and corres- ponds to the area obtained in the previous throttling area calculations. In general "I **!**!* T when we have a drop of pressure due to throttling through various orifices in series. With only two throttling drops, we have *x "v w e * * ~~-~ and x +w y * constant. / m + ** w x "y Prom these two equations, we have at the maximum value of w a , "x " *y Hence in laying out the holes in a sleeve valve, we place the piston at its displacement corresponding to naximum throttling, that is at the point of the 411 maximum retarded or constrained Telocity, making the throttling drop on either side equal. The process of laying out the required orifices and corresponding holes is as follows: (a) At max. throttling displace* ment corresponding to max. retarded velocity in the recoil, P x P_ - and w x * w y but since we have a void in back of the piston due to the displacement of the piston rod, P = P i.e. the total drop 3 the pressure in the recoil .. v cylinder. c A 7 13.2 2 and 2 C A V xm 13.2 (b ) Next move the piston from the position of max. velocity, a unit distance equal to the width of the piston in the direction of recoil. The area to the rear - w_ = "c -r , since no openings have been uncovered m in the rear. The area to the front is obtained from the equations, ~ - + where w ce , w xo etc. are c t x i y the throttling areas at max. velocity and w c ^ w x ^ etc are the throttling areas at a distance from the position of max. velocity equal to the first unit displacement, hence 412 (sq.in) (c) Next move the piston another unit distance in the direction of recoil, the area to the w w rear, c f c hence w c w *x * * (sq.in) (d) Hence for all succeeding points in the recoil, w-g w c - w yj - and y& "xg ~ (e) ID the powder pressure period, we move the piston backward towards the battery position from the position of maximum velocity succeeding units to the rear and the process is exactly similar as moving forward in the direction of recoil. THROTTLING THROUGH A With dependent recuperator SPRING CONTROLLED systems, as in the St. Cbamond VALVE. recoil system, the drop of pres- sure between the two cylinders (i. e. between the recoil brake and recuperator cylinders* may be obtained by throttling through a spring controlled orifice between the two cylinders. A spring valve, however, may be used with an ordinary recoil brake cylinder, the throttling taking place through a spring controlled orifice in the piston. 413 Let p the pressure in the recoil cylinder (Ibs/sq.in) a = the area at base of valve (sq.in) p a pressure in receiving chamber or recuperator (Ibs/sq. in ) p a i = initial pressure in recuperator (Ibs/sq.in) Paf = final pressure in recuperator (Ibs/sq.in) A a = effective area at top of valve (sq.in) a t * area of valve stem S = initial spring compression (Ibs) Sf = final spring compression (Ibs) A * effective area recoil piston h lift of valve in inches c = effective circumference at valve opening Then, at the maximum opening, giving a lift h, ire have pa- Pai^a = ^f ^^ s ^ (approx) and when the valve is about closed, pa- p a fA a = S Q (Ibs ) (approx) When A a = A, as with valves in which the spring is entirely enclosed in the recuperator chamber, we have (p p a ^)a=S when open (approx) and (p-p a f)aS when closed (approx.) When the spring is outside of the recuperator chamber, and a valve stem passes through a stuffing box to the outside of the chamber, we have pa- P a A a * P a -Pa< a ~ a i> = (P~Pa> a+ Pa a i (lbs ^ Further at maximum opening of the valve we have for maximum throttling ! P-Pai * - T~* "bere C = - to - 175 C h 0.6 0.8 hence C A V h = - which gives the lift of the valve at max. opening and corresponding to a spring reaction = S f Ibs. Therefore knowing p, p ai and p af and solving for the total lift h, we have, for the spring required: 414 Initial load ................ 3 (Ibs) Final load ................. 3 f (Ibs) Total lift ................. b (in) Spring constant ............ S f" S o (lbs P er in > b which completely specifies the spring required to properly function the valve during tbe recoil. Now the pressure in tbe recoil cylinder, is K+ff r sin t! R t p - (Ibs/sq.in) A v o * and in the recuperator cylinder, p_ p a j ( ) VAX (Ibs/sq.in) The maximum throttling opening occurs, at dis- placement X m in the recoil, that is at the point during the powder pressure period, where the powder reaction just balances the recoil reaction. This is slightly before tbe end of tbe ponder period and for an approximation we have, where a t * 2 approx. Further tbe maximum constrained velocity may be taken at, V r * g 7 f where g * 0.88 at short recoil * 0.92 at long recoil. Therefore at maximum opening of tbe valve (lift h") we have, K+ sin 6 - R V o and at tbe end of recoil, sin ? - R 415 Now due to the hydraulic throttling. C A V r /K+W r sin - R t V ( 1.2 / - 13.2 / p ai Thus we have a complete specification for the design of the spring. If now, p s = K s + W r sin - R t = pull at short recoil, max. elev. (Ibs), p h =K h~ R t = P U H a* long recoil, zero elev. (Ibs) F v j = initial recuperator re- action, required to hold gun at max. elev. in battery (Ibs), F v f Pyi * final recuperator reaction at the end of recoil (Ibs) We have, at short recoil, max. elevation, at the be- ginning of recoil, p p . T- a -- I A A Sf " T- a -- a * ; - a + a (Ibs) with A A * springs functioning outside recuperator chamber, and at the end of racoil, n P vi (lbs) a -r a t (Ibs) with springs functioning out- side recuperator chamber. The corresponding max. lift at short recoil becomes, 3 Cl tl E 1 O A v F vi now S f -S * a a (-l) 13.2 c / P 3 ~F vi and the spring constant, Ibs. per linear inch, becomes, S f -S 13.2 c A a (m-l)F yi / p g -F vi S = 3 - (Ibs. per in) h i. C A V 416 From the above equations, we see, therefore, that the load on the spring is large at short recoil and proportional to the difference of the pull at max. elevation and the initial recuperator reaction and this load is increased proportionally to the valve stem area and load on the air. Therefore to decrease the load on the springs, the valve stem should be made as small as possible, only sufficient to carry the spring load. The lift varies inversely as the square root of the difference of the pull at max. elevation and the recuperator reaction, and when this difference is large as in short recoil, the lift is proportionally small. Finally the spring constant (that is the slope of the load - deflection chart) increases with the load on the air and with the square root of the difference of the max. pull and the initial air recuperator re- action. On the other hand, if the compression ratio is low, approaching I, or if the annular area or the effective area on top of the valve is small, that is, using a large valve stem, we must have a spring of considerable deflection for a given change in load. When A a 0, or F yi 0, we have no change in load in the spring and the valve would open a given lift h, with a corresponding spring reaction. As the gun recoils, if the lift and corresponding throttling area remained constant, the pressure would drop pro- portionally to the square of the velocity. This, there- fore, causes a gradual closing of the valve since the spring reaction must decrease, and we have a throttling in between an ideal spring controlled orifice and that with a constant orifice. Even with this arrangement we have a vast improvement over that of a constant orifice and the peak in the throttling ia greatly reduced. Now, at long recoil, horizontal elevation, at the beginning of recoil, p p . S f = a - A a (Ibs) A A 417 a (Ibs) with spring functioning inside recuperator chanber as is usually the case at long re- coil (See St. Cbaaond Chapter), at the end of recoil, P b - p v S = a A A bvi * a (Ibs) with spring functioning in- A side recuperator chamber. Tbe corresponding max. lift at long recoil, be- cones, 3_ C a" V h a ^ (inches) 13.2 c / P n -*vi Further B . vi Sf-S = a (m-1) (Ibs) and the spring constant, Ibs. per linear inch, becomes, S*-S A 13.2 c a(i C a V Prom these equations we see the load on the springs is relatively snail as compared with short recoil, the deflection b large and the spring con- stant snail. Thus, in comparing the requirements of spring characteristics at short and long recoil respect- ively, we have, (1) Short recoil and max. elev. = A large spring reaction and small de- flection with a spring constant having a steep load deflection slope. (2) Long recoil and horizontal elev.* A small spring reaction and large deflection with a spring constant having a snail load deflection slope. 418 To meet the requirements of (1) in the St. Chanond recoil system we find Belleville spring used; and in (2) the use of a weak spiral spring. When a spring valve is used without a recuperator, the spring valve is usually located in the piston of the hydraulic cylinder. In the design and working of this valve the following points are important: Let P hi * the initial hydraulic pull (Ibs) P hf the final hydraulic pull (Ibs) A * the effective area of the recoil piston (sq.in) a * the area at the base of valve (sq.in) P ai = initial recuperator reaction P a f * final recuperator reaction R t * total recoil friction Then P h ^ + P a j + R t - H r sin K at the beginning of recoil, and p hf * p af * R t ~ w r sin ^ * K at the end of recoil, hence P ni K + W r sin - R t - P ai : P hf K+W p sin At the beginning of recoil, p r- a (Ibs) the pressure in the back of the valve being negligible. At the end of recoil, S - a (Ibs) The throttling at the beginning of recoil, be- comes a (Ibs) and the spring 419 13.2 a c /P b i(Paf- p ai> C A * V The above equations show that the maximum load on the spring depends upon the maximum hydraulic load, the assembled load on the minimum hydraulic load at the end of recoil, the lift varying inversely as the square root of the maximum hydraulic load and the spring constant or the compression deflection slope of the spring being proportional to the difference between the final air and initial recuperator re- action and the square root of the maximum hydraulic reaction. The spring throttling valve has been used success- fully with an ordinary hydraulic recoil brake cylinder, designed for approximately constant pull throughout the total recoil as in the lower brake cylinders of a double recoil system or in the brake cylinders of a gun or sliding carriage mount. Of course it is impossible to maintain an absolute constant braking resistance throughout the recoil as previously discussed but a sufficient approximation can be obtained to justify its use. In the design of constant braking with a spring control, we have a spring valve seated in the piston. If the throttling takes place mainly through the valve seat, we have p a S o * S h where p * pres- sure in the recoil cylinder, (Ibs/sq.in) S = initial spring load (assembled load)(lbs) S = spring constant (Ibs/in) a * the effective area at the base of the valve. h = lift of valve (inches) Now c o A 7 l l h * , C a to 13.2e/p 0.6 0.8 If the valve is bevelled the throttling area becomes in place of c h, w * D h sin 420 whore D * >ean diai. of the bevel portion of the valve (in) tf angle of bevel leasured with respeet to the central axis of the valve, beneo C Q A 7 j 1 h - ===-, C = to 13.2 n D sin p 0.6 0.8 To design the spring we may adjust S o to give a suitable value of the spring constant S, by the formula, - S > RECOIL THROTTLING WITH When a buffer or regulator A "PILLING IN* COUNTER is desired to act through- RECOIL BUFFER. out the counter recoil, the counter recoil buffer chamber must be filled during the recoil. The filling of the counter recoil buffer chamber during the recoil, affects the recoil throttling in two ways: (1) The total oil displaced by the recoil piston does not pass through the recoil throttling grooves: a part passing into the buffer chamber in the process of filling it in the recoil. (2.) In the buffer chamber, we have more or less pressure during a part of the recoil, since if the throttling into the buffer chamber is just sufficient to fill during the max. vol. of recoil, we will have if the pressure in the recoil cylinder remains constant an over filling during the latter part of recoil and therefore pressure in the buffer chamber, since the throttling drop is decreased due to the decreased velocity of recoil. 421 Therefore tbe total hydraulic reaction eo the piston rod is somewhat modified. Let p * intensity of pressure in recoil cylinder (Ibs/sq.in) A * effective area in recoil piston (sq.in) A b * effective area of buffer (sq.in) V x * recoil velocity (ft. sec) w x = recoil throttling area (sq.in) a o - entrance throttling area for filling buffer chamber in tbe recoil (sq.in) P b * intensity of pressure in buffer chamber (Ibs/sq.in) Then, during the recoil, we have, for the tension in tbe rod "P h " P h - p A - p b A b (Ibs)U) Tbe drop of pressure due to tbrottling through tbe filling in bole to tbe buffer chamber, becomes for continuous filling, _ .1 . _ c o A b v x Ph s P ~ Pb = - (Ibs/sq.in) (2) 175 a hence C'*Au V* Ph = p(A-A b )* X (Ibs) (3) 175 a 175 a (Ibs/sq.in) (4) A - A b Further, ifith continuous filling, we have, for tbe velocity through the recoil tbrottling orifice, u-V v x YX (ft/sec)(5) and therefore, c a (A-A b )* V* P ' (Ibs/sq.in) 175 "x (6) 422 Combining U) and (6) we have, w a C (A-A b ) * / C'*A> 7* 13.2 /P b - n o i 175 a which gives the required recoil throttling area (assuming a density of the liquid = 53 Ibs. cu.ft.) in terms of the total pull P h , the recoil constrained velocity V x and the constant filling in entrance area to the buffer chamber a o . If the density of the liquid is different from that of hydroline oil * 53 Ibs/cu.ft.we have, (sq.in) ni*ni? v* 288g(P h - 288 g a C V X /D(A - A b )' / (sq.in) 12 C r DA 7 -* *-*-) 288 g a where D weight of liquid per ou. ft. If we have several contractions in the filling in passage to the buffer chamber, we have approximate- ly assuming tne same contraction factor for the flow ' 1 1 1 1 _ = + + _ _ _ _ a * x Determination of a o : If we desire a continuous filling of the counter 423 recoil buffer chamber daring the recoil, with a constant entrance throttling area for filling the buffer chamber, we must design a b for throttling at maximum velocity of recoil, since the throttling drop varies with the square of the velocity and is a maximum at maximum velocity, and the pressure in the recoil cylinder remains approximate* ly constant during the recoil. If now, the throttling drop is just equal to the pressure in the recoil cylinder at maximum velocity, since the throttling drop is less at all other velocities and the pressure head the same, we have a pressure in the buffer chamber continuously rising during the latter part of the recoil. If the throttling drop at maximum velocity is less than the pressure head in the recoil cylinder, we have a void in the buffer chamber daring the first part of recoil when the velocity of recoil is large, and there- fore, not continuous filling. For continuous filling, therefore p max > p b at max. vel. of recoil and therefore max. recoil pressure, that i '*. ,, c A b v max C A b V max Pmax i a , hence a Q > - ,(sq.in) 13.2 / p max which gives the proper entrance throttling area re- quired for filling the buffer continuously during the recoil. Since, however, the buffer over fills during the greater part of the remainder of recoil, a o can be made smaller than required for a continuous filling through- out the recoil and yet have a complete filling of the buffer chamber. In order that the buffer chamber may completely fill, (though not continuously throughout the recoil) we have, for the time of recoil, roughly approx. PA 424 and assuming the pressure in the buffer chamber at any tiae of the recoil snail, * p = D (Ibs.per sq,ft) 2g For the filling of the buffer chamber, = A b b, where b = length of recoil (ft) a vt hence / ^ > t * A b b and A Q C o A b A b /-* (sq.in) CQ D *g where b * length of recoil (in) A b = area of buffer (sq.in) C n * contraction constant of orifice )= to 0.6 0.8 p = pressure in recoil cylinder (Ibs/sq.in) A 3 effective area of recoil piston (sq.in) D * density of liquid (los/cu.ft) Since the pressure in the buffer is probably small by this method of filling, we may neglect the total buffer reaction in modifying the tension or pull in the rod. Further the throttling in the "filling in" buffer, becomes, C^AvV 8 b x approx 175 a* hence A b V x * Q b constant - that is, the flow into the buffer may be assumed constant throughout the recoil, hence for the main recoil throttling, we have > C*(AV-Q b )* 175 wj C (A7 x -Q b ) "x ~I 13.2 /p Since, however, p fe (the pressure in the buffer) actually rises even in this method somewhat towards the end of recoil, Qj, decreases with AV X and there 425 fore by slightly modifying the true contraction constant C o , we have, c AV w x * 13.2 /~p~ which is sufficiently exact for ordinary design. For correct filling of the buffer chamber, the filling throttling area to the buffer should be made variable. We nay plot this variable area against recoil and take its mean value as an ap- proximation for the proper throttling area for filling the buffer chamber. The condition for ideal filling of the buffer chamber are, that C O x a x * A b V x and P b throughout the recoil, where u x = the throttling velocity into the filling in buffer, p b * the pressure in the buffer chamber c = the contraction constant of the orifice. a x = the variable buffer filling throttling area (sq.in) By Bernoulli's theorem, we have, DU* D *x p and p * (Ibs/sq.in) 288g 288g where v x = the velocity through the recoil throttling orifice. D = the weight of the fluid per cu. ft. A " * and since P n ~ P A D Yg N w is a variable in the recoil, and therefore b the recoil throttling areas become modified at any instant, such that, 426 _ ~ v bcnee W , _ LJLJL A*V* c x * A """^ 288 g P n * C Constructive difficulties make it impractical to vary A X according to the above theory in an ordinary design but by making a x = a o a constant, and assuming p^ small, we have from the above formula, that the recoil throttling area equals the throttling area computed as if no buffer existed in the recoil, and lessened by a constant area Q VARIABLE RECOIL:- Stability consideration: As the VARYING THE RECOIL gun elevates the overturning AS THE GUN moment decreases, since the per- ELEVATES. pendioular distance from the spade point or the point where the mount tends to overturn on reooil, to the line of action of the total resistance to recoil decreases on elevation. Therefore, since the initial recoil energy is practically constant, it is possible to decrease the length of recoil considerably as the gun elevates and yet maintain stability. When the line of action of the resistance to recoil passes through the spade point, the overturning moment is independent of the magnitude of the recoil reaction, and therefore theoretically the recoil can be made as small as the strength of the carriage can stand. 427 Therefore, the recoil limitations on elevating the gun are clearance at maximum elevation, as well as clearance considerations at intermediate elevations, and the limitation imposed by stability for various elevations of the gun. The recoil may be cut down in any arbitrary manner provided, that consideration be given to strength, clearance and stability at all angles of elevation. The maximum length of short recoil depends upon clearance considerations at maximum elevation, while the minimum length of long recoil depends upon stability at horizontal elevation. To investigate the stability limitations on the length of recoil at low angles of elevation, let C = constant of stability = Overturning moment , Stabilizing moment 0.85 A r = initial recoil constrained energy = - r V r (ft/lbs) V r =* 0.9 Vj restrained recoil velocity (ft/sec) w v + w 4700 Vf = = free velocity of recoil *r (ft/sec) u * travel up bore (in ft) E r 'displacement of gun. during powder period 3 (w+ 0.5 w)u 2.25 (in ft) *r d = moment arm to line of action of total re- sistance to recoil (ft) b = length of recoil (ft) Then > A r C0f s l s - W r b cos 1) b-E, and solving for b, we have / a dA r 00 )- * (W s l g +T* r E r cos 0) -4W r cos(W a l s E r * 2 W r cos $ (ft) 428 which gives us the limiting recoil consistent with stability for low angles of elevation, with a con- stant resistance throughout the recoil. When the resistance to recoil is made to con- form with the stability slope, we have, b A s ~ *r * cos P ) <*X * Solving, we have -EW s l a (b-E P ) Hence, we have, the quadratic equation in terms of b dA p W r cos t a ^fr^lg ft * H s l s E r - ^ B P b w r cos & lf p cos CT Solving for b: we have, W p cos/) / A P EW 8 1 S - /("sis) ~ 2W r cos 0( d+W s l s E r - which gives us the limiting recoil consistent with stability for low angles of elevation, with a variable resistance throughout the recoil conforming with the stability slope. MITHOD Of D1CBKASIHG THE LENGTH OF RICOIL; In the layout design of varying the recoil on elevation, it is highly desirable to maintain a con- stant recoil equal to ihat at horizontal recoil for the first few degrees of elevation and then begin cutting down the length of recoil, to the minimum recoil at max. elevation, since by this method the margin of stability increases as the gun elevates and therefore exact stability at horizontal recoil is 429 00 L 430 431 I / \ C (- -* . - ~ s' ^ $ ? ^ CJ TOM JO 01 (D V) \L 432 no longer of vital consideration as horizontal fire in seldom used. In certain types of recoil systems as in the St. Chamond recoil, the size of the re- cuperator may be decreased by increasing the pull at horizontal elevation and therefore in this type of recoil it is highly desirable to design to the exact stability at horizontal recoil, as the gun elevates with constant recoil we therefore will have ample stability even at low elevations. Therefore, unless limited by clearance , it is desirable to maintain a constant recoil from to 20 elevation, and then cut down proportional to the elevation to the minimum recoil length at maximum elevation. MECHANISM FOR REDUCING Variable recoil is obtained THE RECOIL ON ELEVATION- by decreasing on elevation the initial throttling areas by turning, the counter re- coil buffer rod which contains sets of the recoil throttling grooves, as in the Pil- loux recoil mechanism; or by turning the piston and its rod with respect to the rotating valve, and thus changing the initial openings in the Krupp recoil mechanism; or by rotating a perforated sleeve as in the American sleeve valve. Two methods for rotating the throttling rod, valve or sleeve are used, (1) by a sliding bar linkage as in the Pilloux mechanism or (2) by a four bar linkage as in the Krupp or sleeve valve recoil mechanism. With a sliding bar linkage in the elevation of the gun, a cross head or bar is moved in translation. The bar contains a pin which engages in a helical groove of the rotating cylinder, thus giving the necessary rotatory motion. With a four bar linkage the valve 433 is turned directly in the movement of the linkage during the elevation of the gun. (1) In a layout of the sliding bar linkage, the distance of the translation of the bar or cross head is fixed by the pitch of the helix on the rotating cylinder and the angle turned to be turned by the cylinder. The pitch of the helix may not be constant that is the slope of the helix may vary in the revolution. With a uniform pitch or slope of the helix, the decrease in the length of re- coil against elevation may not be uniform but for constructive reasons it may be sufficiently satisfactory. Knowing the length of the translation of the slide we may layout the valve mechanism. In the sliding bar linkage of the recoil mechanism, the crank with center at the trunnions is made the fixed link, while the frame of the mechanism rotates on elevation. If now we draw two circles with centers at the trunnions and crank pin respectively, the relative displace- ment of the crosshead or bar is the distance between the intersection of these circles and a line drawn through the center line of the slide bar. Constructive- ly, it is convenient to draw a secondary constructive circle tangent to the projectile center line of the initial position of the slide bar, i. e. usually at horizontal elevation. Then at any elevation the center line of the slide bar must be tangent to this circle. Hence the intersection of these tangents with the base circles of radii at trunmion and crank pin respectively gives the relative displacement of the slide. The proper position of the crank pin with res- pect to coordinates with origin at center of trunnions can practically only be determined by successive trials for the proper movement of the slide bar. 434 (2) In a layout of a four bar linkage the angle of rotation of the valve during the elevation of the gun is as- certained from the design of the re- coil throttling. The gear turning the valve may mesh with another gear and from the gear ratio and the maximum turning of the valve the angle turned by the valve crank can be determined. Knowing the angle turned by the valve crank or valve arm we nay then layout the valve mechanism. The four bar linkage consists of the frame connecting the trunnion and valve center; the fixed trunnion crank connecting the trunnion and connecting rod; the connecting rod connecting the fixed trunnion crank and the valve crank or arm; and finally the valve or arm connecting the connecting rod with the valve center. Tbe fixed member of the four bar linkage is the'fixed trunnion crank" joining the trunnion to the connecting rod. If ire draw two circles from the fixed centers of the trunnion and trunnion crank pin respectively, the center of the valve travels along the circular path with center at the trunnion, while tbe crank pin of the valve arm moves in a cir- cular path with center at tbe fixed trunnion crank pin. It is important to note that the relative position of tbe valve crank arm should be measured from tbe tangent to the circle with center at tbe trunnions. Tbe relative angle turned by the valve crank is therefore the difference between the final angle with respect to tbe tangent of tbe trunnion circle when at maximum elevation and the initial angle with respect to the tangent of the trunnion circle when at minimum, usually horizontal elevation. Constructively, it is convenient to draw a secondary constructive circle tangent to a horizontal line through the center of the valve arm. Then the position of the valve center at any elevation is the intersection of the tangent to this secondary 435 circle at the given elevation with the base trunnion circle of the valve. If He lay off from this intersection the length of valve arm to the intersection of the trunnion crank pin base circle, we have the position of the valve arm for this elevation. For the angle turned we note the angle made by the valve arm with the tangent to the trunnion base circle at the valve center, and the initial angle of the valve am with the tangent at horizontal elevation. The difference between these angles is the angle turned by the valve arm, which multiplied by the gear ratio gives the actual angle turned by the valve. ON THE LENGTH OF RECOIL As before for a grooved WITH A STATIONARY SPRING orifice we have from the CONTROLLED ORIFICE. equation of energy: K(b-x)* m B v* (1) where b = length of recoil (ft) x = recoil displacement (ft) v x = recoil velocity at displacement x (ft/sec) m R = mass of recoiling parts and for the total resistance to recoil, for a dependent recoil system K = p A + R - W r sin where p - pressure in the recoil cylinder (Ibs/sq.ft) R = total friction (Ibs) A = effective area of recoil piston (sq.ft) "0* D A*V X D A 3 ?! C V* P~Pa * - and (P~Pa) A * - = - " 2gc*w 2gcw "x then since ,, N /0 . Combining (1) and (2), w x = + R - f r sin (2) 2KC(b-x) _ m_(K-p a A-R+W r sin 0) 436 the ratio C * ~ is approximately con- K-p a A-R+W.sin stant, since the variation of the weight component H r sin amd the recuperator reaction p a A is small compared with K. Then 2 C Q wj =* (b-x) where C o c'c. Therefore the orifice variation is a parabolic function of the recoil displacement and is independent of the initial velocity and therefore variation in the ballistics, and is practically independent of the eight component and therefore of the elevation of the gun. In general, independent of the method of throttling the length of recoil is practically independent of variation in the ballistics of the gun or in the variation of the elevation of the gun. ON THE LENGTH OP RECOIL During the retardation WITH A GROOVED ORIFICE, period of the recoil, we have, from the equation of energy, - K(b-x)= ; m r V* where b = length of recoil (ft) x * recoil displacement (ft) V x = recoil velocity at displacement x (ft/sec) T. r mass of the recoiling parts K = total resistance to recoil (Ibs) hence 2K(D-x). V (1) HOTB: Rot confirmed by observed data. Bditor. 437 D A'V! (Ibs/sq.ft) D A*V* p , p A , (2) 2gC 2 W W P h * total hydraulic pull (Ibs) A effective area of recoil piston (sq.ft) D weight per cu.ft. of fluid (Ibs/ou.ft) C = contraction constant of orifice " herC DA C 2gC K 2C Combining (2) with (1), we have W, (b-x) (3) p h r If now we assume - to always remain a constant C 1 P h a 2C and placing c C 1 - C o , we have W x a (b-x) (4) m r which is an equation of remarkable physical significance We find the orifice variation to be a parabolic function of the displacement and is quite independent of the initial recoil velocity. Therefore with the same weight of recoiling parts, the recoil displace- ment is practically the same for all values of the initial recoil velocity. Since the initial velocity depends upon the ballistics of the gun, we may com- pletely change the ballistics of the gun and yet with grooved orifices the length of recoil remains practically unchanged. In the following discussion the ratio -r- was as- sumed to remain constant; the change in the length of recoil depends therefore on the change in the ratio H p b' Let us examine this ratio for the change under two conditions, (I) As the gun elevates where the weight component is brought into effect. 438 (2) For different ballistics of the gun, where tbe initial velocity is changed. Now for case (1), a V* K * 0.45 ^ - and assuming the same length of recoil, K is a constant and independent of the elevation. If K is to remain constant, its reciprocal p h must remain constant for all elevations. Since K * P n +F y +R t -ir r sin 6 where P h = total hydraulic pull (Ibs) F v = recuperator reaction (Ibs) R t = total friction (Ibs) Hence P h K-F v -R t +lf r sin F v +R t -W r sin 1 * = 1 - ' ' K K K Since F v and K remains a constant for all elevations, in order that K p h or its reciprocal remain a constant, we must p b K have R t -W r sin # t = R t -W p sin g To consider extreme conditions, let us consider, horisontal and max. elevation, then where B = the angle of elevation at max. elevation. Now R t R- + Rp where R* = the total guide friction Rp * the total packing friction Now Rg is proportional to the total braking * K+lf r sin due to the pinching action of the guides, and the packing friction remains practically constant since Pj, does not change greatly. Hence on elevation, R t0m > R t0o usua lly except for large guns with balanced palls. From actual numerical calculations on a series of guns, tbe term R to was found to be slightly greater than 439 S t0 ~*r sin ^M* Therefore, remains practically p h constant. (1) The length of recoil with the sane grooved orifices is practically in- dependent of the elevation of the gun. In case (2) with different ballistics, we have roughly, K t =* 0.45 m r V* K * 0.45 m r V* g and as before the reciprocal of the ratio , becomes, p h ^fy '*r sin h =- a 1 -- therefore for a constant ratio, we should have, p v + fi t -W r sin P y +R t -lf r sin which obviously is K K i * impossible. But F v +R t -W r sin Of is always small compared with K, hence the difference of the above terms must be cor- respondingly smaller. Hence though the ratio changes with different D ballistics, the change h is very small. (2) The length of recoil with the same grooved orifices is practically independent of the ballistics of the gun. ROTI: Tha above disoussion on length of racoil ia retained as a point for discussion. The author's conclusion* are not however well confirmed by observed data. Bditor. 440 COUNTER RECOIL: In the design of a counter recoil ELEMENTARY system, ie are concerned with either DISCUSSION. counter recoil stability when the gun enters the battery position or with the buffer pressure in the counter recoil regulator. In the former, we are con- cerned with the overall force, that is the total force towards the end of counter recoil, while in the latter, with the c'recoil buffer or regulator re- action. Let K v - total resistance to counter recoil (Ibs) P 7 total recuperator reaction (Ibs) B^ = counter recoil regulator or buffer force (Ibs) R t = total friction (Ibs) w x = throttling area of c'recoil regulator (sq. in) C 1 = throttling constant Afc = area of buffer (sq.in) v = velocity of c'recoil (ft/sec) The critical angle of elevation for counter recoil functioning is at horizontal elevation. Then K v =B+R t -F v and for horizontal c'recoil stability in a field car- riage, we have w , + w (b-x) K v h where lg = distance from total weight of system to forward overturning point, usua lly the front \_^L """"" wheel base (ft) x * displacement in c'recoil from out of battery position (ft) b * length of recoil (ft) h = height of center of gravity of recoiling parts from ground (ft) We may express W s lg in terms of the static load on the spade then, T 1 = H g lJ where 1 = distance between spade and wheel contact with ground. Then T 1 + W r (b-x) where T = 150 to 200 (Ibs) 441 If the ground is assumed to exert a downward pressure on the spade comparable with the load T, K y = 0.85 2T 1 +W r (b-x) h which gives the limitation of the magnitude of the total unbalanced force towards the end of counter recoil* For simplicity in the following discussion a constant regulator reaction will be assumed acting throughout the counter recoi I. This method of con- trol was used by the Rrupp and the earlier material of the Schneider in France. SPRIHG RETOHH. Let S = initial or battery load on spring column (Ibs) Sf = final or out of battery load on spring column (Ibs) C t = spring constant Tben p .8 IT -9 F vi a s o> F vf ' s f and the recuperator reaction, in terms of the c 'recoil displacement x, becomes, F- S * (b-x)=S +S(b-x) where S = the b b spring constant, dv then m r v -- = -K.. dx = -(B + B t -P y ) therefore B r v i Sx = - B x x - R t x + (S +S b - ) x 2 * which is the general equation of c 'recoil, with a con- stant regulator reaction and spring return. When x * b, v = 0, hence ~ B x b ~ R t b + ( s o +Sb "~ 2""^ b=0 hence _. B' = S ft + ;: - R t (lbs) 2 This same value may be obtained by a consideration of 442 the potential energy stored in the recuperator. The potential energy of the recuperator, becomes b b S f -S *o s f S dx + / x dx o o b s f s o b* V * T- b (ft.lbs) 2 We have, then, from the principle of energy, , V S f W R^b+B x b = b since S f = S b + S o ou hence B, = S_ + - R* x o g L Substituting this value in the energy equation * 2 r and siaplifying, we have n r v = Sx(b-x) hence S = ' i m r v* b (b-x)x ..d B x -(8 -R t )* J^^- which gives the value of the constant regular reaction. B x * - Clbs) where C = the reciprocal of the contraction factor of the regulator orifice. A b * effective area of buffer w x = variable regulator orifice, and since, S(b-x)x v a r C*Aw s(b-x)x B x 175. r B x and therefore w 2 = _____ (fcx-x* ) ; Value of of regulator (sq.in) 443 C Ag /-s where C Q = * 13.2 / m r B' r x BX - V ^ - R t The unbalanced force of c'recoil, becomes, dv ,_ m r v - (B x + K t - F v ; dx = - (S + - S - Sb + Sx) 2 = - Sx = S( - - x) 2 2 Hence the unbalanced force decreases with the dis- placement of c'recoil, reverses to a negative value at mid stroke. The initial unbalanced force at the beginning of c'recoil, equals Sb , s f" s o s f" s o *2 = ( ~!b~ ~ The overturning force at the end of c'recoil, becomes Sb _ s f~ s o .2 2 GENERAL EQUATIONS The functioning of counter recoil OP COUNTER RECOIL, may best be studied by a consideration of the physical aspects of the dynamic equation for counter re- coil. Let p a = intensity of pressure of the oil in the air cylinder (Ibs/sq.in) "ax a counter recoil throttling area between air and recuperator cylinders (sq.in) A y = effective area of recuperator piston (sq.in) K V = total resistance to counter recoil (Ibs) F v = actual or equivalent recuperator reaction at any displacement "x" from the out of battery position (Ibs) w x = variable buffer orifice at c'recoil dis- 444 placement x for buffer counter recoil throttling (sq.in) Then during the counter recoil for a spring, pneumatic or similar recuperator system, we have, (1) the recuperator reaction acting to displace the gun forward into battery F v (Ibs) (2) the weight component resisting F y - - W r sin (Ibs) (3) tlie guide friction Rg = n W r cos approx. since the pinching action of the guides is small on counter recoil and we therefore have an approxination of pure sliding friction throughout the greater part of counter recoil. This reaction also resists F v . , (4) the packing friction K s+p resisting F v (Ibs) (5) fhe throttling through fhe return of the recoil apertures together with the counter recoil buffer throttling. The throttling through the recoil is small as compared with the buffer throttling and may be neglected or else included with the buffer throttling. The throttling is proportional to the square of the velocity of counter re- coil and inversely as the square of the throttling orifice, that is, the buffer braking becomes, I I C O V H. = - (Ibs) and resists F w * a ^________ % A Therefore, w have n F v -W r (sin * n cos 0)-R 8 +p " "" which is the differential equation of counter recoil. 445 With a hydro pneumatic recuperator system it is possible to regulate counter recoil by lowering the pressure in the recuperator cylinder for the greater part or the entire recoil, by throttling the oil through an orifice between the air and recuperator cylinders. Introducing a buffer chamber in the air cylinder with a buffer attached to a floating piston, gives a simple means for varying the orifice and thus reducing the pressure in the recuperator cylinder or in the recoil cylinder to a value consistent for the proper movement of the recoiling parts in counter re- coil . The pressure in the recuperator cylinder due to throttling through the orifice between the air and recuperator cylinders, becomes, iii a , c o v Pv s Pa - W 2 w ax Hence, for the motion of ths recoiling parts in counter recoil, we have, ' 2 Pv A y - W r (sin + n cos 0) - R s+p - " g = m p v - w x dx or substituting for p v , we have Cn i . C 2 dv P a -A y - W r (sin J0+n cos &)- R s + p ~ ( * * ~~ ^)v = ai r v (2 w?., w? dx where C o = A^o' ' i Now p A V may be regarded as the equivalent recuperator reaction, that is F v = p a A y and further assuming the regulation to be entirely effected through the throttling in the recuperator, we have, for eq.(2) n C rt A ,r F v -W r (sin0+n cos 0)-R s+ _ - v ? =m r v (3) "ax dx which is exactly similar to the previous equation of counter recoil for a simple spring recuperator system. The external force on the total mount, is 446 i dv K v r v , together with the weight of the recoiling parts W p . During the acceleration, K v = m r v - acts towards the breech, and during the subsequent retardation, KV * Br v d7 aots towards the nuzzle. During the acceleration K y is necessarily always less than K the total resistance to recoil since, ~Cv* K = F..+R + -T - W_ sin 0, for the recoil and w* _ . cV K y = F v -R - j - W r sin 0, for the counter recoil, w x ' ' therefore 2 i 2 Cv C v K-K V = 2R + -j- + , roughly assuming total friction the sane on recoil and counter recoil. Hence, in the design of a counter recoil system we are only concerned with counter recoil stability, and not at all with the re- action during the acceleration. If we let, further, W s = weight of total system (Ibs) l s - horizontal distance fron front hinge or con- tact of wheel and ground to the center of gravity of the total system in battery (ft) C = constant of counter recoil stability Overturning counter recoil moment. Stabilizing counter recoil moment. i d = perpendicular distance from front hinge or contact of wheel and ground to line of action of K y through center of gravity of recoiling parts (ft) then, for stability at angle of elevation 6, we have s + Dr cos0)-F v = (2) 447 dv w- 3 l 3 +W r (b-x)cos and - - v = C t 3 (3) dx d 1 which gives us the velocity curve against displace- ment consistent with counter recoil stability. Sub- stituting v in (2) enables us to determine the variable orifice w x consistent with counter recoil stability, since F y is a known function of x. During the acceleration, we have ;>* t , P v -W r (sin + n cos 0) - R s +p ~ " s B r v and since we are not concerned with stability, for inisuiB time during the acceleration K y should be made a maximum, that is the hydraulic braking tern should be made zero, hence cV dv F v -W r (sin 0+ n cos 0) -R s + p = r v - Let further v m = aaximun velocity of counter recoil (ft / sec) x m = corresponding displacement to maximum velocity from out of battery position (ft) Then, for ideal counter recoil, that is the counter recoil functioning in nininun time and consistent with stability, we have, o i b ~ / r v dv = / [W s lg*W r (b-x) cos 0)dx (5) from which we obtain, a B r v m C * To determine x n , we have 448 / x v m ' F v " w r (sin Of + n cos 0)- R s+p ]dx = / ni r v dv hence X . IE P F v dx - [W r (sin0 + n cos 0)+H 8+p l - cos ] (6) 2 and knowing F v as a function of x, we may solve for x m . Substituting in (5') we easily obtain v m which gives the maximum velocity of counter recoil. Thus we see during the acceleration it is de- sirable to make, K v a maximum, that is K v = F v -W r (sin + n cos 0)- Ro +n v max and during the retardation K y should be consistent with counter recoil stability, that is dv cos which can be obtained by increasing the buffer or counter recoil regulator, such that, G'V* W g l'+W (b-x) cos 0+W cos 0)-F = C[ - - - ] A simple graphical solution of the above analysis may be made as follows: Lay off the recuperator reaction F v f-F v ^ and from the ordinates of this curve subtract W r (si"n J0 W cos 0)+R s+ _ which gives the unbalanced reaction proportional to the ordinates to AB, during the ac- celeration period. Draw in below 00', CD parallel to the counter recoil stability slope Q R, such that = = C , the constant of counter recoil stability assumed. Then we locate M such that the area OABM = area M O f D C. Since OABM is pro- 449 w/m +VJC0S t) y- >PJ + p C 'XECO/L ENEffGY PL OT5 COS MO 'DC 450 oortional to the work done during the acceleration, we have Area A P M = - M p V m The velocity curve may be constructed graphically since any increment area abed is proportional to the change of kinetic energy, that is a. i* a Area a b c d I - m r (v t -v j ) and thus knowing the previous velocity, we may con- struct a velocity curve directly. The energy equation of counter recoil: The dynamic equation of counter recoil, is cV F v -(n cos 18 + sin 0)N r -R s +p * m r v - w x dx where F v = the recuperator reaction R g+ = total packing friction. & = hydraulic buffer resistance "* x C Q v v Integrating, we have / (F v - r sin0-R t - y )d**/ m r v dv o w x o where R t = a W_ cos + ^s + n , , x x C v y 2 Separating, we have / F y dx-(W r sin0+R t )X-/ - dx=m r r- o o w x 2 FJow since the relative energy in the recuperator, de- pends only on the position in the recoil, we have, dW F w - since v dx = - dW de where W is the relative potential energy of the re- cuperator, which is equal to the work of compression (approximately) for a displacement in the recoil (b-x)(F y ) If W = the potential energy of the recuperator in the out of battery position, 451 i i a r * dW C o v m r v - / . dx - (ff-sin 0+R t )x - -2 dx = - W dx w 2 from which we obtain _ C Q V (*t- W x )-(r r sin 2T+E t )x - / - dx )= - "x 2 which is the general energy equation of counter recoil Obviously at any displacement in the counter recoil x, - ' * C Q v ra r v If, +(W r sin 0+R t )x+ / - dx + - = W. a constant ** ** 4 f\ O MH^HWMMM^^HMM^^^K x 2 That is, the total energy at any recoil x, is divided into the potential energy of the recuperator, the work done by friction, the work done by buffer throttling and in the kinetic energy of the recoiling mass. Between any two displacements in the counter re- coil K^ and x a we have, approximately, provided the points are sufficiently close: which gives us a method of computing v x knowing v x from the previous point. COMPUTATION OF With a given set of counter COUNTEE RECOIL. recoil orifices, the velocity and force curve of counter re- coil may be calculated by either of the two following methods: If F v = actual or equivalent recuperator reaction at any dis placement "x" from the out of battery position (Ibs) F ? i = initial recuperator reaction (Ibs) w x = variable orifice for counter recoil throttling at displacement "x" from the out of battery position (sq.in) C Q = counter recoil throttling constant 452 n = coefficient of guide friction R s+ p = total c'recoil packing friction (Ibs) A y = effective area of recuperator piston (sq.in) V Q = initial volume of recuperator (cu.in) x * counter recoil displacement (ft) METHOD I - LOGARITHMIC METHOD. The dynamic equation of c'recoil, becomes cV F v - W r (sin * n cos 0)-R s+ = m r v "x d * If we let, R = ff r (sin 0+ n cos 0)+H s+p G o vZ dv then F - R = m_v - w x dx Now F y and w x are both functions of x and therefore the equation cannot be readily integrated. If, however, we take a small interval F v and w x may both be assumed constant during this interval. Considering any two points x^ and x a in the counter recoil, we have *a v 2 n r v dv dx = / where A = F tf -R Rearranging, we have C v d(A- -2 x * w* i 2 i o!,' A hence integrating, we find 453 and * log e (A - m r "x. therefore . . o 2 t log (A -- )' log (A -- ) -- w x' "x 1 2.3n r wi 2 a *s where A = F v -W r (sin 0+ n cos 2J)-R s+p (Ibs) fro which ire nay construct the velocity curve. The advantage of this method is that a small variation of F v and v x has a negligible effect on the equation of motion and therefore fairly intervals nay betaken provided the throttling orifice of counter re- coil is not changing rapidly. During the buffer period where the throttling changes rapidly small intervals oust be taken. The total unbalanced force acting on the recoiling parts during counter recoil, is dv Av , m r v = m r v (approx.) From this the unbalanced force (total accelerat- ing or retarding force) F v -W r (sin + n cos (? )-R g+p x ay be calculated and plotted. To compute the recuperator reaction at any point, we have for spring recuperators, s r s o F v = S + -^--(b-x) b 454 where S Q * initial or battery spring reaction (Ibs) S f = final or out of battery spring reaction (Ibs) and for, pneumatic or hydro pneumatic, V o k FV ' Pai V V -12A V (b-x) V k V where b= length of recoil (ft) x = c 'recoil displacement from out of battery position (ft) V Q = initial volume (cu.in) To compute R s+p , we have, R S+D =100 to 1-50 Zd for ordinary packing where d = iiam. of any one of ths various recoil rods (in) R s+p =Z(C t +C a p)Z[0.15(.05 *Wpd pPlBa3t )+0.75(.05TiWpdpp)] (Ibs) where w = width of the various packings (in) dp = diao. of the annular contacts of the various packings (in) p max = the design pressure, usually the max. pressure in the cylinder to which the packing is subjected to (Ibs/sq.in) p = actual pressure during the various points in the counter recoil to which certain parts of the packing are subjected to (Ibs/sq.in) Obviously since p is variable, R s + p must be variable daring the counter recoil but aq average value of p ay be assumed and the corresponding H g4p can be used with sufficient accuracy. t I V Computation of the throttling resistance C 455 (1) with a filling in buffer, the counter recoil regulation being effective throughout the counter recoil: we may neglect the small throttling through the apertures of the recoil orifice, and then, * C f2 A!v* i v A b v c o T (Ibs) 175w where C' = the reciprocal of the throttling constant A b * area of the buffer (sq.in) w x - buffer throttling area (sq.in) (2) with some form of spear buffer, the buffer action being effective only during the latter part of counter re- coil, we have three stages: (a) the void displacement with no regulation. (b) throttling through the recoil apertures which cannot be neglected due to the much higher velocity of c 'recoil than in case (1). (c) throttling through the buffer orifice, the throttling resistance being large as compared with the resistance due to throttling through the recoil orifice, the latter being neglected. In (b), we have, * f . 3 * (A+a r ) v c o _7 175 w$ xr where A = effective arc of recoil piston (sq.in) a r = area of recoil rod (sq.in) w xr ~ area of recoil throttling grooves (sq.in) In (c), we have, as in (1) 456 * ' * , / B C V w 175 w where A b = area of buffer (sq.in) w x = buffer throttling area (sq.in) With a hydro pneumatic recoil systen, In this type it is possible to loner the pressure in the recuperator by throttling through a constant orifice. Now it has been shown, that At the end of recoil if a spear buffer in the recoil brake cylinder also functions, o y = the effective area of the recuperator piston (sq.in) w a the c 1 recoil throttling area between the air and recuperator cylinders (sq.in) METHOD II - THB HHBRGY KUTHOD. From the energy equation, we have, for any arbitrary interval, a V x 2 ~ v x (W x -W x2 )-(W r sin 0+R p )(w 2 -x i ; -- - (x,-x t )m r ( - ; i a Cov where W xn - the recuperator potential energy at the point "n" in the counter recoil (ft.lbs) To compute W xn we proceed as follows, With a spring recuperator, o O "xn = -^ [s o * ^ (b-x)J d(b-x) (ft.lbs) b 457 = S (b-x)+ 5f (b-x)' (ft.lbs) 2b where S o = initial spring recuperator reaction (Ibs) Sf = final spring recuperator reaction (Ibs) b = length of recoil (ft) x = displacement in counter recoil (ft) With a pneumatic or hydro pneumatic recuperator, b-x b-x V k **n = * F v d(b-x)=F i / ( - ) d(b-x) (ft.lbs) o o V -A Y (b-x) where k = l.Koil in contact with air) = 1.3 oil and air separated by floating piston or pure pneumatic) A O = effective area of recuperator (sq.ft) V Q = initial volume (cu.ft.) P vi = initial recuperator reaction (Ibs) Integrating, we have xn A v (k-l) y"- 1 V*- 1 where V = V o -A v (b-x). Further since, Pai Vj = P a V k or ^.. A" Pai V then, p a i , , k k 1 P V - Pi V aio Hence, when V is in cu. ft., A v in sq. ft. and b-x in ft, we have V = V -A v (b-x) (cu.ft) V o k F v = F> vi^ (lbs) W x = / ;; (ft.lbs) A v (k-l) 458 Usually it is more convenient to express V is in cu. in., A y in sq. in and b-x in ft. V = V -12A ? (b-x) (cu.in) *V F yi ( ) (Ibs) V 12A v (k-l) (ft.lbs) To compute F y , we have log = k log , a linear F vi v logarithmic equation and therefore may be readily plotted. There- fore, vie may make a table for computation of the potential energy of the recuperator as follows: V w x X 12A v (b-x) V V o F r v F V-F V r V T r Vl V O K lOg ^ 12A v (k-l) 459 We have, from the energy equation W -W x i-(W r sin0+R p )x t +m_ s , O 2 2 3 H r t-W_s-W_sinjft+R D )(x -x )- + m r = m. XX r 3 * _2 . *0 O n-1 'xn The solution of these equations, may be put in a table form: X , W X ,-V X 2 (W r sin0+R p )Ax "x ^y 2 V n-l X n v o , W x i -. (W r sin - 3.62 /Vi. (ft/,,0) Knowing v b we may estimate the proper size of the counter recoil constant orifice. Actually the maximum velocity of counter recoil is attained shortly after the out of battery position and at this position the acceleration is zero. But since the retardation is very slight until the variable orifice is encountered, we may assume the recoiling mass to move with uniform velocity at the entrance to the buffer or variable throttling. Therefore at horizontal recoil, =;* P. - n W_ - ~ , W Hence, the constant orifice becomes, w ( where v^ * 3.62 /- where for a spring or "r ^ pneumatic return re- cuperator system ( t s i b CQ = - 175 C = reciprocal of orifice contraction factor and Afc = area of buffer (sq.in) F v * recuperator reaction at displacement Xbj,-d (ft) and for hydro pneumatic recuperator system, 472 -- C = C = reciprocal of orifice con- traction factor. A y = effective area of recuperator piston (sq.in) FV Pa A v lbs - p 3 pressure of oil in air cylinder. For second period of counter recoil: During this period it is customary to maintain a constant total re- tarding force which at horizontal elevation becomes, c o v * P- n 1- , where Rh =c s"~h" (lbs) Since the counter recoil reaction is constant during the retardation, the velocity is a parabolic function of the displacement, that is v = 8.03 / - (ft/sec) "r Substituting this value of v in the following equation, we have (sq.in) , b C A where for a spring or pneumatic recuperator, C o = 175 C * reciprocal of orifice contraction factor Ak = area of buffer (sq.in) 2 F y = recuperator reaction at displacement x, ^' ^ s for a hydro pneumatic recuperator system, C = "" C = reciprocal of orifice contraction factor A T = effective area of recuperator piston F v = pi A y p a * pressure in oil in air cylinder (Ibs/sq.in) 473 COUNTER RECOIL With a variable recoil, the re- FUNCTIONING WITH quirements of proper counter recoil VARIABLE RECOIL, functioning for all elevations are more difficult to obtain. At hori- zontal recoil we must meet the con- dition of counter recoil stability, whereas at maximum elevation, the time period of the counter recoil, for rapid fire, oust not be too long. Since the recoil at maximum elevation is a fraction of that at horizontal recoil, the recuperator reaction at the beginning of counter recoil at maximum elevation is necessarily smaller than that at horizontal elevation. Further at maximum elevation we have the weight component resist- ing motion. Therefore, the accelerating force is necessarily considerably smaller than at horizontal elevation and the velocity attained at maximum ele- vation becomes a function of that at horizontal re- coil. In the design of a counter recoil system in order to obtain sufficient velocity in the counter re- coil at maximum elevation, it is important that a proper compression ratio be used. This in turn effects the initial volume of the recuperator and therefore the entire layout of the recuperator forging. It is here important to emphasize that proper functioning of counter recoil can not be attained by increasing pres- sure where an improper ratio of compression is used. The following analysis gives a rough approximation as to the requirements to be met for proper counter recoil functioning at all elevations with a variable recoil. It will be assumed that the recoil at maximum elevation is reduced to one half that at horizontal recoil and that a constant orifice is maintained until the latter third or fourth of the counter recoil. We have therefore a constant orifice which is the same for the accelerating period of counter recoil at max- imum elevation or horizontal recoil. 474 If now F y j = initial recuperator reaction F v f = final recuperator reaction (Ibs) F ya) = recuperator reaction at middle of hori- zontal or long recoil (Ibs) v s = maximum velocity of counter recoil at maximum elevation (ft/sec) v h = maximum velocity of counter recoil at horizontal elevation (ft/sec) Q = area of constant orifice (sq.in) C o s throttling constant R gif p = stuffing + packing friction (Ibs) B = maximum elevation As a first approximation, we will assume, the maximum horizontal counter recoil velocity to be attained after a displacement equal to one half the recoil. Hence cV F vm ~ n W r - R s +p - 7 * (1) "o At maximum elevation, the maximum velocity of counter recoil will be attained somewhat after a displacement equal to half the recoil, but we are not greatly in error in assuming the same re- cuperator reaction F vin . Hence W-[sin0-(l+ cos 0)nl * i hence C W r [sin0-n(l+ cos0)l 475 We have therefore for required recuperator reaction at the middle of the recoil W r [sin0-n(l+coslB)] If we assume values for v h and v s for design ap- proximations, Me may take,v h = 3.5 ft per sec,. v g = 2.5 ft per sec. then, F VJB =n W r +R s+p +2W r [ sin0-n(l+ cosfl)] If we take a large coefficient of guide frict ion we neglect R s+p ; hence if n - 0.3, F vm *0.3 W r +2W r [sin0 B -0.3(l+ cos0 )l To obtain the minimum allowable ratio of compres- sion, for spring recuperators, we have 2(? vm -F y j)= F ) hence F fv F vi F vf sF vi+< F vf 2(F vm -0.5F yi ) m : F vi F vi With a pne-umatic or hydropne-umatic recuperat- or, we have 2.5 (F vm -F vi )=F yf -F vi (approx.) and F yf = F yi + 2.5(F 9m - vi ) hence 2.5F vm - 1.5F vi 1.5(1.66F vm -F vi ) = n = F vi F vi F vi RKCUPBRATOBS. GENERAL CONSIDERATIONS . After the recoil the recoiling mass must be brought into battery and this must take place at any elevation of the gun and held there until the next cycle of the firing. Obviously sufficient potential energy must be stored during the recoil to overcome the counter recoil friction and the weight com- ponent at maximum elevation throughout the count- 476 er recoil. Further in order that the counter re- coil nay be made in mimimum time, an excess potential energy is required over that required for friction and gravity, in order that a rapid acceleration at the beginning of counter recoil may be attained. Finally in the battery position an excess recuperator reaction is necessary over that for balancing the weight component and over- coming the friction in case of a slight slipping back of the piece in the battery position. Therefore a satisfactory recuperator must satisfy the following requisites: (1) The initial recuperator react- ion should have a marginal excess over that requirad to balance the friction in battery and the weight component at maximum elevation. (2) The potential energy of the re- cuperator at the end of recoil must be sufficient to overcome the work of friction and gravity at maximum elevation during the recoil and rapidly accelerate the gun at the beginning of counter recoil. INITIAL RECUPERATOR In general the size or REACTION. bulk af the recuperator whether spring or hydro pneumatic depends upon the magnitude of t'he initial re- cuperator reaction. It becomes, therefore, im- portant to estimate the required initial recuperat- or reaction to a considerable degree of accuracy. This is especially true in certain types of recoil systems where the size of the forging, especially for guns of high elevation, depends directly upon the magnitude of the initial recuperator reaction and it becomes very important to make this a min- imum. 477 Let Rg = guide friction (Ibs) R v = packing friction of recuperator (ibs) F v i = initial recuperator reaction (Ibs) e y = distance down from center of gravity of recoiling parts to line of action of P v (in) Q t front normal clip reaction (Ibs) Q f = rear normal clip reaction (Ibs) x 4 and y t - coordinates of front clip reaction (in) x a and y a = coordinates of rear clip reaction (in) n = coefficient of guide friction = 0.15 approx, S6 m - angle of maximum elevation. 1 = distance between clip reactions (in) Considering the recoiling mass at maximum elevation in battery, case of slight slipping back from the battery position, we must have (see fig ) or Fy i =n(Q 1 +Q 8 )+W r sin2f m (1) and normal to the guides, Q z -Q i =W r cos(? (2) and taking moments about the center of gravity of the recoiling parts, %i * v -Q t x t -Q 8 x a + nQ x y x -n V, = (3 > Substituting (2) in (3), we have, F vi e v~ Q t x t~ Q 2 x 8 ~ w r cosgfx * n Q t y t " n a 4 y, " n W f cos/C y a = F vi e v -W r cos0(x t +n y a ) hence Q t - (Ibs) (4) x t +Vn(y 8 -y t ) and solving for Q 2 , nQ f y 2 = v a hence CL - - F vi 478 Hence, with sleeveguides, *_ "\y* y* ' I x 9 \ With .grooved guides y t becomes negative and Since with grooved guides, y^y^ approx.,also *!+*, * 1 3 distance between clip reactions, and 7 t a y t * e r lean distance to guide friction, we have, v vrBt R ff - n (with grooved guides) 2F vi r , 1a n (with sleeve guides) 1 (9) Substituting in eq.(l) we have t 2f' 0_+ff r coi0(x -x..) - - - S *- n + , 1+2 n e hence n cos0_(x -x ) W r [sineJ m + - " * * ] 1+2 n e r - (Ibs) (10) 2 e..n l+2n e r and for the initial recuperator reaction, n cos L(x -x ) ' 3 l+2n e, ^ n 2 e v n 1 - 1+2 n e, 479 n cot B (x t -x t ) 1+2 n e r 2 e v n > where k 1.1 to 1.2 (11) 1+2 n e r 1 distance between clip reactions (in) with 3 clips 1 - 2 with 4 clips: 1 b b * length of recoil (in) Estimation of Recuperator Packing Friction 8 p : With hydro pneumatic recuperator systems, the packing friction is usually a linear function of the recuperator pressure. Assuming a given in- itial intensity of pressure p v fflax Ibs/sq.in. in the recuperator, we have, Rp 3 C p p v Bax . The packing friction in the recuperator is divided into the suffing box friction plus the re- cuperator piston friction. To estimate these fric- tions wftmust know the diameter of the recuperator piston rod and recuperator piston. To roughly estimate these diameters, we have for the effective area of the recuperator piston, 1.3H r (sin5 - +0.3 cos ffl ) A T - (sq.in) PV max for the required area of the recuperator rod, 2.6W r (sin0 m +0.3 cos m ) a v (sq.in) where f B allowable fibre stress in rod material. Then the diameter of the piston, becomes, (in) 0.7854 480 and the diameter of the rod becomes, d ' v '0.7854 (in) If w gv = width of stuffing box packing of recuperat- or(assumed ) (in) w pv * width of piston packing of recuperator (assumed Kin ) then assuming the pressure normal to the cylinder or surface of the rod to be made equal to the hy- drostatic pressure in the cylinder, we have R p -(.06 w pv D v + .05 n w 3v d y )p y nax< - .05n(w py D v + w sy d v )p y max! (Ibs) where .05 approx. coefficient of friction of the packing. Approximate Initial Recuperator Reaction: For preliminary calculations, especially when the type of packing and arrangement of cylinders has not been considered we may neglect the re- cuperator packing friction by increasing the co- efficient of guide friction. Without pinching action of the guides in bat- tery the guide friction, R- 0.15 W r cos (approx) (Ibs). To account, for a possible pinching action, as well as the packing friction, for elevations up to 65, approx. R g = 0.30 W r cos (Ibs) and the required initial recuperator reaction, to al- low for possible variations, should be increased from 20* to 30* over that required to hold the gun in battery. Hence F yi * 1.3 W r (sin0 m + 0.3 cos5) (Ibs). With guns of very high elevation, Rg = 0.3 cos t becomes negligible. However, the pack- ing friction remains the same whereas the guide friction is comparable with that at horizontal re- coil due to the pinching action of the guides at maximum elevation. Therefore, it is desirable to use an approximate formula taking these factors 481 2n into consideration. We have, appro*. R e = 1 where n = 0.1 to 0.2. If we take n * 0.3 to ac count for the recuperator packing friction, we have at high elevations, 0.6 F vi e b p vi - 1.3(W r sin m + *Y* ) (Ibs) where e b * distance from bore to line of action of F v j (assumed ) (in) 1 * distance between clip reactions (in) with 3 clips with 4 clips 1 = b b - length of recoil (in) ENERGY REQUIREMENTS The initial recuperator FOR PROPER reaction is designed to be RECUPERATION. somewhat greater than that required to hold the gun in battery at maximum elevat- ion, against the guide and packing frictions. Further, the recuperator reaction, being necessarily derived from a potential function, must therefore increase with the displacement out of battery. The work done by the recuperator, therefore, is in excess of that required and we have, always, an excess potential energy over that required to bring the gun into battery. This excess energy is dis- sipated by the counter recoil regulator. We have, i therefore, merely a transfer of part of the re- coil energy, dissipated by Beans of the recuperat- or, ultimately in the counter recoil. The total heating or rather the average in a recoil cycle is quite independent of the magnitude of the com- pression. However, with high compression ratios, we have extreme local heating where the radiation is small and therefore injurious effects are like- ly to result with the air packings in hydro pneu- 482 atic recoil systems. Further excessive potential energy stored in the recuperator, requires care- ful counter recoil regulation, and as stability on counter recoil is far more sensitive than on recoil, we have more difficulty in meeting the rigid requirements of counter recoil stability. Finally with excessive recuperator energy to main- tain low counter recoil regulator or buffer pres- sures requires a cumbersome and large counter re- coil regulator whereas it is far simpler con- structively to dissipate the recoil energy during the recoil. Therefore excessive recuperator energy is un- desirable for the following reasons: (1) Localized heating resulting with hydro pneumatic recuperators, is in- jurious to the packing. (2) Difficulty in counter recoil regulation and meeting counter re- coil stability requirements. (3) Constructive difficulties due to a bulky counter recoil buffer or regulator required to maintain moderate pressures in the buffer chamber. On the other hand, the mean recuperator re- action must be sufficient not only to balance the weight component of the recoiling parts and frictions, but enough to accelerate the recoiling parts to a given minimum velocity for counter re- coil at all angles of elevation. Since it is con- structively complicated and more or less impractical to introduce varying counter recoil regulation as the gun elevates in the majority of the types of recoil systens are designed on the bases of given maximum velocity at horizontal elevation consist- ent with counter recoil stability and a given minimum velocity at maximum elevation, consistent with reasonable time of counter recoil at maximum 483 elevation. Usually the recoil is shortened at aximum elevation. We are not greatly in error in assuming the respective velocities to be at- tained at a displacement corresponding to the oean recuperator reaction, whicb is roughly from one half to two thirds away from the battery position. We have, then, with a variable recoil, if P vm = mean recuperator reaction (Ibs) R s +p = total packing friction in counter re- coil (Ibs) C Q = throttling constant of regulator w throttling orifice of regulator (sq.in) v b velocity of horizontal e 'recoil (ft/sec) v s = velocity of c'recoil at maximum elevation (ft/sec) n - coefficient of guide friction, for the notion of the recoiling parts at horizont- al recoil, <*; % - " r - R s+p - = for the motion of the recoiling parts at maximum elevation, _ i * C v s F VB -W r (sinJ0 m +n cos m )+R s+p jj fr w o Subtracting, we obtain C W r tsin0 m - n(l 2 v b and X tf h FVIB* n w r +R s+p + W_[sin y _ v a r v h v s cos We see, therefore, that the mean recuperator re- action required depends greatly on the square of the horizontal c'recoil velocity and inversely as the difference between the squares of the borizont- 484 al and maximum elevation, o 'recoil velocities. Since v n is nore or less fixed by c 'recoil stability limitations, whereas v g depends upon the time allowed for counter recoil functioning at maximum elevation, F ym becomes more or less fixed and therefore the required excess potential energy of the recuperat- or. Assuming design values of v h = 3.5 ft/sec. and v s = 2.5 ft/sec, with an increased coefficient of guide friction to compensate for the packing friction, n = 0.3, we have F ym = 0.3W r +2W r [sin0 m -0.3(l+cos m )] which gives a rough approximation as to the value of the mean recuperator reactions required. CALCULATION OP THE MEAN RECUPERATOR REACTION AND THE BNERQY STORED IN THE HBOUPBHATOR. SPRING RECUPERATORS. With spring return recuperators, we have the recuperator re- action increasing pro- portionally with the recoil. If F vi * S Q = the initial spring re- cuperator reaction (Ibs) F vf * s f = the final spring recuperator re- action (Ibs) b - length of recoil (ft) Then Sf+So F yi *F vf (lbs) hence T vf * 2F ym -F vi (Ibs) The potential energy stored in the recuperat- or for s displacement x, becomes x S f -S W / (S + - x)dx 485 c _ c b f b O 2 = S x + - x (ft.lbs) 2b and the total potential energy required at the end of recoil, becomes W = (S + S f )| - (P yi * F vf )| (ft.lbs) With hydro pneumatic or pneumatic recuperat- ors, we have the recuperator reaction increasing as an exponential function of the recoil displace- ment. If p a a intensity of air pressure in recuperator at any displacement in the recoil X (Ibs/sq.ft) * initial pressure in the recuperator (Ibs/sq.ft) * final or maximum pressure in the re- cuperator (Ibs/sq.ft) ^. - ratio of compression Pai A y * effective area of recuperator piston (sq. in) V * volume of recuperator at displacement x (cu.ft) V Q = initial volume of recuperator (cu.ft) Vf = final volume of recuperator (cu.ft) x = recoil displacement (ft) b = total length of recoil (ft) Then, Pa^ a Pai v o where k 3 1.1 for oil in contact with air = 1.2 for oil separated from air by a float- ing piston. Since V = V Q - A y x, for a recoil displacement x, we have y ^ p a = p ai (tr-^T - ^ or ln ternis of tne v total recuperator re- action 486 r O T Tba work of coapression, becoaes V V dv "x - / P. d 7 - p ai Vj / 21 tft.lbs) V V Vk 1-k Since F y = P a iA v , we have for the work of com- pression in terms of the total initial recuperat- or reaction F vi V o , 1 1 . w x ' I- 1 k-1) ^yk-i v*- 1 *o where as before V = V Q -A y x. At the end of recoil, we have substituting, for V, 7f = * 77-) the ratio of compression. now Tbe total work of eoapression in terns of "a" be- comes - y *;i W b - -2 S( H k - 1) (ft. Ibs) k-1 It is custoaary to measure the pressure in Ibs. per sq.in. rather than Ibs. per sq. ft. and the volume in cu. in. The above formulas, be- come Pai 7 o I 1 12C1C-1J 487 -V (B - 1) (ft.lbs) or in terns of the initial re- cuperator reaction F vi and the effective area of the recuperator piston A v (sq.in) we have 12A v (k-l) (ft . lbs) P - V* ~ w b = 2 ( ' - i) (ft. ibs) 12A v (k-l) v and Pa A v = F v = p ai A v ( - - - J (Ibs) W where x = recoil displacement (inches) A y = effective area of recuperator pistomsq .in; V o = initial volume (cu.in) p a j=initial recuperator pressure (Ibs/sq.in) V = V -A v x (cu.in) The mean recuperator reaction, becomes, F vi v o *** ^ m - 1 ) (Ibs) where A y is in sq. > ft., b in ft., and Paf V V o in Cu ' ft ' Since - m = (--) Pai V f and V* hence 1 m - 1 V Q (1 -- )A v b and A y b V Q ( - ) therefore m* at* 1 k-l F v * F vi ( ~ - H* " 1 ) (Ibs) which gives the mean recuperator reaction in terms of the initial recuperator reaction and the ratio of compression 488 Since P v j * 1.3yf r (sin0 a +0.3 cos m )(approx.) (Ibs) we will have 0.3W r ( , t )W r [ 3 in0 B -0.3(l-cosg),)] k-1 1.3W r (sin0 B +0.3 cos0 m ) 1.3(8in0 B +0.3cos0 B ) If we assume v n - 3.5 ft/sec, and v s = 2.5 ft/aec,, then hence 1_ Jt=i 0.3+2[sin0 m -0.3(l-cos0 B )] ^T" - )( n^r^ * - 1 From the above equations, we note that the proper ratio of compression depends on the angle of elevation and is entirely independent of the weight of the recoiling parts. The compression ratio does depend upon the value assumed for the initial recuperator reaction, the higher the initial re- cuperator reaction the lower ratio of compression. The compression ratio increases with the elevation for proper functioning of counter recoil at max. elevation. If now we construct a table with values of m, and the corresponding values 489 ( ) and a ~ and their product for 1 k-1 k = 1.1 and 1.3 res- pectively, we >ay de- termine by inspection and interpolation, pro- vided we know the max. angle of elevation. If we let, 1 k-i B* - * - 1 A - - ; B k-1 r= - T ) [ s i n0 m -0 . 3 ( 1-c osf B ) ] then, where k =1.1 A 1.3 4.71V o. 23 1.O84 1.5 3.24'7 0.37 1.201 1.75 2. 08 0.51 1.279 2.00 2. 138 0. 64 1.368 2. 3O 1. 883 0. 78 1. 468 and where k - 1.3 m A 1.3 5.464 .296 1. 129 1.5 3.732 .326 1. 219 1.75 2. 840 .456 1.306 2.0O 2. 420 .577 1.395 2. 30 2. 113 .703 1. 486 490 from the above tables, carves were plotted with values of C against for k * 1.1 and 1.3 respect- ively. la order to compare the probable velocities obtained in the counter reeeil at maximum and hori- zontal recoil for a given ratio of coapression m, or on the other hand if given values of velocity at borisontal and naximua elevation are wanted the following method enables us to determine the proper value ef the ratio of compression . If we plot for various values of m, the cor- responding value of for a lean aax. elevation rg at 63, against v h as horizontal abscissa and v s as ordinates, we obtain, a series of curves for the various values of a, which having decided upon the ratio of compression to be used enables us to determine immediately the velocity of c 'recoil at max. elevation for any given velocity at horizontal recoil. How, ^ 0.3+ - - [sin*H5.3(l- cos J)] 1.3(sin0+0.3 cos 0) * angle of elevation. (In this series of cal- culations, the angle of elevation will be con- sidered only at 65). .-. - 65. Sin t - .906308 Cos * .422618 Sin t - 0.3(l-cos0) .906308 - .3(1 - .422618)* .7330 1.3(sin0+.3cosO) 1.3(. 906308+ .3 x .422618)1.343 Various values ef C (equation fl) are given in table on preceding page. The only unknown in the equation 111 is the ex- pression 491 Vu v h y" Let = K. Taking the various values V n~" V s of C as given in the pre- ceding table and sub- stituting in formula #1, we get the following values of "K", for the given values of "C": 1.1 1.084 1.5*76 1. 201 1.790 1.2-79 1.933 1.368 2.096 1. 468 2.279 1.3 1.129 1.659 1.219 1.824 1.306 1.983 1.395 2.145 1.486 2.312 ..-!L v-v a h s Now to show the relation V h and V s , a curve will be plotted for each value of "K" as calculated and recorded in the table : Vh - KVn - RV s ; K7! = EV* h - ?J ; 492 KV'U-V! V. P T Now for each value of K, assume values of V h , from to 10 and substitute in Formula #2, and obtain various corresponding values of V s . These values of V s plotted against values of V n enables us to plot the curve, the corresponding values of V 8 and V h for each value of "K". 1.1 SET Or CURVES M When K - 1.576 1.3 V h 123*5678 10 V g .604 1.21 1.81 2.42 2.79 3*62 4.08 4.84 6.05 When K - 1.790 1.5 ^h 12345 673 10 V, .663 1.33 1*98 2.66 3.32 3.96 4.64 5.31 6.64 When K - 1.933 1.75 V n 12345 6*78 10 V 3 .693 1.39 2*08 2.78 3*47 4.16 4.85 5*55 6.93 (Then K 2.096 V h 123*5678 10 V g .72.2 1.44 2.17 2.89 3-62 4.34 5.06 5.78 7.22 When K - 2.279 V b 12345 67s 10 V, .748 1.50 2.24 2.99 3.74 4.48 5.24 5.99 7.49 493 1.3 BIT or CURVES. When K 1.659 1.3 V n 12345678 10 V g .63 1.28 1.89 2.52 3.15 3.78 4.41 5.04 6.3 K - 1.824 1.5 V h 12345678 10 V_ .67 1.34 2.01 2.69 3.36 4.03 4. "70 5.37 7.12 K 1.983 1.75 V h 12345678 10 Vg .704 1.40 2.06 2.82 3. 52 4.22 4.92 5.63 7.04 K 2.145 2.00 V h 1-2345678 10 V 8 .734 1.46 2.19 2.92 3.65 4.38 5.11 5.35 7.30 K - 2.312 2.3 V h 12345678 10 V 3 .753 1.50 2.26 3.01 3.76 4.50 5.27 6.02 7.53 SPRING RECUPERATORS. Spiral spring columns, en- closed in cylinders for pro- tection, are extensively used to bring the recoiling parts back into battery from the out of battery position. For small guns, spring re- cuperators are more useful, since they are simple in construction compact and readily adaptable to a gun mount. With large guns, however, the energy required for recuperation is large and there- fore the spring columns become excessively heavy, since the weight of the springs is proportional to the potential energy stored within the springs. 494 495 496 497 Hence for large guns pneumatic recuperators have become almost universally employed. The stresses computed in springs are based merely on their static loading. During the ac- celeration period of the gun, the spring coils adjacent to the attachment on the recoiling parts, necessarily are subjected to a very large ac- celeration, whereas those coils adjacent to their attachment on the cradle remain stationary. Due to the great resilience of a spring column, probably only a few of the front coils adjacent to the re- coiling parts are subjected to any material accel- eration, the spring not being capable of transmit- ting a force sufficient to accelerate the inner coils. Due to the very rapid acceleration during the first part of the powder period we have an im- pact or very suddenly applied loading on the spring which induces a compression wave, the peak of the wave being adjacent to the recoiling parts and the velocity of which depends upon the inertia per unit and elastic constant of the spring. It is possible that some of the failures in the service of re- cuperator springs are due to the dynamical aspects of the loading on the springs during the firing. Since the inertia loading due to the powder acceleration comes practically on the front series of coils adjacent to the recoiling parts, the coils directly adjacent to the recoiling parts become more greatly compressed and correspondingly stressed. We should expect the front coils, therefore, to give the greatest trouble and this has been found the case in actual service. Due to the complexity of the problem in actual calculations of the dynamic stresses in the spring no attempt will be made here to outline a procedure for such calculations, and only the static loading with suitable safety factors based on experience will be used in the preliminary design of counter 498 recoil springs. Lei = diam. of the helix of the coiled spring (in) R * radius of the helix of the coiled spring (in) d = diam. of the wire (in) f s = max. allowable torsional fibre stress used (Ibs/sq.in) N 3 torsional modulus of elasticity (Ibs/sq. in) T = torque or total torsion at any cross section of the wire (in. Ibs) Considering any portion of a spring column subjected to a conpressive load F (Ibs) , along the helical axis, we have at any section, through the wire, A torsional load T = f R A shear S = P If we assume pure torsion at the section, the torsional fibre stress becoaes, f f g (Ibs/sq.in) where r o = - (in) hence r o T F R - / 2*r dr f. s T* o r o and therefore ,3 a nf a d n f s d F = (Ibs) 16R 8D Next consider the twist of any length of the wire 1. We have, for the torsional shear displace- ment of a circumferential annular of the wire, f s r o t * since f a - N hence r- where 9 the N 1 499 angle between two radius of the wire at two sections 1 distance apart. Therefore a ,^!ii The relative displacement between the extremities of the helix lor a load P, producing an extreme fibre stress f, becomes 2f,Rl 9 R 9 but the length of the total wire Nd of the helix, becomes, 1 2nRa approx. * K Dn where n - no. of coils. Bonce 9 We have, therefore, the two fundamental spring formulas, for springs of circular cross section Kf d* uf s d s PR = (Ibs) (1) 8D 16R nf.D f n 9 (in) (2) B d The above formulas apply strictly only to closed coiled springs, no bending being considered; however, for a first approximation, they may be used for open coiled springs with sufficient accuracy for ordinary calculations. For rectangular wire sections, we have semi- empiroal formulas for the torsion, and deflections; * * f 9 .8b 4 o J T 1 where a Q * length of long side of rectangular section (in) b Q = length of short side (in) 500 J a the polar moment of inertia of the rectangle. 1 * length of wire (in) A * cross section of the wire (sq.in) No " ab* ba 3 ab(a*+b*) l xx+ l yy m 12" "12 3 12 Hence for rectangular section spiral springs, we have, a o b o f s ~ ( a o b o } ', 3a +1.8b 1 3a +1.8b D lOnJD'n . p A 4 N 166nD'n a b (a +b ) - -- t a (in) (2) A*N (3a +1.8b ) If now, we let a = deflection at assembled or battery height (in) b"= displacement somewhat greater than the length of recoil (in) F yi load at assembled height ) initial re- cuperator reaction (Ibs) F vfl * load at solid height or at deflection corresponding to (a+b ' ) n * no. of effective coils N torsional modulus of elasticity (Ibs/sq. in) d * diaa. of wire (in) D * diai. of helix (in) H * solid height of spring (in) f 3 = working ax. fibre stress (Ibs/sq.in) then: for circular springs: for rectangular springs: ab f 501 1.66nD*n A 4 tf " (3a Q *1.8b fs (in) (2') vi ~ ' H - nd (4) In the four equations, above we are given f g P ?9 b f N and H o F ve f a ?* b ' N D and - leaving the four unknowns, d D a and n or d a n and H . Therefore a complete solution is possible, and the proper size spring may be iauaed lately arrived at. ENERGY STORED IN SPRING The fibre stress on a helical spring is direct- ly proportional to the axial load, that is f * TS- F (Ibs/sq.in) and the corresponding axial n u _ ns g deflection, becomes, 9 * f (in) hence the de- Nd flection of a helical spring loaded axially is directly pro- portional to the load, that is 6 = f (in) Hd 4 The potential or resilient energy stored in a helical spring becomes, P Hd 4 a t ,. A - 9 = . 9 (in Ibs) 2 16Dn If the spring is to be stressed to a maximum allowable fibre stress f s (Ibs/sq.in) we have 502 n'Dd'n A 16 N f* 8 (in Ibs) The volume of the material of the spring equals approximately n n a 7 - d n * D - D d n (cu.in) 4 4 Hence the total energy in terns of the volume, is f*s A - that is, the energy stored in a spring 4 N for a given max. allowable fibre stress and torsional modulus, is directly proportional to the volume and hence the weight of the spring. Thus with tbe same maximum stresses and same kind of material, the weight of the spring is directly pro- portional to tbe energy absorbed by tbe spring* The weight of tbe spring in terms of the total energy stored in the spring, becomes where W g * total weight of tbe spring (Ibs) w s = weight per cu.in. of tbe material of tbe spring (Ibs/cu.in) RATIO OP COMPRESSION WITH For minimum weight SPRING RECUPERATORS FOR of a set of counter re- MINIMUM WEIGHT OP COUNT- coil springs the com- EB RECOIL SPRINGS. pression ratio is definitely fixed. Let F vi * the initial recuperator reaction (Ibs) F v f = tbe final recuperator reaction (Ibs) P ye = the maximum solid load on the re- cuperator springs (Ibs) a * deflection of springs to assembled height in battery (in) b * lengtb of recoil (in) b" * detleotion of springs from assembled to solid height (in) 503 Since the load on the springs is proportional to the deflection we have immediately, F re a+b* b" - * - ; and P ra - F v i( 1+ > a The total energy stored in the spring column, A - - (a+b") = - U+2b' + - )(i n .lbs) 222 Since b" and P yi are fixed conditions to be not in the design of the carriage, the only variable in the above energy expression is a. Therefore, for minimum weight, w a d(a+2b"+ - ) dA a s hence da da 1 * 5 and therefore a b" a The ratio of compression, becomes, 2 (approx) Fortunately this ratio is nearly ideal for proper recuperation and hence satisfactory de- signed spring column with minimum weight may be used . RECOPERATOR DIMENSIONS With hydro AND LIMITATIONS. pneumatic recuperators we have two or more cylinders, the recuperat- or cylinder and the air tank or cylinder. Let b = length of recoil (in) b = corresponding displacement in air cylind- er 504 A y = effective area of recuperator piston (sq. in) A a cross section area of air cylinder (sq. in) Paf m * ratio of compression Pai A a r * - * ratio of recuperator cylinders. Ay 1 = length of air volume in terms of cross section area of air cylinder (in) j - 3 length of air volume in terns of re- coil stroke V o * initial air volume (cu.in) Vf * final air volume (cu.in) Then V f V Q -A v b U U V P V Q k -i m * ( ) where k * 1.1 to 1.3 Pai v f therefore. v o i v o *r 3 that is V f "T m k 1 1 , k _k VI 1) = >A T b hence V- A V, m " , A V b * i 'A a b' " m k - 1 m k -1 which shows clearly, that the initial volume de- pends only upon the ratio of compression, the area of the recuperator cylinder and the length of recoil. If now, we decrease the effective area of the recuperator piston, for a given recuperator re- action, we must increase the intensity of pres- sure in the recuperator cylinder, that is: r M Y\ A fty| Pyi fly 505 K vi b n hence V Q * 1 -- (6) since p vi p al p ai approx, i K vi b B = V - - i (6 1 ) Pal m* - 1 Now the size of the recuperator depends rough- ly on the initial volume V A ; hence, in pneumatic or hydro pneumatic systems, it is important to main- tain as high air pressure as possible. In recoil systems, where the recuperator and brake cylinder is one and the same as in the St. Chamond and Puteaux brakes, the effective area of the recuperator piston is that of the recoil pis- ton. Now the pressure during the recoil is limited to a given maximum consistent with the packing and therefore the effective area of the recoil piston is fixed. With large guns the recuperator reaction is relatively small as compared with the maximum recoil pressure, and therefore the intensity of the air pressure is small. Hence the recuperator volume and the size of the recuperator is large as compared with a separate recuperator system, using high re- cuperator pressure intensities. Thus for large guns, or guns with low elevation, separate recuperator systems separate from the brake system usually gives a smaller recuperator brake forging. Limitations of the ratio of compression "m". The limitations of "m" are fairly fixed: (1) The minimum "a" is based on & consideration of the proper functioning of counter recoil at all 506 elevations . (2) The maximum "m" is based on a consideration of horizontal stability in the out of battery position for the recoil, as well as heating and rise of temperature caused by the compression of the air. (1) With guns shooting at high elevation, the recoil must be shortened for clearance at high elevations and lengthened for stability at hori- zontal elevation. Thus high angle guns require a variable recoil, the ratio of short to long recoil being usually from one half to two thirds. The re- cuperator reaction at maximum elevation must be suf- ficient to bring the gun into battery with a moderate velocity in order that the time of counter recoil at maximum elevation may not be too long. This feature is of considerable importance. Raising the air pressure in the recuperator, though it will sufficiently accelerate the gun at maximum elevation, will give too great a velocity at horizontal recoil and thus endanger counter recoil stability. Thus in the initial design it is important that the initial volume is such that it will give the proper . . ratio of compression. The mean recuperator reaction, or rather the recuperator reaction at the middle of the recoil, was shown in the discussion on counter recoil to be, t * v * s where v h the max. velocity a+ horizontal recoil (ft/sec) v g - the max. velocity at max. elevation (ft/sec) total recoil packing friction 507 B coefficient of friction from 0.1 to 0.2 For a preliminary design constant, we nay assume, v b - 3.5 ft/sec. v a * 2.5 ft/sec. and taking a large value of n * 0.3 to compensate % > -0.31! r +2W r {sin0 1| -0.3(l+cos0 m )] (Ibs) With a hydro pneumatic recoil system, ire have roughly, 2.5(P VB -P yi )P > vf -F' vi hence the minimum allowable ratio of compression, becomes, *vf 2.5%.-1.5F vi 1.5(1. 66F va - P yi ) m 9 = 3 ? vi p vi F vi of course the ratio nay be decreased by using lower values of v a or higher values of v^, or both but the above assumed values give a satisfactory counter recoil at all elevations. (2) The maximum value of m is based on the following considerations:- (a) Horizontal stability, where a high final air pressure nay exceed the allowable overturn- ing fo^ee consistent with stabil- ity in the out of battery position. (b) The maximum allowable c 'recoil buffer pressure which linits the potential energy stored in the recuperator in the out of battery position. (c) The allowable rise of tem- perature caused by the com- pression of the air. With light mobile field carriages , stability is very often the determining factor for the maximum allowable ratio of compression. This is likely to especially occur when the mount elevates to very high angles and perfect horizontal 508 stability is required as in anti-aircraft material. If the resistance to recoil consistent with stabil- ity at horizontal recoil is small, and the initial recuperator reaction large, a high compression ratio will cause the total resistance to recoil in the out of battery position at horizontal recoil to be greater than the balancing stabilizing moment. Obviously this critical condition will only occur with guns of high elevation and required to meet rigid horizontal stability limitations. In an ordinary recoil system as it is impossible not to have more or less throttling at the end of re- coil, we must have the maximum allowable re- cuperator reaction a fraction of the total pull for minimum elevation of stability 0j. Therefore the maximum allowable ratio of com- pression from a stability consideration, becomes f vf 0.8[K h +W r (sin0 i -0.3cos0 i )] m max * = (min. elev.) 0.8(R h -0.3W r ) K h " = 0.75 (horizontal F vi F vi elevation) Therefore, when F vi is large and K h small as with guns for high elevation and rigid stability requirements "a" becomes small and low ratio of compressions with corresponding larger recuperators are required. Very often in anti-aircraft material "m" becomes smaller than that required for proper counter recoil functioning at max. elevation. In such a case it is preferable to sacrifice horizontal stability somewhat and increase the horizontal re- sistance to recoil. The previous formula may be expressed direct- ly in terns of stability. If W s * weight of the total mount (Ibs) 509 1 3 = horizontal distance from spade to W g (ft) b h = length of recoil at horizontal elev. (ft) . * min. angle of elevation. We have for the max. compression ratio based on stability, 0.8[(W g l s -W r b b cosef i ) m = M and at horizontal recoil, 0.8[W s l s -W r (b h -0.3)) , 0.75 ( r vi In counter recoil systems using some form of a c 'recoil regulator of a buffer type, we bave a necessary geometrical limitation in the maximum area of the buffer. Thus in filling in types of buffers as in the Schneider and Filloux recoil systems as well as ordinary spear buffers which enter the pis- ton rod at the end of c 'recoil, the effective area must necessarily be considerably less than the area of the piston rod. With spear buffers attached to the piston due to void considerations at the beginning of the re- recoil, we again are limited in a large effective buffer area. If P b max * the max. average allowable buffer pressure (Ibs/sq.in) b = length of recoil (ft) A b effective area of buffer d b = length of buffer c 'recoil (ft) Rp ' total packing friction (Ibs) W Q * total potential energy of the recuperator (ft. Ibs) then, when the counter recoil brake comes into action towards the end of c 'recoil, as with a spear buffer, we have, 510 Pb aax * " and where the counter A b d b recoil brake is ef- fective throughout the counter recoil, -(W p sin0+R p ) f Pb max " ' ~^- " h re V * ratio of compression ? o initial volume of recuperator (cu.ft) F vi - initial recuperator reaction (Ibs) k * 1.1 or 1.3 depending whether air is in contact with oil or separated from it by a floating piston. The expansions for p D assume a constant buffer fere* during the buffer action. This however is not always the case and therefore the above ex- pressions should be multiplied by a suitable constant to take care of the peak in the buffer pressure when the buffer pressure is not constant. It is to be particularly noted that the peak buffer pressure may greatly exceed the average buffer pressure as obtained by the above expressions Combining the above expressions, we have .ax , This is a very important limitation for m and is inherent for all direct acting counter recoil buffer brakes. Values of p b max range as high as 8000 to 10000 Ibs/sq.in. with short spear buffers but such pressures should not be tolerated on future designs. In general the c 'recoil buffer pressure should be maintained as low as possible, thus sim- plifying the design of a counter recoil system; therefore, the lower value of m consistent with a satisfactory functioning of c 'recoil at all elevations should be used. (3) Though the total energy dissipated in a re- 511 coil cycia Bust necessarily equal the initial recoil energy, it is important to distribute the energy in the parts of the system where radiation is most effective, if the energy is dissipated entirely in the throttling both on recoil and counter recoil ire have a large nass of oil with corresponding radi- ating surface. With high compression ratios the air in the recuperator rises to a high temperature, which nay cause injury to the packing and lubrication, and therefore it is important to Maintain a low com- pression ratio and thus decrease the localized heat- ing in the recuperator where radiation is the smallest 4 As to the allowable rise of temperature to be permitted, depends greatly upon the type of pack- ing to be used and the packing specification should state the allowable temperature rise. Tne temperature T at the end of a recoil stroke, above the mean temperature T m at the beginning of the stroke, may be obtained, from the relation, T. Pai Assuming a ratio 2, and a mean temperature 25 centigrade, we have T = 298 x 2* 23 = 349, when k 1.3 and therefore the rise of temperature becomes, T-T m =51C or 92F. The temperature rise increases considerably with the ratio , thus when 2.5, T - T B ?aC or ISff'F. RECDPERATOR DIMENSIONS With hydro pneumatic AND LIMITATIONS. recuperators we have two or more cylinders, the recuperator cylinder and the air tank or cylinder. 512 Let b = length of recoil A y = effective area of recuperator piston A a cross section area of air cylinder Paf m = ratio of compression Pai A a r = j[- = ratio of recuperator cylinders. 1 = length of air volume in terms of cross section area of air cylinder j = - a length of air volume in terms of recoil b stroke. Then, the initial volume becomes, n k A a 1 V x A_l = A_b but since r: - s j - A v ,k - l V m I i rj hence k i M* -1 When a floating piston separates the oil and air, k = 1.3 (approx.) Whan the oil is constant with the air, k * 1.1 (approx.) ^a _ 1 * ' * j i k -1 1 i k k (r-1) - r i k r 513 Tables for a and r for various air column lengths when k = 1.3 are given below: r r r- 1.66 -1.66 log 1.3 log r 3 i. 34 2. 239 . 35005 . 4550*7 2. 851 3.5 i. 84 1. 902 2*7921 36297 2. 307 4. 2. 34 1. 209 . 232*74 30256 2. 007 4.5 2. 84 1. 585 2O003 26004 1. 820 5. 3. 34 1. 497 . 1*7522 .22779 1. 69O 5.5 3. 84 1. 432 15594 .20272 1. 595 6. 4. 34 1. 382 .14051 .18266 1. 523 r r r- 1.25 -1.25 log 1.3 log r 3 i. "75 i. "714 .23401 .30421 2. 015 3.5 2. 25 i. 556 . 142O1 .24961 1. 777 4.O 2. 75 1. 455 .16286 .21172 1. 628 4.5 3. 25 i. 385 . 14145 .18389 1. 527 5. 3. 75 1. 333 . 12483 .16228 1. 453 5.5 4. 25 1. 294 .11193 .14851 1. 398 c. 4. 75 1. 263 . 10140 .13182 1. 355 514 r r-1 r (r-1) log ^T 1.31 log ^1 " 3.00 2.00 1.5000 . 17609 . 22892 1.694 3.50 2. 50 1.4000 . 14613 .18997 1.549 4. 00 3.00 1.3333 . 12483 . 16228 1.453 4. 25 3.25 1.3077 . 11611 .15094 1.416 4.50 3.50 1. 2857 . 1O924 . 14201 1.387 4.75 3.75 1.2667 . 10278 .13361 1.360 5.oo 4.00 1. 2500 .09691 .12598 1.337 5.50 4. gO 1. 2222 .os7o7 .11319 1. 298 6.00 5.00 1. 2000 .07918 .10293 1.267 3 2. 156 1.391 .14333 . 18633 1.536 3.5 2. 656 1.318 . 11992 .15590 1.432 4.O 3.156 1.267 . 10278 .13361 1.360 4.5 3.656 1.231 .09026 .11734 1.310 5*0 4.156 1.20} .08027 .10435 1.271 5.5 4. 656 1.181 .07225 .09393 1.242 6. 5.156 1.164 .06595 .08574 1.218 515 516 517 r r-.71! 5 --.715 log 1.3 log i r 3. 2.285 1.313 . 11826 .15374 1.425 3.5 2.785 1.257 .09934 .12914 1.349 4.0 3.285 1. 218 .09565 .11135 1. 292 4.5 3.785 1. 189 .07518 .09773 1. 252 5.0 4. 285 1.167 ,o67o7 .08719 1.222 5.5 4.785 1. 149 .06032 .07842 1.198 6. 5.285 1.135 .05500 .07150 1.179 V o~ l A v b b '.- 7 M p v dV = |n r Vjj+WpCsin^ nax +u cos2f Bax 1U4A HI a A '!- e where V B * 2 ft/sec, i roughly. Now p v V k p vi v|; fcence p y = p vi V k k hence PV1 ^ V -.375A b 7 k where k = 1.1 to 1.3 - (V -.375 A. B 1-k The solution of this expression is com- plicated and trial values of V Q nay be substituted more easily. _ , Knowing V Q and V f = V Q -A y b, we have V Values of m greater than this value are en- tirely unnecessary for satisfactory functioning t counter recoil at all elevations. When the initial value of the recuperator reaction is ade greater 518 than thai required to bold tba gun in battery, the necessary ratio of m decreases in tbe limit if m 1, then * K vi*Pvi A v 4m r v Due to the uncertainty and variation of both packing and guide friction, an excess initial re- cuperator reaction is always used and thus even for very low values of "" we usually have in modern artillery a surplus of potential energy in tbe re- cuperator. glilRAL DlSiaH LIMITATIONS. SURVEY OF LIMITATIONS The design limitations IN CARRIAGE DESIGN. for a gun mount depend primarily of course on the ? . particular use to be obtained from the gun and the general type of carriage to be used. Though each design la a problem by itself, it is however possible to derive and point out certain broad limitations that lust be observed for a satisfactory design. The fundamental requirements and limitations for the various classes of mounts are considerably different. The question of elevating, traversing, etc. certain more strictly to a given mount. Row- tVer, certain broad limitations apply to tbe various olaases of mounts and for good design these limit- ations must be always considered quite independent of tbe requirements for the particular service of the gun. (1) For mobile mounts minimum weight and stability under firing conditions are primary limitations. (2) For caterpillar mounts minimum weight and stability under firing 519 conditions are again primary limitations. (3) For railway mounts, due to size and cost of parts, minimum weight consistent with stability is ia- portant but other factors such as clearance, method of loading, etc. have perhaps more influence on the design. (4) For stationary mounts for defense work stability is easily secured and though it is highly desirable to keep the size and weight of parts as small as possible, the vital factors are accessibility, ease in loading and endurance. LENGTH OF RECOIL The strength of a gun car- AT HA XI MUM ELEVATION riage depends roughly on the AND MAXIMUM RECOIL maximum recoil reaction. REACTION. How the recoil reaction varies roughly inversely as the length of the recoil for a given recoiling mass and ballistics; therefore it is highly desirable, for lower stresses in the carriage, to maintain as long a recoil as possible. But at maximum elevation we are immediately limited by clearance *of the gun striking the ground or platform. As the height of the trunnions and axis of the bore are fixed by stability at horizontal elevation clearance in traveling and accessibility for loading, the recoil at maximum elevation (as well as the maximum recoil reaction) becomes definitely limited. Means for increasing the recoil and thereby diminishing the recoil reaction are as fellows: (1) By digging a pit under the gun. (2) By placing the trunnions as far 520 as possible to the rear adjacent to the breech end of the gun and balanc- ing the tipping parts by the use of a balancing gear. (3) By raising the trunnions as the gun elevates, obtaining a low height of the trunnions above the ground when stability is required and a high position when stability is no longer a requirement and a long re- coil is desired. LENGTH OP RECOIL AT As mentioned before, MINIMUM ELEVATION howitzers are designed for STABILITY. high angle fire, ranging roughly from 20 to 70 de- grees. Therefore, stability is not of great importance up to 20 degrees ele- vation. At this elevation the moment arm of the overturning force , about the trail support be- comes small, and therefore it is possible to con- siderably raise the trunnion and thereby lengthen the recoil at maximum elevation than with guns. Further for a given height of trunnions" the length of recoil can be shortened for an elevation of 20 consistent with stability. Thus with howitzers, it is possible to maintain a constant recoil length for all elevations. This is of more or less ad- vantage in simplifying the recoil system. With a gun, the elevation ranges roughly from to 50. At elevation the overturning moment about the spade support is a maximum, and the stabilizing moment a minimum. (See Chapter III). Hence a long recoil is, essential in order to reduce the recoil reaction and overturning moment. The maximum horizontal recoil however is limited, due to the fact that at the end of recoil 521 though the overturning moment is decreased by lengthening the recoil, the stability moment is also decreased in the out of battery position due to the recoiling mass being displaced to the rear. Thus we arrive at an initial length of recoil where further increase causes a decreased stability. If H s = weight of carriage and mount together (Ibs) R h = horizontal recoil reaction (Ibs) Vf * max. velocity of free recoil (ft/sec) w r = weight of recoiling parts (Ibs) b height of axis of bore above ground (ft) Then w 0.47 "r T R b * ~ T~ M (approx.). then R h h + W r b = H g l s at critical stability. Now the actual overturning moment, becomes, . ,f r b and the corresponding stability b g moment = "s^s If we differentiate the actual overturning moment with respect to b and equate to zero, we ob- tain, the maximum allowable horizontal recoil for a given recoiling weight, hence 0.47W.VJ h 0.47W.VJ h d( -1 + w b ) , L_ + Wp d b b*g hence b b max - 0.121 V f /~h~ Another limitation on the length of recoil at horizontal elevation, is due to the fact that as the recoil lengthens, the distance between the clip reactions decreases, and the clip re- actions and the guide frictions become excessive in the out of battery position due to the over- hanging weight of the recoiling parts. Such ex- 0.471 r V h 522 cessive guide friction caused by the moment of the overhanging weight combined with the recuperator reaction at the beginning of counter recoil may prevent satisfactory return into battery, Further the bending moment at the rear clip reaction of the gun becomes excessive due to the large overhang in addition to the recoil pull on the gun lug. Thus the length of horizontal recoil is limited by the minimum allowable distance between clip reactions when the gun is out of battery. If, with this maximum recoil the mount is unstable, either the weight of the mount oust be increased or outriggers reaching further out must be used. But for mobile mounts minimum weight is essential, hence extended outriggers or increase of trail length must be resorted to. As the gun elevates, stability increases and the recoil may be shortened consistent with clearance and stability. Kith anti-aircraft guns, it is desirable to shoot from to 80 since the piece must be inter- changeable for field work if necessary. Therefore the limitations on anti-aircraft material are more pronounced and the change of length of recoil is greater from to max. elevation than with other types of mounts. RECOILING WEIGHT FOR The weight of carriage MINIMUM WEIGHT OF GUN proper not including the CARRIAGE. recoiling mass is more or less proportional to the necessary strength re- quired in the carriage. Now the strength of the carriage is roughly proportional to the maximum recoil reaction. Further the weight of a car- riage depends upon the type or configuration of the mount. Hence, for any given type of car- riage the weight is roughly proportional to the maximum recoil reaction. If, therefore, a given 523 type of carriage is designed to withstand a given recoil reaction, the higher the carriage is stressed, the smaller becomes the ratio of the weight of the carriage to the recoil reaction. Therefore the weight of efficiency for a particular type of mount is increased by decreasing the weight of the mount per given length of recoil. Obviously if a given type of mount was designed so that all its parts were stressed to the elastic limit for the maximum recoil reaction we would have the minimum possible weight for the given type of carriage. Let w c - weight of the carriage mount not including the recoiling mass. R * the maximum recoil reaction. c = weight of the carriage mount proper when stressed to the elastic limit, k = the weight constant for the carriage mount proper. k 1 * the weight constant when stressed to the elastic limit. Then ' c > i ' c k r and k ' r Obviously the weight efficiency in a given design pertaining to a given type of mount, becomes, k "c weight off. * 3 k' Wg Now the weight efficiency varies considerably with the type of carriage used, certain types hav- ing considerably more dead weight than other types. Further the weight efficiency depends directly on the factor of safety recommended in the design. A table for the constant "k" for various types of mounts is given below: 524 Weight Re. Max. Length Weight Weight of coil- Recoil of Re- of Con- Sy- ing React- coil. Mount stant te. wt . ion. not in- of clud- Car- ing riage. Recoil- ing Wt. Carriage W . r R L W e K 3'Model of 2520 960 4923 45 1560 .317 1902. 75 -/ 265*7 1050 5250 49 1607 .306 Tr e no h M. 189*7. 75/ v.of 3045 911 12100 46 2134 .176 1916. and 18 3.3" gn 43*72 1435 2100 45 2937 .146 Carriage. and 30 3 . 8 " How. C r- 2040 935 13750 4O 1105 .08 r iage, 1915. a ad 22 4.7" Sun, 7420 2745 17500 70 5675 .324 v. 1906. 4. 7'How.Or- 3988 1372 19430 52 2616 .135 r iage, 1908. and 24 155/m He*. 7600 3498 390OO 51.4 4100 . 105 8 o ha. 155-/-1' il- 19860 9050 66000 43 10810 .164 leaz . and 71 8" Vioker, 2OO48 9356 11730O 52 1O692 .091 Mk. Til. and 24 24O / 41296 15790 15OOOO 46. 25526 .171 8CHIIIDBR. To give a farther physical conception of the meaning of "k" we note from previous calculations that the 155 m/m Filloux is extra strong, most of the fibre stresses not exceeding 10,000 Ibs. per sq. in. Comparing it with the 3.3 inch, a somewhat similar type of mount we would expect the 3.3 inch to be well stressed. This is actually the case. Two very similar types of heavy field trail car- riages are the 8" dickers and 155 m/n Schneider, both having the same type of trail. Both car- riages are well designed, having in the various parts about the same maximum fibre stress. Therefore as we would expect the constant "k* is approximately the same. The 3" Model 1902 is not efficiently de- signed as compared with similar types such as the 75 m/m M.1916. We thus see that "k" when compared with types of similar carriages gives us a crude idea as to the efficiency of the design of the carriage itself. Now the weight of the system is the recoiling weight plus the weight of the mount proper (i. e. the stationary parts), that is w g w r + w c where w s = the weight of system w r =* recoiling weight w e = weight of stationary parts, or mount proper. Per a given type of mount, the weight of carriage may be assumed roughly proportional to the recoil reaction, that is, w c = k R Now from the principle of linear momentum, neglecting the small effect of the recoil reaction during the powder pressure period, and the air resistance, then m v + I 4700 m ? wnere m and v - mass and muzzle vel. of projectile. I 4700 = the momentum effect of the powder gases, hence + - V = 526 but R * approximately. 26 (v+I 4700)* 2 r b k u 1 <<** 4700)' hence R where k * - therefore w ft s k R 2b k k ' Now for minimum weight of the total system, recoiling parts together with carriage mount, dw d(w r +-jU^) s "r kk r that ia - - 1 - d "r ir *. w*-kk' or w_ = r "ft where It * s" obtained from table Jot) ballistic constant To use the above fornula ia a new design we take the value of k fren a siailar well designed type of carriage, using a somewhat lower value of "k" according to the judgment of a designer IB improving the weight efficiency of the mount proper over a similar previous design. Knowing the ballistics of the new mount, we find a very definite weight for the recoiling mass. It is interesting to note that usually the strength curve of a gun say be considerably increased if the proper weight of recoiling mass consistent with minimum weight is used. C H A P T B B VIII. This chapter contains a discussion of some of the types of hydro-pneumatic recoil systems with calculations of characteristics of service designs. It has been found desirable to print this chapter separately. 527 CHAPTER IX. HYDRO-PNEUMATIC RECOIL SYSTEMS. (Continued) SCHNEIDER RECOIL The Schneider recoii SYSTEM. system consists of an in- dependent recuperator sys- tem of a hydro-pneumatic type. The cylinders are in one forging and are secured to the gun. The cylinder forging is known as the sleigh or slide and recoils with the gun. The brake and recuperat- or rods are held stationary and attached at their ends to a yoke on the cradle. The hydraulic brake piston rod is hollow and contains a filling in buffer chamber. Attached to the sleigh and sliding within this buffer chamber is a counter recoil buffer rod. The throttling during the re- coil is effected through an orifice formed by the difference in areas of a circular hole in the pis- ton and the area of the buffer rod. For varying the throttling, the areas of the buffer rod are tapered, i. e. the diameter of the buffer rod varies along the recoil. The recuperator cylinder consists merely of the stationary recuperator piston which moves relative to the forging on recoil. The recuperat- or cylinder communicates by a large passage way to the air cylinder partly filled with air. The air cylinder is placed forward and is made shorter than the recuperator and brake cylinder. This is necessary in order that at maximum elevation the oil in the air cylinder covers the passage way communicating with the recuperator and air cylinders. It is very important in the initial 529 530 531 lay out of the Schneider recuperator system that at a maximum elevation the oil completely covers the communicating passage way in the air cylinder and the recuperator initial volume should be reckoned in the air tank beyond this oil cover- ing. The passage way is made sufficiently large so that we have practically no throttling in the recuperator system. During the recoil, figure ( I ), the brake throttling is effected primarily through an orifice formed by the counter recoil rod in a circular hole in the piston. The simultaneous compression of the air recuperator during the re- coil takes place practically along an isothermal curve, due to the fact that oil and air are in direct contact in the recuperator. It has been found by careful computation, however, that an ex- ponent equal to 1.1 gives a close approximation in the compression curve of the air and the com- pression of recoil in the brake cylinder. The buffer is filled by the pressure head in the re- coil cylinder, the oil passing through fairly large orifices in the buffer head FF, the slide of the buffer head being away from the counter recoil buffer rod, see figure ( I ). During the counter recoil the slide on the buffer bead is pushed in contact with the buffer rod, and the apertures which filled the buffer chamber during the recoil are thereby closed and the throttling now takes place through new orifices of a very small magnitude. The buffer chamber having been completely filled during the recoil enables us to have a continuous regulation through- out counter recoil. The counter recoil throttling is effected through a constant orifice for over half of the counter recoil. We then have a taper- ing orifice until the gun nearly reaches the in battery position. 532 In the Schneider system the recoil is designed constant at all elevations or practically so, a slight variation taking place with the elevation. The recoil system is made to vary according to the stability slope at the minimum firing angle of ele- vation. The primary advantages of the Schneider sys- tem are: (1) An increased recoiling mass due to the recuperator sleigh contain- ing the cylinders, recoiling with the gun and thereby decreasing the reaction on the carriage. (2) The simplicity of the recoil mechanism, especially from a fabrication point of view. The disadvantages of the Schneider system, are: (1) due to the fact that the primary element of simplicity, the throttling effected through a simple tapering counter recoil rod, inherently pre- vents any possibility of a variable recoil. (2) the massive sleigh or slide at- tached to the gun, though reducing the reaction on the carriage, lowers the center of gravity of the recoil- ing parts below the axis of the bore so that on firing a large load is thrown on the elevating arc. To off- set this, on snail caliber guns a counter weight has been mounted on top of the guns. On the larger caliber guns as in the 240 m/m howitzer, a brake clutch was intro- duced on the shaft of the elevating pinion which slipped during firing. 533 534 Further tba air cylinder, "being necessarily placed forward of tbe recuperator brake cylinders with a long recoil gun requires a very long forg- ing and corresponding guides on the cradle. On tbe wbole the Schneider recoil system has proved one of the most satisfactory recoil systems used during the late war, being simple to fabricate and tborougbly rugged, due to its simplicity in de- sign. Example and calculation of tbe Schneider recoil system for the 240 m/m Howitzer: As an example of a satisfactory recuperator brake especially adaptable for a howitzer, calculations in tbe design layout of tbe 240 m/m howitzer recoil system are given in tbe following:- BIOOIL CALCULATIONS 240 M/M SCHMKIDBR HOWITZER. Type of gun - 240 m/m howitzer Total weigbt at recoiling mass * 15,790 Ibs. * K r Muzzle velocity - 1700 ft/sec. * V m Length of recoil B" 44,833 46,73 Angle of elevation 10 60 -i p a Intensity initial air pressure P al - * 576 7854 "a ik./ Initial air pressure - .7854P ai D* 18800 Ibs. sq.in, Height of axis of bore from ground 43" w V* Mean constant pressure P a * 1,189 x 10 Ibs. 64.40 Weight of powder charge if 40 lb. Travel of projectile in bore - u - 160" Maximum powder pressure on base of projectile P m = 2005 x 10' Ibs. 535 Maximum pressure on breech P b 1.12 P m Initial air volume V = 2970 cu.in. Final air volume Vf = V cu. in - A e b" = 1510 Vi t Final or maximum air pressure p.* * p a 4 ( ) V f INTERIOR BALLISTICS. e =' twice abscissa at maximum pressure D - i 16 P e 16 P e muzzle pressure on base of breech Z.e* ^ m p fe 622,000 4 (e+u)* Velocity of free recoil w V + 4700 if 50.25ft.sec. Velocity of free recoil - projectile leaving muzzle w V m + .5w V_ y o , 2. 40.15ft.sec. Time of projectile to muzzle t , i-JiL * * 12 V m .01175 sec. 536 Time of expansion at free gases 2(V f -V ) W r t .01538 sec P b 32.2 Free movement of gun while shot travels to muzzle * 12(W r +w+i) Free movement of gun during powder expansion p ob <* X a - + V t 8 .7179ft. W r 3 Total free movement of gun during powder pressure period Z = X t + X, 1.0279 Time of pressure period T t t + t^ .02713 sec. Total resistance to recoil in battery m r vj + m(b-E) a K ' " ' Variable recoil m T* 2[b-E +V f T- - (b-E)] 2 m r where K * total resistance to recoil during powder period (Ibs) b * length of recoil (ft) E * free displacement of recoil during pow- der period (ft) 537 T = total powder period (sec) c r m = cos stability slope d c = constant of stability d = distance from line through center of gravity of recoiling parts parallel to bore to center of pressure exerted on spades. , 490 - f g 32.2 C \f r cos cw r 0.85 x 15780 m = - - - = (approx. ) = - - - d h : :*-f.-iMiri-!^ f **.-: "rV- -* 3760 E = 1.0279 ft. T - .02713 sec. b = 44 ' 833 3.736 ft. 12 Hence 490 x 50755* + 3760(2.708)* K 3760 .02713* 2 (2. 708+50. 25x. 02713 x x 2.708) 2 490 1264660 8.13 155000 Ibs. (approx) Total resistance to recoil out of battery Rt 8 k - K - m(b- E + ) 2m r 155000 - 3760(3.736-1.028+ 15500 x - 0271 ) 2 x 490 538 155000 - 3760 x 2.824 144,000 Ibs. CALCULATION OF THE VARIATIOH OF TH1 HI- ACTIOB AIB PRESSURE IK TH1 HBCOIL. Initial air volume - 2970 cu.in. Initial air pressure = 576 Ibs/sq.in) Length of recoil (10 elevation) = 44.8 inches. Length of recoil (60elevation)46.73 Effective area of recuperator piston = 35.766 Ibs. Effective area of hydraulic piston = 31.2 Final Pressure (Initial volume) ___^ .^. _ ___^_ , Initial pressure (Final volume) Final volume initial volume * area at recuperat- or piston x length of recoil. .-. Final pressure (10 elevation)- 576( - - )** 2970-35.766x44.8 ' = 576 x 2.345 - 1350 Ibs/sq.in 1368 Final pressure (60 elevation) 2 970 576( 2970 - 35.766 x 46.73 ,,2970.1.1 - 576 x 2.49 - 1434 Ibs/sq.in. 1.299 For 40" Recoil 2970 * " * Final pressure * 576( ) 2970-35.766 x 40 576 x 2.065 * 1189 Ibs/sq.in. 539 1389 x 35.766 42525 (Plot these values above fric- tion) For 35" Recoil Final pressure * 576 (- ) 2970 - 35.766 x 35 576 * 1.815 = 1045 Ibs/sq.in. 1045 35.766 37375 For 30" Recoil 2g?Q Final pressure 576 ( ) 2970 - 35.766 * 30 576 x 1.643 946 Ibs/sqiln. 946 x 35.766 - 33835 For 25" Recoil 2970 ** Final pressure 576 ( ) 2970 - 35.766 x 25 576 x 1.483 - 854 Ibs/sq.in. 854 x 35.766 = 30544 For 200 Recoll ^^ Final pressure * 576 ( ) 2970 - 35.766 x 20 576 x 1.35 788 Ibs/sq.in. 778 x 35.766 27825 For 15- Recoil ^ Final pressure 576( ) 2970 - 35.766 x 15 576 x 1.22 * 702 Ibs/sq.in. 702 x 35.766 25107 For 10" Recoil Final pressure 576( - ) 2970 - 35.766 x 10 576 x 1.155 = 665 Ibs/sq.in. 665 x 35.766 23784 540 For 5" Recoil 2970 ** Final pressure 576 ( ) 2970 - 35.766 * 5 576 * 1.072 617 Ibs/sq.in. 617 x 35.766 - 22067 Calculation of Velocity Curve (During Powder Pressure Period) Point #1. Coordinates V o and X o V o ' V o Kt o m r Kt l o's; When the projectile leaves the muzzle, K = 155000 total resistance to s recoil r u * travel of projectile in bore 160" v o = muzzle velocity = 1700ftsec. w weight of shell = 353 w = weight of powder charge * 40 W r * weight of recoiling parts 15790 m - - = 490 32 (w * .5w)V 3 u - - ; t -^- (353 + .5 40 x 1700) =40 - 15 3<16 : .01176 2 x 12 x 1700 ... , .-40.1S- 155000> - 01175 . 40.15-3.70! 490 36.449ftsec. 541 w + .5if (353 + .5 * 40)160 X fo * w - u s - 15790 * 12 155000 x .01175 "2 x 490~ .32 - .0221 = .2979 ft.= 3.57 inches. Point #2. Maximum restrained recoil velocity and correspond- ing orifice. T * K(T-t Q ) t m .02713 - .01175) 155000 (.02713- 622000 t m .02329 V fn 40.15 * 6220 [.02329 - .01175]!- 490 62000(. 02524 - . 01175) 4x490(50.25-40.15) 40.15+9.328 v = 49.478 f m s 49 . 478 . 15500X.02329 m 490 . see . Xfm " 2m. 542 fo + (*-*(>> - , > m r 6m r (V f -V ) Xf * : x u .32 (see Point II) .32+[40.15+ 6220 (.02329-. 01175)- 490 (.02329 - .01175)' 6x490(50.25-40.15) = .32*. 632 = .952 Ft* 11.42 in. 155000 * .02329* X. .952 = .952 - .0855 2 x 490 .866ft. -10. 39 in. Point *3 (At end of the powder period) 155000 .02713 490 V r 4l.7ft.sec. X r 155000 x . 02713* X r 2 x 490 1.0279 - .1155 - .9124ft 10.94in. Velocity Curve (during retardation period) /2CK- "- (b+X-2X r )0>-x)] V x / m r For x 1.5 feet. 543 3760 [155000 <3. 89+1. 0279-2x. 9124)] (3.89-1.5) 490 37.4 ft. per sec. For x 2 feet 3760 21155000 (3. 89+1. 0279-2 x. 9124)] (3.89-2) P. 490 33.6 For x 3 feet [155QOO ^(3. 89+1. 0279-2x. 9124)3 (3. 89-3) v x 22.8 For x - 3.73(total recoil) Velocity _^_^^^___^____ Calculation of Guide and Packing Frictions. g 2oKd b Guide friction R g - - - - approx. u - .15 K = 155000 dfc * 15 .5 '(in) distance from center of gravity to resultant pull. 1 = 37 + 48=85"(in) mean distance between clip reaction. 2*. 15x154725x15. 5 .-. Guide friction - - = 8450 Ibs 85 Stuffing Box Friction Recuperator stuffing box Diani. * 2.169 Bear sleeve - .5"+. 875" contact Inner packing ring - .787 Gland - .87 544 Recoil stuffing box diam. = 4.728 Rear sleeve - .75*. 5 > Inner packing ring .787 Inner gland - .866 (Spring pressure + 0.1 pressure)(.75 diam. x H .09 x length of contact) Formula. 1058 Spring pressure from drawing 10.124 .785 (6.4375-5 .3437* ) 104 Ibs/sq.in. Oil pressure in recuperator = 576 + 1350 A . Initial Final - - - = 963 Ibs/Sq.in I Oil pressure in recoil 2222+1670 = - 1946 Ibs/sq.in. 2 Recuperator stuffing box diam. = 2.169 length of contact (dermatine )=.787 Friction - . 75x2 .169x3 . 14" (963 +104)x.09x. 787375. Recoil stuffing box diam. =4.728 Length of contact .787 Friction . 73x4. 728*3. 14* (1946+104)* .09x .787=1572 Total stuffing box friction = Recoil stuffing box friction + Recuperator stuffing box friction 1572+375*1947 Ibs. Total stuffing box friction. Total friction guide * stuffing box. - 8450 _____ * 1947 10397 Ibs. Calculation of Throttling Areas . 545 2[K- -(b+X-2X r )](b-x) C A 13.2 /K-p a -rR t +W r sin But '[K- 7 (b+X-2X r )J(b-x) V, C A* V x W, 13.2/K-p a -R t +W r sin Pa = Pai x Ar^PP 1 " *) 3 * 11 ^* 4 ! pressure x ef- fective area of piston = 576 x 35.766 = 20600 C = 1.39 (constant) A = 35.766 H r sin = 15790 x .0848 = 15550 R t =guide friction + stuffing box friction = 10,000 Its. K = 155000 V x = take the values as calculated for vol. curve, From calculations: when x = 3.57" V = 36.449ft.sec. w x = .061 x 36.449 = 2.223 x 2 = 4.446 When x 10.39in. V = 42.118ft.sec. w x = .061 x 42.118 - 2.5691 x 2 * 5.1382 When x = 10.94in. V = 41.7ftsec. *r x *.061 x 41.7 = 2.5437 x 2 = 5.0874 When x * 1.5ft.or 18i n . V x = 37.4ftsec. 3 W L39 x 3S.766 2 x 13.2 /154725 - 20600-10397+15550 .061x37.4=2.2814 sq.in.*2 rods=4.5628 546 l& ^ t * 5- 3&W6? *- _ 547 546 When x V, 2ft.or 24in. 33.6 .061 x 33.6 2.05 sq.in. x 2 = 4.10 When x - 3ft.or 361 n. V x - 22.8 w x . 061x22. 8-1. 28 sq.in.x22.76 When x 3.75ftor 44.8intotal recoil) w - Comparison of Throttling Areas. I no b a a Recoil Calculated Area of Orifice (2 rod.) Frenoh Value 3.5-7 4.46 4. 413 10.39 9. 13** b. 129 10.94 5.08 I 74 5.084 18. 4.5628 4.54 24. 4. 10 4.08 36. 2.76 2. 69 44.8 0. 0. SCHNEIDER COUNTBR RBCOIL. The counter recoil is divided into three periods: (1) The accelerating period, the : counter regulation being controlled by a constant orifice through the buffer in the recoil rod. (2) The retardation period, the count- er recoil regulation being controlled by a variable throttling orifice through the buffer head. 549 (3) A constant orifice period at the end of recoil, the throttling orifice being very small and the displacement a very small part of the recoil. The displacements corresponding to (1), (2) and (3) are 1 Q , l b and 1 Q respectively. Counter Recoil Data. Length of constant orifice 1 Q = 31.3 inches Length of variable orifice 1^ = 7.85 inches Length of constant orifice at end of recoil 1 = 5.68 inches b = Total c 'recoil 44.83 inches Constant orifice period 0.7 b Variable orifice period 0.175 b Constant orifice at end of c 'recoil 0.125 b where b length of recoil There being 2 recoil brakes, we have for the buffer reaction: . *;; B - * where A^ = area of one buffer = 9.859 a Q area of constant buffer orifice = .0664 sq.in. a o = area of constant buffer orifice at end of recoil - .022 sq.in. Considering the c 'recoil at horizontal elevation, during the constant orifice period, we have o o 2 - 3026 where A = load on air - friction = P y - I R 550 and C ZR 2c A b ^^MM 175 6290 Ibs. 490 2.78 x 940 175 15 A x Total L OT* 2o(a-*M V Buffer in. * in. * 2.3 * force t . A . 4 397oo 3415V* . 198 1.16 4550. . 8 1. 2 37860 .386 2.67 243OO. . 8 2. 36860 .386 3.05 31700. 1. 3. 35810 .493 3.3 37100. 1. 4. 34910 .493 3.23 35500. 1. 5. 33910 . 493 3.18 34400. 1. 6. 33160 .493 3. 13 33500. 1. 7. 32510 .493 3.09 32900. 5. 12. 28710 2.5 2.91 29000. 5. 17. 25710 2.5 2.745 25720. 5. 22. 22710 2.5 2.5 22700. 5* 27. 20710 2.5 2.46 20750. 4.3 31.3 18910 2.13 2. 3 6 18950. Beginning of Variable Orifice. After this period the unbalanced force was assumed constant. * ib * r M*o - *;> .3-. 25 245 (- .655 -) = 1880. # = unbalanced force 175 ** 551 I n o b Reooil. x in 1880 x 1300-1800 x V Total lb. Buf f r 245 32.3 33.3 .0833 . 1666 157. 360. 4.6 1 ? 3.94 2. 16 1.99 20500 20200 34.3 35*3 36.3 37.3 39. 15 .25 333 .416 5 .656 47o 627. 780. 940. 1230. 3-38 4.75 2. 12 1.47 .286 1. 84 1.66 1.45 1. 215 .52 9900 19500 190OO 18500 18000 40. .5 I750u 17000 42. 44. 83 5 .0 16500 16000 3 o i n t X Foot 45 80 z 323.5- 4580x ^215 V.I Le d X 458OX 245 f.. on Air 1 .0833 382 2833 11. 5 3. 39 28400 116 2 .1666 762 2453 10. 3. 16 27600 108 3 .25 1145 2070 8. 45 2. 91 26800 . 102 4 333 1525 1690 6. 9 2. 63 26000 .092 5 . 416 1900 1315 5. 38 2. 32 24600 .083 6 .5 2290 925 3. 78 1. 945 24400 .065 7 .656 3000 215 8. 76 * 94 21000 .044 ftHKttlDgB X any interval unbalanced force M r = mass recoiling parts V o * max. velocity of o 'recoil V x velocity at any point V x * velocity at beginning of period l e 1 > length of constant orifice period in feet 552 1^ s length of variable orifice period in feet l c = length of final period for c 'recoil in feet P a = load on air in Ibs. R Total friction 2K*AV = total "buffer force 175 W b = length of c'recoil (ft) Period 1 Q 2K 2 AV a P a - R - = acceleration 175A Assume velocity of 3.5 ft. per second and solve for orifice W, Period l c p a - R = 175 W* Assume velocity of 1 ft. per sec. and solve for *x Period lu x = Knowing V solve for K x for various points 553 175 P a = R Solve for W x ST. CHAMOND RECOIL. ST. CHAMOND RECOIL The type of St. diamond brake SYSTEM. here discussed, consists of three cylinders; a hydraulic brake cylinder, a recuperator cylinder containing the floating piston which separates the air and oil, together with a regulator valve for throttling the oil between the hydraulic and recuperator cylinders, a third cylinder serving as a part of the air reservoir and therefore communicating with the recuperator cylinder air volume, the remainder of the third cylinder being used for storing oil for the brake mechanism. One of the peculiar features of this type is the regulated spring valve where the main throttling occurs. The valve functions somewhat as a pressure regulator or governor, since if the pressure falls, the spring reduces the valve opening tnereby in- creasing the throttling drop and the pressure in the hydraulic cylinder. The pressure in the recoil cylinder, (i .e. the hydraulic pressure) is the sum of the air pressure, plus ttie floating piston friction drop, plus the throttling drop through the regulator valve. At short recoil the air pressure is necessarily small compared with the throttling drop. The resistance to recoil is large and therefore the recoil pressure large. This requires a large throttling drop and the air pressure becomes necessarily small compared with the throttling drop. The large throttling drop requires a very small valve opening, with a large pressure reaction against the valve. To balance this reaction a very stiff spring is re- quired. Such spring characteristics have been ad- 554 mirably met oy the use of Belleville washers. At long recoil the resistance to recoil ia small, there- fore the throttling drop is small, requiring a large orifice area. Since the pressure in the recoil cylinder is small together with a large orifice opening, a weak spring with large deflection is desirable. Such spring characteristics are best met with an ordinary spiral spring. Hence, at long recoil, low elevation, a spiral spring functions alone, while at short recoil maximum elevation the belle- ville and spiral spring function in parallel. The regulator ia so designed that at low elevation only the spiral spring functions. To modulate or regulate the velocity of count- er recoil to a low velocity, the pressure in the recoil cylinder is lowered just sufficiently to balance the total friction during counter recoil. At the end of counter recoil the recoil cylinder pressure is reduced to zero and the recoiling mass is brought to rest by the total friction alone. To reduce the pressure during the first part of counter recoil throttling through a constant orifice la effected in a separate passage way or channel leading from the recuperator to the recoil cylinder. At the end of counter recoil additional throttling around a buffer rod and its chamber, is effected reducing the pressure in the recoil cylinder to zero or nearly so. DESCRIPTION OF THE Referring to figure (10) is OPERATION OF THE ST. shown a schematic diagram of CHAMOND RECOIL. the operation of the St. Cnamond recoil system for both recoil and counter recoil. Recoil:- During the recoil a flow or stream of oil passes by the regulator valve from the hydraulic to the recuperator (oil side)cylinder. The pres- sure p of the oil against the recoil piston is re- 555 1 556 RECO/L REGULATOR Fig- 6 557 duced by throttling through the regulator to a pres- sure (p a ) against tne oil side of the floating pis- ton. Due to the friction of tbe floating piston the air pressure p a is less than tbe pressure on the oil side of the floating piston p^. The tension in the recoil rod is balanced by tne total pressure on the recoil piston plus the hydraulic piston friction plus the stuffing box friction in the recoil cylinder. The valve in the counter recoil orifice remains closed during the recoil. REGULATOR VALVE. The throttling during the recoil is controlled by tbe regulator valve. See figure (11). The regulator valve consists of two parts: an upper stem and the lower valve stew. The lower valve stem is seated very carefully on a circular seat at the top of the entrance channel. As the valve lifts, the throttling area becomes the vertical cir- cumferential area between the valve and its seat. The spiral spring reacts on the lower valve stem. The Belleville washers at the top of the upper stem, react only on that valve stem. The upper stem rests in a valve box or housing. To move the upper valve stem (other than the slight deflection possibly compressing the Bellevilles) the whole housing or valve box is moved by a cam as shown in diagram. The diameters of the upper part of tne lower valve stem and the lower part of the upper stem, (that is the diameter of the stems of the regulator valve,) are tbe same. At short recoil the reaction of the Belleville on the upper stem is transmitted by the mutual reaction between the upper and lower stems at their surface of contact. The valve opening and consequent throttling drop of pressure depends upon the deflection of the spiral springs or Belleville washers, the spring reaction balancing the hydraulic reaction on the valve. Neglecting the small dynamic reaction, the 558 hydraulic reaction on the valve is fhe product of the intensity of pressure in the recoil cylinder and the base of the regulator valve, minus the product of the intensity of pressure in the re- cuperator cylinder and the effective area on the upper part of the regulator valve. At long re- coil, since the loner valve stem comes in con- tact with the upper stem, the effective area on the upper part of the valve is obviously equal to the area at the base of the valve. Hence the hydraulic reaction at long recoil is merely the product of the difference in pressures between the recoil and recuperator cylinders and the area at the base of the valve. At short recoil the upper stem of the regulator is brought down by the cam at its top, until its lower surface is in con- tact with the top surface of the lower valve stem. The effective area, therefore, on the upper part of the regulator valve equals the difference in areas between the area at the base of the lower valve stem and the area at the upper end of the lower valve stem, or the area of the upper valve stem; the two latter being always equal. Hence the hydraulic reaction at short (or intermediate recoil for the greater part of recoil) equals the product of the recoil intensity of pressure and the base of the valve, minus the product of the recuperator intensity of pressure and the difference in areas between the base and middle stem of the valve, when upper and lower stems are in contact. At long recoil the hydraulic reaction is balanced above by the spiral spring reaction. At short or intermediate recoil the hydraulic reaction is balanced by the combined reaction of the Belleville washers and the spiral spring though the latter is negligible compared with the former. COUNTER RECOIL. The regulator valve it closed during counter recoil. The oil flow during counter recoil, therefore, is different from that in recoil. The valve is seated, 559 but the oil is allowed to pass through a very small hole in its center. This orifice is constant through- out the whole of counter recoil. There is another channel for the oil leading from the bottom of the buffer chamber in the regulator body. This oil passes through a ball valve. As the floating pis- ton returns to its initial position at the end of counter recoil, the regulator rod enters the buffer cavity, thus obstructing entrance of oil to this cavity. This rod is tapered so that when it has fully entered the cavity there is no clearance between the rod and the entrance, and tne oil in returning to the recoil cylinder nust all pass through the central opening in the valve. By neans of this regulation it is possible to allow the gun to return to its "in battery" position quickly, but its final movement is so controlled that there is no ehock. The throttling areas in the counter recoil channels are so designed as to cause sufficient throttling to lower the pressure in the recoil cylinder that it nay practically balance the total friction, during the counter recoil. At the end of counter recoil this friction alone brings tne recoiling mass to rest when it reaches the battery position. GENERAL THEORY OF THE Figure (11) shows the ST. CHAMOND BRAKE. regulator valve stem for both long and short re- coil. Let R v reaction on base of throttling valve p * intensity of pressure ia recoil cylinder p a * intensity of air pressure. p a intensity of pressure in recuperator cylinder (i. . on oil side of floating piston) a entrance area of valve or effective area at base of valve. 560 a = area of valve stem S b * spring constant of belleville washer S s = spring constant of spiral springs C = effective circumference at base valve h = lift of valve from initial opening h the initial compression of the spiral 3 valve spring at initial opening h b * the initial compression of be Seville washer at initial opening w = throttling area v = velocity of flow through entrance area "a" V = velocity of recoil A = effective area of recoil piston d = density of oil The hydraulic reaction, at long recoil becomes i R v - p a a, and at short recoil, we have, the value R y - p a (a-a t ). The belleville washer reaction, oecomes R b = S b (h b +h) and the spiral spring reaction, becoraes, R s = S s (h s +h). Hence at long recoil, we nave R y -p a a = S S (h s + h) + F (1) and at short recoil, we find K v -p a (a-a t )*S s (h s +h)+S b (h b +h)+F (2) that is, R v -p^(a-a t )=S s h s +S b h b +h(S s +S b )+F Now at intermediate recoil the upper valve stem is separated from the lower valve stem by a distance h o when the latter is just about to leave its seat. If h Q is tne separation between the two stems before recoil and if e = the initial lift of lower valve stem required to clear the valve, then h o =h Q = e Hence at intermediate recoil, we have R v -p a (a-a t )=S 8 (h s +h)+S b (n-n +h b )+F, that is R v -p a (a-a t )=S s h s *S b (h b -h )+h(S s +S b )+F (3) Where F is the valve stem friction and will be neglected, let C S s h 8 and C Q =S g h s +S b h b c o* s s n s * 3 b( h b- h o> 561 The reaction against the base of throttling valve, in terms of the pressure at the entrance to valve, becomes, 2 R v -p a = -^- (4) g where PJ = the pressure at a mid section in the entrance channel of the valve. Further 2 * = E + h t (5) d 2g d Neglecting the friction and accelerating head, h t as snail we have, therefore dv* p = p - wnich gives the pressure in the "& entrance channel in terms of the recoil pressure (i.e. the pressure against the hy- draulic piston) hence dav dav dav R v-Pt a+ -J P a 2 g g or dav dv . R=pa + = (p + )a (6) 2g 2g Therefore at long recoil, we have dv (p+ )a=C + S s h + p a a 2g and at short recoil we have, dv* (p+ )a=C +(S s +S b )h+ Pa U-a t ) (8) dv* (p+ )a=C +(S s +S b )h+p a (a~a ) (9) 2g Considering now the main throttling through the circumferential section, around the effective 562 circumference of the valve, we have, for the ef- fective throttling area, oh 1 w where K ' Contraction factor of K o 0-775 orifice , tbe corresponding pressure drop through the valve becomes, K!A"V' p - 175(ch) Further av AV. hence V*=(T)* V* hence lp+ $() b * long recoil (10) S g +S b recoil (11) and ll f $-(-)* v * _ c _ if S 8 + Sjj mediate re- coil (12) Considering only the main throttling, or ratber designing the recoil flow channels to have throttling as compared with tbe throttling through the regulator valve, we have P * P * Pa hence we have the three fundamental equations for the recoil pressure, in terms of the velocity of recoil and tbe pressure in tbe re- cuperator cylinder: K S S 8 AV p, - - - - +p a (13) 175C a [(p+ -(-)" ]a-C- P 4a)' ** 2g a 144 recoil d A V* 175C*[(p+ (-)* )a-C'-p a (a-a t ) a at 2g a 144 short recoil. AV + p a (15) 175C*[(p+ (-)* a-C '-p a (a-a )]' _ 2g a 144 mediate recoil. where the units are obviously, p' a and p in Ibs. per sq.in. V in ft. per sec. A i a i and a t in sq. in. d in Ibs. per cu.ft. If further J * I a~ * 144 then 9 uati n8 (13), (14) and (15) reduce to the simpler form - :; - ; - z (16) at long re- 175C [(p-p a )a+J V -C ] coil p _p - . - .. - 175 C [(p- Pa )a +Pa a + J V -C Q ] recoil K*(S s +S b )A*V* ;; (18) at in- 175C [( P - Pa )a +P ;a + J V -C Q "] ter.ediate recoil To compute p for any given displacement and cor- responding recuperator pressure and recoil velocity, we find the solution in the form of a cubic equation The solution is as follows: From equations (16), (17) and (18), KV.AV ' 175C a a p ~ p a " 564 or P-Pa VV* 175C 8 a 2 lot B , K A ' V ' S * or 175Ca* 175 Ca* p-Pa * z J o vic o PaW'-Co and or = a a a Then from the above equations, we have or B=Z +2Z m+Zm To eliminate the 2nd degree term, substitute, 2 a * 4 4 Z X - r m hence Z -X - - X + - n O 39 and , 3 3 9 * 9 H * Z V^ O m Vo. V A G ifl A ~ Ql A" 1 " ~" T ID Expanding, we find 4 8 s x ,,a 82.. B = X 3 2? 3 * 8 - "' * , 8 x - f .' = X 3 - - m*X - m 8 Further let N 3 * + -7 then X* - - m*X-N% 3 ^7 m 3 Solving by Cardan's method ^ + / / / ) m 2 4 730 2 4 730 565 2 X a Z + ~~ Hi + - 3 a 3 hence * N 8 . 3 / /i! _1_ 2^ p (/ 2 * 4~ " 730 + 2~ " ' 4 " 730 "3 } During the greater part of recoil except at the very beginning and towards the Very end of re- coil it has been found by actual calculations, that the term *- becomes negligible in comparison with 730 ^ e and nay be omitted without appreciable 4 error. The above equation, therefore reduces to the simple form 2 p<= m(N - g)+ p a Another and far simpler method for the com- putation of p established by Mr. McVey, consists in the construction of a table, with assumed values of P ~ P a . The table is based on the two following equations (neglecting dynamic bead as small) (p-pa> A *Pa A i s c o + ( s s +s b) h short recoil (a) (p-p a )A=C +S s h long recoil pa AY a and p-pl - 175C , h , If ve assume a mean air pressure throughout, (the error thus introduced having been found small), we have or = s s Assuming a series of values of (p-p a ) e obtain a series of values for h, now from (b) 566 567 568 569 from which a series of values can be established for corresponding values of p-p a and h. Knowing the retarded velocity for any given point, the corresponding value of (p-p a ) and h can be picked from the table and knowing p a for the given point in the recoil, the recoil pressure p is ob- tained. It is to be noticed, that substitution of (b ) in (a) gives a cubic with a second degree term, as before. Thus no direct simple solution is possible. The table method is recommended even for short re- coil since the error introduced by assuming the air constant is relatively snail. GENERAL PROCEDURE FOR Due to the complexity of CALCULATION OF RECOIL, the general equation of re- coil no mathematical solution is possible, except by ex- panding into a series. Such a solution of a recoil equation is known as the "point by point" method and has been used before in this text. The object of actual computation of rec-oil curves for a given type of mount is to ascertain the ratio of the peak to the average resistance to recoil at maximum and zero elevation. The average resistance may be readily obtained in the preliminary layout of a design and knowing the peak ratio for a given type of mount, enables the peak resistance to be obtained and the consequent stresses in the carriage. Let Vf = free recoil velocity at point "n" (i.e. the velocity generated in the recoil- ing mass by the powder pressure). V rn corresponding retarded recoil velocity R n = total friction, stuffing box and guide friction. $ = angle of elevation of gun. Now, the end pressures at the beginning and end of 570 <*JU- recoil, becomes p-pi * for ft a -a t C p* Pa( - )* for short recoil A & Since = at long recoil, we have for the re- sistance to recoil, K p n A +R n for long recoil K p n A+R n -W r sin0 for short or intermediate recoil Long recoil: For 1st point long recoil, Pao + + R o V r t " v f t and knowing y ri 3 , A* A* i A* A* i 2. a ( / + / + / J _\ m + D l 4 4 730 2 4 730 3 Pa , m = 175C 2 a 2 a For 2nd point long recoil, and knowing V r a 730 {>AtV r.S 8 J V a -C B V-- ; m = f 175C a a a After a very few intervals the valve opens sufficient ly so that the term 571 rr may be omitted, then for point "n" at long re- I o(j coil, (p m A+R n ) V n -V< v rrr v rm> ' At and knowing V n m r 2 P n - m(N - -) + p a Obviously after the powder pres- sure period V fn -V fm = and we have the simple dynamic equation of recoil. Short Recoil: The procedure for the calculation of the velocity and pressure curves for short recoil is exactly similar as- for long recoil. For 1st point short recoil, P ( ) + + R -W r sin0 c* 1 a a 1 * r, "f , -< ~ T 5 > 4 * and knowing V_ a * i , /N 8 /N~ 1 /N A*~ 1 2. a( / + / + / J )m+Dl 1 2 4 730 2 4 730 3' Pa t P where 175C 2 a a a for 2nd period short recoil, p'A+R -W sin V ~ v =v f ~ v f - C - ii and knowing V r2 , p 2 can be obtained by a solution of the previous cubic equation. The greater number of points of recoil excepting a few points at the beginning and end may be solved with suf- ficient accuracy by the expression, 2 p n =m(N- -gO+Pa "nere, as before, 572 V^- KAV(S s +S b ) N - / + B 27 " 175Ca and Calculation from constructed table of (p~Pa) h, and V~ The procedure here is exactly similar as above; each preceding interval establishes a new retarded velocity which from the table establishes a new recoil pressure. This recoil pressure substituted in the dynamic equation in turn establishes the re- tarded velocity at the end of the interval under consideration. Judgment must be aged in the proper increments of time to be used. The closer the intervals the more accurate the velocity and pressure curves. At the beginning and end, the time intervals should oe taken smaller. During the major part of re- coil the time intervals can be fairly large. As a check during the powder period the retarded velocity should be roughly 0.9 of the free velocity of recoil. CALCULATION OP THE VARIOUS In the calculation FRICTION COMPONENTS DURING of the vertical pres- RECOIL. sure and retarded velocity curves for the St. Chamond brake, the frictions vary as a function of the pressure. At long recoil the pressure variation is small and we are not in great error in assuming constant friction: with short recoil, however, a peak value is obtained and with it a change in friction. The frictional resistance opposing recoil are: (1) Guide friction which is function of the total pull. 573 (2) Stuffing box friction which is a function of the recoil pres- sure. (3) Recoil piston friction which also is a function of the recoil pressure. (1) The guide friction during recoil has been previously expressed by the following equations: 2(p b e+Bb)+W r cos0N R = - n * n where p b is the powder reaction on the breech c is the perpendicular distance between the axis of the bore and a line through the center of gravity of the recoiling parts parallel to the axis of the bore. B = pA, the hydraulic reaction of the re- coil piston n = coefficient of friction, from 0.15 to 0. 20 "b * distance down of the line of pull from the center of gravity of recoiling parts where X A and y t are the coordinates of the front clip reaction and x^ and y g are the coordinates of the rear clip reaction having axis and origin through the center of gravity of the recoiling parts Considering the somewhat inaccuracy of a "point by point" method of computation, it is be- lieved the following formula is sufficiently accurate, 2nBb + nW r cos 0(x t -x a ) R = - - C - 2nr or when *~x * s small, where r is the mean distance from the center of gravity of the recoiling parts to fhe line of action of the guide frictions. 574 (2) The packing friction formulas have been already considered in more or less detail in Chapter VIII. The stuffing box friction, R 8 c^ + c; P where c t C t - nd r (bf+af+a t f t ) (3) The hydraulic piston friction, Rp= C " * C ndp (bf+af i ) From the above formulae p * recoil pressure in Ibs. per sq.in. P =intensity of pressure caused by Bellevilles or packing springs in Ibs. per sq.in R b * belleville or packing spring reaction on annular area of packing spring at as- sembled load in Ibs. d r * diam. of piston rod in inches. d o = outer diam. of stuffing box packing ring in inches. d * diam. of recoil cylinder in inches. d i =inner diam. of piston packing ring in inches. b = width of leather contact of packing in inches. f * corresponding coefficient of friction * - - silver contact of flap of one flange of packing ring in inches. f t coefficient of silver friction .09 Then p o becomes, for (Dp 575 In summing up the component frictions, we have nW r cos 0(x t -x f ) C-2nr R S = c; + c t ' P R p - c t " + c; P hence R - (C*C|[*K t )+(CJ + C; 1 +K 2 ) p c 4 +c s p showing the total friction resisting recoil is a linear function of the pressure in the recoil cylinder. Floating piston: The oil pressure in the recuperator cylinder during the recoil is greater than the air pres- sure by the drop of pressure caused by the float- ing piston friction. In the previous recoil equations, the recuperator oil pressure has been used in place of the air pressure. To compute this pressure knowing the air pressure, it is only necessary to compute the floating piston friction drop. In the discussion of the floating piston in Chapter VIII, we have Rf s C t +C a p a where C t - Kdl(bf+af t )(p +p )+20 t f lPq ] C a = nd[2(bf+af t )+2a t a For symbols see discussion of floating piston in Chapter VIII. All dimensions may be expressed in inches and pressures in Ibs. per sq.in. in place of the center of gravity system as used previously. The resulting friction Rf is therefore in Ibs. The drop due to friction, becomes, R f p a - p a - where A a is the area of the floating A a piston or recuperator cylinder in sq, inches . 576 577 GENERAL THEORY OF Counter recoil is divided in- COUNTER RECOIL. to two periods, (1) the first period or constant orifice period and (2) the second period or buffer period wnere the main re- tardation takes place. The second period is the critical period in the design of a counter recoil system, since with field carriages the stabilizing force of counter recoil is relatively small, there- fore too rapid retardation of the recoiling mass will cause the mount to be unstable on counter re- coil. Let A = effective area of recoil piston K o = contraction factor of constant orifice K t * contraction factor for variable orifice. W Q = area of constant orifice x = variable area of buffer throttling R = total friction p a = recuperator oil pressure Pa * Pa^ equivalent recuperator pressure on recoil piston C * = throttling drop constant for con- 176 stant orifice. K*A* C x = - * throttling drop constant for buffer 175 orifice. R o =l? r sin0 + R a resistance constant. For 1st Period of Counter Recoil: Considering the notion of the recoiling mass, from tne initial displacement of out of battery, we have dv pA -W r sin0-R=m r v but P ' Assuming the throttling drop is entirely through the constant orifice during the first period 578 .3 2 K Q A V dv of recoil. Hence p^A - W r sin 0-R=ro r v 175w* dx or p v * Since p a is a function of x t the equation is not possible to integrate directly, but by divid- ing the constant orifice period into several in- crements, and taking a constant air pressure equal to the mean air pressure for the interval, we get a very close approximation of the true velocity of counter recoil by the following solution, 9 ro r d v dx = Integrating, we have m r w o CjjV* C o v Substituting for the base 10, we have C o v* C rt v* logf(p a -R ) -]=logUpa-R )' as 2 iv z V>'4miu z "o o ^.om r w o From this equation, knowing the velocity at the beginning of any arbitrary interval and with the mean recuperator pressure we can obtain the velocity at the end of the interval. It will be found that fairly large intervals may be assured with considerable accuracy, providing the air pressure does not vary greatly. The velocity curve for the first period should be continued from the out of battery position to x = b-d, where d= the length of the counter re- coil corresponding to the buffer length. Let vj, = velocity of counter recoil at entrance to buffer. For 2nd Period of Counter Recoil. The recoil displacement is d, and the initial 579 velocity v^ . In order to be assured that the c 'recoil is completely checked, the counter re- coil energy of the recoiling mass at entrance to buffer should be dissipated in a distance somewhat less than d, from 0.7 to 0.9d, depending upon the design constant of the recoil system and gun. Let k * the proportional distance of d that the recoil energy is to be dissipated along. For counter recoil stability the minimum force during the buffer period is obviously obtained by using a constant force during the entire period. There are two methods consistent with counter recoil stability: (1) When the total friction is small compared with the overturning force permissible with counter recoil stability, by "bringing the recoiling mass to rest into battery with t"he friction alone. (2) Khen the total friction is greater than the overturning force permissible with counter recoil stability, by bringing the recoiling mass to rest into battery by a force equal to the total friction minus a recoil pressure exerted on the recoil piston. In method (1) obviously, for a given kinetic energy of the recoiling mass at entrance into buffer, the recoil displacement during the buffer action is fixed. Method (1) We have for the required recoil displacement during the buffer action: 1 2 -m_v b where k = 0.7 to 0.9 kR where R is the total recoil friction (guide, stuffing box and piston friction). 580 The length of the buffer in the recuperator cylinder becomes, d' - d I The velocity curve is evidently a parabolic curve against displacement, that is dv R * - m_ v dx x v / Rdx = - m r / v dv b-d v b R(x-b+d) = -(v - v) hence / 2R v / v* -- (x-b+d) m Since it is assumed that p = 0, we have substituting for v, we have w x in terms of the dis- placement x from the out of battery position, vg - U-b+d) "x M "o " -- (x-b+d)] where b = length of recoil in ft. and d = length of buffer recoil in ft. x - counter recoil displacement from out of battery position in ft. The throttling drop 581 through the constant orifice has been found 1 7 i M by calculation to be small as compared with the throttling drop due to the buffer. Therefore a simplification in the calculation may oe made by omitting this term, hence . 7- and substituting for v, we have 13 2 K t A /m r v b -2R(x-b+a) H X m / approx . which gives 13.2 m r p^ the required throttling area in terms of the displacement of counter recoil, (x is measured from out of battery position in ft; b - recoil displacement in ft. and d = recoil buffer displacement in ft). Method 2 If h = height of center of gravity of recoil- ing parts above ground (practically from ground to axis of bore) and w g = weight of entire carriage and gun and C' s - distance of w s from wheel contact, then critical stability (at elevation) we have, (R-pA)h=N s l^ using a factor of 0.8, we have W 1 ' s s R - p A * 0.8 where R is the total friction, h but now a function of the re- coil pressure, let R = C t +C a p then W s l' C +P(C -A)=0.8 1 h W 1 ' or _ w s x s 0.8 C h b l p = where C - guide friction as- (C t ~ A) sumed independent of p + that part of packing friction in- dependent of p. C * that part of packing friction dependent upon p. The counter recoil velocity curve, "becomes during the buffer recoil, 582 v - (x-b+d) and for the pres- n\ r sure p in the recoil cylinder, we have K Pa - 175w P hence KI A v/"" 1 w x - / 13.2 K*A 2 v 2 (p a -p>- or in terms of the displace- ment x, (R-pA) (x-b+d) 2(R-p A) 175(p a -p)w*-KA* [vg (x-b+d)] m r Neglecting the constant orifice throttling drop, we have the following approximate formula 12.2 It should be carefully noted that if v^ is allowed to become too great it may be found very difficult to prevent the final check of counter recoil without shock sure with even prompt throttling by the regulator the kinetic energy ot the gun may overcome the the opposing friction and cause bumping. DESIGN FORMULAE ST. In the preliminary design CHAMOND RECOIL SYSTEM, of a St. Chamond recoil sys- tem, we must consider the following. (1) The proper weight of recoiling mass wit"h given ballistics and allow- able recoil at maximum elevation for minimum weight of the total mount, gun and carriage. 583 (2) Tha length of recoil at zero ele- vation consistent with stability. (3) The total resistance to recoil at short recoil maximum elevation and at long recoil, zero elevation. (4) An estimation of the guide frict- ion and packing frictions for both recoil and counter recoil. (5) The recuperator reaction re- quired to hold the gun in battery at Maximum elevation. (6) Limitations of recuperator dimensions . (7) The calculation of initial air pressure and air volume, final air pressure and air volume. From this the equivalent air column length. (8) The calculation of strength of cylinders and proper thickness between walls. (9) The layout of the recuperator forging distance of center lines of cylinders with respect to axis of bore, location of trunnions, etc. (10) The calculation for maximum and minimum throttling areas. (11) The calculation for entrance chan- nel area to regulator valve, regulator dimensions and channel areas around and leading from the regulator orifice. (12) The reactions on regulator valve corresponding to deflections at maximum and minimum opening and the design of spiral springs and belleville washers. (13) The design of cam mechanism for changing the initial opening to regulator valve for decreasing the recoil on elevation. (14) The design of packing for float- 584 ing piston, recoil piston and stuffing box. (15) The design of the counter re- coil and chamber, throttling grooves and constant orifice with its chan- nel leading from the inside end of the buffer to the recoil cylinder (16) The layout of gauge and pump details and all other details. (1 ) Proper weight of recoiling mass: From "General design Limitations" we have _ * / kk 1 where "c k T and ., _ gUv + m 4700.)' 2b m = mass of projectile I * mass of charge b = length of short recoil in feet R recoil reaction in Ibs. g = acceleration due to gravity (ft/sec) w c = weight of carriage excluding recoiling mass in Ibs. k may be obtained from table in Chapter VII or the ratio w c may be computed from a similar well designed type of carriage, using a somewhat lower value of "b" according to the .judgment of the designer in improving the weight efficiency of the mount proper over a similar previous design. (2) Length of recoil at zero elevation, Prom pressure curves obtained experimentally it was found that the resistance to recoil at zero elevation is practically constant throughout the recoil. 585 Let b = length of horizontal recoil consistent with stability in feet wv+w4700 Vf = - = free velocity of recoil. w = weight of projectile in Ibs. w * weight of charge in Ibs. W f = weight of recoiling parts in Ibs. v = muzzle velocity (ft/sec) V r 0.9 Vf(approx.) = velocity of restrained recoil. u * travel of projectile up bore in feet A recoil constrained energy = 7 Jt i r v r E * recoil displacement during powder period = 9 06. * .64 "r Overturning moment C * constant of stability = - . . , . Stability moment W g * weight of total mount, gun and carriage l g * moment arm of w g about spadepoint. d = perpendicular distance from spade point to line of action of the total resistance to recoil. Usually 5 = 0, cos = 1 and d = h = height of axis of bore above ground. then W r cos and when * 0, we have, 1 / 2AW_h b - E + [W s l s /Ofjjlp 8 ] Tir P V *^ Ordinarily the constant of stability will be as- sumed at C * 0.85. For rough estimates, especially wnere the length of recoil is comparatively long, C s V r cos where A 2A 586 1 -7- " r m r v ! (3) Resistance to recoil, short and long recoil, For design calculations, Bethel's formula for the total resistance to recoil is sufficiently accurate. Let w = weight of projectile, Ibs. Vf r = weight of recoiling mass, Ibs. M = travel upbore in inches d = diam. of bore, inches. v = muzzle velocity of projectile (ft/sec) Vj = max. free velocity of recoil (ft/sec) b s = length of short recoil at max. elevation in ft. bj, = length of long recoil at zero elevation in ft. w = weight of powder charge Now for the free velocity recoil, we have w 4700 + wv v = - then at maximum elevation and short recoil, we have K s '1.05[ 8 u + (. 096*. 0003d )m D s v (where 1.05 accounts for the peak effect due to throttling) and at horizontal elevation, long re- coil, we have y the peak effect 2g h h +(.096 +.0003d)m- being zero (4) Guid*e and packing frictions - Recoil and Counter Recoil:- In the calculation of guide friction during the recoil consideration must be given to the pinch- ing action at the clips due to the pull being 587 usually below the center of gravity of recoiling parts. Failure to consider this fact will give erroneous results for the friction at high elevation. Also the recuperator must be de- signed not only for the weight component at max. elevation but the friction just out of battery. Since at the end of counter recoil we have the full air pressure acting on the recoil piston, the pinching action and corresponding guide frict- ion being a factor of importance. Let B = the total braking including the packing friction of recoil piston and stuffing "box, Its. d^ = distance down from center of gravity of recoiling mass of line of action of B in inches. pA = the recoil reaction, Ibs. n = coefficient of guide friction x^ and x a = coordinates in direction of bore of clip reactions measures res- pectively from center of gravity of recoiling parts in inches. 1 = x + x = distance between normal clip reaction, in inches. _ R fi = total guide friction. From a similar previous design, a value of b and 1 may be assumed. Prom ChapterlV, we have 2nB d b+nW r cos and for a first approximation assume x then R = 2n8 a b but K = B+Rg- W r sin (K+W r sin 0)1 1+2 n 588 Knowing K and assuming 1 and b, we have B s pA+R s +R p where R g - stuffing box friction R = recoil piston friction. Since the design and estimation of packing friction is in greater part based on previous empirical data, the width of packing and cor- responding dimensions being entirely an empirical matter, we must estimate the proper value from data on previous satisfactory packing used on other guns. Now in general the packing friction may al- ways be represented by the following equationt- R CY+C p. where C t is the friction component caused by the springs or Belleville washers. Since the object of the Believilles is to compensate for the deficiency of the oil or air pressure normal to the cylinder due to Poisson's ratio, if we know tne maximum pressure and assume dimensions for the packing we may compute C t as well as C ? and thus estimate the friction at any other pressure. Maximum pressures normal to cylinder should be taken as follows: Pn k Hydraulic piston - 88 Pmax. 0.88 Stuffing box 0-86 p max . 0.86 Floating piston: Air side i- 20 Pa max. 1.20 Oil side 1-38 Pa max. 1.38 Knowing the maximum pressure normal to the cylinder "p n " we have, max. n dC.05 b+.09(a + T^Pn approx. 589 s s 1 I tl 590 where d * diam. of piston rod or cylinder in inches, b * width of leather contact of packing a = total depth of one silver flange of pack- ing cup in inches. a t = total depth of outer silver flange. Prom the above equation, we have C i* c 2 Pmax. = nd[.05 b+.09(a+ p n where Pmax. tne max i raum fluid pressure, Further, K d 0.731 .05b + . 09 (a+ r^) )p roax % hence o a C K d[ .05b+.09(a+ )](p n - 0.73 Pmax ) a C = n d 0.73[.05b+.09(a+ )] 8 2 As a guide for suitable values of a, a t and b with given maximum fluid pressures, the following table has been constructed of values used in cer- tain experimental recuperators. 75 m/m Model of 1916 MI. Pmax Ibs/sq. in. Recoil piston Stuffing box Floating piston 0.14" 0.18" 0.19" 5120 0.14" 0.18" 0.19" 5120 0.09" 0.18" 0.29" 1270 3.3" Model of 1919. a a b Pmax lbs/sq in. Recoil piston 0.137" 0.233" 5500 Stuffing box 0.137" 0.233" 5500 Floating piston 0.137" 0.194" 1850 591 4.7" Model of 1906 Pmax Ibs/sq.in, Recoil piston 0.128" 0.128" 0.335" 3820 Stuffing box 0.128" 0.128" 0.335" 3820 Floating piston 0.128" 0.200" 0.335" 1200 4.7" Model of 1918. a a t b Pmax Ibs/sq.in, Recoil piston 0.156" 0.218" 0.218" 4500 Stuffing box 0.156" 0.218" 0.218" 4500 Floating piston 0.125" 0.218" 0.35" 2300 The dimensions a, a i and especially b increase somewhat with the diameters of the cylinders (that is with the caliber of gun) as well as with the fluid pressure. Let C^ and C^ be the packing friction constants for the stuffing box. C" and C" be the packing friction constants for the recoil piston 4- U n to D ^R ^~ V ** ^O / \ V W / P tnen Kg v p i t x a = C o +Q p i 2 P Therefore, the recoil reaction, becomes, for any pressure p t (K+w r sin0)l pA = (C +C p) 1+W n d b If we assume the maximum recoil pressure p max corresponding to maximum elevation # roax we have C. d[.05b+.09(a+ where p n = k p max , k being obtained from the previous table. Therefore the effective recoil piston area, becomes (K+ r sin0_ ax )l a k n d[ - 05 592 - . (1*2 n d b )p max In general p max = 4500 Ibs. per sq.in. but as we shall see in 6, with guns of low elevation and with reasonable horizontal stability, the max. pressure may be necessarily smaller than the pack- ing limit pressure of 4500 Ibs. per sq. inch. The assumed max. pressure for calculation of packing friction and in (5) the recuperator reaction is at this stage a questicn of experience. The guide friction when the gun is in battery becomes, 2 n B r d b R = l+2nr where B v = p a A-(R- s + R p ) i.e. the tension of the rod in battery r = distance down to mean friction line 1 = distance between clip reaction in battery. n = coefficient of friction = 0.15 to 0.2 b = distance from center of gravity of re- coiling parts to line of action of B y Further the value of Rg^R in battery becomes, To compute the drop of pressure across the float- ing piston friction, we have Rf *+C"'+C ' approx. while p a ~P a * hence 2C"' + (C Pa = h : , _ (2A a -C g )p a -2C t C +2A . which gives the air pressure in terms of the re- cuperator pressure (for recoil computations) or 593 the recuperator pressure in terms of the air pres- sure in terms of the air pressure (for counter re- coil computations). In a preliminary layout, we are not greatly in error in assuming either Pa-Pa = approx. drop due or to floating piston. "a (5) The recuperator reaction required to hold the gun in battery To ensure a sufficient margin for the holding of the gun in battery and overcoming the friction in battery, an excess of 20* to 30* is used over the minimum recoil reaction, hence 2nK y d b _ R g = - ; k = 1.2 to 1.3,n0.15 1+2 nr K v and B 8 +R p =C,+ C, where C t =0. 15* (d r +d) (d p +d)[.05b + .09(a+ T a C =0.73n(d r +d)l .05b t and for a trial value, Pmax = 4500 Ibs. per sq.in p s A = - 4500 d * diam. of recoil piston d r =diam. of piston rod dfc 38 distance down from center of gravity of re- coiling parts to line of action of K v Substituting in (1), we have, 2nk K v d b IT V ~ - +C t +C 2 ~ 1+2 nr A 594 kW r smn0_ ax +C bence K = - - - Z* i - (2) 2nk d C l+2nr A (6) Limitations of Recuperator Dimensions: In the design of a recuperator layout, we must consider the proper ratio of area of re- cuperator cylinder to effective area of piston, the limitations of areas based on this ratio and on the maximum allowable packing pressure in the recoil cylinder as well as on the difference between the horizontal pull and recuperator re- action at maximum elevation. If now, w n = max. area of orifice at horizontal re- coil in sq . in. A = effective area of recoil piston in sq.in. A a * area of recuperator cylinder or floating piston in sq. in. V = max. recoil velocity in ft. per sec. P n = horizontal pull in Ibs. p s = pull at maximum elevation in Ibs. K v = recuperator reaction at maximum elevation in Ibs. 1 K = - = reciprocal of orifice contraction 0.773 constant. w c = channel or port area at cross section beyond regulator valve in recuperator cylinder in sq. in. W s * area of channel section through diameter of regulator valve in sq. in. a = entrance area to regulator valve. d = diam. of regulator valve. r = ratio Floating Piston Area of Effective Area of Recoil Piston Now the maximum throttling area w^ i the limiting throttling area, since all constant port 595 or channel areas in the recuperator should bear given ratios with respect to this area and must be suf- ficiently large as compared with w h so that there is no appreciable throttling through them, or loss of head due to friction or acceleration. The following table gives ratios of channel or port areas in the recuperator with respect to the .maximum throttling area w^ and the area of the recuperator cylinder or floating piston. Model "h "c w e C = 1 a, w h *c C *T *a -. *>*-. 4. V "-M. 19O6 0. 38O 1.61 4. 235 0. 126 0. 608 4.7"M. 1918 0. 854 3.6-7 4. 3OO O. 207 O. 64O 3. 3"-M. 1918 0.267 1. 11 4. 160 0. 186 0. 470 75-M. 1916 0. 1*75 o. 7s 4. 47O O. 120 0.371 4. 3OO o. 160 From the above table, the following design constants will be used based on satisfactory layouts: C = = 4.3 1 w h Now = f = 0.16 C. c C . , *' a - 0373 r " Considering the throttling through the regulator orifice, we have 2 h KVv* (1) 175(p n -K v ) and substituting for w h =.0373 r A, we find 4.11 AV 2 that is r = 2.62 V Pb'Ki (2) Hence, the ratio of the recuperator area to ef- fective area of recoil piston varies as the square root of the effective area of the recoil piston, 596 and for minimum recuperator area we must have min- imum effective area of recoil piston. Hence the recuperator area always varies as the - power of the effective area of the recoil piston. Now for minimum neight of the recuperator forging it is important that the cylinder areas be made as small as possible. Therefore the re- coil piston area in general is limited by the maximum allowable pressure in the recoil cylinder consistent with the packing pressure limitations. If the packing limiting pressure is taken at 4500 Ibs. per sq. in., then A = -i- 4500 Substituting for A, we have .039V / - ^ Pb~ K v How r is limited by the following considerations: (a) In an ordinary layout, the initial volume of the air, may be represented as the sum of _, air column in the recuperator cylinder plus the air column in the air cylinder, that is where k t = initial air column length in re- cuperator length of recoil and k ~ air column length in air cylinder length of recoil k may be obtained from the fol- 597 lowing table: b X 1 b b '..x x b ' k b wax 4. 7"-M. 1918 40 28. 27 1. 41 o. 7o ; ; 4. 7"-M. 1906 7o 56.50 1. 240 0. 807 75m-. 1916 46 43*08 1. 07O 0.937 3. 3"-M. 1918 60 47.67 1.260 o.795 1. 246 0.811 Therefore e may assume K i =0.8 and k f will be assumed k a = 0.7, hence k t + k s = 1.5 The initial volume as shown in (7) may be represented in the recoil piston displacement and the ratio of the final and initial air pressures. Paf If o = and k = (1 to 1.41 use 1.3) then when m = 1.5 (6) = 3.73 . k - 1 2.0 3.0 = 2.42 - 1 1.75 - 1 Using a ratio of m = 1.5, and combining Eq. 4 and 5, we have 3.73 Ab=1.5A a b hence = r = 2.5 598 Paf If a lower value of m = - is used, then r > 2.5 Pai where as with a higher ratio, r < 2.5 If we continue increasing m for the minimum permissible value of r we soon arrive at the limitation where the maximum possible recoil of the floating piston limits the ratio r. since A a a min b max - A b - Then a min b max On the other hand to obtain a value of r m j n = 1.25, would require a high value of m, approx. a 3.0, and the temperature rise of the air would be excessive and very injurious to the floating pis- ton packing. Further, at horizontal recoil, where a stability slope for the total resistance to recoil is highly desirable, we have the minimum throttling drop due the small value of the pressure in the recoil cylinder at horizontal recoil. The peak effect of the throttling drop is thereby reduced, and since the pressure in the recoil cylinder is the throttling drop plus the recuperator pressure, a large increase in the final air pressure over the initial will overbalance the decrease in the throttling drop towards the end of recoil. There- fore, a large value of m = -& must result in a rise of the total p ai resistance to recoil towards the end of recoil which is entirely inconsistent with the re- quirements for horizontal stability If, therefore, at horizontal recoil, we limit the ratio m =s to that value which gives us a constant resistance throughout the recoil, we have neglecting the slight throttling drop at the end 599 of horizontal recoil. Substituting for w h .0373 r A K 0.773 and we have, 2 0.145(m-l)r 2 =6.87 - (8) Pai From the following table, knowing V and p a ^ and from the above equation (m-l)r a , we can readily obtain r or corresponding value of m. 3 CYLINDERS 2 CYLINDERS r r 2 r 2 6. 88 (m-1) V 2 Pai m-1 V* *ai 1.5 2. "75 .3273 1. 147 .3787 1.547 .5063 1.6 2. 6 .3*724 1.015 .3780 1. 345 . 5001 1.7 2. 89 . 4204 . 910 . 3824 1. 188 .4994 1. 8 3.24 . 4714 . 824 . 3884 1.064 . 5018 1.9 3.61 .5252 . 752 .3950 . 960 . 5041 2.0 4. OO .5819 . 694 . 4038 . 880 . 5120 2. 1 4. 41 . 6416 . 645 . 4138 . 810 .5197 2.2 4. 84 .7041 .599 . 4218 .750 .5280 2.3 5. 29 .76 9 6 . 560 . 4310 . 7oo .5287 2.4 5.16 .8380 .527 . 4416 .653 .54*72 2.5 6. 25 . 9093 .497 .4519 .612 .5565 As a check, we may obtain the ratio m con- sistent with horizontal stability from another point of view. At the end of recoil, we have, neglecting a small throttling drop, P a f* = Ph wnereas the initial reaction, we have, A s K hence Paf Ph m = - = which gives immediately the a * v maximum ratio tor m or the minimum ratio for r, consistent with the require- 600 nents for horizontal stability. Again the maximum value of m - depends n further upon the heating limitation (i. e. the permissible rise of tem- perature of the air in the recuperator forging). Since the question of heating depends upon the various factors, as radiation through the cylinder walls, the frequency of firing, etc. we must as- sume by experience for the given type of gun, the maximum allowable temperature use during the com- pression of the recuperator air. Thus, using a ratio m = 2, we have k-l T Paf k 0.23 ( ) = 2 Assuming a mean temperature T m Pai at 25 Centigrade, we find T - 298 * 2'* 8 = 349,(k=1.3) therefore, the rise of temperature during a recoil stroke becomes, T -T n =51C or 92F. This rise of temperature is not excessive. If the rapidity of fire is great the mean temperature will rise. The quantity of heat generated is the work done on the air divided by the mechanical equivalent of heat. Now if the mean temperature has risen to its constant maximum value, then the heat generated during the firing stroke, must be dissipated by radiation through the cylinder walls during the period of loading, the temperature gradient varying, decreasing during the process of radiation through the cylinder walls. Thus we see we have two aspects for the maximum ratio of m and the corresponding minimum value of r: (1) To maintain at best, a constant resistance to recoil throughout the recoil at horizontal elevation, rather than a rise in the over- turning force at the end of recoil which would be entirely opposite 601 to the requirements for the proper stability slope at horizontal recoil. This limitation for r is of special importance when p n ~K y is small, we have r fflin determined by the equation, (m-l)r = 6.87 where we may obtain d r ai for various values of (m-l)r* from the previous table. (2) On the other hand, when Ph~K v is large as with guns of low elevat- ion and where we have a good margin of stability, the peak effect in the throttling becomes larger or the ratio Pv> IL is larger, therefore, a higher "v value of m can be used. In such a case we become limited by the rise in temperature of the air during the firing. (a) When (2) becomes the limitation we may use a higher value of m and therefore a lower value of r. (b) The expansion of the oil varies considerably with temperature, different viscosities, the rapidity or frequency and continuity of fire, etc. Therefore the floating piston will have various initial displacements, resulting in different initial air pressures and most important in unsatisfactory functioning of the buffer on counter re- coil, the buffer action start- ing at various different points 602 on counter recoil, unless the ratio is made sufficiently large, since with a large bulk of oil, temperature changes and consequent expansion is less. (c) Due to the wear of the pack- ing on the floating piston it has been customary to limit the maximum velocity of the float- ing piston to not over 12 ft. per sec. though it is believed that the packing may be designed to withstand a surface velocity of 20 ft. per sec. Therefore, in conclusion, from a consideration of (a), (b) and (c) r min ordinarily should be limited to: r min =1.3 to 2.0 In difficult designs, however, the proper minimum value of r should be determined more from a consideration of aspects (1) and (2) in (a) rather than (b ) and (c). Limitations for the maximum value of r: For very large values of r, slight changes in the quantity of oil due to leakage, will have a marked effect on the relative initial positions of the floating piston and recoil piston. Further it would be difficult to gauge slight variations in the quantity of oil due to the relatively small motion of tbe floating piston which moves the gauge rod . But most important from a point of economy in the weight of recuperator forgings and very often in a satisfactory layout, a too large value of r becomes prohibitive. fle limit the max. value of r to, r max =3.5 Rence the design limitations for r becomes for an ordinary layout with 3 cylinders r =>1.8 and r-^3.5. When only two cylinders are used we have 603 tbe following considerations: (1) If the length of the recuperator air cylinder has the same total length as the recoil cylinder, we x have k i =0.8 and k a *0, hence k A=0.8 A hence r = 0.8 i Ab=0.8 A b hence r . m k - 1 m k - 1 If m = 2 a heating limit on the ratio of com- pression, we have 1 k - - = 2.42 for k = 1.3 I m* - 1 and 2.42 r - = 3.025 0.8 On the other hand (in consideration of a constant recoil reaction for stability at horizontal elevation) if we decrease m to m * 1.3, we have 3.73 r * - 4.67 which gives a very o bulky recuperator with too snail relative displacement of the floating piston as compared with the recoil displacement. Therefore, if only two cylinders are to be used and of tbe same length we are peculiarly limited by bulk and a small movement of the floating piston as compared with the recoil piston, on the one hand, while with a decrease in the ration r, the final air pressure is increased and overbalances the peak of the throttling plus the initial air, thereby giving a rise in the recoil reaction towards the end of recoil at horizontal elevation. Therefore as a compromise, if two cylinders must be used of the same length we may take r = 3.5 and m = 1.8 and r is to be considered 604 constant for this combination. Thus we see by the use of two cylinders only and of the same length, the ratio cylinders become excessive if aoderate compression ratios are maintained, whereas for moderate cylinder ratios we must maintain high compression ratios which cause undue heating and a rise of the recoil reaction at horizontal elevation. A more satisfactory combination for two cylinders only, is by use of longer recuperator cylinder than recoil cylinder. This is usually feasible especially for guns, where the tube is long. A satisfactory approach to three cylinders may be had by the use of a sufficient overhang of the recuperator cylinder. The air column in the overhang can be reasonably assumed at 0.5 the horizontal recoil. Therefore in the equation V Q =A a (k t +k 2 .)b . We may assume as before k t =0.8 k 0.5 2 = 1.3 hence k t +k s Hence with a ratio of pressure expansion in the recuperator m = 1.5, we have 3.73 Ab=1.3A a b hence A a = r = 2.87. By increasing the ratio of ex- pansion we may limit the minimum r to: *n)in = 2'5. From the above it is evident that though it is not feasible to use a recuperator air cylinder as long as the combined length of a separate recuperator and air cylinder, on the other hand the minimum ratio of r is greater than with an ordinary layout and with only two cylinders, the total weight of t"he forging may exceed that of these cylinders. Hence if two cylinders are to be tried in place of three, preliminary separate layouts for the two combinations should be worked out in view of minimum weight and 605 satisfactory layout before either plan is adopted. Therefore, in a design layout we start with which determines the Ph~ k v 'a ratio providing it falls within the limits A r min and r max* Hence the recoil area, becomes, A = 4600 Anti-aircraft Guns; Anti-aircraft guns are the most difficult to design without having excessive bulky re- cuperator forgings, P n -K v becomes small, since K y is larger to hold the gun in battery at max- imum elevation and p^ is small to satisfy stability at zero elevation, further p s is large, there- fore r is in general large, usually 3 or above. If r exceeds 3.5 using 3 cylinders, we must increase Pt]~K v either by reducing K y and then al- lowing the gun to return slower into battery at maximum elevation and with a smaller margin of excess battery reaction of the recuperator at maximum elevation or preferably increase p n at the slight sacrifice of stability at zero elevation, - in this case we have P h =K v +. 000912V 8 3.5 2 It will be rarely found, however that r ex- ceeds 3.5. Howitzers: With Howitzers, we again meet the condition of a large K v but since horizontal stability is not a consideration, p n may be relatively large, and therefore p n ~K v still may be maintained large. 606 r is generally medium or small at the sacri- fice of horizontal stability. Guns : With guns, r is in general small, usually from 2 to 3. If r decreases below, 2.0 to 1.8 using 3 cylinders, or 2.5 using 2 cylinders with a longer recuperator cylinder than tfce recoil, the effective area of the recoil piston is now determined by the formula: x A = 0.2425 (pu - K w ) and the maximum pres- v* _ sure in the recoil P g cylinder, becomes, p max = A Howitzers and Guns on Same Carriage: When Howitzers and guns are adopted for t"he same carriage, K y must be large for the howitzer and p h small for the horizontal stability of the gun. Therefore p n ~~K v is small, p s large and we meet exactly similar conditions as with anti- aircraft guns. Hence, for this combination, r is in general large, usually 3 or above. If r exceeds 3.5,pj, must be increased or K v decreased, these values being connected by the relation K V +.000912V (7) Calculation for air pressures and volumes: From (5) we have for the initial recuperator reaction K v - k W r sin ^m a X +C i where k Pmax ma y at 4500 in. a +.09(a+ s) B = 1.2 to 1.3 be assumed Ibs. per sq. JPmax 1 C 2nkK v d b c z A +d)[.05b ^O.lSn nr WP C =0.73n(d_+d)[.05b + .09(a + )] 2 PS A = = effective area of recoil piston 4500 a, a t and b are contact lengths of the pack- ing in inches. d^- distance down from center of gravity of recoiling parts to line of action of K v in inches. 1 = distance between clip reactions in inches n = 0.15 r = distance down from center of gravity of recoiling parts to mean friction line in inches. For guns of low elevation and reasonable stability where r falls below, r we have A = 0.2425 ( p - Ky ) and V Id = /0.7854A+dr 2 r A = 0.2425 (p h -K v ) and the maximum pressure ^* in the recoil cylinder is thereby reduced to: PS Pmax = * and the area of A the recuperator cylinder becomes, A a =r min A. With an ordinary layout using 3 cylinders, r min =2 and with 2 cylinders and an overhang, r m j n =2.5. Knowing A a , we have for the inside diameter of the re- cuperator, D a : 0.7854 To determine the diameter of recoil cylinder, we must know the area of the piston rod,A r . Then /A+A r D * / where A_ is determined as follows: 0.7854 r 608 If 6 is the total hydraulic braking including the joint frictions at the stuffing box and re- coil piston, we have B+Rg=K+W r sin# where 2nd b )=K+W r sin l+2nr The maximum stress in the rod is at a section at the lug, at maximum acceleration of the recoil- ing parts and at maximum elevation. If T = the tension in the rod at the lug L K+W r sin0 ffiax 2nd b 1+ (1- =-)+ Pb r- l+2nr where W e weight of rod and recoil piston p b = total maximum powder pressure on base of breech. If f Dax is the maximum allowable working fibre stress in the rod, we have f * max and A a * rA. Now with guns of low elevation, K y is snail and p h relatively large, hence the difference p n ~K v is large and p s is small. There- fore r becomes small. In (7) tables and a chart has been con- structed giving values of m and r for different air column lengths, the air columns being expressed 609 in terns of a ratio of the length of air column divided by the length of recoil, that is j - where 1 = length of ai* column, b b = length of recoil. A maximum limitation of m, based on a moderate temperature rise of the air during the recoil and a constant reaction throughout the recoil at horizontal elevation (i. e. no increase of the recoil reaction at the end of recoil, ) will be taken at 1.8. Evidently for different air column lengths, we will have different minimum values of r, corresponding to a waximum value of m = 1.8. The longer the air column lengths, the lower the ratio r. If r falls below the minimum allowable value of r (i.e. the r corresponding to m = 1.8) for a given air column length, r becomes a constant, and the area of the recoil piston must be in- creased according to the formula*- then K v p a ^ = -j- Ibs. per sq.in. initial recuperator pres- sure intensity. Hence the initial air pressure becomes, Pai s Pai + ' a * d Paf A a t C|| l = 1.46nd a [.05b + .09(a + y)] Paf Next to determine the proper ratio of m = - Pai we must consider the following: The initial volume V o is expressed by either memoer of the equation, 610 A g l where k 1.3, b = length of recoil 1 length of air column, reduced to an equivalent cross section A. Since r * , we have A r - rj, where j b The following tables give the relation of o and r for various values of j. Pax Pai 0.8 r 1.3 .77 m r 1. 2 9.575 1.4 5.476 1.6 4. 116 1.8 3. 442 2.0 3.002 2. 2 2.746 2. 4 2. 546 2.6 2. 401 2. 8 2. 282 3.0 2. 188 3.2 2. Ill 3.4 2.047 3.5 2.018 611 Values of m and r for j = 1.1 1. lr 1. lr 1. lr r 1. lr 1. lr log... Klog m -1 1. lr-1 1. lr-1 1. lr-1 1. 8 1. 98 .98 2.020 . 30535 .39696 . 2. 494 2. 2. 2 1. 2 1.833 . 26316 .34211 2. 198 2. 2 2.42 1. 42 1. 704 .23147 . 30091 1.999 2.4 2.64 1. 64 1. 61 .20603 . 26888 1.857 2.6 2.86 1.86 1.54 . 18686 . 24291 1.750 2. 8 3. 08 2. 08 1. 48 . 17056 . 22173 1.666 3.2 3-52 2. 2 1. 4 . 14520 .18876 1. 544 3.4 3- "74 2. 74 1.36 .13513 . 17567 1.499 3.5 3.85 2. 85 1.35 . 13066 . 16986 1.4^9 Values of m and r for j 1.3 1.3' 1. 3r 1.3r r 1.3' 1. 3r-i Klog 1.3'-1 1.3*-1 1. 3r-l 1. 8 2.34 1.34 1. 746 . 24204 .31465 2.064 2.0 2. 60 1. 60 1. 625 . 21085 .27411 1. 880 2. 2 2.86 1.86 1. 538 . 18696 .24305 1. 850 2. 4 3. 12 2. 12 1. 472 .16791 . 21828 1.653 2.6 3.38 2. 38 1. 420 . 15229 . 19798 1.577 2. 8 3.64 2.64 1.379 . 13956 . 18143 1.519 3.0 3.90 2. 90 1.345 . 12972 .16734 1.470 3-2 4.16 3.16 1. 316 . 11926 . 15504 1.429 3. 4 4. 42 3.42 1. 292 . 11126 . 14464 1.395 3.5 4.55 3-55 1. 282 . 10789 . 14026 1. 381 -sol *i fcfl 612 Values of n and r for j = 1.5 l.Sr l.Sr l.Sr r l.Sr l.Sr 1 Klog - m 1. 8 2. 70 i.7o 1. 588 . 20085 . 26111 1. 824 2. 3. oo 2. 00 1. 50O .17609 . 22892 1. 694 2. 2 3. 30 2. 30 1.435 . 15685 . 20391 1. 599 2. 4 3. 60 2. 60 1.385 . 14145 . 18389 1.527 2.6 3- 90 1.345 1.345 . 12872 .16734 1. 470 2. 8 4. 2O 3. 20 1.313 . 11826 . 15374 1. 425 3.0 4. 50 3.50 1. 286 . 10 9 2 4 . 14201 1.387 3.2 4. 8O 3. 80 1.263 . 10140 . 13182 1.355 3.4 5. 10 4. 10 1. 244 . 09482 . 12327 1. 328 3. 5 5. 25 4. 25 1.235 .09167 . H917 1. 316 Values of m and r for j = 1.7 1 *7 1.7r 1.7r 1.7r r i . /r 1.7r-l log m 1. 8 3.06 2. 06 1. 485 . 17184 . 22339 1.672 2. 3.4 2. 4 1.4166 . 15124 . 196612 1. 572 2. 2 3-74 2. 74 1.3649 . 13510 .17563 1. 498 2. 4 4.O8 3.08 1. 3246 . 122084 . I587o 1.441 2. 6 4. 42 3-42 1. 2923 . 11139 . 144807 1.395 2.8 4.76 3-76 1.2659 . 102O5 . 13266 1.357 3.2 5. 44 4. 44 1. 22522 . O8820 . 11466 1. 302 3.4 5. 7s 4. 78 1. 2092 . 08249 . 107237 1. 280 3. 5 5.95 4.95 1. 2O20 .07997 . 103961 1. 2704 The values in these tables have been plotted, with the chart, fig. (12). The ordinates in this chart give values of n corresponding to values of r in the abscissa. Each curve represents the relation of m and r for a satisfactory layout, with a given air column, the air column length being expressed as a ratio with respect to the length of recoil. The air column lengths are taken at 0.8b, l.lb, 1.3b, 1.5b and 1.7b where b = length of recoil Values of m and r for air column lengths inter- mediate between these values, may be easily ob- 613 -xo'r'* : U**v.U ,? at 614 tained by interpolation. Solving for the proper value of r, and as- suming an air column length, NO immediately ob- tain ID and therefore p af since p ai is now known. To prevent a rise of the recoil reaction during the recoil at horizontal displacement as well as to minimize the temperature rise, during the recoil, we will limit m to 1.8. Therefore r is definitely limited for various air column lengths. Its upper limit is more or less arbitrary, it being desirable to prevent a too bulky forging and obtain minimum weight. The upper limit of r will be assumed at r * 3.5. Thus, when we have but two cylinders of the same length where the air column is somewhat shorter than the length of recoil * 0.8b, we find r very definitely limited to a constant value 3.5. (8) Strength of Cylinders: Strength of cylinders should be based on maximum pressures. As shown in Chapter IV. 3 D t = D o where p t * - x elastic limit of the Pt~P material used (Ibs. per sq.in) p- maximum pressure in cylinder (Ibs/sq in.) * Pmax usua lly 4500 Ibs.per sq.in. in recoil cylinder * p a f final recuperator pressure in recuperator cylinder D =outside minimum diameter in inches. D - inside diameter which is given in inches, Thickness between cylinders should be P d +Paf d a w * where w = minimum allowable tnick- 615 ness between cylinders (inches) d = diara. of recoil cylinder in inches d a diam. of air cylinder in inches (9 ) - Calculation of maximum and minimum throttling areas. Since all port areas are constant multiples of the maximum throttling area, the exact deviation of this area is of prime importance. From (6) we find, for the throttling through the regulator orifice, "h " TTT7 - Z~~\ (max. throttling area) I/O vp n ~K v ) V* A 3 TT* w 8 K A v (min. throttling area) 175(p a -K v ) where w h - the max. throttling area usually at horizontal recoil (sq.in) w s the rain, throttling area at max. elevation (sq.in) A = the effective area of the recoil piston (sq.in) V - the max. restrained velocity = 0.9 V_ approx. K = - the throttling constant 0.773 pj, the minimum pull, usually at horizontal recoil (in Ibs) p s = the maximum pull, at maximum elevation. (10) Layout of Recuperator Forging: In the layout of a recuperator forging, we must decide as to the arrangements of cylinders. Depending upon the value of "r"* _^a_ , we have three possible arrangements: 4 o jr. pi'SKfT'O r ' s< j 616 (1) Three cylinders, the two re- cuperator cylinders symmetrical with the recoil cylinder. (2) Two cylinders, the recuperator cylinder having an overhang with respect to the recoil cylinder. (3) Two cylinders, the recoil and re- cuperator cylinder being of the same overall length. Paf From the chart giving values of n = for values of r = *a for different air Pai colunn lengths, we see that the values of "r" for the curves giving lengths of various air columns are limited on the one hand by the maximum value of m = 1.8 consistent with stability at zero degrees elevation and normal use of temperature in the re- cuperator, thus giving the various mini Bum limit- ing values of "r" for vari ous air columns, while on the other hand the maximum allowable value of r = 3.5 depends upon proper counter recoil function- ing and layout considerations. If now r is obtained by the formula, r * .0309 The possible lengths of air columns consistent with the limitations are determined. If r from the above equation is low, then we must have longer air column lengths and there- fore usually three cylinders, whereas if r is large, short air column lengths are possible and two cylinders may be used. With arrangement (3), r becomes practically constant and unless r from the above equation falls in the neighborhood of 3,5, it will be necessary to increase the effective area of the recoil cylinder with a consequent larger recoil cylinder. 617 Having decided upon the arrangement and number of cylinders from a consideration of the proper air column consistent with "r" we have now to obtain the exterior dimensions of the forging. Exterior Dimensions: The primary exterior dimensions of importance are: (1) A cross section of the recuperator, giving the location of the piston rod with respect to the center line of the bore, the axis of the several cylinders, and the position of the guides, thus determining the external 8-iv . } - - if ^*v -> - - ' T; contour of the cross section of the d V,*n lu forging. (2) A longitudinal section of the recuperator, giving the overall length, location of the trunnions, elevating arc, etc. In a satisfactory exterior layout, the follow- ing points must be observed: (a) The center of gravity of the recoil parts should be made in a vertical plane through the axis of the bore and at a minimum perpendicular distance below the axis of the bore con- sistent with a satisfactory layout. (b) The center line of the en- trance channel to the regulator valve, (that is for the passage- way between the recoil and re- cuperator cylinders) should pass through the center of the recuperator cylinder. Preferably the center line of the entrance channel should be in a horizontal 618 (c) plane. If < the connecting channel cross section, and D the diameter of the recoil cylinder, the distance between the center line of the recuperator cylinder and recoil cylinder must not exceed D To meet condition (a) the recoil axis is usually nearer to the axis of the bore. To overall lengths of the recuperator forging nay be estimated roughly from the following table: a a inches. 2 4. 7"-M 19O6 94" "TO" 4. 7 "-. 1918 68.75" 40 3.3'-w.l913 86* 60' 75 /-M.19l6 72.83" 46' 1.52 Therefore ordinarily the total length of re- cuperator forging trill be taken at 1.5 the length of max. recoil. It should be shorter if practicable. (e) Without a balancing gear, for guns of moderate elevation, the trunnions should be located in the horizontal direction at the center of gravity of the tipping parts plus one-half weight of projectile and charge 619 when the gun is in battery. More or less error in the location of the trunnions as respects the center of gravity of the tipping parts in the vertical plane per- pendicular to the axis of the bore will not effect the balance, unless the angle of elevation is considerable, and the center of gravity of the tipping parts is considerably above or below the trunnions. Therefore, in order to prevent % : T- j , r a reversal of the reaction on the elevating arc and pinion during recoil and counter re- coil, it is highly desirable with guns of moderate elevation to locate the trunnions on or below the center of gravity of the recoiling parts which are usually below the axis of the bore. When a balancing gear is introduced, as is sometimes necessary when the gun fires at high elevation, the trunnions are placed axially or in a longitudinal direction, farther to the rear in order to have as long a recoil as possible at max. elevation. Further with a proper design of the balancing gear location of the trunnions with respect to the center of gravity of the tipping parts in a direction perpendicular to the axis of the bore is no longer so restricted except that in order to avoid reversal of stresses on the elevating arc it is desirable to locate the trunnions slightly below the center of gravity of the re- coiling parts, but the distance must be quite small or the arc reaction will become large. 620 In gone designs it may be necessary to locate the center of gravity of the recoiling parts above the bore, and the ponder pressure couple will then be in the opposite direction. If P^e is the powder pressure couple, and K the resistance to recoil and S the distance down from the axis of the bore to the center of the trunnions, in order that there be no reversal of stress on the elevating arc, we oust have (P b -K)e K(S+e) = > P b e hence S * > In K determining the final values of e and S, the weight components, out of battery and conditions existing in counter recoil must be considered. (f) With guns above 155 m/m, two separate recoil systems *% L d "' symmetrically placed above and below the gun should be used. The gun should recoil in a sleeve and the trunnions should be located slightly below the axis of the bore. Interior Dimensions; The primary interior dimensions of importance, are: (1) The port area or channel leading from the regulator towards the floating piston in the recuperator cylinder should bear a constant ratio to the maximum opening of the valve which occurs for minimum pull, usually at horizontal elevation. If c the constant channel area from the regulator valve w h the max. recoil orifice. Then w c = 4.3 w h (2) The area of the channel or port connecting the recoil and recuperator 621 cylinder, w a should have the fol- lowing relation with respect to wjj, that is, w a =* 3.5 to 4.3 w b (3) The entrance channel to the regulator valve froa the recoil cylinder a, which is also the area at the base of the regulator valve, should be: a = k h where the limits of h are 2.3 to 3.5 If we pass a cross section of the recuperator through the center of the regulator valve, the channel area on either side of the valve, that is w c -(the vertical section through the axis of the valve normal to the recuperator axis enclosed within the area w c )= w^ and w -> c If h represents the depth of the section w c and d a the diameter of the regulator valve at its base, we have roughly, w c - d a h c = 0.5 w c = 0.55 w c approx . Hence "c w b "h d a ' 0.46 = 1.935 and a = 2.94 b c h c h* Mow in a suitable layout h c - 0.2 D a where D a the diameter of the recuperator cylinder, hence -& a - 73.5 a (4) The length of the buffer chamber s c: *;;-. - : i - ' tflk _n. ^ ,. is based on a consideration of -.: counter recoil. If d fa = the recoil length during the buffer action d = the length of the buffer From a consideration of counter recoil, 622 7 r V* d b (0.15 w r +R p ) If V - 3.5 ft. per sec. 0.8 as a maximum value, we 0.238W r have d b * - 0.15W r +R p * nd 0.238W r A d b ' * ( ) -T * min * length of buffer where R p -0.15 K (d+d r )[ .05* + .09(a and p max =4500 usually. The buffer chamber should be made d=1.2 to 1.3 d^ (11) Regulator Dimensions: Referring to fig. (11) let, a = area at base of regulator valve (sq.in) a t = area of upper and lower valve stem (sq. in) *< d a diameter of regulator valve at base (in) ^ t a diameter of regulator valve at stem (in) c - effective circumference at base of valve (in) wfi From (10) we find that, a - 73.5 and D *a h d a * 9.675 -- where w h maximum throttling opening (sq.in) D a diameter of recuperator (sq.in) Now d a *0.6d a approx. and a t 0.7854d| =0.2825d| hence a t 0.36a. The opening of the valve is the effective lift multiplied by the effective circumference of the valve at the valve seat. Extension guides or "flaps" to ensure proper seating of these valves reduce the effective circumference at the valve seat. It is customary to use three flaps of a circumferent- ial length each, equal to the arc of 60 angle, de- creased by two millimeters on either side, making the linear length of flap atthe circumference equal to the arc of 60 minus 4 millimeters. Hence 623 nd a 12 c * - + " 2 25.4 -0.3925d a +0.4725 In the throttling or lower valve and its stem equalizing pressure ports should be bored within. In the stem itself, the inside diameter or diameter of the vertical port should be dl 0.5 to 0.6 d. t *t Pour equalizing holes just above the seat in the regulator valve, in a horizontal plane, meet- ing at a common opening at the center should be inserted. From the center opening there should be a very small vertical opening leading to the recoil cylinder, this acting as a pressure equalizer between the recoil and recuperator cylinders. The opening, however, should be made negligible as compared with the throttling open- ing and small as compared with the counter recoil constant orifice. (12) Reactions on Regulator Valve: Let P b = reaction of Belleville washer on regulator valve (in Ibs) < reaction of spiral spring on regulator (in Ibs) p = pressure in recoil cylinder (Ibs/sq. in) p a = pressure in recuperator (Ibs/sq.in) a = area at base of valve (sq.in) a - area of valve stem (sq.in.) h * lift of valve from initial opening (in) h =lift of valve from seat of initial opening (in) c = effective circumference at base of valve in inches. S b = spring constant of Belleville washer (Ibs/in) S g * spring constant of spiral springs (Ibs/ in) 624 hjj = initial compression of Belleville washer at initial opening (in) h ^initial compression of spiral spring at initial opening (in) Then, at short recoil, or intermediate recoil, we have pa -p^(a-a t )=F b +R s (approx ) hence (p-p a )a + Pa a t sR b +R s and at long recoil, we have (p-p a )a=R s (approx) < 2 ) : . / Further, we have the following lifts of the , valve, K A V (3) At short recoil I loom U) At long recoil where p s and p h are the values of p at short and long recoil respectively, 0.773 The spiral spring should be designed on the following basis: (1) The maximum compression should be taken at from 2/3 to 3/4 the solid load of the spring. (2) The initial compression should be taken at from 1/4 to 1/3 the solid load on the spring. Hence, using the maximum limits, the compression from free to solid height h fs =2b (5) and therefore ,f n S3 h = - n (6) where f 8 3 max. allowable torsional fibre stress (Ibs/sq.in) (Usually = 120,000 Ibs.per sq.in) 625 D s = diam. of helix in inches. d g = diam. of wire N = torsional modulus (taken at 12,000,000 Ibs/sq.in) n = number of coils of the spring Proa previous design layouts, the total height of spring column, at assembled height should not exceed p a inches. Hence the solid height H 2 . becomes, - - h = H Q but H =d g (n+l) hence 0.5D a -h d = - r^ (7) n+1 Combining (6) and (7) we have nfD 0.5D-h n (8) 2N" h n+1 The load at assembled height = - R g 9 hence 3nf,d| R = - ^ (9) 32D Combining with (7), we have 3nf_ 0.5D a -h and with, nf a Dn 0.5D -h (Eq.8) we may determine 2Nh n + 1 , _. n and D. The solution may t>e simplified by assuming -, as a first approximation, we u art* lo aoUo 0.5D a -h 0.4D a ^^ = 0+1 n then have f s D t R S .01885 0.8 D a 626 Solving for D, d,and n, we have 2.62 / : inches If we assume N=10,500, 000 Ibs/sq.in. f s = 120,000 Ibs/sq.in. then 0.216 If D is too great for a satisfactory layout, we may increase the height of the spring column slightly or let the maximum working load on the spring move closely approach the load at maximum compression. Solving for the diameter of wire "d" and the number of coils "n", we have s and if f s =120,000 Ibs per sq. in. then d = .0305 y R,,D S o o Test pressures are usually at double the service pressure, hence the material will be strained up to 3/4 the elastic limit. (13) Design of Cam Mechanism and Layout; Briefly, the action of the cam is to control the motion of the upper valve stem which reacts against the Belleville washers. At long recoil the valve displacement (i . e. the displacement of tbe unoer valve stem) is sufficient so that the lower valve stem, is never brought into contact with the upper stem, and the lower stem is controlled entirely by the spiral spring. At an intermediate 627 recoil, the lower stem is brought ultimately in contact with the upper stem and the valve is con- trolled by the compound characteristics of the two stems. As the upper stem initial position is brought closer and closer to the lower valve stem, the valve opening depends more on the characteristics load deflection slope of the Belleville washers. Finally at short recoil, where the upper valve stem is brought into initial contact with the lower valve stem and the displacement of the cam is zero, the valve opening depends practically on the Belleville characteristics alone, the effect of the spiral spring being negligible. It is to be noted that the throttling at intermediate recoils approximates that if a constant orifice, with however the characteristic p ea k effect in the braking with a constant orifice eliminated. The throttling, there- fore depends upon the displacement of the valve and the characteristic load deflection curves of the Belleville and spiral springs. Let g = ratio of cam movement to valve movement (usually taken at 5) X = distance valve should lift to engage Bellevilles (in) h s - initial compression of spiral springs (in) b o = clearance of valve (in) h = lift of valve (in) h b 3 initial compression of the Bellevilles (in) S b 3 change in load per unit deflection of the Belleville washers, i.e. the Belleville spring characteristic (Ibs) S g * change in load per unit deflection of the spiral spring, i.e. the spiral spring characteristic (Ibs) Then at an intermediate recoil, the reaction of the spiral spring, becomes, R g =S s h s +S s (h+h o )(lbs) The reaction of the Belleville washers becomes R b =S b b b +S b (h+h -X) (Ibs) while the hydraulic re- action becomes, (p~P a i)a + P a i a t (Ibs) 628 where p = the intensity of pressure in the recoil brake cylinder (Ibs/sq.in) p a i = the intensity of pressure in the re- cuperator (Ibs/sq.in) a = area at base of valve (sq.in) a t - area of valve stem (sq.in) Then for equilibrium of the valve, S s (h s +h+h )+S b (h b +h+h -X)-[ (p-p a i>a+Pai a t}* Therefore, for the distance of valve lift to engage Bellevilles, is ]> J X = JsgOvho+hJ+Sba^+bo+hM (p-p ai )a+p ai a S b L The variation of the length of recoil against ele- vation may be made in any arbitrary way, but, how- ever, the following method is usually employed. In general, assume the le ngth of recoil that of horizontal recoil from Q to # t degrees, (usually from to 20 elevation), then, decrease the re- coil proportionally with the elevation, (i.e. from 20 to max. elevation, the recoil length decreases uniformly to short recoil at maximum elevation). Thus if, b = length of an intermediate recoil * corresponding angle of elevation b h = length of recoil at horizontal elevation b s = length of recoil at maximum elevation Of Q- maximum elevation 0^= initial elevation where the recoil is shortened . , then b b b = - ?.-0)+b s (ft) 0m-<*i The resistance to recoil corresponding to the length of recoil "b" is given by: K = 1.03[ 2g uV f b+ (.096+. 0003d) a u 629 where * - weight of projectile (Ibs) W weight of powder charge (Ibs) * W r = weight of recoiling parts (Ibs) u = travel up bore (inches) d = diam. of bore (in) v * muzzle velocity (ft/sec) Vf* max. free velocity of recoil (ft/sec) and *v+W4700 V f - (ft/sec) * r 0.47W r V| For a rough approximation, K = (Ibs) gb The required recoil braking is given by (K+W r sin B = - R n or B = - - -- R_ 2ue b l+2ue b approximately where 1 = distance between guide clips (in) e b 3 distance from center line of bore to center line of brake cylinder (in) r = aean distance to guide contact (in) R = brake cylinder packing friction (lb) For the lift of the valve, we have a .098A*V,. V.in) where A = the ef- fective area of recoil piston (sq.in) K v = recuperator reaction (Ibs) V r = velocity of retarded recoil, about 0.9Vf (ft/sec) c = effective circumference of lower stem (in) we may also express the lift in terms of the pressures, then >098A VF h * - Un) where p = - a the pressure intensity in the brake A cylinder (Ibs/sq.in) 630 SPRING* & CAM MOTION (x) DIAGRAM Fig. 13 631 K y P a i= - = the initial recuperator pressure intensity (Ibs/sq.in) (14) Counter Recoil Design: The function of the counter recoil buffer is to reduce the pressure in the recoil cylinder to a very low value practically zero. The recoil- ing parts are therefore Drought to re^st by the combined packing and guide friction in a displace- ment corresponding to the buffer length in the recuperator cylinder. For a preliminary design layout, the entrance velocity into the buffer may be taken at a counter recoil velocity of 1 meter or 3.28 ft/sec, but preferably less than this. To allow for a margin in variation of counter re- coil friction the buffer displacement will be re- duced in the counter recoil to 0.7 its actual value. Then, we have 0.7d b ( .l5W r +Rp-)= ^-^ 3728* hence 0.238W r The corresponding displacement of the buffer in the recuperator, is A db where A = effective area of recoil piston *a A ft cross section area of recuperator cylinder The length of the buffer rod will be male about 20* greater. Hence for the length of the buffer rod, we have 0.238W- A The length of buffer chamber is usually constructed from 20 to 30* greater than the buffer rod, hence d - 1.2 to 1.3 d b ' (ft) for length of buffer chamber 632 The ID ax i BUB allowable counter recoil velocity at borisontal elevation should not exceed 3.5 ft/sec. The counter recoil velocity for a satisfactory design ranges from 2.5 to 3.5 ft/sec. The velocity used in counter recoil should be such that with the expression g 0.7 i b <0.16I r .B{>. ^ 55=15.. d b ranges from 1/4 to 1/3 the short recoil b g The packing friction for the recoil may be expressed by the relation, R -C^+C^p (Ibs) where p = Ibs/sq.in. in the recoil cylinder. On counter recoil during the buffer action p = approx. hence R'C t approx. Now C t is that part of the packing friction due to the Belleville compression of the packing and is designed for the maximum recoil pressure p aax (Ibs/sq.in) If D r = outside diameter of packing ring, (in) d r = diameter of rod (in) a * depth of silver flange of packing (in) a 1 depth of outer silver flange (in) b= packing contact (in) then t RC t * R(D r +d r )[.05b + .09(a+ )]0.15 p >ax (Ibs) The guide friction on counter recoil may be taken at RJ-0.15 to 0.2 W r (Ibs) For the total recoil friction, we have Constant Orifice Opening: at max, elevation, * A V (sq.in) 13.2/p a A-R f -W r sinIa - 0.41 ft. Constant of stability (assumed) Overturning moment Stability moment Horizontal distance from spade point to line of 2700x81+300x34 action of W 3 (ft) l g = 30QO ' = 6.35 ft. S = angle of stability 20 d = moment arm of over- turning force = h t cos ef+ds-1 sin ax 3.74 ft used b h sax.3.75 * 3.74 ft APPROXIMATE DimiBIOi OF RICOPIRATOH FOROIB08: Max. resistance to re- coiKat max. elevation) (Ibs) K. Min. resistance to re- coil (at horizontal elev.Mlbs) Max.pull (nax.elev. ) Ubs) P,-K 8 + r sin 0-2R 2R*W r sin m (approx) 0.46 1260 2.6 32.2 18700 Iba. 51.5 0.47 1260 3.75 32.2 13100 Ibs. 18700 Ibs. 51.5 Min.pulKO 8 elev.) (Ibs) P b -K h -0.3W r * Initial recuperator reaction (Ibs) K y - 1.3W r (sin0 m +0.3 cos B ) 1.3"1260(. 9848*. 05"0. 1736) = 1700 637 Ratio of recuperator cylinder area Effective area of re- coil piston A a r S -A = 18700 = 0.039x46.35 12730-1700 2.35 From chart - assume air column = 1.36 r in 2 -5 Total weight of recoil piston and rods (Ibs) 30 Ibs Effective area of re coil piston (sq.in) If r > r min A. = 0.243 ff Corresponding max. pressure (Ibs/sq.in) PS P.ax - = Approx .max. tens! on rods at horizontal (Ibs) 18700+1260x0.3420+ .30 24000 Xl3 ' 4 27000 Ibs. Assumed max. fibre stress (Ibs/sq.in) max I elastic limit 60,000 = 40,000 Ibs/sq.in 638 Area of recoil rod (sq.in) max Diani. of recoil rod d r 1.127 Total area of recoil cylinder (sq.in) A r A+A a * Inside diam. of recoil cylinder (inches) D r =1.127 Area of recuperator cylinder (sq.in) A. = rA * 27000 40000 .676 1.127 /.676 = .925 in, use 1 inch. 4.16+0.676 = 4.836 sq in. 1.127 /4.8S6 2.48 inches. 2.5 x 4.16 * 10.40 sq, in. Inside diam. of re- cuperator cylinder(in) D= 1.127 1.127 /10.40 =3.63 inches . COMPUTATION OP PACKING FRICTIONS, Recoil friction Width of leather contact of packing (assumed ) (in) b 0.18 in. to 0.25 in. 0.21 inches Depth of one silver flange of packing cup (in) a * 0.14 in. to 0.16 ia. - 0.14 inches. 639 Depth of outer silver flange a 1 = 0.18 to 0.22 in.= 0.18 inches Constant spring com- ponent of total pack- ing friction (Ibs) C - w(D r +d r )t.05b+.09 (a + J n (2. 48+1)0. 05x0. 21+0. 09 (0.14+0.09)0.15x4500 = 230 Ibs. Pressure constant for fluid pressure com- ponent of total pack- ing friction (Ibs) C f = n(D r +d r )[.05b+.09 ( a + | A )]0.73 = fi n (2. 48+1) [0.05x0. 21+0. 09 (0.14+0.09)]0.73 - 0.250 Total recoil packing 'friction (Ibs) 230+0.250 Ibs. x 4500 = 1350 Floating Piston Friction Constant spring component of floating piston friction a CJ1.12nD a [ .05b Paf s G Paf G * Pressure constant for fluid pressure com- ponent of total packing friction (l&s) C'=1.46nD. a [.05b+.09(a+ -;)] * . 63f0.05xO. 21+ 0.09(0. 14+0. 09)]p af - 0.4P af 1.46xx3. 0312 0.52 640 CALCULATION OP THI D I H K 8 I N 8 OF TUB RECUPERATOR FORCING: Max. resistance to recoil (ax .e lev at ion) (Ib s ) uV f b g + (.096+. 0003d v 1.05 1260*51.50 64.4 2. 5+ (.096+0. 0003*4. 134) 19,700 Ibs, 80*51.5 1500 Win. resistance to recoil (horizontal elevation) (Ibs) - * rVf h 2g uV f b b +(.096+. 0003d) 1250*51.5 64.4 3.75+(0.096 Maxinum recoil packing friction (Ibs) packing friction) Kg (P aax 4500 or approx. ) +0.0003*4.134) 12900 Ibs. 80*51.5 1500 230+0.250*4500*1360 Ibs. 641 Distance between clip re- actions (inches ) (assumed) 1 - -; 60 inches gravity of recoiling parts to mean friction line (inches) r 1 = Distance from center of gravity of recoiling parts to axis of piston rod(inches) d = 6.5 inches Coefficient of guide friction, n=0.1 toO.2 = 0.15 7.5 Pull at max. elevation (Ibs) ""I* p max l-2nr Pull at horizontal elevation (Ibs) 19700+1260*0.8848 2x0.15x7.5 1+ 60-2x0.15x6.5 - 1360 18800 2nd; l-2nr C t hence w i" 12900 2x0.15x7.5 60-2x0.15x6.5 11,550 Ibs. - 230 Excess recuperator bat- tery reaction constant (recuperator constant) K=l.l to 1.3 = 1.2 642 Recuperator reaction in battery at max. elevation k(W r sin0 m +C t ) 1.2(1260x0.9848+230) 1.1. 2( 2x0.15x7.5 60+2x0.15x6.5 = 1980 4.16 Max. restrained recoil velocity(f t/sec) V r = 0.92 V f = Patio of Recuperator area Effective recoil piston area. 47.40 ft/sec 1.625 V r /- 2. 625x47. . / 18800 4500(11550 2.6 - 1980 If r < r m ^ n (see chart and assume air column) 51.6 inches Effective area of recoil piston(sq.in) P s 18800 A = * * 4.18 sq.in. 4500 4500 Area of recuperator cylinder (sq.in) A a = r A = 2.6x4.18 = 10.87 sq.in, If r < r B i n (8ee chart and assume air column) 643 Effective area of recoil piston 0.2425 v r If r > 3.5(Tio short cylinders - see chart) Effective area of recoil piston(sq.in) P s A = oo ^ f Area of recuperator cylinder (sq.in) A, = 3.5 A a Horizontal recoil pull P h =K v +. 000912V' 2 12.25 Where length of Air column is assumed Length of air column in terns of length of max. recoil (assumed) 1.445 Ratio of final air pressure initial air pressure M ( . ,)* ' (see chart of rj tables) = 1.5 644 Initial recuperator pressure (Ibs/sq.in) v 1980 * 473 Ibs/sq.in. A 4.18 Final recuperator p re ssure( Ibs/sq.in) Paf * Pai (approx) = 1.5 * 473 710 Ibs/sq.in, Initial air volume (cu.in) *~ Al = 10.87x51.6 = 580 cu.in. When ratio of final to initial air pressure is assumed. Assume Paf Paf m = = r- = 1.5 = 1.5 Pai Pai Initial air volune(cu. in) 0.77 0.77 V o - ( m ". 7 7_ 1 > A . b h (4=^^)10.87x45 560 cu.in. Length of air column (inches) 560 a 51.8 in. A, 10.87 Initial recuperator pressure (Ibs/sq.in) ' . ~ i222 , 473 Ibs/sq.in, 4.18 645 Final recuperator pres- sure Paf 4.73 710 Ibs/ sq.in INITIAL AND PIMAL AIR PRESSURE AHD AIR VOLUME. Initial recuperator pressure (Ibs/sq.in) 473 Ibs/sq.in Floating piston friction(initial)(lbs) C t +C Pai = .4x710 +0.52 x 473 530 Ibs. Drop of pressure across floating pistondbs/ sq.in) [* c s Pai ai Final air pressure (Ibs/sq.in) Paf = ra p z Final air volume (cu in) z V - o - A b h = 473 530 10.87 3-, = 140,000 J Torsional modulus (Ibs/sq.in) :i? .f* y 11,500,000 1 N = 10,500,000 f = 10,000,000 J SeS ^fii -vyfc ri^i H5i ilOO* Diam.of helix spiral spring regulator valve 1 * C * pinches; f& 3 /N*h"^R ; i 't) bo' io ci-tgn; D s - 4,.G^/ ; 8 < If N = 10,500,000 f s = 120,000 (Ibs/sq . in ) A"*S S D =0.216 / = a 3 D! Diam. of wire of spiral spring (inches ) 3/CD7 i =1.503 7 ? ? = f s If f s =120,000 Ibs/sq.in. i =.0305 / R 3 D S - 5 c c? . ' - % A J "fcT "-**/ 1 9w . 652 COUETiK BICOIL DgSIGH - BUFFER DBSIGH COHSTAMT ORIFICg AMD PORT AREAS. Packing friction at end of counter recoil (Ibs) 23 lbs Recoil length during buffer action (ftl 0.238H r 0.238x1260 d b * r> isw +R' - = - 715ft 0.15W r +R p .015^1260+230 l!L"' to ' bs) _ 8 - 61n - Length of buffer rodlft) 0.238W . A = 1.2 - - f = 0.15W r +R^ A a 2 d b 1.2x.715 = - = .33 ft.= r 2.6 3.96 in. Length of buffer chamber (ft) d1.2 to 1.3 d b ' = 1.25 x 3.96 = 4,95 in. Win. allowable counter recoil velocity (ft/ sec) at max. elev. Max. allowable counter recoil velocity (ft/ sec) at horizontal elev . v0. * 2.5 - 3.5 ft/sec? 2.5 ft. sec 653 Total counter recoil friction (nax. elevation) (Ibs) W_(sin0_+0.3cos0 m ) +0.3W r cos0 m 230+.25[ 1260( ' 9848 * 4.18 .3X.1736, ] +.3xl260x .1736 = 290 Ibs. Total counter recoil f riction(min. elevation) (Ibs) 230+ .25(^1H^ ) 4.18 +.3x1260=700 Ibs. Recuperator mean pressure (Ibs/ sq.io) 473+710 =590 Required constant counter re- coil orifice at max .elevation 8 1.25x4.18 (K=1.25) Required constant counter recoil orifice at elevation (sq.in) V ol 13.2/p'A-W t 13.2/590x418-290 -1260X.9848 .0645 sq.in. 1.25x4.18 x3.5 12.2/4.73x4+18-700 =.0765 sq.in. 654 0.0645.0.0765 Entrance buffer area (sq.in) KAV / 1 "b ' TTT / .00894A 2 v* 1.33x4.18x3.5 Pa 13.2 .00894x4.18 *3.5 590 (.0705) a v = 3.5 f tXsec(approx) Layout entrance area of buffer, with re- quired depth in groove. Decrease depth of groove to zero at end of buffer du . Deflection from free to solid height of spiral spring (in) h so s 2h " * 2x0.1862 * 0.373 in. _ _ ____ ___ ___ _ Spiral spring constant (Ibs.per in. ) R sc a 2280 83 " h s0 3 .373 This spiral spring will be too bulky for practical purposes. Therefore we will let the Belleville spring washers take care of all conditions at different elevations and design the spiral spring strong enough to keep the valve closed when gun is in battery. Spiral spring reaction at short recoil R; -s,* 8.45 t fc" k , .in* K + p -k (p- (p- h sin0 p*i> .!> 20 45-.OO 13O5O . 342O 13430 120OO 10OOO 2390 48. 89 . 1725 25 43.75 134OO . 4226 13930 12470 10470 2500 50.OO . 1680 25 41. 25 142OO .5736 14920 13370 11370 2710 52.O6 . 1624 50 37.50 15650 . 766o 1662O 14900 12900 3030 55. 50 . 1520 65 33-75 17400 .9063 18540 16640 14640 3500 59.16 . 1425 75 31.25 18750 .9659 19970 17930 15930 3800 61.64 . 1370 BO 3O.OO 19550 . 9848 20790 18690 16690 3980 63.09 .133fi Linear notion of cam rod against elevation. sidering spiral spring reaction negligible. Con- h + h 8 - ( P -p ai )a+p ai a t 0.175+h- (p-p ai )0. 747+127 10700 657 a b o d e f X Vfax.of cam rod inches 3 b" h cor- h + (P- 3 + 127 f d-g rected 0.175 0^747 10700 20 45^00 . 1867 . 3612 1785 1910 . 1784 . 1828 .911 25 43.75 . 1800 .3550 1965 1990 . I860 . 1690 . 845 35 41. 25 .1110 . 3460 2030 2160 . 20 20 . 1440 . 720 50 37.50 . 1600 . 3350 2300 2430 . 2270 . 1080 . 540 65 33.75 . 1500 . 3250 2610 274O . 2560 . 0690 .345 75 31.25 . 1450 . 3200 2840 2970 . 2780 .0420 . 210 80 30.00 . 1405 .3155 2970 3100 . 29OO .0255 . 127 105 M/M HOWITZER 75 M/M GUN (Double Charge) MOUNTED ON SAME CARRIAGE. Given: d - diameter of the bore (in) 75 m/m Gun Normal Super 105 m/m How. 2.953in. 4.134in v = nuzzle velocity (ft/sec) 1500 2175 1500 w = weight of charge (Ibs) 1.401b3. S.OOlbs. 3. 25 Ibs u = travel of shot up bore (in) 109.50in. BO.OOin m = max. angle of elevation 80 0^= min. angle of elevation w - weight of projectile (Ibs) 151bs. 331bs. Pjj=max .powder pressure on base of breach (Ibs/sq. in. ) 34,000 24,000 b g = length of recoil at max. elevation 1.3ft. 2.5ft. t> h =length of recoil at elevation 2.4ft. 3,75ft. 3.75ft. 658 WBIQRT Of GUN AND CARRIAGE. Similar Guns W V E=Muzzle Energy *g E/wg *t w= X* wt. 75 BB. French 16 1700 716000 1050 705 2657 39 75 mm. U.S. 16 1700 716000 750 956 3045 25 75 mm. British 16 1700 716000 995 720 2945 29 3.8 How.M. 1906 50 900 378000 432 876 2040 22.6 4.7 Gun M. 1906 60 1700 2690000 2688 1000 8068 33.6 4.7 How. M. 1908 60 900 755000 1056 716 3988 27. 6" How.M. 1908 120 900 1510000 1925 785 7582 25.7 155 m/m How. (Sch) 95 1420 2970000 2745 1080 7600 36.5 155 m/m Gun (Fil.) 95 2300 8400000 8795 960 25600 34.5 155 m/mHow. (St. Cham) 95 1520 3400000 3040 1120 7700 25.3 8" How. VI 200 1300 5250000 6652 790 19100 35.0 8" How. VII 200 1525 7200000 7730 933 20050 38.7 Average E/wg of 888 E/wg=1000 E/wg=888 v 7 normal gun super gun howitzer 15x1500 64.4x1050 15x2175 64.4x1000 33*1500 64.4x1000 1100* 1240 *1150<( 1290 1100+30 1130 W r How. 1180+30 = 1180 Average 1155# 1260# 659 Using highest % of * r to W t (39*)W t - - = 2970* 397 2970 - 1155 1815f W r Weight of recoiling parts 1230* and 1260* for gun and howitzer respectively are the minimum weight that could be used on account of stresses. The condition being to get the minimum weight. These values are used: W. 1230* gun r * 1260* how. Gun super 38.00 105 m/m How. 51.50 7 '.,,. i. r,.tL -in-' -; "I-- 5 " _r ^ i-ll.inji 5m/m ormal 23.60 1 Vf max. free velocity = wv+4700 w 15x1500+4700x1.4 1230 15x2175+4700x3 1230 33x1500+4700x3.25 1260 " sec K s Resistance to recoil at 80 elevation w r = 1.05[ V2* 2g r 1 uV, 1 b_ + (.096+. 0003d) * 2 v 1230x23.60 1500 660 1.05U0640* 1 1.5(. 096*. 0009)1. 724 1 . 1.05x10640 fl __- '1.05(10640* )= * o720 1.5+.167 1.662 _ f 1230x38. 00 __ 1 64.4 " X 1.5 1.05x27600 17600 1.645 1260x51.50 64.4 V 2 .5+ ( . 096+ . 0003x4 . 134 ) 1500 1.05*51900 . 19700 2.761 h = height of axis of bore above ground. Assumed 36" b h = max. allowable horizontal recoil / 2g , / 23 ' 60 x3 51", 82 ",1004" 64.4 V r * max. velocity of constrained recoil ,9Vj(app) 21.20,34.20,46.35 A recoil constrained energy = - 1230x21.20 8580 i 1230x34.20 64.4 22400 64.4 1260x46.35 42000 64.4 661 E * recoil displacement during powder period. , 1B n."^~ 1230 12 1230x12 .15+.5X3. 109.5 2.24x16.5x109,5 2 ' 24( } - * ' 275 ft 1230 1230x12 33+.5x3.25 x 80 2.24x34.63x80 2.24( - ) = - = .41 1260 12 1260x12 STABILITY l s - 3000 .. 6 . 35 . n . c = constant of stability = ' .96 stabilizing moment ef g * angle of stability 20 d - moment arm of overturning force h t cos0+d8=l,,sin0=36xO. 9397+7. 5-81x0. 3420 V O 13.60 in. =1.13 ft. b n = length of recoil at the angle of stability s l s +W r E cosg/|* s I s +W r Eco8g)*-4W r cos0(W s l s E -) 662 3000x6. 35+1260* 0.4lx.09397+y^H s l 8 +W r Ecos0) 11 - 2x1260x0.9397 4xI260x .9397(3000x6. 35x .41) .96 19, 540*/119, 540) '-4740 (7800-49500) 2370 19. 540+/110,000, 000 19.540+10490 2370 2370 3.75 ft, RBCUPgRATOR FOHGIBG8. Approximately = maximu!n resistance to recoil .45 r .45 1230 . . . nn -- VI * - - 23. b g 1.5 32.2 .45 1230 60 1.5 32.2 * 38. 2.5 32.2 * n in. resistance to recoil 0.47 ^r y , m 0.47 J[r 3.75 32.2 75 Gun 105 * Mora- Super How. 6380 16500 18700 2660 663 0.47 1230 3.75 32.20 0.47 1260 3.75 * 32.2 -^ &1.5 P g = max. pull = K S approximately P fi = nain. pull = K h ~0.3 W r K v = initial recuperator reaction* 1.3W r (sin0_+.3cos0_; X ill til = 1.3xl260(. 9848+. 3x. 1836) 1.3x1260x1.037 r recuperator cylinder area eff.area of recoil piston A. ~ .039V, .039x23 .60 / 2290-1700 10.1 P h -1700 75 or* al Gun -Super lOg KB HOB. 6740 13100 380 16500 18700 2290 6370 12730 1700 5.35 664 _ 19000+17150 P h = 3560 p af final air pressure n ' - . . . . : (generally) 1.5 P a ji initial air pressure e = length of air column from chart 1.25b A. = effective area of recoil piston = p s 19000 4500 4750 4.00 sq.in. (Usually packing is designed to stand a pressure of 4500 to 5000 Ibs) P S * max. pressure corresponding to r = 3 P^ = I222? = 4750 Ib3< 4 W Q = total weight of recoil piston and rods, 30 Ibs. T. * max. tension on the rods at elev.K<- + ff r 30 19300+13.45x24000* =27000 1260 t = assumed fibre stress = - elastic limit max 70000 35000 Ibs/sq.in. 27000 A r - area of the recoil rod = = .772 sq.in. 35000 d r = diameter of recoil rod = 1.127 /a r = 1.127/772 + .99in.(nake 1 in. A r total area of recoil cylinder = 4. +.781 - 4.781 sq.in D r 3 inside dianeter of recoil cylinder - 1.127/~T r 1.127/4.781 = 2.46 in. A r = rA * 3x4 = 12 sq.in. D a * 1.127 /TI = 1.127 /T2 = 3.9 in. diaicster a a of float- ing piston. 665 CALCULATION OF PACKING FRICTIOH. b = O.lSin.to 0.25in.use 0.21in. a = depth of outer silver flange of packing cup 0.14 in, to 0.16 in., use 0.14 in. a 1 = depth of outer silver flange 0.18" to 22", '0.18" c t = constant spring comp.of total packing friction a 1 c t = n(D r +d r )[.05b+.09(a+ -)]0.15 P max 45 = n(2.46+.?81;r.05x.2H-.09C.14+.09)).15 P max 10.2(.0105+.0207).15P max =10. 2x.0312x. 15x4750 c 1 = 226 a 1 C_ = n(D n +d_) [ .05b+.09(a+ ) ] . 73 JT I I Q = n(2.46+.781)[.05x.21+.09(.14+.09)].73 = 10. 2x. 0312". 73 C = .232 2 Rp = total recoil packing friction - c t +c p(p= Ibs/sq.in) 226*. 232 x 4750 1326 Ibs. FLOATING PISTON FRICTIOH *.i>3 c ls cast(spring constant) of floating piston = a t 1 12 ^Do I 05b^*09(.L +~"~*) j p o f = G P a ^ 3 ' d I o 1 = 1.12 x n x 3.9[.05x.21+.09(.14+.09)]p af = GP af = 1.12 x n x 3.8 x .Q312P af = ,428P a =GP af (in Ibs) P af = final air pressure ^ = pressure constant for fluid pressure corap. of total packing friction. - = 1.46nD a [ .05b + J = 1.46nxD x. 0312=1. 46 x n x 3.9 x .0312 = .558 666 D2SI6N OF RBCUPSRATOR 75m/m Gun 105m/m Normal Super How. 6720 17600 19700 b_+(.096+. 0003d) K h = min. resistance to recoil B V! i uV b + (.096+. 0003d) - v 1230x23.60 64.4 1500 10640 b h +.167 - 1230x38 (b h =2.4 ft. =29 in.) 1 27600 - - 3.925 1260x51.5 4140 64.4 3.75 +.267 ^ t 1260x51.5 - 64.4x4.017 max 226+ .232 6720 7030 Ibs. 12,900 Ibs, 616 1236 1366 1 * distance between clip reactions 60in. - 616 - 616 7044 1.0388 17600*1210 - 1236 16900 1.0388 19700--1240 1.0388 - 1366. P b =pull at horizontal elevation in Ibs. 3560 667 r 1 = distance from center of gravity of recoiling parts to mean friction line. 6. Sin. n coefficient of guide friction (.1 to .2) .15 d^ * distance from center of gravity of recoil- ing parts to axis of piston rod 7. Sin. PULL OH THB ROD 75 m/m Gun Normal Super P g =pull at max. elevation K 8 +W r sin0 2nd" L 1-2 nr 6720+1230X.9848 ___^_^_ _ 2x.l5x7.5 60. 2x. 15x6. 5 6720+123QX.9848 1.+.0388 7930 18800 2nd K l-2nr - C 668 p ,( - D -- 226). 945 1.0388 (3560x1. 0385+226) 1.0388=K h =4140 7030 1.0388 - 226)x.945 6180 12900 P h =( -- 226)*. 94.5 11550 1.0388 R * excess recuperator battery reaction constant 1.2 (1.1 to 1.3) K v * recuperator reaction in battery at max. elevation R(W r iinB m *c ) 1.2(1260. 9848*226) K T . , 2w "1 C *7 C OOO 1-R ( + _^j l 1>s ^ *.1&*7.5 ^ .233 1+2nr A 60+2x.l5x6.5 4 1.2x 1466 1760 1760 ___^______^__^___ _______ ____^^_ 1-1. 2(. 0363+058 1-.1128 .8872 K y a 1980 V r - .92V f =. 92x23. 6=21. 7=V r ; 35.00 = 47.40 _ _ o enc tr / S _ o coc^oi " / 188QQ A C r - - 2.625 V r / '-* 2.625x21, A P (P h -K v ) 4750(3560-1980) 57 4750x1580 r 2.85 669 r . la . 2.625 V r 168 . 81.6 4750(6180-1980) 4750*4200 = 2.79 af - 1.5 = B 1 - l.Sb p s 18800 A - - - 3.96 4750 4750 A a = rA = 2.85x3.96=11.30 sq.in.A a Pjj=nin. pull on the rod V r = velocity of recoil corresponding R 1.295 w ? =U0373 rA>2 h "v r .2435 175x. 00139 (P h -K v ) Pp = pressure the packing should withstand P*V 6.9 S- 1 - Pp(P n -K y ) r 2.625 V. j = length of air coluran in terns of max. recoil 1. 1.3b 670 j - 1.3 *( r '1 ) ' from the chart 1.5 r.J-1 Ky 1980 p a j = initial air pressure (Ibs/sq.in) = - * A o <7w P ai * 500 Ibs/sq.in. (approx.) P if * * P ai = !- 5 x 500 s 75 x p a f approx. VQ * initial air volume = A a x i a = 11.30 x 45 662 cu.in. = V & - 1 INITIAL AMD FINAL AIR PRBSSURB AMD AIR VOLUME. 50 R t floating piston friction initial 3 C i +C P ai = -428xP af +.558^500 = .428x700+. 558x500 = 321+279 B t 500 P Pli + - 1 - 2 ai = 500 = - 500+44.2 ai A a 11.30 P ai - 544 Ibs/sq.in. P ai = 540 Ibs/sq.in. p af = * P ai l. 5x544=816 P af = 810 Ibs/sq.in. V =662 sq.in. V f = final air volume = V Q -Ab h =662-3. 96x45= 662 -178 V f = 484 cu.in. 671 P a average drop of pressure across floating pis- ton = C t +C a (P al +P af )0.5 540+810 540+810 * .428 x + ,558x = (.428+. 668)675=. 986x675 P a = 665 Ibs. W- = 30 Ibs. V T L * tension on the rod =K s + ^ 30 r = 188000+126Qx. 9845+13. 45x24000* 1260 = 18800+1240+7700 T L 27750 Ibs. F fflax = assume fibre stress 1/2 elastic limit = 32500 Ibs/sq.in = F m&x 27750 A_ area of the recoil rod = .853 sq.in.=a_ 32500 d r diameter of recoil rod 1.127/.853 = 1.04in d r A'* area of the recoil cylinder = 3.96+1.04 = 500 sq.in.= A' D r * 1.127/5 2.52in.=D r D =1.127/11.30 =3.78K=D a diameter of air cylinder W* - 30 Ibs. STRENGTH OF CYLINDERS; Test pressure = 2 x service pressure P t = max. allowaole fibre stress for cylinders = 3/8 elastic limit = 3/8 x 60,000 = 22500 Ibs/ R aft min. outside diameter of recuperator l ao 1 . S9 A 22500+810 P t -P... 22800+810 21700 U d * - --T 676 P" 496 ~ 3.19 A = 3.19( ) =* 3.19 x .0643 d s f 120000 .205in. D 8 * diameter of helical spring 4 x d D_ .82in. deflection per coil nf s D I Gd g ia500.000x.205 f = .118 496 - = 4200 Ibs. per inch of deflection required .118 248 - = 1162 spring const. .213 4200 - 3.61 effective coils 1162 n = no. of coils = 3.61 + 1 = 4.61 use 4.5 coils P h R s = load on spiral springs at elev. = - - P a *l o c c r\ = ( -- 500). 93 = 400 x .93 3.96 F s = 372. Ibs. Wp ^422 h" lift of valve at long recoil = - - = .193 c 2184 h" * .213 inches Valve seat clearance = .02 "W 25 .213in> h" - lift at short recoil = .1144 c 2 . 184 h" =.1144Jn. R sc = load at solid height of apiral spring -j- x 373 496 Ibs. h gc = deflection from free to solid height of spiral = 2h" = 2x.213 h sc = .426in. 677 g S g * spiral spring const. = 1162 # * S_ h__ 426 ~_ O C *-^ -.^ii^^MiP-^Bta- g * spiral spring reaction at sbort recoil S s (h" + .02)1140(.1144 + .02) R' - 153 Ibs. INITIAL DEFLECTION WORKING * N <0 DEFLECTION SOLID HEIGHT .1OG9 -.2.13 .O65 * load on Belleville at sbort recoil 18800 ( - .500). 93-. 355x500 3.96 (4750-500). 93+177. 5 3770 # at solid beigbt of Belleville washers ' I R b = ! x 3770 R bo = 5650* hu * deflection of Belleville from free to solid height 3h'=3x.H44 h bo .343* 678 S b * Belleville spring const S b - 16080 6512 .343 16080 *o h- Si CU 10 -.ie *-.1064- -.646* DESIGN OF CAM MECHAHISM AND LAYOUT. g = ratio of can movement to valve movenent usually 5 X = distance valve should lift to engage Bellevilles S^ = working deflection h g - initial corap.of spiral spring h = clearance of valve h = lift of valve hjj = initial compression of the Bellevilles. X = {s s (b s +h *h)+S b (b b -i-h +h)-[(P-P ai )a+P ai a l ][' R s =S 8 h s +S s (h+li ) (Steins of two springs are in con- tact) (P-P ai ). 93+500*. 355 1162*. 1066+11. 62 (h+. 02) +16080*. 1144*16080 (h+. 02) 679 124 +1162h+23+1840+16080h+322=(p-p al ). 93 + 177. 2 17242h+2132= (p-p ai ) .93 18500 .098AV r 2660 360 .0194 51.58 1.00 5. 60 3000 7oo .0378 54.77 2.07 11.62 3250 950 . 0512 57.01 2.92 16. 40 3500 1200 .0648 59.16 3.83 21.50 3750 1450 . 0781 61. 24 4. 78 26. 90 4000 i7oo .0917 63.25 5. 8O 32. 60 4250 1950 . 1050 65. 19 6. 85 38.50 45OO 2200 . 1188 67.08 7.95 44. 6O Normal LENGTH Of RECOIL 80 Elevation V r = 21.70 corresponding (P~P a i) from curve 3500 P s = (3500+. 500)3. 96=4000x3. 96-15820=P s 680 -226+ . 232x4000-226+930 R Bajt -1160 Ibt. K g + r sin0 P s * 2nd 8 R l-2nr K a +1230x.9848 15820 - 1160 1+.0388 16450-K.+1210-1200 N y. K g 16440 - 1.05 [ ' 2fi uV f ' b.+ (.096+. 0003d) v ,,. 1.05x10640 11200 16440 - b+.167 b+.167 16440b+2750-11200 8450 b .513 16440 b - 6.17ia. Super V r 35 corresponding (p~P a i )*4100 Ibs P g - (4100-500)x3.96=4600x3,96 P, - 182001 K 8 +W r sin0 L s 2nd, 1 + l-2nr 226+. 232x4600-226+106 R max ' 130 K a +1210 P. 18200 = - - 1300 1.0388 18880K S +1210-1350 681 W_V| K, 18020 lb. 1.05 I - 1.05x27600 X -^ ^ b+.145 19020b+276029000 b 16.55in. Howitzer V f 47.30 ft/s.c. (P-P ai )-4620 Ibt P 3 = 5120 " 3.96 = 20250 Ibs. K s +W r sin0 2nd b 1- 2nr ~ R max R aax s C t ' t ' C t P * 226+. 232*5120 226+1190 R max ' 142 IT 8 -1240 P g 20250 - -- 1420 -2100OK.-1240-1470 1.0388 K g 21230 lbs.= 21230 = i- b+.267 21230b +5630=54500 b'- -2.3 feet 2123 b"27.6 inches. b s = (80-0)+30 60 1(80-0) +30 682 1(80-0) +30 4 1.05x51900 b+.267 655000 54500 - +.267 12 655000 b"+3.21 b"+3.21 C 3.96 K s +W r sin0 2nd - C. 1-2 nr K g +1260 sintf - 226 P c = K_ + 1260sin) 9 1.098 ,178 V r 1.0388 - 212. .178x47.30 8.43 .178 20 25 35 50 65 75 80 b " 45 43.75 41. 25 37.5 33.75 31.25 30 *, 13600 13950 14710 16050 17720 19000 19700 s inflf . 3420 . 4226 . 5736 . 7660 .9063 .9659 . 9848 K .* r in^ 14030 14480 15430 1702O 18860 20220 20940 K m * r int r 12800 15200 14100 15500 17200 18900 19100 1.098 P . 12600 13000 13900 15300 17000 18700 18900 (P.-P.i) 1060O 11000 11900 13300 15000 i67oo 16900 P^-K V 3.96 2680 2*780 3000 3360 3790 4220 4270 683 (p-p ai ) 51.77 52. 73 54.77 57.97 61.56 64.96 65.35 h . 1628 . 1596 . 1536 . 1452 . 1363 . 1297 . 1286 1162(.1065+.02+h)-H6080(.1144 + . .93+500X.355] 1 P s -1980 (1162h+147+16080b+2150l ( ) .93+178] 16080 3.96 (17240h+2300-.235P s -290)= - (17240h-235P s 16080 16080 +2010 1.072h-(.00001463P 8 -.1251> 16 h 1.072h PS 1.463 105 8 b X 5X 20 .1628 .1749 12600 .1842 .0591 .1158 .580 85 .596 .1715 13000 .1900 .0649 .1060 .530 35 .1536 .1648 13900 .2034 .0783 .0870 .435 50 .1452 .1559 15300 .2390 .1139 .0420 .210 65 .1368 .1469 17000 .2490 .1264 .0205 .103 75 .1297 .1392 18700 .2735 .1484 .0008 .004 80 .1286 .1380 18900 .2765 .1514 .0000 .000 Counter Recoil Buffer, constant orifice and port Areas. R' = packing friction at end of counter recoil C t = 226 Ibs. d recoil length during buffer action T"" 684 0.238W r .238x1260 300 300 z ^ = i a 15^1260+226 189+226 415 d b - .723 ft 8.7 in. A d b d^ length of Duffer rod * 1.2 x d fa x , 1.3 A a r 1.2 x .723 , .3 ft d b - 3.6 in. 2.9 Length of buffer chamber * 1.2 to 1.3 d b ' dg 4.5 in. allowable counter recoil velocity 2.5 to 3.5 ft/sec. total counter recoil friction - aax elev. W r (in 0t0.3 cos a C t + C t E - - ]+0.3W r 3.96 .1736 ..3X1260 226 +.232(.2615 )+66 R t " 290 Ibs. Max. elevation , +C ( - ) +.3H P 226+96+378 A ' * 700 Ibs. Win. elevation. p ai +p ap max. recuperator pressure = P^ 500+750 2 625 685 c'recoil orifice at 80 elevation KA X (K1.25) 1.253.96 x 2.5 24.6 24.6 ^^ - ^ ^^ ^^^ ^^^ - = i 3 13 . 2/625x3 . 96-290-1260x9848 13.2/670 411 .06 sq.in i 34.5 34.5 'o " = 13.2/P a A-R{ 13.2/2470-700 13.2x38.35 34.5 506 * .0682 sq.in. = .0641 KAV entrance buffer area = 13.2 0.00894AV 1.33x3.96x3.5 13.2 .00894x3.96 x3.5 625 - (.0641) 2 / 1 1.395 1.395 / * .097 sq.in. 625-418 Lay out entrance area of buffer, with required depth of groove, decrease depth of groove to zero at end of buffer dJ.. CHAPTER X. RAILWAY GUN CARRIAGES. TYPES OF MOUNTS. For coast defense or other use of heavy artillery, it has been accepted that mobility is of great importance. Materiel in permanent emplacements is more readily subjected to attack. Further with long coast lines it is impracticable to supply enough permanent batteries for adequate protection. By introducing heavy mobile artillery ire increase the protection and develop the advantage of concentrated fire at any one point when needed. Railway artillery meets the demand for mobility in a very satisfactory degree. Very heavy weights, as occur with large caliber guns and their corresponding mounts, are most readily transported by rail. Hence there has been a tendency of development along two lines; first, a mobile railway carriage that is entirely self contained and fired directly from the rail and (2) a mobile mount, transported by rail but set up on a semi-fixed emplacement. For extreme Mobility the first is most useful, wherein for coast defense work the second plan offers many advantages. Railway carriages have been developed along the following lines. In their methods of firing. (1) Sliding carriage type with no recoil mechanism , the carriage merely sliding back during the re- coil along special constructed rails or guides, trucks being disengaged. (2) Railway carriages with a recoil system, the whole carriage in ad- dition recoiling on special ways on 887 688 rails, the trucks being disengaged, or the trucks being engaged and the secondary recoil being direct- ly along the rails. (3) Fixed or platform mounts. With light railway artillery, the car is held stationary by suitable out- riggers and we have usually a bar- bette type of mount, mounted on the car. With heavier types, the girder which supports the tipping parts is placed on a large pintle bearing with sometimes additional support at the tail of the girder with a circular way or track for all round or sufficient traverse. In this latter type the trucks must be disengaged and the main girder run on to the permament emplacement. The sliding carriage type (1), was developed successfully in France and was considered satisfactory during the late war. This mount, however, is sub- jected to the direct firing stresses with consequent requirements for a very heavy girder and trunnion support. It has on the other hand the advantage of doing awaj with a recoil system. At best, how- ever, it can be regarded merely as an emergency type of carriage that might be developed under great stress of war pressure and not suitable for use against moving targets. In railway carriages of type (2), we have virtually a double recoil systew. However, since the recoil is designed for stationary service as well, or for the condition at max. elevation where the secondary recoil is small, the maximum reactions at the beginning of the recoil are the same as in a. stationary mount, with a single constant recoil. When the trucks are disengaged a specially built track must be laid, and the 689 girder slides back on friction shoes, which are lowered to engage with the track. Mounts of this type are illustrated in our 14" railway mount ME. When the trucks are not disengaged and the secondary recoil takes place on the track, the bearing reactions of the truck wheels must be suitably designed to sustain the additional firing load and the trucks must be suitable braked to resist the secondary recoil, and bring the mount to rest after the firing. When a built up track, trucks disengaged, is used the successive firings must necessarily take place along the tangent of the track, whereas firing directly from the rails, permits the use of a curved or Y track, and con- siderable traversing is thus possible by the firing taking place at different points on the curved track. With railway carriages of type (2) very little traversing is possible on the mount itself and therefore the track must be laid very closely in the direction of firing. In railway carriages of type (2), we are greatly limited by road clearance. For clearance, the trunnions must therefore be in the traveling position in a low position. On firing however, at maximum elevation, the recoil becomes limited. To pro- vide for a suitable recoil at maximum elevation the trunnions are raised and a balancing gear throwing the trunnions to the rear may also be introduced. With fixed or platform mounts of type (3), the special features are the methods of erection on to a serai permanent emplacement and the disengagement from the traveling con- dition of the mount . We may have a center turn table which serves for the pintle in traversing and the tail of the girder is supported by a suit- able circular guide which balances the overturn- ing moment and thus releases the otherwise bend- ing or overturning moment on the pintle bearing. 690 With this type of mount large traversing is complete- ly possible. SPECIAL PIATURS3 IN THB DESIGN. Recoil System: (1) The recoil should be simple and rugged. (2) A constant recoil or approximate- ly constant for all elevations should be used. (3) A constant resistance to recoil is satisfactory since questions of stability are not usually of prime consideration, and the recoil is thus simplified. (4) The counter recoil should be sirople, an ordinary spear buffer being usually satisfactory although other control may be sometimes necessary. Bere again counter re- coil stability is no longer a con- sideration and high velocities of counter recoil are not objectionable provided there is no shock at end of counter recoil. (5) With very large guns used at high elevations, high pressure pneumatic recuperator systems should be used in place of spring columns, since the weight and bulk of springs be- come excessive. (6) Sleeve guides for the gun have been found most suitable and tne various pulls should be so far as possible symmetrically spaced about the axis of the bore, thus reducing tne bearing reactions in the sleeve and making it also possible to keep 691 the center of gravity of the recoil- ing parts close to the axis of the bore. Tipping parts: (1) The cradle should be of the sleeve type thus reducing the bearing pressures over guides and clips. (2) The recoil and recuperator can be strapped on with suitable shoulders for bearing surface to take up the recoil load from the cyli nders . (3) The trunnions should be located near line through the center of gravity of the recoiling parts and parallel to the axis of tue bore. This reduces the elevating re- action during the pure recoil to merely that due to the moment ef- fect of the recoiling parts out of battery. (4) Great effort should be made to locate the center of gravity of the recoiling parts as near the axis of the bore as possible either by symmetrically distributing the re- coil rods and attachments or if necessary introducing counter balancing 1 weights. Thus the whipping action during the powder period is reduced with a correspond- ing reduction in the elevating arc reaction during the powder period. (5) With high angle fire ^uns or howitzers, the trunnions may be thrown to the rear, and balancing 692 gear introduced, thus making long recoil possible. Another plan for accomplishing the same results is to raise the trunnions before fir- ing. (6) The trunnion bearings should be supported on springs during travel- ing, though compressed so we have solid contact during firing. (7) To reduce the friction during the elevating process, ball or roller bearings should be introduced in the trunnion bearings, or in an inner trunnion should be introduced of smaller radius than the main trunnion for reducing friction on rotating the tipping parts. LIMITATIONS IN With heavy artillery mounts, BRAKE LAYOUT, either railway or lor permament or mobile emplacements, counter re- coil stability is not a consideration. On the other hand we are limited to a maximum allowable buffer pressure in the counter recoil. With counter recoil systems which come into action towards the end of counter recoil, practically the entire potential energy of the recuperator most be dissipated by the buffer over a relatively short displacement. Now since the potential energy of the recuperator is a con- siderable fraction of the energy of recoil, we see that the buffer reaction is of a magnitude comparative with the brake resistance during the recoil. Further the effective area of the c'recoil buffer, due to constructive limitations, is necessarily considerably smaller than the effective area of the recoil brake. Hence the buffer pressures with a short c'recoil buffer, become very great. This is especially pronounced with a short buffer and high 693 angle fire gun where the unbalanced recuperator energy is necessarily great, when the gun c'recoils at lovi elevations. As to the limiting allowable buffer pressures, no hard and fast rule can be made, but it is certain that the buffer pressures in many of our mounts are rather too high for light construction, requiring heavy and strong buffer chambers . With recoil brakes having a continuous rod extending through both ends of the cylinder, the effective area of the buffer must be necessarily very snail and the stroke of the buffer short due to the fact that during the recoil it is important that the void displacement be not too great. Hence this type of brake with continuous rod and enlargement for c 'recoil buffer ram, has inherent- ly excessive buffer pressures. It is very important with such mounts to maintain a minimum recuperator energy, that is to use the minimum recuperator re- action combined with a low ratio of compression, consistent with proper c'rscoil at maxiuum elevation, To reduce the -buffer pressure, the c'rscoil regulator should be effective throughout the recoil, and thd effective area of the buffer should be as large as possible. This actually has been obtained constructively in our 16 inch railway mount, the buffer area being equal to that of the recoil brake and c 'recoil regulation taking place throughout the rscoil. The buffer pressures are therefore compar- able with the brake pressures during recoil. DESIGN LAYOUT OF Assuming a preliminary layout RECOIL SYSTEMS. has been made, the weight and the ballistics of the gun given, we may estimate from previous mounts, the probable weight of the recoil- ing parts and tipping parts. Therefore, we will assume the following data given or estimated from previous mounts: 694 W r * neigh! of recoiling parts (estimated) (Ibs) d * diameter of bore (in) v nuzzle velocity (ft/ec) w * weight of projectile (Ibs) weight of charge (Ibs) p bm 3 maximum powder pressure (Ibs/sq.in) b = mean length of recoil (ft) m maximum angle of elevation * minimum angle of elevation u travel up the bore of the projectile (ft) Calculation of E and T: From the principle of Interior Ballistics, we have, R PU, = - d" p b|n = max. total pressure on breech (Ibs) wv 1 P e - average force on projectile (Ibs) _ e - u[(~ ^ - 1) /Pv max. where d v = diam. of recuperator rod D y = diam. of recuperator cylinder Wy = width of stuffing box packing Wy = width of piston packing then R p = Z R ph + ZR pv = total packing friction If P n = the total hydraulic reaction P = the total tension or poll in the brake rods F V = the total recuperator reaction .Fy = the total tension or pull in the re- cuperator rods then P n =P n +2R ph ; F v =FV2R pv Guide Friction We may now estimate, more exactly, the guide friction. We have two cases, (1) When the resultant pulls are symmetrically balanced around the axis of the bore (2) When the resultant pull is off set from the axis of the bore. In (1) we have simply R- = n!T r cos Gf (Ibs) In (2) we have 2n(B+R )e b +nW r cos0(x i -x a ) R = (Ibs) 1+2 n e b where n = 0.15 x t and x a are the front and rear clip reaction coordinates with respect to the center of gravity of the recoiling parts. 702 1 distance between clip reactions and length of sleeve in cradle, e^ * distance down from bore to resultant line of action of mean total pull (B+R p ). In general, however, we may neglect R_ as small compared with 8, and 2 n e b as small compared with 1, then, 2n(K+W r sin0)e b +nW_cos0(x -x ) R = x ' (Ibs) 1 The term n W r cos 0(x t -x 8 ) is usually small com- pared with 2n(K+W r sin0)e^ and further very often we may assume x t = x z approx., hence, 2n(K+_ sin0)e b R g - (Ibs) which is usually sufficiently accurate for ordinary calculations . It is to be particularly noticed, that when the pulls are offset from the axis of the bore, the guide friction increases on elevating which is exactly opposite to the condition of sym- metrically and balance pulls about the axis of the bore, when Rg = nW r cos 0. [nitial Recuperator reaction, The required initial recuperator reaction is given by the following formula: n cos (x. -x ) , 1+2 n e r 2e v n 1 - 1+2 ne r here R py = 2 .05 * (d v v +D tf W v )p vi = assumed initial recuperator pressure n cos m ) r 703 A y assumed effective area of recuperator piston d ? * dia>. of recuperator rod (in) D v = diam. of recuperator cylinder (in) w y = width of stuffing box packing (in) W v = width of piston packing (in) 1 * length of sleeve or distance between guide reactions (in) e v = distance from center of gravity of re- coiling parts to resultant line of action of F v e r = mean distance from center of gravity of recoiling parts to guides ( = 0, for sleeve cradles) x and x a = coordinates of front and rear clip reactions from center of gravity of recoiling parts in battery (in) n = coefficient of guide friction (=0.15) m = angle of max. elevation The above formula is complicated and the fol- lowing formula is usually sufficiently accurate and takes into consideration as well the pinching action between the guides and clips, vi = ^ W sin^gj+R -)(lbs) where k = 1.1 to 1.2 i 1 when e v is small as with symmetrically balanced recuperator pulls, then F v j = k[ W r (sin0 m +n coB/) B )+R p ] where k = 1.1 to 1.2 If we include R p with n W r cos 0, we may in- crease k, and we have the elementary formula as before used, P yi = 1.3 (W r sin0 B +0.3 cos0 m ) (Ibs) 704 Counter Recoil Buffer or Regulator Design Counter recoil regulators may be divided in- to two general types, (1) Systems which are effective only during the latter part of counter recoil. (2) Systems which fill themselves during the recoil and are effective throughout the counter recoil. In type(l) we have a short spear buffer or plunger entering the buffer chamber towards the end of recoil. Type (1) buffers may be further sub- divided into:- (a) Plungers attached to a continuous recoil rod, the re- coil rod passing through a stuffing box at either end of the piston. (b ) Ordinary spear buffers with- out a continuous recoil rod. In the design of a counter recoil system, we are primarily limited to a maximum allowable buffer pressure, counter recoil stability in heavy artil- lery being of no great importance since the stabil- ity limit on a counter recoil is usually as great as on recoil. Since, however, a considerable part of the recoil energy becomes at the end of recoil stored in the recuperator, we have this energy absorbed in the counter recoil, by the counter re- coil regulator in a short buffer displacement, with a consequent large total buffer reaction. We are limited in tbe counter recoil brake usually to a smaller effective area than in the recoil brake; consequently the buffer pressures become, due to constructive limitations, very large. Hence it is highly desirable to maintain as low a buffer pressure as possible. With any form of spear buffer of type (1), to reduce the buffer pressure, the effective area 705 of the buffer plunger should be as large as pos- sible and the length of buffer as long as possible. In the design of a spear buffer of type (1) we have the following limitations :- (1) The diameter of the buffer, should not exceed a value, that due to the sudden withdrawal of the buffer, the void displacement in the recoil brake should not be greater than the free recoil dis- placement during the powder period E. (2) The length of the buffer should not exceed a value that during the counter recoil before the buffer enters its chamber the buffer chamber should be completely filled. Let A * effective area of recoil piston (sq.ft) A'= effective area of recoil piston on counter recoil plunger side (sq.ft) L b =length of plunger or buffer (ft) A b * effective area of buffer (sq.ft) du =diam. of buffer chamber D - diam. of recoil brake cylinder i_ = diam. of recoil brake rod Now A 5 0.7854(D a -d) ; A'=0.7854 (D*-dg ) sq.ft. A b =0.7854(dg-d) (sq.ft) type (l)(a) buffer, A b =0.7854 dg (sq.ft) type (1) (b) buffer. Now for condition or limitation (1), we have (ft) or A(L b -E) g A 1 = A(l- - )sq.ft. L b L b In terms of the diameters, we have E D-dg 706 / E )* T + d*(l- ) which gives us the b limiting value of d b . It is interesting to note that when 6=0. d]j*d r or in other words when the diameter of the buffer a plunger is made equal to that of the rod, no void is required in tbe recoil cylinder. From the above expression, we note that in- c re as ing the length of the buffer decreases the diameter of the buffer and thereby increases the buffer pressure. On the other band the c 'recoil energy is ab- sorbed over a greater distance with a longer buffer, thus reducing the total buffer reaction, and it is probable, that this cause more than effects the slight increase of the buffer pressure dueto tbe decrease of the buffer diameter. Further the value of d|> is very often entirely limited by constructive considerations alone; hence a long buffer is highly desirable. In a type (1) (a) buffer due to the relative- ly large value of d b required to give a sufficient buffer area, the length of the buffer depends en- tirely on the limitations (1). This type of buffer will be considered in detail later. For the limitation (2), with a continuous rod, we have a void produced at the end of recoil on tbe buffer side of the recoil piston. To compute this void, we have, with an initial void in the battery position AE, for tbe void on the buffer side of the recoil piston at the end of recoil, or the out of battery position. Void c =A r b-A(b-E=(cu.ft) where A r = area of the recoil cylinder (sq.ft) Therefore, Void c = (A r -A)b + AE = a r b+AE(cu.ft) Now in the c 'recoil, the spear buffer chamber is evidently not filled until tbe void displace- ment has been over run, and this displacement a^> +AE . v becomes, X a = r 707 D* AE Since - is small, for a close approximation, the r buffer length should not exceed L b -0.8b(l- Jt )=0.8b(l-|f) ft. The mean buffer pressure may now be computed, knowing the potential energy of the recuperator. The potential energy of the recuperator is given by either of the following expressions: p vi v o **o * fv ^ B ~ D(ft.lbs) (k=1.3 approx.) V HO - (ft.lbs) A r (k-D where Vf=V o -A v b A v = effective area of the recuperator piston (sq.ft) V o the initial volume (cu.ft) F ? i = the initial recuperator reaction (Ibs) %i m = the ratio of compression Fvi Then, the mean buffer pressure, becomes l -(W r sin0+R )b P b *r - - - (Ibs/sq.ft) where R o =total packing and guide friction (Ibs) b = length of recoil (ft) d ? L b = 0.8(1- ^-) (ft) A b = 0.7854 dg )sq.ft) r-T' (ft '> 708 In type (1) (a) buffer, where we have a con- tinuous rod and enlargement back of the piston for the c 'recoil plunger, in order to have a sufficient effective buffer area, the diameter of the plunger must be necessarily large as compared with a spear buffer. Therefore, to maintain a void displacement in the recoil not exceeding the free recoil displace- ment during the powder pressure period, we must have a very short buffer. Hence if A - effective area of recoil piston (sq.ft) A'-effective area of recoil piston on c'recoil plunger side (sq.ft) L b = length of plunger or buffer (ft) effective area of the buffer If further d = diam. of recoil brake rod D = dia. of recoil brake cylinder d b = diam. of buffer chamber we have d b * C b D where C b depends upon constructive considerations [(l-Cg)D-d r ] and A b = 0.7S54(dg-d)-0.785 (C D-d) Now to reduce the buffer pressure it is de- sirable to make L b as long as possible and A b as small as possible. To do this we must make d r as small as possible as compared with D. This re- quires a large effective area for the recoil brake. Hencs in type (1) buffer we may reduce the buffer pressure by reducing the recoil brake pres- sure. If H O = the total potential energy of the recuperator we have 709 F - V ( m k _ x) (ft> lbs) (k=1 . 3 a pp rox .) = the ratio of compression where Pyf A v = effective area of the recuperator piston (sq.ft) V Q = initial volume of the recuperator (cu.ft) then for the mean buffer pressure, ire have W -(W r sin0+R p )b A b L b = 0.785 Now (D*-d)(CgD-d*) - 2 L (l-Cg)D*-d If we assume C b = 0.7 roughly, we have A b L b =0.785(D 2 -d* hence H o -(W r sin0+R p )b 6 0.785(D*-d*)E where b = length of rscoil (ft) E free displacement in the recoil during powder period (ft) H Q = potential energy of the recuperator (ft. Ibs) R Q = total friction (Ibs) since we have p b = [B +(W r sinJO+R Q )] - P h E where P h = total hydraulic brake pull (Ibs) p b = assumed intensity of pressure in hydraulic cylinder (Ibs/sq.in) Therefore, to decrease the buffer pressure, with a type (1) (a) c'recoil regulator: (1) Lower the max. pressure in hy- draulic braka cylinder during the 710 recoil. (2) Decrease the length of recoil (3) Decrease the potential energy in the recuperator. We see that the above expression is fixed by the free recoil displacement E during the powder period. BY PASS PIPES In order to loner the buffer USED WITH LARGE pressure on counter recoil, when SPEAR BUFFERS. the c 'recoil regulation is by a short spear buffer or plunger, it is often necessary to in- crease the diameter of the plunger materially over that of the rod. By the introduction of a by pass and valve (closing on the counter recoil Heading from the buffer side of the recoil cylinder to the outer end of the void chamber of the buffer, the pressure back of the recoil piston (on the buffer side) can be effectively lowered without a full void by being required in the recoil cylinder to take care of the sudden withdrawal of the buffer plunger during the first part of the recoil. Let w a = required area of the by pass pipe T * total powder period (sec) tg=tirae of travel through void during the re- coil (sec) E = recoil displacement daring powder period (ft) A = effective area of recoil piston (sq.ft) A'=effective area of recoil piston on plunger side (sq.ft) S * recoil displacement during void (ft) Lfc - length of buffer (ft) Pg = mean pressure in the rear of the recoil piston (Ibs/sq.in) p 1 - max. pressure in the rear of the recoil piston (Ibs/sq.in) 711 Now the total quantity of oil that Bust pass through the by pass pipe, becomes, Q A(L^-S)-A'L^ (cu.ft) After the gun has recoiled the void displace- ment, the void bade of the piston, i.e. the plunger side of the recoil piston, becomes gradually filled with the further recoil. The pressure in this rear chamber however is zero until the chamber be- comes completely filled. If X s is the displace* ent in the recoil when this chamber is just filled', obviously, A(X S -S)=A'X S hence X s = A^T 7 "" Let t xs = the corresponding time in the recoil. We have two cases: (1) IThen X s < E: (2) Where X s > E. For case (1), t xs and X s are connected by the equation, p ob (txs-to)* XsX fo *[V fo - f- ""-'o)- 7-77-7-; Kt M -t )(ft> r 6 "r (V f~ 7 fo ) (approx) from which by trial values we may estimate t xs For case (2) we aay compute t xs from 2K(X S -E) T * "here V X8 V? -xs K ~r V r Vf approx. = max. recoil velocity (ft. sec) K * total resistance to recoil T = total time of powder period To calculate the mean pressure in the chamber back of the recoil piston, - Dvl pj = where D = density of the fluid- 53 Ibs/cu.ft. v m = the mean velocity in the Q pipe and v. = ; r ft/sec. 713 here Q A(L g -3)-A'L b w a area of pipe (sq.ft; t b ~t xs = tine of travel through the recoil displacement Lb~X g Now t b is the time for the recoil displacement L b , hence m r (V r -V b ) here / 2K(L b -E) V b /?| "r To calculate the maxifflum pressure in the chamber b.ck of tb. pl.t..,- D( ,- A , )V . "* "' " lbs/s <- ft - where, when x g E _ __-.^ _ / 2K(X g -E) v xs = ' V t _ From the constrained velocity curve, we may cal- culate p' during the displacement (L g -X s ). Since p >3 p o approx. we may assume p 1 constant and use the previous expression for p n . It is important here to note that the recoil throttling must be modified to maintain a constant pall on the brake. Ph-pA-p'A 1 (Ibs) D A v oi "* p-p' (Ibs/sq.ft) 2gC'. Combining these two expressions, we may solve for the required modified recoil throttling area w x (sq.ft) in terms of the known values p' and P b . It is important that the recoil brake function at least at the end of the powder. We place, there- fore, S B, then X s = 713 2K(X a -E) = ./ u* _ xs m r and t b = T+m r (V r -V b ) 2K(L S -E) V b = A - usually p 1 should not m r exceed a. few hundred Ibs/sq.in. and the value of w a and A 1 should therefore be corresponding. In such a case no material effect in the recoil throttling is obtained and a modification of the grooves is unnecessary. DESIGN OF SIDE PRAMS The loading on the GIRDERS. girders and the correspond- ing stresses depends upon the method proposed for firing. These methods may be classified as follows: (1) Firing from semi fixed base plate, with a large pintle bearing and the girders extending to the rear supported at their end by an outer circular track. The horizontal and a part of the vertical reaction is transmitted to the pintle base a. o i i -- ; ~J : ;; plate, the horizontal reaction being taken up by a vertical spade extend- ing below into the ground from the base plate and the vertical load being balanced by the upward re- action of the ground on the base plate. No balancing moment is assumed to be exerted by the ground on the base plate. This assumption, makes it possible to readily determine the upward normal reaction of the outer circular track. We have, therefor*, 714 with this method of loading the horizontal and vertical reaction at the pintle bearing and a vertical reaction at the tail of the girder balancing the trunnion reaction due to firing. This loading should be considered at both horizontal and maximum elevation. (2) Firing from the pintle base plate assumed bolted down to a concrete base. In this method no outer track for supporting the tail of the girder is necessary. We have therefore at the pintle bearing a horizontal and vertical reaction, together with a bending couple balancing the firing reactions at the trunnions. This loading should be considered at both horizontal and maximum elevation. (3) Firing from a special layed track, the nount recoiling in translation on this track. By this method the vertical load is somewhat distributed by several shoes brought down in con- tact with the track. The horizontal component due to firing at the trunnions is balanced by the total sliding friction equal to the weight of the mount plus the vertical firing component times the coefficient of track friction and the inertia resistance of the mass below the trunnions to ac- celeration. Though the horizontal reaction on the trunnions is theoretically slightly reduced due to the acceleration of the cradle in which the gun recoils, we may practically consider that the total firing load is brought on to the 715 trunnions, since the acceleration of the total mount backwards is relatively small and the mass of the cradle quite negligible as com- pared with the large mass of the main girders, trucks, etc. below the trunnions. The vertical com- ponent due to firing at the trunnions is balanced by the upward reactions on the various shoes. Finally the couple produced by the horizontal reaction at the trunnion and the resultant of the inertia resistance and the shoe frictions, is balanced by a couple produced by the vertical reaction at the trunnions and the resultant normal or vertical reactions of the track or guides on the various shoes. This requires a uniforaiity increas- ing upward reaction on the various shoes towards the rear. The load- ings should be considered at horizontal and maximum elevation. (4) Firing directly from trucks riding or recoiling back on the rails, This loading is similar in character- istics to (3) except now the sup- porting reactions are concentrated at the truck pintles. Again the loadings should be considered at horizontal and maximum elevations. When a girder is designed to meet all four requirements in the methods of firing, we have for the two elevations, eight types of loading to be considered as applied to the girder. Knowing then the loads brought on to the girde, we have, the following points to consider in the layout of the girder as regards its strength. 716 (1) The proper flange area to carry the requisite bending at a section of given depth. (2) The proper depth of girder for all other sections. (3) The proper cross section of the webs for carrying the total shear. (4) The proper pitching of the rivets for carrying the longitudinal shear . (5) A careful study of web reinforce- ments or stiffeners. (6) The distribution and design of cross .beams or transoms connecting the two girders. (7) The detailing and design of the pintle bearing. (8) The reinforcement in the web re- quired for the elevating pinion bearing. Reactions between tipping parts and girder trunnion reactions: 2H=K cos 0+E cos 6^ (Ibs) (1) 2V-K sin 0-E sin 6^ +W t (lbs)(2) and for the elevating gear reaction. Ks+P b e E = ; In battery (Ibs) (3) J Ks+ r b cos E * Out of battery (Ibs) (4) j where H and V * the horizontal and vertical com- ponents of the trunnion reaction (Ibs) K = total resistance to recoil (Ibs) E = elevating gear reaction (Ibs) j radius from trunnion axis to line of action of elevating gear reaction - with rack and pinion = radius of each (in) 717 6 = angle between j and the vertical. S perpendicular distance from line through center of gravity of recoiling parts and parallel to bore to center of trunnions (in.) e = perpendicular distance from axis of bore to center of gravity of recoiling parts. With a balancing gear introduced between the tipping parts and girders, we must modify the trunnion reaction to include this reaction. The elevating gear reaction is not changed, since the moment of the tipping parts about the trunnions is always balanced by the balancing gear in the battery position of the recoiling parts. Since it is usually customary to locate toe trunnions along a line through the center of gravity of the recoiling parts parallel to the bore, S =0, and therefore, p e E = "-; in battery /X X>0 \ W r b cos 9 g m . out of battery j Now since e is usually made very small, P b e W r b cos and may be neglected as compared with J j H and V. Hence, we will assume the elevating gear reaction to be negligible, and we have the total firing load brought onto the girders at the trunnion. Then 2H= K cos : approx. reaction between 2N=K sin0+W t : tipping parts and girder. Reactions between base plate and girder. Considering the reactions on the base plate, if it is considered that the ground can offer no bending resistance as in assumption or method (1) 718 OF 9.1-f . e> Fig. 719 1 A/ ^-; //-* ta \*-*7^Vv~ ...... 1 \AS. 1 S^ VVy X ^ VA> rfr, , 3 ^^ i -c i -c J J7^^ ."" L fl- T i..l Fig. 3 720 of loading we have the reaction between the base plate and girder as equivalent to:- (1) A vertical reaction through the center line of the pintle bearing * V (Ibs) (2) A horizontal reaction at the pintle Hp (Ibs) (3) A couple H p (h-h p )(in.lbs; where h = height from ground to trunnions (in) hp = height from pintle bearing to trunnion (in) In method (2) of loading we have the reaction between the base plate and girder equivalent to:- (1) A vertical reaction through the center line of the pintle bearing? V P (2) A horizontal reaction at the pintle H P (3) A couple resisting the over- turning moment * H h P Constructively, only the horizontal reaction is taken up at the pintle bearing, the vertical or normal reactions being taken up at the travers- ing rollers. Thus, the roller reactions are equivalent to a couple H h p and a resultant vertical or normal reaction V To calculate the individual traversing roller reactions we proceed as follows: Consider the rollers equally spaced around the periphery of the roller path. Then, taking loooents about the front outer or end roller in the direction of the axis of the bore, we have, for the various roller reactions, see fig. (2). Assuming "n" chords passing through a pair of rollers and perpendicular to the axis of the bore projected in a horizontal plane, then, 721 p t =k(x t +y ) p a k(x a +y ) ---- p n =k(x n +y) Taking moments about the front roller k[2x t (x t +y )+2x 2 (x 2 +y ) --- 2x n -i+y =+x n(*n + y> = H h p +V p r Simp lif ying, we have ky(2x t +2x 2 -- 2x n _ 1 +x n )+k(2x*+2x 2 --- 2x*_ 1 +x*) - H h p vV p r and for the summation of the vertical reactions, ky+Wk(x t +y)+2k(x ? +y) -- 2k(x n _ a +k(x n +y )=V p 2k ny+k(2x t +2x 2 + --- 2x n _ 1 +x n )=V p To solve, we note that, A(ky)+B(k)=H h p +V p r C(ky)+D(k)=V where A =(2x i +2x s --- 2x n _ x +x n ) B = C = 2n D =(2x Knowing x^x --- x n we may readily obtain P o P t --- p n To compute x^x -- x n for the rollers, we bave for the angle to the various chords, 2n 360 6 * radians or - degrees n _ 2n 360 . 9 -2 rad. or 2 - degrees a n n n 2* n 360 9 n = 2 ~ rad ' r 2 ~ tbereforexr(l-cos 6^) (in) x a =r(l- cos 9 2 ) (in) x n = r(l- cos 9 n ) (in) where r = radius to the center line of the roller path. 722 Proa the previous equations we nay now compute P , P t - P n (lbs), the individual roller reactions. The previous formulae, assume contact betiieen each roller and the roller track under maximum firing conditions. If the roller path has a small diameter, we nay have the condition, when, only the rear roller is brought into contact, the over- turning moment on the girder being balanced by a couple exerted by the base plate an upward re- action at the rear roller contact and a downward reaction at the front circular clip contact. If the circular clip has a radius approx. equal to that of the roller path, then we have for the sax. roller reaction Hphp+Vpr=2p max r H VV .av* - where r = radius of the roller path (in) P >ax = max. roller reaction (Ibs) V p =aax. upward reaction at pintle (Ibs) External forces exerted on the girder during firing: The external force or the girders are shown in plates A and B for the four methods of loading. In method (1) of loading, we have the re- actions of the tipping parts H and V, the reaction of the base plate H and V together with the couple Hp(h-h_) and the reaction of the outer track on the tail of the girder N. Further we must include the total weight of the girder which though actually distributed we will assume con- centrated at its center of gravity at horizontal Ig from the axis of the trunnions. 723 Taking moments about the pintle bearing, H h +H(h-h_)-Nl =0 hence v * Hh N = ^ (Ibs) *n where H = K cos 5 and h = the height of the trunnions from the ground (in). Knowing N we may compute for the strength of the tail of the girder, for method (1) of loading. In nethod (2) of loading since we are detail- ing the strength of the girder in the region of the trunnion and pintle reactions, we must take the actual components of the reaction into con- sideration. These consist of the trunnion and elevating arc reactions of the tipping parts, that is the reactions H,V, and E and the reaction of the base plate consisting of the various roller reactions and the horizontal reaction of the pintle as shown in "Reaction of Base Plate on Girder" diagram, In method (3) of loading, where the mount slides back on a special constructed track, we have for the reactions on the girder. (1) The H and V components of the trunnion reaction of the tipping parts. (2) The inertia resistance of the girder, resisting tne acceleration of the girder acting at the center of gravity of the girder = dt (3) The weight of the girder acting at its center of gravity Wg (4) The normal reactions of the track shoes N a and N b (5) The frictional or tangential com- ponents of the track shows n(N a +N b ) 724 In calculating the stresses on the various portions of the girder we must of coarse consider both the weight and inertia as distributed forces, but for dealing with the overall reactions, we may assume their resultant effect as concentrated force passing through the center of gravity of the girder. When the trucks are entirely disengaged in this method of firing, we have, d 2 x H-nUa+M-"^ dT 1 = 3nd N a +N b =v+V *g when tlie trucks are not disengaged but hang from the girder, we must consider both their weight and inertia reaction, hence if W t jj = weight of trucks (Ibs) M t - = mass of truck we have, H-n(K a -KT b )-(o) g +m tk ) - and N a +N dt* To compute N a take moments about N b (see fig. (4). d*x *.(Vl b ) + H h-Vl b -Vl b -W g (l b -l b )- m g (h-h g )-0 hence H*X )+ ~ (h ~ " H * N = - -- - (Ibs) a l a +lb and for Nb talcing moments about N a , we have - . (h-h g ) hence , - - (Ibs) In method (4) of loading, we have the mount recoiling back directly on the rails, and the trucks react on the girder with reactions H a , N and Hjj, N b , at the truck pintles a and b . The 725 tipping parts react on the girder with components H and V at tne trunnions. In addition we have the inertia resistance d 8 x nig - - resisting the ac- celeration and the weight of the girder both acting through the center of gravity of the girder. For a horizontal motion back along the rails, tie have d x H -(H.+Hv)- m- - - = and normal to the ** U t %*.* I rails, V+W g -(VN b ) = To calculate H a and H^ the horizontal components of the truck reaction we must consider the trucks separately. In firing directly from the rails the trucks are usually braked. If W^jj and M^ = weight and mass of either truck W w and f w = weight and mass of a pair of wheels I= w k a = moment of inertia of a pair of wheels about the center line of the axle. d = diameter of a car wheel k = radius of gyration of a pair of car wheels (=0.7 d approx.) N W = normal reaction at base of car wheel N. = normal reaction of brake shoe on wheel * per pair of wheels f w = coefficient of rail friction * f s = coefficient of brake shoe friction R w =tangential force exerted by rail on base of car wlieel Now for the motion along the rails, we have, d a x H a - S R w = m tk j^ Considering the rotation about the csnter of gravity of a single wheel we have, 2m w k 2 d 2 x 4m w k 2 d 2 x 2N s f s wbere n = no. of pair of wheels per 726 (3) Of LOAD/NG 1 te. v< j I M A//* MfT/iOD (4)0f O4D/MG 1 I/ ^I^K ^^-= A/- s-\ ^T f X -^HH -9++ Fig. 4 727 SfCT/OA/ A-B o o o O O O T I e-j ^* I: Fig. 5 728 4ra w k d*x truck and likewise H b =(m tk +n )j^+SN s f s The tera 2N g f s is difficult to calculate since it depends upon bow hard the brakes are set. If the brakes are set to skid the wheels, no rotation occurs, and we have ( ja x ZN f Assuming f B =0.2 and 2N w =Hg+V+W tk we have, d a x H a +H b =2n tk + 0.2 ("g+V+W tk ) and therefore d * x H-0.2(W t +Y+ tk ) - = - froffl which we may easily dt 2m tk +m g calculate the horizontal inertia loading for any position of the girder. The reactions at the truck pintles, become res spectiveiy, b* g^ b 1 g' + d a x m g (h-h g )-Hh /i u _ \ d 2 x ' B tk * ~+ Q - 2 N a dt 2 Ubs; (Ibs) *b Hh+Vlj+Wgdg+1 5 * d t 2 _ . flVio^ \liOS ) +0.2 M b (Ibs) Comparison of Truck Pintle Reactions. In method (3) and (4) of loading we find N b -N a = -___ Now in general the horizontal resistance is small as compared with the inertia resistance nu r Hence we may approximately assume H=m g 7 8 dt 2H(h-h g ) Kdt therefore N b -N a = Further, we are not la + ^b greatly in error in 729 assuming h g = - then we have, N b -N a = " i 1 a* 1 b That is the difference of load thrown on the rear and front truck respectively equals the horizontal trunnion reaction times the height from the trunnions to the horizontal center line through the truck pintles, and divided by the distance between the trucks. Obviously as the gun elevates H decreases, while V increases; therefore at max. elevation the loadings on the trucks are more nearly equalized. With railway carriages, since at maximum elevation h H is relatively small compared with N a or N b , for all practical purposes we may consider that the re- quired strength of the girders must be equally strong on either side of the trunnions. CHAPTER XI. GUN LIFT CARRIAGE. Single recoil systems where the recoiling mass does not translate in recoil parallel to the axis of the bore, appear in various types of mounts. Illustrations of such types nay be found in our model 1897 Barbette mount, where the gun and top carriage fora a single recoiling mass, recoiling up an inclined plane. Railway carriages especially in France bare been used, where the recoiling mass, (gun and top carriage ) recoil on a gravity plane mounted on the car. The object of the inclined plane is to return the piece by gravity into battery. Carriages with no recoil except the slid- ing back of the gun and top carriage as a single mass on rails have also been extensively used, the resistance to recoil being merely the friction offered by the rails or slides. CHARACTERISTICS OP Due to the fact that the INCLINED PLANE recoil is not along the CARRIAGES. axis of the bore, during the powder period, a component of the total powder force normal to the inclined plane or slides is introduced. This component therefore introduces large stresses in the carriage, the component increasing with the elevation. The ex- cessive stresses thus introduced at high elevation, prohibits the use of this type of mount for firing at high elevations especially for large calibers. The type of mount is useful for where the elevation is not great. With large size howitzers this type of mount would necessarily produce a very heavy mount for strength and, therefore, from the point of view of mobility alone could be regarded as none else than poor design. 732 Since the gun recoil is not along the axis of the bore a reaction on the projectile normal to the bore is introduced. This reaction reaches a maximum closely at the maximum elevation. It possibly introduces unequal wear on the rifling in the gun tube itself. This reaction further introduces a slight spring during the powder period on the elevating arc and pinion. APPROXIMATE THEORY OF Even, for a very close RECOIL, NEGLECTING approximation the reaction NORMAL REACTION OF of the projectile normal PROJECTILE ON BORE. to the bore during the powder period has a very snail effect on the recoil, though it is of importance in estimating the maximum elevating arc reaction during the powder period. If, then we let Pfc = total powder reaction on base of projectile, in Ibs. B = hydraulic braking of recoil cylinders parallel to inclined plane in Ibs. R = total friction of the recoil in Ibs. w r and m r = weight and mass of recoiling parts (in Ibs) = the angle of elevation of the axle of the bore 6 * the inclination of the inclined plane. E = displacement of free recoil during powder period (in ft.) T * total time of powder period (in sec.) Vf= velocity of free recoil (in ft/sec) K = the total resistance to recoil, in Ibs. b = length of recoil, in ft. Then considering the recoiling parts during the powder period, we have, u V P b cos(0+6)-(B+R+W r sin9)= r r and since dt KB+R+W r sinO 733 734 P b cos(2J+9)dt then / " r but P h cos(0+9)dt / - = V f cos (0+9) n> r therefore at the end of the powder period, we find KT V r =V f cos(0+9)- (1) m r KT 8 and X r 'J) cos (0+9)- - (2) 2n j, During the remainder of the recoil, we have j m r V* = K(b-X r ) (3) Substituting (1) and (2) in (3) and simplify- ing we have m r Vfcos 2 (0+9) K = - (4) b-(E-V f T)cos(0+9) Obviously V f cos(0+6) and E cos (0+9) are the component free velocity and displacement parallel to the inclined plane. EXTERNAL REACTIONS ON THE If we consider the sys- RECOILING PARTS AND TOP tern, of the gun w g and CARRIAGE ROLLER RE- recoiling top carriage ACTIONS. w c , we have by D 1 Alemberts ' principle, considering inertia as an equilibriating force, the following external reactions:- (1) The powder reaction along the axis of the bore P b (2) The inertia force of the re- coiling mass, opposite to the motion during the acceleration, and in the direction of the motion during the retardation and parallel to the inclined plane d*x r 735 (3) Weight of the total recoiling parts W r (4) The normal reaction of the rollers E N (5) The braking pull exerted along the axis of the hydraulic brake cylinder B (6) The total friction along the roller track R These forces are shown in fig.(l) Pesolving (1), into a couple and a single parallel force through the center of gravity of the recoiling parts and combining with (2) we have,(l) and (2) equivalent to, A powder pressure couple P b d where d = the perpendicular distance between the center of gravity of the recoiling parts and the axis of the bore . A component parallel to the inclined plane through the center of gravity of the recoiling parts dy P b cos(0+9)m r =B+R+Y r sinp=K and a component normal to the inclined plane through the center of gravity of the recoiling parts P b sii)(0+9) Thus (1) and (2) reduce to A couple P b d and the parallel and normal com- ponents through the center of gravity of the re- coiling parts, K and P b sin(0+9) To reduce the couple P b d and the consequent stresses, the center of gravity of the recoiling parts should be located at the axis of the bore, or slightly below to ensure a positive jump. Since the center of gravity of the gun is at the axis of the bore, the top carriage center of gravity should also be located at the axis of the bore. This is impractical, but if the top carriage is made light as compared with the gun, its effect in lowering the center of gravity of 736 B I 1 'I' T 'i 737 the total recoiling parts is small. To compute the roller reactions on the' in- clined plane, we proceed as follows: Taking moments about tlie front roller reaction "0", we have Kh r +P b d+P b l r sin(0+9)fW r (l r cos9-h r sin8)-Be=N i l j +N 1 ---- N n l n (5) where h r and l r are the coordinates normal and along the inclined plane of the center of gravity of the recoiling parts with respect to the front roller "0" e = the moment arm of B with respect to "0" N n l n = the moment of the n th roller reaction about "0" When the top carriage is light as compared with the gun, the center of gravity may be assumed approximately at the trunnions and therefore P^d^O Hence (5) reduces to, Kh t +P^l t sin (flf+6 )+W r (l t cos6-h t sin 6 )-Be* N^+N^ ---- N n l n (6) where h t and l t are the coordinates of the trunnion with respect to the front roller "0". Further, we have, P b sin (0+0) +W r cos 9 =N o +N i +N 2 ---- N n (7) If we assume the roller base is rigid, we "have Bk(l+c) N Therefore if, SM Q = H t l t = N 2 l g ----- N n l n ZN = VV*. -------- N n we will have, M Q =T<(1*+1* + 1 ------ 1) kc(l i + l f + l 3 -- l n ) (8) ZN*k(l 1 + l a ------ l n ) + (n+l)kc From which we determine "k and c EXTERNAL REACTIONS ON THE If we consider the MOUNT AND TRAVERSING ROLLER system consisting of REACTIONS. the gun, and top car- riage, that is the recoiling parts, to- gether with the "bottom carriage which rests on a circular base plate supported by traversing rollers, 738 we nay eliminate the Mutual reaction between the recoiling parts and bottom carriage since it has no effect on the equilibrium of the system. Further by the use of D'Alembert's principle we may again regard the inertia resistance of the recoiling parts as an equilibriating force. We have therefore as before, (1) The powder pressure couple P b d (2) The total resistance to recoil through the center of gravity of the recoiling parts and in the di- rection of the recoil K (3) The weight of the system W s (4) The pintle reaction balancing the horizontal component of (2) (5) The traversing roller reactions. Let W s = weight of system 1 3 - moment arm of W s in battery about rear traversing roller lg = moment arm of H S at recoil X or b from battery W bc = weight of bottom carriage lb c = moment arm of tf^c ?bout rear traversing roller W r = weight of recoiling parts l^. = moment arm of W r in battery about rear traversing roller b = length of recoil The moment of the weight of the system changes during the recoil. If we take moments about the rear traversing roller, we have for the weight during the recoil WpdJ-Xcos 6 )*W bc l bc = s l s hence W s l g = W s l s - H r X cos 6 and when X = the length of recoil b, we have W s lg = W s l s -W r b cos0 Further if, h^ and 1^1 are the vertical and horizontal battery coordinates of the center of gravity of the recoiling parts with origin at the rear traversing roller then the out of battery coordinates become and (l-b cos 9) respectively. We have 739 for the nonents about 0, in battery W_l_-P K d-Khi _-_._--- 8 S D cos 6 - K l r sin 6 + P b sin(0+e)(icos 6- h r sin 6)+2N i l j +2N s l t N n l n (11) and in the out of battery posit.ion W g l s -W r b cos e-K(b r +b sin 6) CO s 6-K(k r - b cos 6)sin 6 * 2 N i 1^2N 2 l a N n l n (12) If we assume the center of gravity of the recoil- ing parts at the trunnions, then P b d disappears, and h = h{ and 1^ 1{ As before N Q kc, N =k(l t +c) N n =k(l n +c) hence 9 II 1 j. O W 1 Ml -l*fO12j.Ol2 1\ / 1 *5 \ 2N i 1 t +2N a 1 . W * k(2 1 i* 2 l l ' ~ 1 V (13 ^ We also note that PbSin(0+6)cose-Ksin 6+l 8 = EN (14) where EN=k(2 l t +2 1 2 (21 n _ 1 +l n )+2kcn From equation (13) and (14) we may solve for k, and c and thus eonpute the roller reaction NQ,^ INTERNAL REACTIONS With gun lift mounts the TRUNNION REACTIONS, trunnions are a part of the gun itself and are located at the center of gravity of the gun. Neglecting the normal reaction of the projectile, and taking moments about the center of gravity of the gun, that is about the trunnions, we have, E j = 0, (j-eonent arm of E about the trunnions), there- fore the elevating arc reaction E = 0. If X t and Y t are the components of the trunnion re- action, parallel and normal to the inclined plane, respectively, w- = the weight of the gun alone. We have, considering the gun alone, fig.( dV )-WgSin 9 - m g - I (15) 2Y t =P b sin(0+6 )+W g cos 6 dV but Pb cos (0+8) - K = m r - dt 740 dV "a a g hence n g = P b cos (0 + 9)- K 1 dt m r m r Substituting in (15), we have "ff 2X . p. cos (0+9K1 ^)+K -* - W e sin 6 1 flj ID \ (16) 9 which gives us the components of the trunnion reaction. The resultant trunnion reaction, "becomes, S t - / X+Y$ (17) The elevating arc reaction is zero, except during the first part of the powder pressure period. To compute this "whipping action" during the powder pressure period, we must plot the moment of the normal reaction of the projectile about the trunnions as the projectile moves along the "bore. The normal reaction of the projectile, equals, N =m sin (01-6) (18) dt The weight component normal to the bore "being neglected since we will assume a fairly large breech preponderence, but dv P^cos (0+6)-K P b cos(0+9) dt m r m r If U = the travel up the bore U t * the distance from the center of the projectile in its initial position to the center of the trunnions. Then, the elevating arc reaction becomes, N (U-U t ) E -* j 741 P b (U-U t )sin2(0+9) = - - - (19) 2<" r j From a plot, the maximum moment was found to oc- cur, when the shot reaches the muzzle, and we then have for the maximum elevating arc re- action, mP ob (U -U t )sin2(0+e) E = - (20) breech when shot leaves muzzle) n iic 27 P| ax , /,, 27 p max. C = U( --- 1)+ / (1- -~ - )-i * (twice 16 p e 16 p e abscissa cf max. pressure) P e = (pjj =total max. powder force on breech) V Q = muzzle velocity; P e = mean powder reaction on breech Ifr - travel up bore in feet REACTIONS ON TOP Neglecting the elevating arc CARRIAGE. reaction during the powder period, the reactions on the top carriage reduce to the following: (1) The trunnion reactions divided into Xt an: * ^t an< * equal and opposite to the component re- actions exerted on the gun. (2) The weight of the top carriage acting through its center of gravity --- W c (3) The braking pull reaction --- B (4) The roller reactions of the in- clined plane. Assumng the center of gravity of the top carriage at the trunnions for convenience, we have dV 2X t -W c sin 6 - B = m c 742 2Y t +W c cos 6 = N (22) and taking moments about the front roller reaction, we have 2EM Q = 2X t h t -2Y t l t +W c cos 6 l t -W c sin 6.h t dV - Be - m c h t (23) dv [P b cos(0+9)-K] where " s ~ " at ro r h t and l t are the coordinates of the trunnion with respect to the front roller "0", and normal and parallel to the inclined plane. 9 = the perpendicular distance from the front roller to the line of action of B If, as is usually the case, the center of gravity of the top carriage is not located at the trunnions, we have equation (21) and (22) the same, but equation (23) modified to:- P b cos(0+8)K M .2X t h t +2T t l t +W c coa 6 l c -W c sin 9 h c -[ ] c h c -Be (24) where l c and h c are the coordinates of the center of gravity of the top carriage parallel and normal to the inclined plane and with origin at the front roller. As before, the moment of the roller re- actions 2M o =N t 1 i +N 8 1 a' l ' N s 1 3 N n 1 n therefore 2M o =k(l+l|+l* l*) + kc (l i + l a +l a l n ) SN = kl t +kl a kl n +(n+l)kc and N n =kc, N k(l + c), N =k(l +c) N-^kdn+c) O ' I 1 ' 2 fc that is solving for k and c we determine N t S^ N n knowing the total normal. Substituting in Eq.(24), "g m g m r m r 2Yt a Pb sin (0 +e )* w g c o s e and noting that, nigl t +m c l c m r l r 743 m g h t +ID c h c = m r n r we have,P b [ (h t -h r )cos(0+e) cos 6 - Be = 2 M Q (25) Now (h t -h r )cos(0+9)+(l t -l r )sin(0+9) is evidently equal to the perpendicular distance between the center of gravity of the total recoiling parts and the axis of the bore. Hence (25) reduces to P b d+P b sin(0+9)l r +K h r -W r h r sin 8 + W r l r cos 6 - Be = Z M (26) where d =(h t -h r )cos (0+6 )+(l t ~l r )sin(0+9) This is evidently the same as equation (5) obtained in the consideration of external force on the re- coiling parts. REACTIONS ON BOTTOM CARRIAGE. The reactions on the bottom carriage consist of the following:- (1) The braking pull exerted along the axis of the hydraulic recoil cylinder. (2) The roller reactions normal to the inclined plane. (3) The horizontal reaction exerted by the pintle bearing. (4) The supporting reactions exerted by the traversing rollers in a vertical direction. Evidently (1) and (2) is the reaction of the top carriage on the bottom carriage, which is divided into the components (1) and (2). Thus in battery, the moments of (1) and (2) about "0" the point of contact of the front roller reaction of the inclined plane reduce to ZM o +Be but ZM +Be=P b d+P b sin(0+9)l r +Kh r - W r h r sin 8+ r l r cos 9 where i r and h r are the coordinates of the center of gravity along and normal to the plane of the re- 744 coiling parts with respect to the front roller. Therefore during the powder pressure period the reaction of the top carriage on the bottom carriage is equivalent to, (1) A powder pressure couple "P b d" (2) A component of the powder force normal to the inclined plane and through the center of gravity of the recoiling parts n P b sin (0+0)" (3) The total resistance to recoil parallel to the inclined plane, and through the center of gravity of the recoiling parts "K" (4) The total weight of the recoil- ing parts through the center of gravity of the recoiling parts ii ui " w r During the pure recoil or subsequent retardation, we have, 2M +Be=Kh r -W r h r sin 9+W r l r cos 6 and therefore the reaction of the top carriage on the bottom carriage, is equivalent to (1) The total resistance to recoil parallel to the inclined plane and through the center of gravity of the recoiling parts K. (2) The total weight of the recoil- ing parts. To compute the horizontal pintle reaction, we have H K cos -P b sin(0+9)sin the total normal reaction on the traversing rollers, be- come ZN*P t sin(0+9)cos 9 - K sin 9 + 1 "r' f *bc where W^ c = weight of bottom carriage If further 1 * moment arm of r in battery about rear traversing roller x recoil displacement from battery l bc = "oment arm of W bc about rear traversing roller W s = weight of entire system above traversing rollers 745 l g = moment arm of W g about rear traversing roller Then, for the moment of the weights about the rear traversing roller, we have, W r (lp- x cos 0)+*bc 1 bc = lf 's 1 s ~ w r x cos ^ Therefore, for the moments about the rear travers- ing roller, we have ZM =W s l s -W r x cos 0-P b d+P b sin(0+6)[ (l-x cos6)cose -(h+ x sin8)sin Q] -K( (h+ x sin8 )cos9-(l-x cos6)sin8] When P b is a maximum x is negligible; therefore for the maximum roller reaction, we have ZM = W s l s -P b d+P b sin(0+9)[l^cos8-h;sine]-K[h^cos6- EXACT THEORY OF RECOIL Doe to the normal reaction CONSIDERING NORMAL of the powder charge and REACTION TO BORE OF projectile during the travel PROJECTILE. up the bore, the recoil is more or less effected, de- pending of course on the weight of the shell and powder charge as compared with the weight of the recoiling parts. Let Pb = powder reaction on breech of gun P_= powder reaction on base of projectile P e - mean powder reaction in bore of bin N ^normal reaction of projectile to axis of bore N" t = normal reaction of powder charge to axis of bore N = N + N = the total normal reaction of powder charge and projectile to axis of bore. B + R = total braking resisting recoil parallel to inclined plane, w and B = weight and mass of projectile w r and m r = weight of mass of recoiling mass w and m = weight and mass of powder charge = the angle of elevation of the axis of the bore 746 9 * the angle of inclination of the inclined plane x 1 and y* = coordinates along and normal to tbe axis of the bore. a = travel of the projectile along tbe bore or relative displacement along tbe axis of tbe bore x = tbe projection or component of the absolute displacement of the projectile parallel to the inclined plane Considering the motion of the projectile, we have d*x ' P p - n + ng sin N = - sin (0+6)+g cos 16 (2) dt where dx' du dx ,., -. ,_. T - T - ? <*) ) for the motion of the powder charge, " P P = [ ~ 2 cos(0+e > 1+5 sine) N * i r d*x' d*x = ytr; -- r-T CO8 (0*8)kg sin J (4) * dt* dt 5 ^ sin(0+e)*Ig cos A (5) 4 1 where . 2 (6) Is the resultant acceleration of the center of gravity of the ponder charge, and for tbe action of tbe recoiling parts, P b cos(0+e)-N sin(0+e)- r g sin 6 -(B+R)- P ^ Nbere N H t +N, Combining the above equations, we have 747 I.dx' I d*x ()*-; ... cos(0+o;- - Trcos 2 (0+6 ) + (m+m)g sin0cos e, at* 2 dt* d*x (0+9)-(m+I) sin* (0+9)-(m + [B)g cos0sin (0+6 )-ni_g sin9 dt dx -(B+R)= r Expanding and simplifying, we obtain d'x 1 5 d g x _ I d'x . d a x -(m+I+iB r )g sin 9 - (B+R) * m r - - U t- that is n . ,d*x * d*x . m. d fl x =B+R*(B+I+ r )g sin 9 (8) It is to be noted that d*x' d*x [ cos (0+9)- -rsin* (0+9)] is the projection or dt * dt component of the re- suotant acceleration of the projectile parallel to the inclined plane, and B+R+(m+m r +in)g sin 9 is the total external force parallel to the plane. Neglecting gravity and with free recoil(B+R=0) f that is no extraneous force acts, hence we have Ma In terms of the relative acceleration * * d t dx' du dx . since i.d"u . dXT . m.d*x [ (.*- Jcos (0+8 ) + ] (+) hence (*n + ")T-T cos(0+9)=(m+B+ r )7f7 (10) 2 dt 2 dt 1 and by integration <*? ) J7T cos (0+9)=(m+iii+in r )^ (11) cat a t 748 (a+-)u cos (0+9)=(a)+m+m r )x (12) Hence (10, (11), and (12) gives us the free ac- celeration, velocity and displacement up the inclined plane with respect to the corresponding function up the bore of the gun. Again considering equation (8) and substituting dx' du dx = - - cos 10+9) dt dt dt we have m d*u d*x ( m+ )T~7 cos(0+9)-(B+R+lm+fii+m r ]g sinQ )-(m+m--in..) r = 2 at* * d t hence d t x dt u B+R(m+S+ni )g sin9 (13) dt* n+I+m p Integrating, + du B+R+(m+m+m r )g sin 6 =( - )-- coa(0+9)-[ - = - ] t (14) dt and m B+R+(m+i+m r )gsin 9 x = ( = - )u cos(0f9)- -[ - = - ] t * (15) B-HB + nij. m+m+m r which are the general equations of constrained re- coil during the travel of the projectile up the bore. Neglecting m and ro as small compared with m r and if we let m then dx B+R+m P g sin 9 V f co 8 (0+ot)-( - - - )t dt a B+R+B p gsin6 x = E cos (0+) - i ( - 1 - ) t * (15') "r which are sufficiently approximate for ordinary 749 calculations . HUMERICAt COMPUTATION. 10" Gun Barbette K - braking force cos (0+6)* (0+8) * 190 2 b+T V f cos(0+6)-E cos(0+6) 89.820(29.76x0.9455)' 2 (+0.0446x29. 76x0. 9455-0. 9455)32. 2 L & 247000 Ibs. 89 . 820 (29 . 76x0.9976)' 5Q 274000 Ibs, 32.2x2( +.0446x29.76x0.9976-0.9976) 1 o W r 89820 Ibs. V f * 29.76 ft/sec, b * 50 in. T 0.0446 sec. E 1 ft. Zero elevation 9 4 TRUNNION RIACTIONS. 0*19 750 Y - F sin(0+6)+W g C o S 6 fl K x F cos (0+e(i- -SO+K -*.r f i sin e M r m r K (W r sin 6+B) - 247000 Ibs. K = P m xA -32000*78.54 = 2513000 Ibs. W g = 76830 Ibs. As a check, we may consider the forces external to the system above the rollers. F cos(0+9)h t +F sin(0+e)L t +W r cos 8L t -W r sin h t -[P cos (0+e)-K]h t -Be * ZM 2376000 "31 73,656,000 2513000 " .3256 * Ib * 12,273,500 89820 x .9976 x 15 = 1,344,000 87,273,500 89820 x .0698 x 31 * 194,000 2376000-247000 x 31 66,000,000 240700 x 12 2,889,000 69,083,000 ZM 18,190,500 moment of the rollers ZN=F sin(0+6)+W r cos 6 = 2513000* . 3256+89820* .9976 907600 Ibs. total normal load on the 7fift^0 X * 2376000(1- 2)-76830x. 0698^247000 343570-5.363 + 211284 = 549500 Ibs. 751 Y = 818000+77000 895000 Ibs. 5Z5.000 in C\J GUN Sectional Modulus=195 Force on the trunnions * * 10 s X30.2+80.1 * 10 /1. 103 = 1.050000 Ibs. 525000*. 3. 375 S = - = 9080 Ibs/sq.in. fibre stress 195 ROLLER RBACTION. about front roller, F cos0-K 752 B K-W r sin9 B 240700 2M (o) Xh t +YL t +W c cos 9 L t -W c sin 9 n t - c ^f b t - Be d'x dt 549500x31+895000xl5+13000x.9976xl5-13000x .0698*31 - (2376000 - 247000) 130 x 31 - 89820 240700x12 * 18185000 moment on V+W c cos 9 the rollers. 895000+13000X.9976 - 907960 Ibs. total normal load on rollers jigACTIOM OK TH8 TRAVgRSING BOLLIR8. F sin(0-e)+W t W s -Ksin 9 (P cos 0-K)cos 6 +F sin (l--6) wt. of reooiling part 89000 Mt. of the rest 53000 2513000X. 2588+89000+53000 (-247000). 0698 650000+89000+53000-17200 = 774,800 (2513000x9455-247000). 9976+2513000X. 2588 2124000 + 650000 * 2774000 667,600 53,000 AO"O^V ,- I r^r ~r~r T L V V )jh L i4 l n 20 rollers 117,000" 753 SECTION MODULUS 537, 50,000 M( )-722000x80+53000x65 61450000 inch/lbs. moment on the rollers. M K(2 1J+2 If lg)+Kc(2 l t +21 t 2 ln-x+1,,) V XN-K(2 l t +2 l f 2 l n _ 1 +l n )+2Kc 6145000K(2x2"73.+2x974 +2x2072 +2x34.4 +2x49,. 5 +2x64.9 +2x78.8 +2x89.6 +2x96.4 +99 )+Kc (2x2.3+2x9.4+2x20.2+2x34.4+2x49.3+2x78.8 +2x89.6+2x96.4+99) hence 6145000=73600K+990Kc 774800" 990K+ 20Kc 61430000 34000000 73600K+990KC 49000K+990Ko 27450000 24600K Hence 8 - 1180 Reaction on the last roller 99x1180=117000 Ibs. Force due to rifling and its effect on the travers- ing chain. F r t = Iw rt MK*n rt 2 M 754 v 5 in. t - .0162 . 60S 32.2 R = .8 r = 4 rifling 1 turn in c5 caliber 1 turn in 250 inches * 20.83 ft. " * 2 " 606x4*x770x2 77 "" 606x16x770 32. 2x5x. 0162x12 238500 Ibs. 32. 2x5x. 0162*12 7465920 - 312984 Torque 238500x4 95400 ft. Ibs, 95400x sin 15 * 9540Qx.2588 = 24700 ft. Ibs. " TR sin = 24700 T D x 24700 - 2500 Ibs. 49 Tension on the chain at pinion 2500 500 Ibs. * . 755 VELOCITY OF FHCC HKOL mnfi offnojfcn.f-X3.7i TfVnfLfD BY RECOLIH(,f*KTf DURING z.ssr re. KccotLiNs mxrs OfPHOJECr/tt /O-INCH BARBETTE CAfff?/AG MODEL OF /893 THE EFFECT OF. THE TRAVEL OF THE PROJECT/LE UP THE BGHE ON THE ELEVAT/MG ARC CHAPTER XII. DOUBLE RECOIL SYSTEM. OBJECT In order to reduce the reaction of recoil on a carriage to a moderate value when the caliber is large a long recoil is necessary. A long recoil requires long guides and in addition is usually prohibitive due to breech clearance necessary to avoid a great loss in stability due to the overhanging of the recoil- ing weights at low elevation when the gun is out of battery, etc. A long recoil may be avoided by the use of a double recoil system and the stability of a railway or a caterpillar carriage at the same time increased. This latter factor is the real distinctive value of a double recoil system over a corresponding single recoil system. It is important to note that a caterpillar or railway car braked with a single gun recoil system is essentially a double recoil system, the ground or rail offering a tangential reaction which corresponds to the reaction of the lower recoil system. Obviously when a top carriage moves up an inclined plane under the recoil reaction of the gun and the resistance of the lower recoil system or when with a single recoil system the cater- pillar or railway car runs back on the ground or rail under the recoil reaction of the gun and the resistance of the ground or rail, the recoil reaction of the gun becomes different and the throttling grooves must therefore he necessarily different, then with a single recoil system when a constant recoil reaction is imposed between the gun and top carriage. 757 758 CLASSIFICATION. In the design of a double recoil system it is desirable in order to simplify calculation and secure uniformity of stresses throughout the recoil to have both the upper and lower recoil reactions constant throughout recoil. However, in ordnance design it has been customary to mount single recoil mounts, gun and top carriage together on caterpillars, etc., and for augmenting the stability to allow the top carriage to recoil as well up an incline plane, the inclination of the plane being sufficient to bring the systeu into battery after the recoil. The recoil reaction of the upper system can there- fore, with a double recoil no longer be constant since the recoil reaction is the sum of the air reaction, a function of the relative displacement between the gun and top carriage, and the throttling reaction which is a function of the relative velocity. Therefore, with a constant braking on the lower re- coil system, to ascertain the displacement of the top carriage up the incline plane, it would be necessary to carry on a somewhat elaborate point by point integration for the various dynamical equations and displacements at each point of the recoil. Hence in the following discussion we will con- sider the dynamical relational- CD With a constant resistance for both upper and lower recoil systems. (2) With a given upper recoil system and a constant resistance for the lower recoil system. APPROXIMATE THEORY FOR (1). Reactions and velocity for double recoil systems: Let P = resistance of gun recoil system W or w r * wt. of recoiling parts (upper) 759 O/V SYSTEM Fig.l 760 WgOr w c = wt. of top carriage and cradle (lower) V = initial velocity Z * displacement of gun on carriage, i. e. = relative displacement N = upper normal reaction between recoiling parts and top carriage M - lower normal reaction between top car- riage and inclined plane. X = total run up on inclined plane. / or v = velocity of combined recoil t = corresponding time for combined recoil = angle of elevation of gun 6 = inclination of inclined plane. Since during tbe powder pressure period, there is no appreciable movement of the top carriage up the inclined plane, and the timeaction of both the upper and lower recoil reactions is negligible as compared nith their time actions in tbe pure recoil period after the ponder period, we may as- sume the recoiling mass to have an initial velocity V at the beginning of the recoil, where wv + 4700 V = 0.9 ( ) "r where w = weight of projectile w = weight of charge v = muzzle velocity and 0.9 is a constant to allow for the effect of the recoil reaction on the recoiling mass during the powder pressure period. Consider now fig.(l) Tbe retardation of the recoiling parts is the vector difference of tbe velocities at the end and beginning of tine t divided by "t", that is v-V a - hence assuming axes parallel and normal to the guides of the upper recoiling parts, we have the following equations of motion for the recoiling parts, 761 g snd ,_ ^ x ^ w r v sin(9+0) N-W r cos0 = -5 ' (2) g t Since tliere is no roation the cou pie between the recoiling parts and top carriage need not to be considered. Next considering the motions of the carriage above, we have, along the inclined plane: H, i O D _ . C 1 P cos(0+9)-N sin(0+6)-W c sin 6 - R = - (3) and normal to the inclined plane, N cos(0+6)+W c cos 9 +p sin(0+9)-M = (4) If, after the recoiling mass and top carriage are brought to a common velocity, we consider both as a single mass in motion neglecting the effect of the further motion of the gun on its slide, the common mass brought to rest by a constant force H. g Hence the retardation after time t, becomes, R g a r = -j *r and the interval of common retardation, r c becomes, ^ + ^ t r = and the corres- yj +w "g ponding displace- t. r c ment - a r t r = v * Therefore, the total dis- "g placement (since the top carriage is uniformly accelerated to a velocity v at time t)becomes. Since the relative displace- ment equals the absolute displacement of the gun parallel to the guides minus the displacement of the top carriages parallel to the guides, we have V+v cos (0*9) v cos (0+9) V Z = t t hence Z * - t 22 2 762 763 ENSRGY SQUATION FOR Let x 1 and y 1 = the co- DOUBLE RECOIL. ordinates parallel and nornial to the gun axis. x and y = the coordinates parallel and normal to the top carriage inclined plane. v x t = -j- t where x x = the displacement of the top carriage up the inclined plane at the instant when the re- coiling mass and top carriage icove at common velocity v, hence x t = - t . Then for tne recoiling parts, we have, (P-W r sin0)x'= i [ V*-v 2 cos 2 (0+6 )] (!') (In direction of upper guides) and (N-W r cos)x t sin(0+9) = ^ro r v 2 sin 2 (0+6) (2 1 ) (at right angles to upper guides), and for the top carriage alone, we have [P cos(0+9)-N sin(0+6)-W c sin 6-R]x t = \ M c v 2 (3 ' ) (Top carriage up plane) Subtracting (3 1 ) fros (I 1 ), we have P[x l -x t cos(gJ+e)]-w p sinefx' + N sin(0+6)x t +W c sin 8.x t +Rx in.fV'-v^os 8 (0+6)]- - v 2 S f 2 t and substituting (2) in (4), we have P[x'-x t cos(0+9)-W r sin0.x'+ ^m r v a sin 2 (0+6) +W r x t cos 0sin(0+Q)- ~ m r tV*-v a cos 2 (0+6)]+ i m c v 2 + w c sin 9.x+R x t = Now the relative displacement between the gun and top carriage becomes, Z = x'-x t cos (0+6). Hence the above expression reduces to PZ-W r [x' sin0-x t cos0sin(0+6)]+ |m r v a + % c v*+W c sin e.x+R x t = | m r v a (7) Now W r [x 'sin0-x t cos0sin (0+6)] is evidently fhe work done by gravity on the recoiling parts and W c sin 9.x is the same for the top carriage. In terms of the relative displacement Z, the work done by gravity on the recoiling parts may be obtained by consideration of fig. (2) From fig.( 2), we have, x 'Z+x t cos (0+6) 764 and the work done by gravity on the recoiling parts becomes, W p (Zsin0-x t sin 6) = W r tx'-x t cos (0+9)sin0-x t sin 6] *W r (x 'sin0-x t sin0cos 0cos 9+x t sin0sin 9-x sin 6) * W r t* 'sin0-x^sin0cos0cos 6+x sin 6 (sin*0-l] = W r (x 'sin0-x t sin0cos 0-x t sin 6 cos 2 0) = tf r (x'sin0-x x sin(0+9)cosen Hence equation (7) reduces to, PZ-W r (Zsin0-x t sin 9)+W r x t sin 9+i(m r -nn v where x t = - t . Further since R(X-X I )= 7( > _ v - 2 ^ 6 ' Usually Z and x are given. Hence, the unknowns are V, P, t, v, K and R; therefore a complete solution is possible. A final check may be made by substitution in the energy equation: PZ -w r (Z sin0-x sin 9)+w c x sin 9 + i(m r +m c )v*+Rx i = ^m r V a (7 1 ) where x t = - t or in the form PZ-w r (Zsin0-x sin 9)+w c x sin 9 +Rx = 7"i r V* In a preliminary layout for a double recoil system, the limitations are usually the length of upper recoil, that is the total relative dis- 766 placement between the gun and top carriage, and the total run up the inclined plane. A direct solution of the various reactions in terms of these given quantities is especially useful. a a h * cos(0+9) b = M g 1 = sin(0+6) n r cos(0+9) c = - g n = i r sin 9 d * n r cos2f "c g * g w r sin(0+6) f - g 2 2Z (6) t = same as (5) gives t directly b v (7) p a+- - c - f(p.V) same as (1) t t (8) N = d + f - f(N.V) same as (2) t (9) hp - IN- N-R - g- f(p.N.R.V.) same as (3) (10) X > + p f(R.V) same as (4) 2 R Elimination N (11) hp - Id - i^ v - n - f v = R (8) in (9) t t 767 f (P.R.7) pV 2 ' i- 5 v from 10 f (R.V. ) o Elimination R (12) If+g (13) P , -- *-- v * -_ 2 n xh- v 2 f(P.V) elimination of P b cv ld+n Id + g ~ - * ""-' ,,_ x aht bhx bhV chx p ch (15) ahx v ~t 2 T T" ~ x < dl+n ) t(dl+n)y x fl+g + (fl+g)V+ -V 2 -pV*=0 2 ch + fl*g _ aht b_h chx _ t(ld--n) ^ x(lf-t-g) 2 2 2 2 t 2 t bhx + ahx + - x(ld+n) = This equation may be put in the form a 1 V 2 +b ' V+c '=0 w_cos 2 (0+9) w r +W r w r sin a (0+6) W P i C w a 1 = + + 2g 2g 2g 2g w r W r +W c W c or a 1 = - + hence a' = 2g 2g 2g Solution of b ' aht Z " t + = w sin cos (0 + 6) - = sin(? cos(0+8) 2 V V 768 bh *r v "r v 2g "r 003(0+8) = r^- cos (0+6) ^ * - " - ;TT - * TT" cos 2 (0+6) a 2gZ 2gZ 0Q t w r cos0sin(0+6) 2Z r - - = ~2Z - - = cos0sin(0+9) 2V V nt sin 9 V v xg _ xVw c t 2gZ hence xV w r Zw r w r V W C Z xVw c - sin9 + cos (0+9) sin9+ - 2gZ V 2Wg V 2gZ simplifying xV Z w V (w r +W p ) sine (w r +W c )+- c -cos (0+9) = b 1 2gZ V 2g 2g + ahx + x* r sin0cos (0+9) bhx w r^ + = x cos (0+9) t 2gZ - dlx = - x w cos 0sin(0+9) - xn * - x W c sin 9 xw r V a xw P [sin0cos(0+9)- cos0sin(0+9)]+ cos(0+9)-xW.sin9 2gZ - xw r [8in(0+9)cos0-cos(0+e)sin0] - xw.sin 9 + cos (0+9)- x W.sin 9 2gZ 769 xw r V 2 C 1 = cos(0+9) - x sin 9(w r +W c ) ..V 2 cos(0+9)- x sin(w r +W c ) (w r +W,J( - - sin 9)+ cos(0+e) r c 2gZ V 2g As an example of the solution of these equations and a calculation of the prime reactions, the 240 m/m Schneider Howitzer was taken with the top car- riage moving up a plane inclined at 6 with the horizontal and with 40" upper recoil and a total of 30" recoil up the inclined plane for the lower recoil. Muzzle velocity V n 1700ft/sec, Travel up the plane x 30in. Length of Recoil L 40 in Angle of elevation 20 Angle of plane 9 6 Weight of carriage W c 11,500 Ibs, Weight of gun w r 15,800 Ibs, Weight of the charge W 35 Ibs. Weight of projectile w 356 Ibs. Relative displacement Z=L- 40 in. wV m +4700 w ^356+1700+4700+35. V 0.9( ) = 0.9( ) w r 15800 6070+1640 7710 = 0.9 = .9 = 44 ft/sec. 158 158 sin(0+9)=sin 26 = .4384 cos(0+9)=cos 26=. 981 sin 9 = sin 6 = .1045 sin = sin 20 = .342 cos = cos 20 = .9397 770 2,22 . 1045) . , .891 44 64.4 = 27300C.5124 -,0081)+9620 13770+9620 b 1 = 23,390 , 64.4x40 .891 - 2.5 x 27300 x .1045 - 317406 - 7132 c 1 = 310,274 23390 V = 310274 2Z 2x3.33 t = = = .1515 from (5) V 44 13.2652 27300 2 2.5 x .1515 + - - from (4) 64. 4R 13-2652 74459.408 R 74459.408 R = = 49800 1.4952 13.2652x.4384 N 15800 x .891 + 490 x - from (2) 1 bib = 14080 + 18810 N = 32890 P - 15800x,342 + 490 44 ~ 13 ' 2652x - 891 from .1515 5404+490x212. P - 109,500 t - 0.1515 ft/sec. V 13.2652 R 49,800 N = 33,000 P = 109,500 771 As a final check on the calculations, the values obtained were substituted in the energy equation. The slight discrepancy between the two sides of the equation is due to numerical approximation. v 8 i PZ+ (M r +M c ;+x .W c sin e +x'R= -M r V+w r (Zsin0-x 'sine ) Ct x 1 = t = 1.0048 365000+74, 5000+1200+50, 000= 490, 700 475,000+16,400 = 491,400 dev. 1.42X EXACT THEORY FOR CONSTANT Let x 1 and y 1 = the RESISTANCE ON BOTH UPPER coordinates along and AND LOWER RECOIL SYSTEMS. normal to the axis of the bore (upper recoil coordinates ). x and y = the coordinates along and normal to the inclined plane (lower recoil coordinates) m r and r = mass and weight of recoiling parts m c and w c - mass and weight of top carriage plus cradle v = velocity of any instant along inclined plane. ?' = absolute velocity of recoiling mass along axis of bore. t = time from beginning of recoil = angle of elevation of gun 9 = angle of plane E = free recoil displacement for upper recoiling parts during powder period T = total time of powder pressure period p = resistance of gun recoil system 772 N = upper normal reaction between recoiling parts and top carriage R - lower recoil resistance parallel to inclined plane n = the coefficient of sliding friction p b = total powder pressure on breech at instant t, Hence t 1 = time of common recoil v = common recoil velocity for both recoil- ing parts. xj = absolute displacement in the direction of the bore to where the recoiling masses move with common velocity x = corresponding displacement up inclined plane at common velocity Z = total relative displacement between upper and lower recoiling mass. Vf = free velocity of recoil (See "Dynamics of Recoil") RlJ = counter recoil buffer resistance for upper recoil system. Considering now, the motion of the upper recoil- ing parts, we have , P b -P+w r sin0 =m r - (1) dt dv N = w r cos 0=m r sin (0+8)- (2) From (1), we have that is, t Pfcdt (p-w r sin6) / --- t (p-w_sin0) V. -- - - t = v' (3) m r now, when t = T V^= Vj hence at any time after T, we have (p-w r sin0) V f i -- t = v ' 773 Integrating again for the upper recoil absolute displacement, we have t (p-w_sin0) Vfdt - 2m r but tit / V f dt = / V f dt + / Vfidt o o T Hence ( P -w_sin0) x' = E + Vft(t-T) t (4) 2 r which gives the absolute displacement of the upper recoiling parts along the axis of the bore. Considering now, the motion of the lower re- coiling parts, ne have, p cos (0+6) - JT sin (0+8) - w c sin e-R (5) ' dt Substituting N from equation (2) into (5) and simplifying, we have p cos(0+9>-w r sin(0+9)cos0-w c sin 9-R=[fli c -m r sin*00+e)~ dt (6) p cos (0+9 )-w r sin (0+9 )cos0-w c sin9-R Hence v = [ ] t (7) m c -m r sin*(0+9) and the corresponding displacement up the plane, becomes pcos(0+9)-w r sin(0+9)cos0-w c sin0-R x - t = ] t . (8) 2[m c -m r sin(0+9)] The relative velocity between the upper and lower recoiling parts, become v r =v'-v cos(0+9) (9) and the corresponding relative displacement x p =x'-x cos (0+9) (10) When the upper and lower recoiling mass move to- gether with a common velocity, v r = 0, hence v'v cos (0+9) hence we obtain the time t 1 for the 774 comaon velocity, from p-w_sin0 pcos(0+9)-w_sin(0+9)cos0-w ft sin9-B ?ff , ( - - ) t * [ n c - r sin*(0+9) cos (0+9) t 1 simplified, we have p-w r sin0 Pcos(0+9)-w r sin(09)cos0-w r 3in9-R r m c -m r sin(0+9) (12) cos (0+9) As a check, the time t 1 for attaining the common velocity of the upper and loner recoil masses, we may equate the components of the absolute velocities of the upper and loner recoiling mass parallel to the inclined plane. Considering the motion of the recoiling parts parallel to the inclined plane, we have v V f cos(0+6) - - cos(0+6)t'+ sin(0+8)t f r r since now the reaction N has a component N reacting on the upper recoiling parts parallel to the inclined plane. Let Nt* = w r cos t 1 = r sin(0+9) v hence D "r +9)- -2- cos(0+6)t'+ sin(0+6 )cos0t '+sin* r "r "r (ef+9) v -- sin 6 = v 0+e) = Vf ,(0+6)- S- cos(0+6) t' + ~ m r m p cos0-sin 9] t Let sin(0+9) cos(?-sin 6=(sin0cos0+cose)cos0-sin9 775 * sinfcos0cos9 + cos* sin 9 - sin 9 = sin2fcos0cos6 + sin e (cos'0-l) = sin0cos0 cos9 - sin 9 sin* = sin0cos(0+9) bence p-w r sin0 v cos(0+9)=Vi - ( )t ' r Substituting for v and reducing, we have, as before t , _ !J _ p-w r sin0 p6os (0+9 )-w r sin (0+9 )-w c sin8-R . -,.. . B r m c -ffl r sin(0+9) Knowing the value of t' and substituting in equations (4) and (8), we obtain the total relative displace- ment between the upper and lower recoiling parts, that is Z = xj- x t cos(0+9) (13) where t 1 is used in the values of x 1 and x res- pectively. The total energy of the system where the two masses arrive at the common velocity v t , becomes Z Z -(ra r +R c )v* + / Pa^Z where / p a dg is the potential energy in the recuperator. Let R^ = the buffer resistance during counter recoil for the upper recoil system, then Z / RjJ d Z = the work done by the buffer in the upper recoil system. If now we assume that the counter recoil of the upper system is completed during the recoil of the lower system, we have Z Z i(m r +m c )v; + / p d Z = B(X-x t )+ / R,J d Z = w r (X-x t )sin0+w r Zsin0+w c (X-x i )sin 9 (14) A physical meaning and relationship of the re- actions in this equation, may be had oy a con- sideration of the component dynamical equations for the parts' of the system. 776 Daring this second period of the recoil, we have for the lower recoiling mass, that [R+w c sine+Nsin(0+e)-(p a -R b ')cos(0+e)]dx = - n c dV and for the upper recoiling parts, along and normal to the boref (p a ~RjJ )-w r sin0)d(x cos(0+9)+Z] ~ m r v x dv x and (w r eos0-H)d[x sin(0+e)}*-m r vjdvj and the above equations and integrating the sum, ire have X X / R dx a + / w r sin6 dx + / (p a -R b )dZ+w r f [sin * t x (0+6) cos - cos (0 + 6)sin0] dx o v - / w r sin0 dZ=(m r +m c )r^ (since v*+ v^ 2 = v* for initial valve) Simplifying we obtain equation (14) In general the potential energy of the re- cuperator is partially divided in overcoming the work of the upper recoil buffer Z / R dZ and in augmenting the run up the inclined plane over that if there were no re- cuperator present. Hence in general Z Z ^ 2 / p a dZ > / R fa ' dZ and / p. dZ - / RJ dZ o o o o is the additional 2nergy over that Kinetic energy at common velocity which augments the recoil up the inclined plane. We nay assume with small error, however, that Z Z Z Z / p a dZ - / RjJ dZ or that / p a dZ - / R^ dZ is negligible. This does not imply that p a -R= 0. Since (p a ~R^ ) (X-x t ) cos (0+6) (roughly) is the agent by which the upper recoil energy is dissipated, When the lower recoil is comparatively short 777 and the resistance of the lower recoil system R is large, we have often a condition, where counter recoil in the upper recoil system be- comes impossible and we even have an over run of the upper recoil system. Thus assuming during the second part of double recoil, that the upper and lower recoil mass move as if one, we have the retardation dv R+ (w r +w c )sine - - * -_ and for the upper recoil- df m r +m c ing mass dv P a - w r sin 0= m r ( -- )eos(0+e) hence dt m P a -w r sin0 = [R+(w r +w c )sin9]cos(0+9) m r +m c Now if p a > w r sin + [R + (w r -rw_)sin0]cos(0+9) m p +m c Counter recoil of the upper recoil system is pos- sible during the second period of the lower recoil system. If however, m r Pa < w r sin0+ [R+(w r +w c )sin 9]eos(0+6) We have a tendency of over recoil of the upper re- coil system hence counter recoil of the upper re- coil system is impossible. For this case the energy equation reduces with exactress to i(m r +m c )v*H(X-x t )+(* r +w c )(I-x t )sin 6 (15) The velocity curve during the second period may be obtained with sufficient exactness by as- suming the two masses to recoil together, then <* R+(w r +w c )sin8a-(m r +m c )v x v and / [R+(w r +w c )sine)dJc ) (m r +ra c )v dv 778 m r +m c [R+(w r +w c )sin6] (x-x t ) = (- )(v a -v* hence /~ a[8*(. r *n )ine]U- t ) ?./.? ' m r +m c , RECAPITULATION OP FORMULAE Prom approximate FOR CONSTANT RESISTANCE TO solution with limited RECOIL BOTH UPPER AND upper and lower re- LOWER. coil, calculate P and R. Then during the powder period, we have p-w r sin9 v' . Vf ~ ( ) t ra r pcos(0+9)-w r sin(0+6)co30-w c sin9-R m c -m r sin 2 (0+9) and the relative velocity becomes v^v'-v cos(0+9) t p-w_sin0 x 1 = / V f idt - ( )t o 2m r pcos(0+9)-w r sin(0+9)cosl-w c sin 6-R and the corresponding relative displacement, be- comes x r =x ' - x cos(0+6). After the powder period during the remainder of the first period of recoil, we have p-w r sine! v'=V i( - )t 3 t and for the relative velocity v r v'-v cos(l+9) Furtber p-w.sinar x'E+V f i(t-T) - ( )t f 2m_ 779 p cos(0+9)-w sin(0+9)cos0-w.sin9-R x .[ - - 2 - ]t . 2a c - r sin a (0+9) and the corresponding relative displacement be- comes x r x'-x cos (0+8). The tine for the common velocity becomes, V f , t'= p-w p sin0 pcos (0+9 )-w r sin (0+6 )cos0-w c sin9-R ., fl . - ]cos(0+9) and the common velocity becomes pcos (0+9 )- Wp sin (0+9 )cos0-w c sin9-R V ** t- m c -H. r sin(0+9) - U ' p-w p sin0 x'=E+V f ,(t'-T)-( - - )t' -oDDj. pcos (0+9 )-w_sin (0+9 )cos0-w_sin 6-R x -f - - - ]t a 2[m c -m r sin (0+9)] and the total relative displacement for the upper recoil system becomes, Z * xj - x t cos(0+9) Oaring the second period of the recoil, ire have v*-2(R+w r +w c )sin 9(x-x t ) v = i m r +m c the upper recoiling mass being assumed locked with the lower recoiling parts. CALCULATION OF THROTTLING GROOVES As a first BOTH UPPER AND LOWER RECOIL. approximation, it will be assumed that the total friction is mainly guide friction and proportional to the normal reaction between the upper and lower recoiling parts. Then Rg* nN where n * 0.2 to 0.3 and N w_cos0 +m p sin(0+9) where v and t 1 have t' 780 been already determined. Considering the upper recoiling parts, we have Pp a +p n +Rg*p a +p_+nN hence p n =p-p a -nN. Further if the ratio of the final to the initial air pressure = m, we have Paf m and if A effective area of upper ai recuperator and b = Z = recoil displacement on top carriage, then for the initial volume we have x 7j * A a b where k = 1 to 1.41 assume 1.3 m k - 1 v i Hence p a =P a i( ) v i *a x r x r = being the relative displacement. There- fore knowing v r and x r and the total pull p, we have / -i \ "n * r Pn s P-PaiV-A > and W x i = Vj_k-nN KAnV i- A a 13. ay A h where A n = the hydraulic piston area and k the reciprocal of the throttling constant. Lower throttling grooves Knowing R from previous data, we have R A v where v is the recoil velocity up 13. 2/- plane. EQUIVALENT MASS OP ROTATING When a double re- PARTS WITH A DOUBLE RECOIL. coil system, con- sisting of two separate recoil systems is used, mounted on a railway car or caterpillar, it is customary to consider the car or caterpillar 781 sufficiently braked to allow no recoil. In fact a salient feature of the design is to make "R* small enough so that the rail or ground friction, induced by proper braking, is sufficient to balance R. Due to the complication of a double recoil, as well as the impossibility with very large mounts of taking up the recoil energy even with a double recoil system without an excessive recoil displace- ment it has been the custom to use a single recoil and allow the railway car or caterpillar to run back a limited distance dependent on the magnitude of the braking. The recoil of the car on very large railway mounts may be considerable. This greatly reduces the stresses on low elevation as well as augments the stability. In fact with such mounts stability is of no longer a consideration. When a single recoil system is used but the car or caterpillar recoils in addition, we obvious- ly have a double recoil system and all the prevous dynamical equations together with the method of computing the throttling on the upper recoil or now the recoil systems holds the same. The lower recoil resistance R is now the tangential reaction exerted at the base of the car wheels or at contact of ground and caterpillar track. In the acceleration of a railway train, railway engineers customarily allow for the rotational inertia of the car wheels by increasing the translatory mass from 8 to 10 percent. Due to the limitations at times of car or caterpillar recoil and the great variation of the magnitude of the rotational in- ertia as compared with standard railway practice it is important to calculate the exact effect of the rotational inertia in terms of an equivalent addition to the translatory mass. Consider a railway car or truck with "n" pairs of axles. Let w c and m c = weight and mass of car not in- 782 eluding wheels. and B W * weight and mass of a pair of wheels. I * D *K* = moment of inertia of a pair of wheels about the center line of the axle. d * tread dies. of a car wheel k * radius of gyration of a pair of car wheels N w = normal reaction at base of a pair of car wheel x Kg * normal reaction of brake shoe on wheel per pair of wheels f w coefficient of rail friction f a * coefficient of brake shoe friction R,, * tangential force exerted by rail oc base of car wheel p * recoil reaction N * normal reaction between recoiling parts and car * angle of elevation Now independent of rotation or any other motion, the translatory motion of the center of gravity of a system depends only on the external forces applied. Hence p cos - N sin > - 2 R w (m c +Zn w ) dt Considering the motion of a single car wheel, we have for rotations about the center of gravity of a pair of wheels hence pcos0-Nsin0-2N 8 f s =(m c *2m lf +Z - ; there- fore the translatory mass is increased by the 4mk which is the equivalent translatory 783 mass of rotational inertia. The a ass of the lower recoil system therefore, becomes 4k* m-+2m_(l+ -) and this value is to be sub- d stituted in the previous dynamic equations. The equivalent resistance for R is now the summation of the brake shoe friction,- that is R = Z Ngfg and this value is to be used in place of R in the previous dynamic equations. It is important to note, however, that the actual tangential force exerted at the base of the * : .. car wheels is not 2N s f s but ZR, - 2N g f s + 2 JT- - 943 '-is '-'-si* ->riJ-afr HoJHi srff Consider a caterpillar track and connector mechanism: Let R t total tangential track reaction between track and ground (in Ibs) R w = total tangential roller reaction on track (in Ibs) r w = radius of roller wheel (in ft) r c * radius of sprocket (in ft) r t * radius of sprocket gear (in ft) r = radius of brake drum gear (in ft) r = radius of drum of brake clutch (in ft) R * tangential reaction between sprocket gear and drum gear (in ft) TQ = torque exerted on brake drum (Ib.ft) E = mechanical efficiency of sprocket mechanism E t * mechanical efficiency of transmission between sprocket gear and drum gear. E fi - mechanical efficiency of brake drum and mechanism E N 3 mechanical efficiency of roller trucks. A = resultant normal bearing reaction of sprocket shaft (Ibs) J * resultant normal bearing reaction of 784 brake drum shaft (Ibs) f t and f f corresponding coefficients of friction B C = total mass of caterpillar excluding recoiling parts m w k = moment of inertia of roller wheel (ft. Ibs) m s k| = moment of inertia of sprocket wheel m gs k s * moment of inertia of sprocket gear = moment of inertia of drum gear 1 Moment of inertia of drum, d * the increment change in the radius to account for friction between gear teeth Considering the motion of the caterpillar, we have dv p cos 0-N sin -Rt * m c 77 The tension in the caterpillar track at the sprocket becomes, T = R t -2R w . R *" k " d * (2) B w r* dt and its moment about the sprocket axis, (assum- ing the upper track tension as nil) becomes, (R t m w k w dt Considering the angular motion of the sprocket shaft we have m w k w dv , m s kg-nn g3 Kg s dy ' (3) Further considering the angular motion of the drum shaft, we have dv (0 hence si, , 785 Where E^ takes care of the friction less in the drum of gear bearing. The friction loss between the drum gear and sprocket gear may be considered, by letting Illl * 1 li r,-d 3 E r, hence Substituting in (3), we have v dv T Q ix Rt ^o- E B r dtoE E E Errt dt where E O takes care of the friction loss in the sprocket bearing. Therefore, the track re- action becomes, k| dy (8) and substituting in the equation of translatory motion (Eq.l) we have T D r t m H k p cos 0- N sin0 = [m c + '."" + BoM. r.r E B rS ^ T D Evidently is the brake torque referred E o E t E 2 r 2 r o to a reaction at the base of the track, considering the mechanical efficiency of the gearing . The translatory mass is augmented 786 due to the rotational inertia of the rotating parts by the terra 2 m s k s* m gs k |s which is the equivalent mass of the rotating ele- ments. It is to be noted that the mechanical ef- ficiency enters in the rotational inertia since the bearing reactions depend upon the external reactions, and the moments of them in turn de- pend upon the rotational as well as translatory inertia. The effect of the translatory inertia on the rotating element in modifying the bearing reactions will be neglected, being small. Hence R in the double recoil equations is now the braking torque referred as a tangential force at the track base, that is, The actual tr ck re.action is R t given by equation (8). As a check on equation (9) ire may note that from the energy equation, we have t t D 9 ' ^dda^nifi^fid ft i p cos0-N sin0 dx = -d ( )w' +d(5- ra c v *) the reaction R t doing no work. Further, we have r, r i v de* %-r 4 - dx K = r r O a V V If W_ = r " r r o r w hence substituting these values, we find 787 T r . :t . therefore the equivalent translatory mass, to ac count for the rotational inertia becomes, gd k| d a m s k s + m gs k| s ' When the caterpillar track is heavy or there is a long space between the driving sprocket and the front idler sprocket, its inertia ef- fect must be considered. Therefore, let r = radius of drive gear sprocket (in ft) r = radius of front idler sprocket (in ft) t fc i i " m^k = moment of inertia of idler sprocket (units Ib.ft) m t = mass of caterpillar track per unit length 1 * length of upper span of caterpillar track (in ft) (ar o m t )r = moment of inertia for that part of track in contact with driving sprocket (units in Ib. ft) (nr^m t )r = moment of inertia for that part of track in contact with front idler sprocket (units Ib. ft.) T = tension at section at point of contact of lower track and drive sprocket wheel (in Ibs) T. = tension at point of contact of lower 3 track and front idler sprocket (in Ibs) T a = tension at point of contact of upper track and idler sprocket T = tension at point of contact of upper track and drive sprocket From kinematics we must have the relative velocity of the track with respect to the frame 788 dv equal to v and the corresponding acceleration where v is the translatory velocity of the caterpillar at instant with respect to the ground. Considering the lower track since at any instant it oust be at rest, we have for the dif- ference of the tensions at its extremities, Za w k w dv TO - T, - R t - j-Jj! g (10) I*** at where the second member is the reaction on the track due to the tangential reaction of the ground and the reaction of the truck rollers . Considering the angular notion of the drive sprocket shaft, we have (To -T t )r r -8 t n^> _ E_ 1 PRIMARY EXTERNAL REACTIONS WITH With a double A DOUBLE RECOIL SYSTEM. recoil system, the 791 first period when the top carriage is accelerated to a common velocity for both upper and lower re- coiling parts and a second period with a re- tardation for both recoiling masses. The reactions should be considered during both periods. External reactions during first period: By O'Alembert's principle we may regard the inertia force as an equilibrating force, then for the primary external forces of a system con- sisting of the upper and lower recoiling mass to- gether with caterpillar or railway car. (1) The inertia resistance of the recoiling mass divided into two components . (a) The inertia force parallel to axis of the tore through the center of gravity of the upper recoiling parts, p 1 or K x t (b) The inertia force normal to the upper guides through the center of gravity of upper re- coiling parts, N 1 or K v *\ (2) The weight of the recoiling mass acting vertically down = V r (3) The inertia resistance of the top carriage and cradle acting through the center of gravity of the top carriage and cradle parallel to the inclined plane opposite to the acceleration up the plane = K x or m c & * dt * (4) The total weight of the top carriage and cradle acting vertically 792 down = c (5) The reaction of the ground on the caterpillar track or the re- action of the rail on. the braked wheels of a railway mount using a double recoil, which are divided into the following components: (a) The tangential reaction of ground or rail. (b ) The normal reaction of ground or rail which is not uniform but distributed so as to produce an upward normal reaction combined with a couple. When the mount is just stable as with a light caterpillar at zero elevation (5) reduces to a single reaction about which moments are taken and therefore would not be considered for critical stability. The primary external reactions are shown in fig. (3). Considering the motion of the upper recoiling parts, we have, during the powder period, Pb-P+W r sin =m r for the acceleration of the upper recoil- ing parts parallel to the guides. And A N - W r cos0 = m r - - = m r sin (0+9) for the ac- dt * dt celeration of the upper recoiling parts normal to the upper guides. The external reaction on the recoiling parts when considered with the total mount, becomes, during the powder period, d 2 x P'=K X *P b -m r - -=P-W-sin! r dt parallel to the guides in the direction of P b . After the powder period during the first period of recoil, we have 793 P-W r sin0=-m r along the guides and H-W r cos0 - dt* dv m r sin(0+9) normal to the guides. Therefore the external forces on the re- coiling parts during the first period after the powder period, becomes. P'=K X =-m reversed =m_ =P-Jf,sin0 along the 1 dt * bore and dv tf'=Kg =m rI7 sin(0+6) = N-W r cos0 normal to the bore. Hence during the first period of recoil either during or after the powder period, we have P'=K X =P-W r sin0 along the bore downward and 1 dv N' = K V =ra r sin(0+6) normal to the bore dowaward * t d t or = N-W r cos2f. We have in the above neglected the powder pressure couple, it being at best small, witb little or no effect on stability. The inertia force of the top carriage is evidently dv C "T~ reversed parallel to the inclined plane. Hence the external forces not including (5) become (1) pl=K y =P-W r sin0 along the bore 1 dv K' + K V a ni r - sin(0+6) normal to the bore t dt (2) U r = upper recoiling weight, vertically down, d v (3) ""cJ" = inertia force of lower recoiling parts parallel to inclined plane. (4) tf c = weight of lower recoiling parts, vertically down. The external reactions during the second period: dv During the second period, m c T~ = K x reverses d t t in direction since the lower recoiling mass now becomes retarded. On the assumption that during this period the upper recoiling parts have the same notion as the lower recoiling parts, we have 794 STABILITY COMPUTATION Y c * Accf/tration of Carriage up jo/an* P * Broking Force -f=>-H/,.5"> t Any/e of C/evo+ion 6 flng/e of inc/ined P/ane s about P r~or any of her point X' r - TC r -ff'Cos 6) , Yr i. 3 795 DOUBLE REDDL SYSTEM STCHAMOND &OMM. HOWITZER (SCHNEIDER, 796 for the inertia force of the upper recoiling parts: if m r r in the direction up the plane and parallel to the inclined plane. Consideration of the forces acting when counter recoil for the upper recoiling parts place during the second period of recoil will be taken in "variable resistance to recoil for the upper recoiling parts". STABILITY FOR DOUBLE RECOIL SYSTEM. Consider- ing fig. (3) let a c = ** c dt = ac- celeration of carriage up plane. N'=ni r a c sin(0+6) P'=P-W r sin0 = resistance to recoil for upper re- coiling parts parallel to guides. W r = weight of recoiling parts * t = weight of caterpillar. IY C = weight of lower recoiling parts Let, x r y r , x c y c , and x^y^ be the respective co- ordinates for the variops weights, from the over- turning point 0, at the end of contact of the caterpillar track and ground. Then for moments about 0, for the various external forces, in battery, we have M Q = (N l eos0+P'sin0+W r )x r + (w c +m c a c sin6)x c + (N l sin0-P'cos0)y r +(m c a c cose)y c +w t x fc and for any other position in the recoil, the various coordinates of the above equation change to x r =x r (3 l cos0+3cos9), y r =y r -(S 'sinfl-SsinS ) x =x c -Scos9, y=y c +Ssine here S'=/ v rel dt, and 3 s / v c dt v rel = relative velocity between upper and lower recoiling parts v c - velocity of carriagrs up inclined plane. Further let A = W t x t +W r X r +W c X < ! B=(N'cos0+P'sin0)x r C =(N'sin0-P'cos0)y r 797 dv c D = (m c sin 6 ) x c dt dv c E = < m c~ cos 8 >yc F=A+B+C+D+E For stability, we jnust have F = 0. The critical position fir stability for the first period is at the end of the first period when the two re- coiling parts begin to move with the same velocity. The coordinates, therefore, become x r =x r -(Zcos0+x t cose ),y^=y r -(Zsin0-x t sin 9) x c sx c~ x i cose > v c =y c +x t sin9 x =dis placement up inclined plane at common velocity Z=total relative displacement between upper and lower recoiling mass. Assuming as before that during the second period the two recoiling masses move as if one, we have, for the condition of stability M o =W t x t +w c x c +W r x r~ ID c a c sin9 x c" a c a c cos0 yc" m r a c sin P 9 x - m r a c cos 6 y' r =0 where a c = ' m c +m r and the critical stability is at the end of recoil and x r =x r -(Zcos0+Xcos9); y ' = y p -(Zsintf-X sin e) x c ax c~ Xcos e > y c = yc +x sine where X is the total run up the plane. If, however, the upper recoiling parts move into battery during the second period while the top carriage still continues moving up the in- clined plane, then x r =x r -J[ cos9. STABILITY WITH A SINGLE RECOIL We have as be- AND CATERPILLAR BRAKED. fore the same in- ertia and weight moments but in addition rotational inertia couples. Since the effect of a couple is entirely independent of the axis about which moments are taken, we merely have to add in the previous 798 799 moment equation the additional rotational inertia terms, taking of course account of the algebraic sign of the inertia couple. The following inertia couples are intro- duced with a caterpillar using a simple mechanism as assumed before: Stabilizing inertia couples: JT* = drive sprocket and bear couple dv = roller truck couples dt m i k ! dv = front idler couple r dt 1 r] = track inertia moment where r = r o +r i - and 1 = total span of track 2 Overturning inertia couple:- m d k 3 +a gd k |d dv ( - *~)*.T? = drum shaft inertia couple r r *d"t Therefore tfce stability equation becomes, F = A+8+C+D+E+G+H+I+J+L and for stability P ^ 0. Where during the first period A=ff r x t +W r x r +W.XI , B =(N'cos(2f+P l sinCf)x' C * (N'sin0-P'cos0)y ' H dv D = ( ffl c"^ sin e >*c> E = (m c~J7 cos s k gs k dv "w k w dv ^ ^ ii dv mk J =[n(r + r)+2 1 r] / 8dd x dv - ( - ' r t~"~ where the coordinates r o r dt refer to point of contact of ground and track at rear end of track. During the second period, the 'inertia couples become 800 f?5 ACT/QMS ON 77PP/MG PdfTTS DOUBLE WML SYSTfM - fff/KT/OMS ON 71PPM6 f>AffT3 W B^TTfffY - ftf ACTIONS ON T/PP/MG P/4ffT3 OVT Of BdTTtffY Fig. 6 801 dv c dv c reversed, therefore, A-D-E-B r sine. x p -m_ dt dt cose y_-G-H-I-J-K=0 where 7*- is determined by dt the relation, T D r i , dv c 3 (ir r --m c +m e ) where m r =mass of recoiling parts, m c =mass of caterpillar and mount excluding recoiling parts, m e = equivalent mass for rotational inertia. ELEVATING ARC AND TRUNNION REACTION In comput- OF TH TIPPING PARTS. ing the various reactions in a double recoil system, we oust consider the inertia effect of the various parts in modifying these reactions over their static values or as would occur with a single re- coil. The primary inertia forces induced by the double recoil are: For the upper recoiling parts: (1) The inertia force of the upper recoiling mass divided into components through the center of gravity of the upper recoiling parts, parallel and normal to the axis of the bore, respectively. For the lower recoiling mass: (2) The inertia force of the top carriage and cradle acting through the center of gravity of this combined mass, opposite to the ac- celeration, and parallel to the in- clined plane. The inertia resistance of the top carriage and cradle may be divided into two parallel coapnents through the center of gravity of the cradle and top carriage respectively the 802 magnitude of the components being proportional to their respective masses. x t and y t * coordinates of upper recoiling parts parallel and normal to the axis of bore. x and y - coordinates of lower recoiling parts parallel and normal to inclined plane . K x = inertia component along bore of upper recoiling mass through its center of gravity (Ibs) K v = inertia component normal to bore of i tipper recoiling mass through its center of gravity (Ibs) K xc = inertia force of cradle through its center of gravity and parallel to inclined plane (Ibs) K xc = inertia force of top carriage through its center of gravity and parallel to inclined plane (Ibs) W r = weight of upper recoiling parts (Ibs) x r and y r = coordinates from trunnions of center of gravity of upper recoiling parts in battery, parallel and normal to bore (ft) H C = weight of cradle (Ibs) x- and y = coordinates from trunnions of t i cradle parallel and normal to bore (ft) W c = weight of top carriage (Ibs) H c = total weight of lower recoil parts (Ibs) W - ~" W A TW 1 * w" t = weight of tipping parts X^ and Y t = components of trunnion reactions parallel and normal to axis of bore of gun. = elevating arc reaction (Ibs) 803 j - elevating arc radius about trunnions or perpendicular distance to line of act on of (Ibs) B = total braking between upper and lower recoiling parts (Ibs) R I = total friction between upper and lower recoiling parts (Ibs) P = total resistance between upper and lower recoiling parts (Ibs) N = total normal reaction between upper and lower recoiling parts (Ifas) Z = relative displacement of upper recoil- ing parts wit}) respect to lower recoiling parts. Pjj = powder reaction on base of breech (Ibs) e = distance from P to center of gravity of upper recoiling parts (in) Then during the acceleration for the upper re- coiling parts, we have dx Ph (B+R -W_sin0)=m r - - along the bore and the dt* external reaction on the upper recoiling parts during the powder acceleration, becomes dx t K x =P b -m r g (Ibs) along the bore 1. U L =B+R-W r sin0 along the bore =P-W r sin0 along the bore During the retardation, B r - = -(B+R-tf r sin0) and the external reaction dt 2 on the recoiling parts parallel to the bore is the inertia force, dx t K tf - -m- - =B+R-tf P sin0 x * P dt = p-w r sin0 Hence either during the acceleration or retardation 804 the external com.ponent parallel to the bore on the recoiling parts equals the total resistance to recoil off the upper recoiling parts.. The in- ertia force normal to the bore, becomes, t t K v 3m r~TT7 (Its) Since - = v sin(0+9), where * t at* at v is the velocity of the lower recoiling parts up the plane . K v = m r & sin(0+6) (ibs) y r dt = N-W r cosd (Ibs) For the lower recoiling parts, we have dv K s = -m (Ibs) along the inclined plane *dt K xc '-rag --> (Ibs) along the inclined plane a 2 u t Elevating arc and trunnion reactions: Let us now consider the tipping parts, that is the recoiling parts, together with the cradle. By the use of D'Alemberts principle the problem in Kinetics is reduced to one of statics, provided, we introduce the proper inertia forces.. Further, the nutual reaction between the upper recoiling parts and the cradle of the lower recoiling parts:, becomes, an internal force for the system consisting of the tipping parts. Therefore, introducing the inertia forces, we have: For the reactions of the tipping parts in battery:- Along the bore: fig. (6) t +Ecose 9 +W t sin0-K xc icos(0+9)-2X=0 dt 8 805 Normal to the bore: W t cos0-E sine-K xc isin(9+ef)+K y i-2Y0 Moments about the trunnion: dx t (0+9)x c -Ej - since in the battery position the center of gravity of the tipping parts is located at the axis of the trunnions. Further, t pu-n- - =B+R-W P sin0*K, i We have, for the dt 2 elevating arc reaction, P b e+K x iy r +K v ix r +K xc i [y c icos(0+6)-x c is in (0+6)] E= - ' - -j and for the components of the trunnion reaction 2X=K x i+Scos9 e +W t sin0-K xc icos(0+8) 2XW t cos0+K y i-E sine e -K xe isin(0+6) For the reactions of the tipping parts out of battery: In any intermediate position, out of battery the entire tipping parts are displaced backwards up the inclined plane but in addition we have a relative displacement between the re- coiling parts and the cradle of the top carriage, equal to Z (in). Therefore, the moment of the tipping parts about the trunnions, become W r (l r +Zcos0)+H c il c i=Mt where l r and l c i are the horizontal coordinates of the upper recoiling parts and cradle in the battery position. Since center of gravity of the tipping parts are located at the trunnion in the battery position, we have V? r l r +W c il c i=0 hence M t =W r Zcos0. Then, the reactions along the bore 806 K x i+Ecos9 e +W t sin0-K xc icos(0+9)-2X=0 Horaal to the bore: W t cos0+Ky i-Esin6 e -K xc i sin (0+9 )-2Y=0 Moments about tbe trunnion: K,iy r -K y iX r +K xc icos(9+0)y c ,-K xc .sin(9+0)x c ,+W r x r cos0- Ej=0 Hence, we have for tbe elevating arc reaction for a relative displacement Z out of battery j and for the components of tbe trunnion reactions, 2X=K X i+Ecos6 e +W t sin0-K xc icos(6+0) 2Y=H t cos0+K y i-Esin6 e -K xc isin(6+0) REACTION BETWEEN UPPER AND In the calculation LOWER RECOILING PARTS. of guide and clip re- actions and the bend- ing stresses in the cradle it is necessary to know the nature of the reaction between tbe upper and lower recoiling parts as well as its distribution. The reaction between tbe two recoiling masses, consists of: (1) Tbe resultant braking reaction acting parallel t o the guides and through the controid of the various pulls. (2) The guide friction acting along the guides. (3) The normal clip reactions, which may be divided into: (a) a normal component per- pendicular to the axis of tbe 807 our or B/trrf/?y Fig.7 808 ft&ICTMN BTWW UPPfft 4N> tOWfff fffCOtUMG FMffrS : POUBL E fffCOfi F>OS/r/OM Fig. 5 809 bore, (b) a couple between the two parts. The magnitude of the couple depends upon the assumed position of the line of action of the normal component; therefore, we may assume the normal component in its most convenient position for calculation. Let N * total normal reaction between upper and lower recoiling parts (Ibs) N t * front normal clip reaction (Ibs) N f * rear normal clip reaction (Ibs) x t and y t coordinates of front clip re- action along and normal to bore with respect to center of gravity of upper recoiling parts (in) x c and y a * coordinates of rear clip reaction (in) M couple or moment reaction between upper and lower recoiling parts (inch- Ibs) P n *total hydraulic pull including packing friction (Ibs) P a * total recuperator reaction including packing friction (Ibs) R * total guide friction (Ibs) d b s distance from center of gravity of upper recoiling parts to P D (in) d a * distance from center of gravity of upper recoiling parts to B a (in) d r distance from center of gravity of upper recoiling parts to R n coefficient of guide friction (0.15 ap- prox. ) B * 2P n +2P a Total braking (Ibs) R = n(N t +N f )=guids friction (Ibs) l v * horizontal distance from rear roller contact of top carriage and inclined 810 plane to line of action of W r d = distance from A, normal to lino through center of gravity of upper recoiling parts and parallel to bore Then B d b ZP h d h +ZP a d a . Considering the re- actions on tbe recoiling mass in battery, we have, d t x P b -m, * =B+R-W r sin0, along tbe bore dt N=Ky i+W r cos#, normal to tbe bore M=P b e+Bd b +R d r , moments about center of gravity. Taking moments about A fig. (7) at the rear roller contact of top carriage and inclined plane, we have, for tbe moment of the re- action exerted by the upper recoiling parts, on the lower 1 M =B(d-d h )+R(d-d,)+M-N( ^ +dtan0) costf Substituting for M, its value M=Bd b +Rd r +P b e and for N, its value N=W r cos0'V? -V?- 1 m r m r . p-W r p-W_sinCf therefore Vj-Vj' 1 + ( Vfn -V f ^^-(-g- - )A t n which gives a "point by point" method for determin ing the absolute velocity of the gun parallel to the bore. Now if we substitute for the normal reaction N in (3) its value dv c N *t r cos0+in r - sin(0+6) u t we have P cos(0+9)-W r sin(0+6)cos0-B r dt dt hence dv c p cos(0+6)-W c sine-W r sin(0+6)cos0-F dt m +m_sin*(0+9) w i and between instants t n -i and t n , we have m c +m r sin(0*9) A t The total braking P, becomes P^P^Pw + P f (Ibs) In the static or single recoil, the top car- 816 riage stationary, we have P ns C o ""* "xn NOII for the same relative displacement between the upper and lower recoiling parts for the double recoil, the throttling area is the same, namely w xn , then _ v v rel v rel* Ph= c o bence P h =P hs "In * Therefore, from a static force diagram, knowing the relative displacement = static recoil displacement, we may determine P hs and v| . If '1 TS 1 has been determined for the point, the hydraulic braking is readily determined from the above equation. The recuperator reaction is determined from the static force diagram when the relative dis- placement is known. When the upper recoiling parts begin to counter recoil relatively to ths lower recoiling parts, we have v " Pf Procedure for recoil calculations We must first construct a static force and velocity diagram for the upper recoil system as would occur if the mount had a single recoil, the top carriage being fixed. Let v * nuzzle velocity (ft/sec.) u* travel up bore (ft) w = weight of projectile (Ibs) w = weight of charge (Ibs) P a =max. total powder reaction (lbs) wv then average pressure on breech P e * ~~* (Ibs) 2gu Pressure on breech when shot leaves muzzle 27 u P ob b 1.18 P B (Ibs) 4 (b+u) .here b-(2Z - l)t /I- g > -1 (ft) ^ 3 Time of travel to bore t o 3 T~ u o (sec) wv +4700w Max. free velocity of recoil V f * - (ft/sec) "r Free velocity of recoil when shot leaves muzzle (w+0.5w)v (ft/sec) Time during expansion of powder gases 2(V f -V ) w r ti = r (sec) Total powder period T = t. +t (sec) *o Free displacement of recoil during travel up bore w+0.5w Free displacement during expansion of gases P ob (T-t Q ) f -o - 3 +v fo*! b-E+V f T for variable resistance to recoil m r V|+m(b-E) 2[b-E+V f T- | (b-E)] 818 At t when the shot leaves the nuzzle . Mo V o~ V fo~ m r At t,jj when ire have max. restrained recoil velocity, Koto v nT v fn -- (ft/sec) r K t a where V fm V f Q +P ob (t m -t Q )tl -- ] (ft/sec) 4ffl r (V f -V ) K(T-t ) t m T (sec) p ob At tine T, the end of the powder period * K T V r =V f - ( ft/sec) m r K T 2 E r =X (ft) 2m r After the powder period, during the retardation, we have for constant resistance to recoil, V = / ( b -X) (ft/sec) m r for variable resistance to recoil, /(K - f(b+X-2E r )(b-X)) v = / 2_2 (f m r where b = the total length of static recoil (ft) 819 a = C s ; C s =0.85 approx.; b= perpendicular distance froa spade to line of action of K. --* Construction of static force diagram: We have, for a constant resistance through- out recoil, K*P bs +P a +Pf-W r sin0 hence Phs* p a* p f = K+W r sin0 (a constant) For variable recoil, in battery K S K O , out of batteryK =k where k = K o -m(b-E r ) and K = K Q during the powder period . =K -m(x-E r )=k+m(b-x) hence P bs +P a +P f =K +tf r sin0-fl>(X-E r ) r r Value of components Pf,P a and P . For a first approximation, the friction component becomes, P f =0.2W r cos0+p (estimated packing friction) and will be assumed constant. The recuperator re action becomes, p _p . P a =P ai + - X for springs a <* jj where P a i=total initial spring reaction P a =total final spring reaction V o k Pa =p ai ( - ) "here k=l.l to 1.3 v -v V rt =initial volume (cu.ft) rt i i 8 A v = effective area of recuperator piston (sq.in) P ai = initial air pressure (Ibs/sq.in) m o = ratio of compression (from 1.5 to 2) The hydraulic throttling reaction, becomes for constant recoil, P ns =(K+W r sin0)-P a -Pf 820 for variable recoil P ns =K +W r sin0-m(X-E r )-P a -P f where the value of P a corresponds to the displace- ment X. Construction of static counter recoil diagram: The counter recoil may be divided into and acceleration period, controlled or regulated by a throttling resistance through a constant orifice, and the retardation period where the recoiling mass is brought to rest into battery by a constant resistance to recoil, with a varying buffer throttling. If P a =tbe recuperator reaction Pf-total friction of counter recoil assumed the same as for recoil and constant. B s =static buffer reaction l o =length of constant orifice period (ft) l b =length of variable orifice period (ft) Then during the acceleration, dv v s p a -Pf-W r sin0-B^=m r v where B^= c o ( = a con- U X M stant) and during the retardation _ f* I TJ X dv c o v s B +W r sin + Pj -P a = - m r v where B^= i "* Now nay be determined by assuming a max. w o counter recoil velocity 3.5 ft/sec. at max. velocity, we have, e i P a -Pf-W r sin0- (-r)vj s = and assuming v* s , 5 c , we readily determine * G * The velocity and force curve during the first period may therefore be constructed as follows: (1) Plot the recuperator reaction against counter recoil displace- ment, that is, 821 V*, V -A v (b-X) b = length of re- coil (2) Assume P =(0.2W r cos0) (estimated packing friction) Con- stant for the counter recoil. (3) Divide the acceleration period into "n" intervals and take the mean air pressure for this interval. Then, knowing the velocity at the beginning of the inta 1 "**!* *e can compute the velocity at the end of the interval by the formula, - log (A- - ) s log(A- -j-^) p where A = P a -P f -W r sin0 c o = G and determined as outline above. (4) .Next construct from the velocity curve a static buffer against counter recoil displacement, that is c o B s '( ) v o The velocity and force curve during the re- tardation period of counter recoil may be con- structed, as follower- CD The total resistance to counter recoil being assumed constant during this period, we have B s +W r sin0+P f -P a =K v whence, / 2K v (b-x) v = / 822 A/ rig. 9 823 - m v* T* m where K y = and v m is determined from 1^ the previous point by point method to the end of the dis- placement 1 . Then, the velocity and buffer force against recoil displacement is determined, since J ra r v m v o k B ^ = J_JL_ +p ( 2 ] K -P f -W r sin0 V -A y (b-X) / 2K v (b-X) and v / where K v Dynamical equations of double recoil for point by point method of procedure for construction of re- action and velocity plots: Let = min. angle of elevation of gun P = total pull between upward and lower re- coiling parts (Ibs) P hg - static hydraulic pull (Ibs) P a = recuperator reaction (Ibs) F= total friction assumed constant (Ibs) Vj=free velocity of recoil (ft/sec) V r =velocity of upper recoiling parts parallel to upper guides (ft/sec) V re j= relative velocity between upper and lower recoiling parts (ft/sec) V* o = velocity of lower recoiling parts up plane (ft/sec) X = displacement of top carriage up inclined plane (ft) B = static counter recoil buffer reaction (Ibs) R = lower recoil reaction parallel to in- clined plane (Ibs; Then, during the powder pressure period, '+P a +P f (1) 824 P-W sin0 Vg = YJT 1 (V - Vg" 1 ) At (2) m r [Pcos(0+8)-W c sin9-R-W r cos0sin (0+8)j At v n = ya- 1 + (3) m c +m r sin a (0+9) V rel =V r -V c cos(CJ+e) (4) - ^- A t (5) + - A t (6) c "c 2 After the powder period, P-W P sin0 v n v n-i _ A t After gun begins relative counter recoil, p=p- a - )_ P ( V 8 In determining P QS v s and P a the relative displacement must be equal to the static dis- placement of the recoil, that is * re l =x s from which we determine P v 825 240 M/M HOWITZER, GAS-ELECTRIC T7PB, DOUBLE RECOIL, 24 Elevation, R 45,000 Ibs. VELOCITY DISPLACEMENT P I T T 6 c R R u n u a e e p i t t t n r 1 1 n e a a r A a t r 1 1 P i t t p a V a I i 1 * T B r g V V a 1 i r a 1 e e e n " e a k 1 7 e a e o 8 i n g e i p f t. o i o F a n o 6 n d r u e c n e 8 1 i d a During Powder Pressure Period I . OO4 151700 15.93 .703 15-332 .0307 .0014 2 .006 142400 32.849 1. 648 31.438 . 1710 .0084 3 .01 140000 41.431 3.310 38. 593 .5217 .0332 4 .012 138300 41. 351 5. 105 36.971 .9751 .0837 5 .016 128900 37.351 7.225 31. 161 1. 5201 . 1824 6 .02 117900 32. 801 9.405 25.051 2. 082 . 3487 7 .02 108000 28. 65 11.195 19.061 2.523 .5548 8 .02 96000 24.99 12. 51 14. 271 2. 856 .7919 9 .02 86500 21. 721 13.45 10. 191 3. 101 1.051 10 .02 77ioo 18.835 14. 03 6. 8O5 3.271 1. 326 11 .02 677oo 16.33 14. 23 4. 12 3.3804 1. 6091 12 .02 61000 14. 10 14.175 1.95 3. 441 1.893 13 .02 56600 12.052 13.941 .092 3. 461 2. 174 14 .002 54900 11. 854 13.911 .076 3. 46 2. 202 Gun beginning to C'Roooil 15 .01 46300 11.03 13. 601 .64 3.457 2.339 16 .01 45430 10.233 13. 263 l. 137 3.449 2.474 17 .01 42300 9. 501 12. 86 1. 519 3.436 2. 605 18 .01 38790 8.841 12. 397 1. 779 3.419 2.731 19 .01 35680 8. 243 11. 868 1.937 3. 40 2.852 20 .0 1 33550 7. 689 11. 297 1. 981 3.38 2. 968 21 . 01 33070 7. 144 10. 717 2.056 3.36 3.078 22 .01 31600 6.63 lo. 10 7 2.03 3.34 3. 182 826 24O M/M HOKITZER, GAS-ELECTRIC TYPS, DOUBLE RECOIL, ' o o r. t i nu ed) 23 .01 31*740 6. 113 9.5 2.037 3.32 3. 28 24 .01 31500 5. 6oi 8. 889 2.0 19 3.3 3.372 25 .0 U 3168O 4. 982 8. 159 2.018 3.276 3.474 26 .014 31350 4. 269 7. 299 1.981 3. 248 3.582 27 .016 31590 3. 446 6.323 1.974 3. 216 3. 690 28 . 18 31360 2.530 5-215 1.94 3. 181 3.794 29 .02 31490 1. O6 3.99 1.914 3. 142 3.886 30 .02 31520 . 481 2.765 1.889 3. 104 3.954 31 .01 31470 .031 2. 153 1.878 3.085 3.979 32 .02 31350 1.O5 1.063 1.962 3.047 4.011 33 .016 30000 1.82 . 036 1.85 3.017 4. O2 240 M/M H01ITZER, TRACTOR MOUNT, DOUBLE RECOIL. ) Elevation, R - 80OOO VELOCITY DISPLACEMENT P I T T G C R U i o o u * e P i t t t n r 1 n e a a r a t r 1 1 i t P s V a i 1 a r B g V a 1 i r e e n g e e k 6 8 i P e 1 n P t c e i. l 5 a r n o e r o e During Powder Pressure Period 1 .004 152000 15. 76o .753 15.011 .O300 .0015 2 . O04 144600 28. 578 1. 425 27. 16O . 1143 .0059 3 . 004 143900 35.505 2.087 33.428 .2355 .0129 4 . 004 14O30O 39.258 2.713 36.558 .3755 .0225 5 .004 137600 41. 235 3.3C6 37. 943 . 5245 .0345 6 . OO4 13460O 42.035 3.869 38. 185 .6763 .0439 7 .003 133500 41. 355 4. 984 36. 400 .9751 .0834 After Powder Pressure Period 8 .02 125800 36. 225 7.303 28.965 1. 6288 .2072 827 240 M/M HOWITZER, TRACTOR MOUNT, DOUBLE EECCIL. ( Cent inaed) 9 .02 110800 31. "705 8.803 22.945 2. 1479 .3683 10 .02 10050O 27.605 9.753 17.905 2.5565 5539 11 .04 91300 20.145 1O . 6 2 1 9.575 2. 1063 .9613 12 .02 74400 17.105 10.138 7.005 3. 2722 1. 1689 13 .02 69000 14.290 9.360 4. 980 3-3920 1.3640 14 .02 64600 11. 650 8.340 3.360 3.4750 1. 5410 15 .02 61500 9. 140 7.150 2. 040 5.529 1. 696 16 .02 58300 6.760 5.790 1.010 3.560 1.825 ft .023 56700 4. 100 4. 120 .0 3.5W 1.939 Gun beginning to C'Eecoil 18 .004 477oo 3.710 s.^so . . 1592 1.955 19 .004 477oo 3. 320 3.340 . .1592 1.969 20 .004 477oo 2.930 2.950 ..O05 . 1592 1.982 21 .004 477oo 2. 40 2. 560 -.010 .1592 1.993 22 . 004 477oo 2. IgO 2. 170 -.010 . 1592 2.003 23 . 004 477oo 1.760 1.780 -.010 .1592 2.010 24 .004 477oo 1.370 1.390 -.010 . 1592 2.016 25 . 004 477oo .980 1.00 .015 .1592 2. 021 26 . 004 477oo .590 .610 .016 .1593 2.021 2-7 . 004 477oo . 200 . 220 .019 .1592 2.026 OP Plane (in) Kx t Ky t K x V s F h <-HH <1 145000 45000 62COO 1180OO 2 124000 32000 46OOO 96000 4 112000 26000 38000 82000 6 10 3 5 CO 21000 30000 7000O 8 9500O 17OOO 24000 58000 10 89000 13000 18000 49000 12 82000 1OOOO 14000 400OO 14 76000 7200 11000 32000 16 7OOOO 5000 77oo 240OO 18 64200 2100 4 COO 13000 20 60000 1000 1000 130OO 22 55OOO - 2000 - 1500 8OOO 24 2000 - 3000 - 2500 5000 26 50000 - 6000 - 8500 2000 828 Gun beginning to Counter Recoil ". rrr "'. , 28 41000 - 8000 - 11000 O 30 36 42 46 38000 2*7000 25000 24800 -10000 -14000 - 1 60 -16000 -14000 -21000 -22000 -22000 12600 12*700 12800 48 24700 -15000 -21800 13000 1 Acceleration up plane. In battery 60 Elevation. v 4 .102 v e =.207 a - .102 .292 average a a =.105 .086 v 4 = .345 a = .053 a * .085 .086 ace. = 43 I/sec. .002 Out of battery 60 Elevation v v t 6.550 6.550 v a =6.350 ace. = .050 .133 a =.150 9 v 4 =6.150 .020av. = .133 13.3 I/sec. a .010 (Reversed ) In battery 30 Elevation v = .318 v = .475 v .600 v = .720 a = .318 a!= .157 a~ .125 a = .120 average * .180 ace. = 180 .002 90 I/sec.* Out of battery 30 Elevation v 1 = 8.630 v a = 8.323 v a =7*970 V 4 *7.600 a = .307 a = .353 a,= .370 123 .343 average = .343 ace. = ' * 34. sec.* .010 (Reversed ) 829 OF 830 Out of battery 30 Elevation (1) Recoiling parts along bore i 50000-15790x.542100.1bs. (2) Recoiling parts up plane acceleration - 34. ft/sec. 15780 x 34 16700 Ibs. normal comp. 16700", 91355= 32 2 15200. Ibs. 5231 Top carriage up plane x 34 3 32 .2 162x34=5510. Ibs. 5513 Cradle up plane x 34=171x34= 33.2 5820 Ibs. Stability of 240 Caterpillar. Moments taken at Elevation, Howitzer out of battery, about a point under of rear track sprocket. (1) Weight of recoiling parts 15790x59. Weight of cradle 5231 116 Weight of top carriage 5513 80 Weight of bottom carriage 5250 45 Weight of tractor 55000 128 Inertia of recoiling parts 58000 93 Inertia of recoiling parts 6700 59 Inertia of cradle 10820 86 Inertia of top carriage 11580 71 931,610 601,565 441,040 236,250 7,040,000 5,394,000 395,300 930,520 822,180 1,703,785 Inertia forces 60 Elevation in battery (1) Along bore 159000-15790 cos 30(Recoiling parts) 15000-13765-145325 Ibs. (2) Up plane acceleration = 43. ft/sec.* 831 15790 x 43=21070 Ibs. normal comp. = 21070*. 81355 32.2 * 19250 Ibs. 5231 (3) Top carriage up plane x 123. =162. x43. =6966. Ibs. 32.2 5513 (4) Cradle up plane x 12. 3=171x43. =7350 Ibs. 32.2 Out of battery 60 Elevation (1) Along bore recoiling parts = 70000-1367556325 (2) Up plane recoiling parts acceleration 13.3 ft/sec.* 790 13.3 = 6517. Ibs. normal comp. = 6517*. 91355 5950. Ibs. 5231 (3) Top carriage up plane H * 13.3 = 162 xis.3 32.2 (4) Crldle^p'plane |fi| * 13. 3 =171 x 2274. Ibs. In battery 30 Elevation. (1) Recoiling parts along bore 147000-15790". 5=139100 Ibs. (2) Recoiling parts up plane. Acceleration 90 ft/sec. 1 S790 x 90=44200 Ibs. Normal comp. 44200*. 91355 32.2 = 40300 Ibs. (3) T.C. up plane |22I x90=162.x90=14600. Ibs. J o . o 5513 (4) Cradle up plane x90=171x90=15400 Ibs. 32.2 (2) About center line rearmost roller. (llOin. from trunnions ) Weight of recoiling parts 15790." 41. 647000. Weight of cradle 5231. 97. 507000. Weight of top carriage 5513. 62. 342000. Weight of bottom carriage 5250. 27. 142000. Weight of tractor 55000. 110.6050000. 832 Inertia of recoiling parts 58000. 93, Inertia of recoiling parts 6700. 59, Inertia of cradle 10820. 86, Inertia of top carriage 11580. 71, -5394000, - 395000. - 930000. - 822000. + 147000 Direct Pads on Rollers. In battery Weight of recoiling parts 16700*. 99452 Weight of cradle 5231x. 99452 Weight of top carriage 5513*. 99452 Inertia of recoiling parts 17500*. 99452 + Inertia of recoiling parts!40000x. 10453 + Hydraulic resistance 58426, f29967-28433+140000x . 1 17500x. 10452-26534. M045J 76200. Out of battery. Weight - Inertia of recoil- ing parts Inertia of recoil- ing parts Hydraulic resistance 26534*. 99452 + 26389. 6700.x. 99452 - 6663. 58000x. 10453 + 6063. 25789. J+22400 58000x. 99453 1 1+6700X. 10452-26534. *. 10452J 78330 60 In battery, Weight 15790+5231+5513 .99452 + 26389. Inertia of recoiling parts (145324 x. 91355) + 132762. Inertia of recoiling parts (19250.x. 40674) - 7838. 166981 [6966+73 50+19250X. 91355 Hydraulic resistance <\ [I45325x.40674+26634x. 10453 24400 Ibs. 60 Out of battery Weight (15790+5231+5513). 99452 + 26389 Inertia of recoiling parts(56325 x .91356)+ 51456 Inertia of .recoiling parts (5950.x. 40674) 2420. 75425 &274+2155+56375X. 40674 "J Hydraulic resistance > >-. |5950x . 91355-26534x .10453] 32800 In. 80000 Out = 30 In. Out 60 In. Out * Weight of bottom carriage 5250 Ibs, At 30 Elevation. Hydraulic resistance. In battery. 13 9000*. 80902-40300*. 58779-14600- 15400-26594 x. 10453 114000-23700-14600-15400-2780=57520 834 /L / 112 , z \ 835 836 837 "7 \ 838 Bfi If] 35 i I >- -* *r > * 839 840 841 \ Fig. IS 842 843 IfOO 1OOO soo o soo IOOO /soo xooo 24-00 3OOO \ 5 F C* VIM - TERPILLAR HOWfT7ER WO HIO HIM mr FRMMC VA f5C LS RK -NE --^. m DEK u Z40 ) . - ' N, N / -; 5t>- ^ * X - 1 f.' tra. / , / / i ' . -- ' :.: . - n ' \ z 3 J C' 3 5 3 ? ' & 1 / /^O /. INCHES ,^ J .'- i 1 ' <0 if C 1 / s .- 10 , a ? Ho t 30 Z*0 ZfO , ; . ' ! : ' 1 \ HOTI :-!h ^. P n n toon. *. 30KHCC OP X COMPRlii - s. .h r.-. . t ! M 1-. f IN H --+ 1 stf,a (*- **1 auraneo OM tmcill fr EQUMJIEH \ * . 1 1 , EC I : 3 3 00 00 i c c c C _ i ! > i - ,il , , 3 1 s H | * " E -?N -1 A. K 9 ,, !/1 1 C , - .a.- FWH CAT M 3 5j oS L O If i II f 1 I M 5 = TOC HIT ITEBl. I. FO 4 ^ ITU) OH VI ; '- tc MUJ : guuau f I 1 844 845 346 IO N 00 L 847 ur> iZ 848 Hydraulic resistance out of battery 42100*. 80902+15200*. 58779+ 5510+5820-2780 34100* 8940 5510+ 5820-2780=51590. Ibs. THEORY FOR VARIABLE RESISTANCE From the point IN UPPER RECOIL AND CONSTANT by point method RESISTANCE IN LOWER RECOIL as previously SYSTEM. discussed in some detail, we find, that the resistance of the gun recoil system varies from its static value in the battery position, to very nearly the recuperator reaction plus the total friction of the upper recoil system, the throttling at the end of the upper recoil being negligible and therefore the hydraulic braking becoming zero in the upper recoil system; further it was found that the gun recoil braking falls off proportionally on the time. Let P S = static braking for gun recoil system = initial braking reaction on gun recoil system (Ibs) P a * final or out of battery recuperator reaction for upper recoil system (Ibs) Pf * final braking of reaction on gun recoil system (Ibs) R t - total friction of upper recoil system (Ibs) lf r = weight of recoiling parts (Ibs) H C = weight of top carriage (Ibs) V = initial upper recoil velocity (ft/sec) Z * displacement of gun on carriage (ft) N * upper normal reaction between top car- riage and inclined plane (Ibs) R * lower recoil resistance parallel to in- clined plane (Ibs) X = total run upon inclined plane (ft) v = velocity of combined recoil Rel. vsls.=0 t 1 * prime P common recoil 349 The mean braking zone for the upper recoiling parts, becomes, p .p .p r s af t 2 Further the distance run up the inclined plane during the time tj was found to be approximately X = - v i t l . The approximate equations for the double recoil, with a variable re- sistance in the upper recoil system and a constant resistance for the lower, become, ff V + to 4700 V=0.g( ) (1) "r W_ V-v cos (0+9) P -ff r sin0= t ; 3 (2) r v iS n N-W r cos(? = 5 t (3) c t P.cos(0+6)-Nsin(0+9)-w.sin 9-R= -- (4) g t 1 w r +w c X = 1 v t f + - - v* (5) 2Rg Z = - t 1 2 DBRIYATIOK OF THB PYNAMICAI. EODATIOHS POINT BY POIMT METHOD COMP UT AT I OH : Total pull between upper and lower re- coiling parts: This reaction is composed of:- v (1) the hydraulic braking pull =P ns ( )* v s (Ibs) (2) the recuperator reaction at the relative displacement under con- sideration P a (Ibs) (3) the friction between the recoil- ing parts Pf (Ibs) 850 v el Hence PPh s - +p a* p f (Ibs) v s REACTIONS ON THE UPPER RECOILING PARTS; If P b the powder reaction, then for the gun along its axis, we have, P b -m r - - P+W r sin0=0 (1) d v and normal to its axis dy' N-m r - -- W r cos0 =0 (2) Integrating equation (1), we have, t m P b dt P-W sin0 - ( - - - ) A t= A v r l - m P-W sin0 v n _ v n-i _( - i - ) A t , v n . v n f f m r r bence P-W r sin0) v n = v n-i +(y n_ v n-i)_ ( - - r r f f m From a somewhat different point of view, we have from (1) r, , , HP*W.ini0 since d * d x ~ni.. dv ' /% + cos(0+9) See acceleration diagram, dt d * x rel dv c then P b -n P t + cos (0+e)]-P + w r sin0+e dt a dt Integrating, we have p b dt P-W r sin0 m -( TO t! * v rel* v c cos(0+9)=v r hence, as before, yn _ yn~i + (yn_yn 1 \ _ m r Fro the vector disgram of acceleration, 851 d a v' dv = sin(0+9) hence equation (2) becomes, dt 8 dt dv c N-m r sin (0+6)-W C os0 = (2a) dt REACTIONS ON THS LOWER RECOILING PARTS These reactions are N and P reversed (the mutual couple having no effect on the translation) of the upper recoiling parts, the braking reaction K of the lower recoil brake and the weight and kinetic reactions of the top carriage. The normal reaction and couple exerted by the plane has no effect on the motion of the sys- tem, then, along the inclined plane, Pcos(0+9)-Nsin(0+9)-W c sin e -R-n o ~jf~ = C dv c Substituting N =m r ; sin(0+9 )+W r cos0 01 T we have ^ v Pcos(0+9)- m --4 sin 2 (f+6)-W_sin 6-tf r cos0sin(0+9) d t c -R-tD c = 0, combining terms and simplify dt ing, we have ii -p r c dt c ' i hence dv c Pcos(0+9)-W c sin9-W r sin(0+9) CO s0-R dt m r sin 2 (0+9)+ai c and between any two intervals, Pcos(0+6)-W sin9-W_cos0sin(0+6)-R j.D-Ttn-l, r . . 1. V GEOMETRICAL RELATIONS, To compute P it is necessary to compute the relative velocity and displacement respectively for any given interval in the recoil. Obviously from a velocity diagram 852 v rel =v r -v c co3(0+9) and the relative displacement v rel~ x ref + ? A l and the displacement C C up the inclined plane x = xj}" 1 + ---' - A t METHOD OF COMPUTATIOH, Knowing v -1 , vp" 1 and vjjgj at the beginning of the interval, we have, v n-i 1*6 1 P=P hs ^ ^"^ +p a* p f at relative displacement xg^J v. then v^^v"' 1 +[ - . . ,^ QX - 3A t c c m+msin 2 (0+9) and _ v n sy n-l - r "" From these values, we have v? e l = v r"~ v c and therefore v n +v n-l tf n _ x n-i v rel +v rel x rel - rel + - - - A l o and 1 After the powder period, obviously the expression for v r reduces to, P-W r sin0 m v n sy n-i _ ( - L - ) A t HELATIVB COUNTER RECOIL OP THE UPPER RECOILING PARTS: In the expression for P, the hydraulic re- action and friction reverses. If B 1 is the c're- coil buffer force in the upper recoil system at a given relative displacement, then v a. S-..-T? 853 P n = - B '("l ) i p f = ~ p f assuming friction the same. v s hence, , y rel P=P a -B'( - )~Pf (Ibs) The remaining expressions are the same as before. This method of computation is sufficiently accurate and was followed in the recoil cal- culations illustrated. APPROXIMATE CALCULATIOMS FOB STABILITT WITH A DOUBLE RECOIL. Reactions and velocity for double recoil system: P = resistance of gun recoil system W r * weight of recoiling parts (upper) W c = weight of top carriage and cradle (lower) V = initial velocity z = displacement of gun on carriage R * reaction of lower recoil system N = upper normal reaction between recoiling parts and top carriage M lower normal reaction between top carriage and tractor. X = total run up on inclined plane. v = velocity of combined recoil t = corresponding tine. = angle of elevation of gun 9 s inclination of plane. Values assumed for computation of recoil: P * R = 9 = 6 W r = 15,790 Ibs. c =ll,570 Ibs. = General equations for double recoil: W r V-v cos (0+9) P-H r sin0= [ - - - ] (1) g 854 w r v sin (9+0) N-W r cos0 (2) g w Pcos(0+9)-Nsin(0+9)-W c sin9-R - (3) [Ncos (0+9 )+H c cos9+Psin (0+9 )-M- 0] (4) v *r* w e B 2 ** 2Rg ** Pz+ 7(-^j^) v*+R | t j m r V a (6) wxv_+4700w V-0.9C ) (7) = | t (8) Energy equation: PX = ^m r [v k -? a cos a (0+e)] Indication of P Nxsin(0+9)= ~ m r v a (0+9) t r [Pcos(0+9)-Nsin(0+9)-R]x jM c v 2 0(X_-Xcos(0+9)]+ i m r v a sin 2 (0+9)+R x = -ra r t z r I i [V-? 2 cos 2 (0+9)]- i m.v 2 2 l a t Rvt i hence Pz+ -m r v a + ^v 2 * m r V 2 Further wxv m +4700* w V 0.9( ) where w = weight H r of shot, w = weight of powder W r = weight of recoiling parts v m * muzzle velocity of shot 240 M/M DOUBLB RECOIL MOUNTED OX UARK III MI CATERPILLAR. SCKMZIDER HOWITZER AT 0* ELEVATIOH OF HOWITZER. Given: W r * 15780 Ibs. weight of recoiling parts H c = 11570 Ibs. weight of sliding carriage 855 9=6 angle of inclined plane. V - 45 ft/ffi max. velocity of upper recoil- ing parts at beginning of re- coil. R * 80000 Its. resistance to recoil on lower recoil system. From static force, Diagram 240 M/W Howitzer, P s = 155000 max. pull Rt+P a f =60,000 maximum recuperator reaction plus friction at end of recoil. Approximate Calculations, P s +P af +R t P - - - whence P s =155,000 (Ibs) P af +R t = 60,000 (Ibs) |215,000 hence P c = 107,500 Ibs. mean reaction = 1.480+0.58= 2.06 ft. Z * 22.5x0.158= 3.56 ft. = 42.7 in. Check on Z by energy method: t, 27360,-- 80,000*10.4 107,500 Z = 2 ( "327F ) = ~~^i - X - I 1579 Tf ~~- ^ 32.2 107, 500Z+46, 000+68, 200=487, 000 Z-3.,56 ft. Cheek R = 80,000 Ibs. For horizontal recoil, -0= e=6 sin 9 0.1045 cos 9 = 0.9945 45-0.9945, 107 . 500 32.2 856 1S790 v N-15790 ~~ * 0.1045 - (2) 11570 v 0,9845x107, 500-NxO. 1045-11570x0. 1045-80, 000 = 32 .2 t 107, 000-0. 1045N-1210-80, 000=359 - v 25, 790-0. 1045N N-15,790 L -- ' - hence 7.11N=136,290 t ooV DJ. . ft H 19,170 Ibs 51.2 v 490 (45-0. 9945v ) 51.2 v N-!5790 107,500 51.2v-692-15.3v 66.5v=692 hence v - 10.4 ft/sec. 51. 2x10. f 3380 - 1575 Total time = T+t*. 032. 158=. 190 sec. 240 M/M POUBH RKCOIL MOOHTgD OH MARK IV MI CATKRPILLAB. "~~ APPBOIIMAT1 CALCPLATIOHS TOR 240 M/M QAS-KLECTBIC DOOBLB HICOIL 8TSTKM AT 24 1LKVATIOK OF HOWITZIB. Given. I r 15790 Ibs. weight of recoiling parts. H c * 11570 Ibs. weight of sliding carriage. 9 7 e angle of inclined plane. V 45 ft/a. max. velocity of upper re- coiling parts at beginning of recoil. R = 45,000 Ibs. resistance to recoil on lower recoil system. From static force, Diagram 240 M/K Howitzer P g = 155,000 max. pull R t +P a f * 60,000 maximum recuperator reaction plus friction at end of recoil. Approximate calculations, P -= whence P a 155,000 Ibs. 857 P R - 6 ' 000 "aft" ~ [215,000 107,500 Ibs. R = 45,000 Ibs., * 24 Elevation of gun, 6= 7 angle of inclined plane 0+6= 31 sin(0+9)=.5150 cos (0+6)= . 3572 sin 9 = .1219 W r sin0=1578Qx. 515=6140 Ibs. 107, 500=6140=490 ( 40 "' 85y2V ) (1) t N-15,790x.8572=4SOx >515 V (2) 107, 500x.8572-N. 515-11, 570x. 1219-45, 000-359 - (3) v 359 - = 93, 200-0. 515N-1410-45, 000 v _ t 46, 800-0. 515 N _ N-13520 t " 359 252 46,800-0.515 N=1.425N-19250 K = 34,000 Ibs. 252 v 490(45-0.8572 v) N-13,520 107,500-6140 252 v 22080-420v SM80 101,360 337v 4480 hence v = 13.23 ft/sec. 252x13.23 * * nn *c, n ~ 0.163 sec. 20,480 Total time T+t0.163+. 032=0. 195 sec, s_ 27360 13.23 x -(13.23xQ.195) + - - 4 2x32.2 45000 x = 1.955+1.65=3.59 ft. 43 (in) v 45 Z = - t = xQ.163- 3.67 ft. =44 in. check. 6 Z 107,500 Z * J(Z2) I3T23 2 * ^221 x 73.23 x 0.163 32.2 2 32.2 74, 500+48, 500=497, 000 858 Z 3.48 ft. The discrepancy between this value of Z of the above is due to the fact that work done by gravity is omitted in the energy equations. Theory of stability not braked. By D'Alerobert 's principle, we have K * P IB ' r dt (1) (2) where KQ * dynamic inertia resistance of recoil- ing parts * 0.9 K (assembled approximate) F = tractive force reaction r * radius of traction rim 859 d 2 9 d 2 9 d 2 9 (F-BF)+r-Rr I TTT * J TE* +! 8(7-78) w v at* dt 2 (2) (3) r' (4) (5) I t =2nr mr 2 Hence we have the following equations: (1) v d 2 x (F-2F 1 ) r-Rr =[I -I +3ar 2 (itr+l)} (2) r -^ ** (3 > ^ A * fl (4) f c\ i- ' r 2 * A t W The reaction of the truck rollers on track, = 40.4^2- i dt 4 The reaction of the clutch shaft pinion, ,i. 1.75 24 20.35 d a x d a x d a x R (-)* -- - - = 19.1 - hence R - V 1.43 5.2 6.00 dt dt 2 dt The reaction of the drive gear pinion, . 7.32 860 d*x d*x /. R(0. 2165 )-(76. 4*0. 846)- =7.32 4 at* dt d*x R (0.2165 )-(64. 6+7. 32)- R - = 332.0 0.2165 dt* R = 331.0 - 331 .-^ dt a dt* d*x and I+2mr*Ur+l) I 35.54 m = ~ = 4.65 1^=18.5 r = 1.43 ft. 1 = 158 in. or 13.2 ft, bence B 35.54*9.30x1.43 (nl. 43*13. 2) d^x 18.5 d a x 1.43 dt a 1.25 dt* (p-40.4 Hi)i.43-331 i = 274.8 Hi dt a dt* 663.4 d*x , d*x bence p = _._ _ = 463 2l!259 + 15000 _ ^ m 838 32.2 32.2 d 2 x d 2 x 50000-463 T-T=838 d a x 50000 - - = 38.4 ft/sec.* dt* 1301 dt * * dt* )V* r R-(ir*i k-*i r kj ~)(v dx 861 Check on equivalent mass of rotating parts: Kinetic energy of rotating parts in terms of translatory mass, +7 lO.l(^j)* + J- 18 -5(^j)*+7 9.3(nl.43+13.2 v t 28*48. 6+430 J 10.1 Sj =20.2 t 18.5 \t 82.2 231.21 I M rot t M rot = 463.0 CALCOLATIOH OF* STABILITY. Evaluation of inertia couples: d 2 x Track rollers 20.2 = 653 dt a d a x Track inertia sprockets 274.8 - = 8900 d t Intermediate gear 7.32 =237 dt Clutch 19.1 =616 dt Resultant couple effect; 653 10,169 8900 237 616 9,932 ft.lbs. stabilizing moment, due to 119,000 ft.lbs inertia couples of 10169 wheels: 862 K d =0.9K=500001bs. Overturning moment. 50000x72=3, 600, 000 ( overturning moment)lbs. Stabilizing moment: 6248x69.5=2,480,000 3,520,000 9396x111* 1,040,000 119,000 3,520,000 3,639,000 M c - = x 32.3 = 36400. 36400x32.5=1,185,000 3,639,000 1,185,000 4,824,000 Dynamic Overturning moment 3,600,000 Ibs. Stabilizing moment 4,824,000 Ibs. Static Overturning moment 4,100,000 Stabilizing moment 3,520,000 d*x 50000-15000-4647-*-= 1270 d t d t d*x d 8 x 3 500 0*1 73 4r T hence -T-T 20.2 ft/sec. at dt' 292x38.61 Ft = M V t = - = 0.226 sec. 50000 , 0.'.86 . 2 . 28 ft . S = S^ S t = 2.28 .87 3.15 CHAPTER XIII MISCELLANEOUS PROBLEMS AND TYPES OF CARRIAGES. GENERAL DYNAMIC EQUATION 01 RECOIL The follon- DURING POWDEF PRESSURE PERIOD. ing theory is perfectly general and specially ap- plicable for types of mounts that do not recoil along the axis of the bore. Let m - mass of projectile i = mass of the powder charge v = absolute velocity of the shot up the bore ft/sec. v x component of v parallel to recoil path ft/sec. v y = component of v normal to recoil path ft/sec. v rel * relative velocity of the shot in the bore m r = mass of the recoiling parts P = mean powder force = angle between axis of bore and path of recoil N g = normal reaction between projectile R = total resistance of the recoil system (Ibs) u = travel up the bore (ft) X = retarded recoil displacement (ft) Xf = free recoil displacement (ft) B = angle between absolute velocity of pro- jectile v and path of recoil. Assume half the charge to move with the projectile and half with the gun. The reaction between the gun and projectile, becomes P cos 0-N g sin0 along the bore 863 864 P sin0-N g cos0 normal to the bore. The equation of motion of the recoiling parts, becomes, along the recoil path, dV Pcos0-N s sin0-R= (m r + 0.5 ID ) Integrating and Pcos0-N s sin0 Rdt dividing by m r , we have ( n c - )dt- . __ = V Now from m_+0.5m m r +0.5m the vector diagram of velocities, we have, adding vectorily, v re j+Vv but since V = Vj approx. that is the retarded velocity of recoil is approx- imately equal to the free velocity of recoil, we have v rQ i + Vf= v (approx.). Now in the free re coil Pcos0-N s sin0 ( - _ - )dt=V# that is the expression r+- 5 m Pcos0-Nsin0 is v m p +0.5m measured by Vf and which assumes, for given intervals of time P and N s are not greatly different in the free recoil as compared with the retarded recoil. If R was sufficiently great to prevent an appreciable recoil N g would disappear but P would not vary even then greatly for given intervals of tine between free and stationary recoil. Further N s is small even in free recoil as compared with P, hence the above expression would be but slightly modified. Next, considering the motion of the projectile in a direction parallel to the recoil, we have (Pcos0-N 8 sin0)dt=(m+0.5)v x but since * x a rel cos0-Vf we have (Pcos0-N s sin0)dt=(m+0.5m) (v pel cos0-Vf ) Combining with the expression for free recoil of the recoiling parts, (m r +0.5l)V f 3(m+0.5m)(v re jcos0-Vf ) Hence, V* = cos0 Since 8 equals the angle v makes with the recoil 865 path, we also have (Pcos0-N s sin0)dt=(m+0.5i)vcosB and therefore (m r +0.5m)Vf>(m+0.5i)vcosB ffl + . Sm p hence V f m +Q.5I Now B differs very little from 0, and assuming 8=0 hardly modifies the recoil effect; further 0.5 m is negligible as compared with m r . Hence in +0.5 IB Vj 3 - v cos approx. The dynamic equations of recoil, become therefore Pcos0-N a sin0 m+0.5m m r +0.5i (m+0,5i)v rei cos0 Et m r +0.5i m+0.5l Rt v cos0 (approx.) m r m r Integrating again X = fv*dt - - = *_ cos0- 2m r m r +m+m During the after effect period of the powder gases, the reaction of the powder is approximately along the axis of the bore and the procedure of computation has been previously discussed in detail. The effect of the reaction N s is to deviate the motion of the projectile, causing the projectile to leave the muzzle at an angle somewhat greater than the angle 0. To compute this angle, we have, v sin B-v re ^sin0 v rel / B r+ a * g v/ m+0.5i ,cosB hence sin B = sin0=( )( ) v m+0.5m n r +0.5i cos0 m r +m+i j m r +m+I tan B= tan0 and B * tan () tan0 m r +0.5l m r +0.5i The increase in ths apparent angle of elevation becomes B-0 and is usually small and may be neglected in recoil problems. On the other hand, to compute N s is important since it causes an additional load on the elevat- ing mechanism during the travel of the shot up the bore 866 H S - dV ^ (m+O.Sro) 2 dv (+0.5i)* m r * dv dt 2 r 0.5l dt o-i n 90(P -i n - vm+u. om ; - (approx) t fKA normal reaction of the projectile when a gun recoils at an angle (6 with the axis of the bore, is always proportional to the powder reaction which varies from point to point along the bore. Though the max. reaction occurs practically at t~he beginning of recoil, the moment is usually found greatest when the shot reaches the muzzle of the gun. REACTIONS AND GENERAL EQUATIONS Consider the IN A FECOILING MOUNT. recoiling parts to be constrained in movement always parallel to the axis of the bore, the constraints being offered by suitable guides or a gun sleeve fixed to the cradle. We will assume rotation possible about the axis of the trunnions. Let Pfr s tbe powder reaction on the breech (Its) Q^ and Q = the front and rear clip reactions (Ibs) tan M =t- the coefficient of guide friction M r and W r - mass and weight of the recoil- ing parts (Ibs) B = total braking force (Ibs) X and Y = the coraponents of the trunnion re- action parallel and normal to tbe axis of tbe bore (Ibs) = elevating gear reaction (Ibs) .) * distance from trunnion to line of action of E (ft) M c and tf e = mass and weight of the cradle (Ibs) 6 e - angle between E and axis of bore v r * relative velocity of recoiling parts in cradle (ft/sec) 867 = angular velocity of tipping parts about the trunnion (rad/sec) I r = moment of inertia of recoiling parts about the center of gravity of the recoiling parts. I tr = moment of inertia of recoiling parts about trunnion axis It c = moment of inertia of cradle about the trunnions. X Q and y Q = battery coordinates of the center of gravity of the recoiling parts with respect to the trunnion. x c and y c = coordinates of the center of gravity of the cradle with respect to the trunnion. d^b= distance from trunnion to line of action of B. T * /7[ a +Y 2 - total trunnion reaction, r' = radius of trunnion bearing n t * friction angle in the trunnion bearing x x y and y - coordinates of the front and rear clip reactions with respect to the trunnions. R8ACTIOHS OH THE RECOILIMG PARTS. The reactions on the recoiling parts, consist of the reactions of the cradle Q t Q a and B, the reaction of the ponder P b and the various inertia forces as shown in the diagram. Referring to fig.(l) and considering the motion of the recoiling parts assuming by D'Alemberts principle, kinetic equilibrium, we have (1) Along the axis of the gun dw dv r P b -B-(Q t +Q t )sin u+l r sin0-m r w a (x o -x)-m r y m r =0 d t (2) Normal to the bore (Q f -Q t )cos u-H r cos0+m r w a y -B r (x -x) +2m r wv r *0 dt 868 Fig. 1 869 (3) Moments about the trunnion, dv r dw P(e+s>Bd tb -m r s-I t +2m r Bv r (x -x)-W r cos0(x -x) u .x i - where I t =I r +oi r r = I r +m r [ (x o -x)* + y and thus a variable with the recoil. REACTIONS OM THE CRADLE. Referring to the cradle, we have the reactions Q Q 8 and reversed, of the recoiling parts on the cradle, the trunnion reaction divided into components X and Y, the elevating gear reaction E and the various inertia forces as shown in fig.(l) passing through the center of gravity of the cradle, together with the inertia cou pie I o . Referring to fig. (1) we have, (I 1 ) Along the direction of the bore or guides B + (Q t +Q t )sin u +W c sin0+m c w f x c 'nigy,, +Ecos 6 e -X=0 dt (2 1 ) Normal to the guides, -(Q -Q )cosu-H_cos0+m,,x r -- m,,vr 2 y-+Esin6 + Y=0 21 t ** c j*. B - ** at (3 ' ) For moments about the trunnion axis, dw -W e sin0.y c -Bd tb -Ej-I tc - where I tc =I c + n! c r c = EXTERNAL REACTIOHS ON THE TIPPING PARTS Assuming the tipping parts to be balanced about the trunnions, which is customary in order that the tipping parts may be rapidly elevated, ve have W r x c -W c x c =0 m r x -m c x c =0 and and for the total weight of the tipping parts W t =W r +W c and M t =m r +m c If now, we combine, 870 (1) and (I 1 ), (2) and <2'),(3) and (3 ' ) and noting the above relations, we will have for the kinetic equilibrium of the tipping parts, (1") along the bore dv r P b +W t sin0+m r w*x-m p -- +Ecos 9 e -X=0 (2") normal to the bore -W t cos0-m r x + 2m r wv r +Esin6 Q +Y=0 Q. t- (3") moments about the trunnion Jv r dw d " P b (e + s)-m r -rr- s-I t -ItcTT +2m r wv r (x -x)+W r x cos dt -Ej-Tr'sinu t Therefore, we have for the retardation, exerted by the top carriage on the tipping parts, For the trunnion reactions, r d t YW t cos0-Esin 9 e -m r (x +2w v r ) For the elevating gear reaction, dw ^ v r P b (e+s)-(I t +I tc ) ' +W r xcos^-Tr l sinu 1 +m r [2wv r (x c -x) -.s] E= j APPLICATIONS OP THE PRECEEDING When the brake FORMULAE. cylinders recoil with the gun as in the slide or sleigh containing the recoil cylinders and rigidly attached to the gun used with the Schneider naterial, the center of gravity of the recoiling parts falls considerably below the axis of the bore. To offset the effect of the large powder pressure couple and reduce the reaction on the elevating arc, we may employ a counterweight at the top of the gun to raise the 871 center of gravity nearer the axis of the bore as was done on the 155 m/m Schneider Howitzer or we may introduce a friction disk at the elevating pinion, this allowing rotation of the tipping parts about the trunnion. In other types of mounts, a spring buffer may be introduced in the elevating gear thus re- ducing the elevating gear to a small finite value, and the moment effect of the powder pressure couple being distributed over a longer period. If now we neglect, w*x and 2w v r as small and during the powder pressure period x being small we may neglect also, x - and ff r x cos 0. The re- actions on the tipping parts, become dv r X-P b -a r +W t sin0+G cos 6 e Y*W t cos0-Esin & 6 dw d *r and PfcCe+Bj-Tr'sinu^Ut+I^) - m r^~ 8 Now P b dw where K= the total resistance to recoil during the recoil neglecting the rotation effect during the powder period. If y is small, that is if the trunnions are approximately on the axis of the bore, we have, Pb=* r JL = R (approx.) dt Assuming the brake disk on the elevating pinion shaft to offer a given torque, we may readily compute E. In other words, the pinion bearing is designed for a given reaction. This reaction should be comparable with the reaction required in the out of battery position of the recoiling parts. 872 Ks+W r b cos0 That is E = c( ) where c = 2 to 3 de- pending upon max. allowable angular dis- placement of tipping parts, where b = length of recoil, K * resistance to recoil, s = distance from K to trunnions. The trunnion reactions become simply, X K+W t sin0+E cos 9 Y W. cos 0-E 6 t e Thus the trunnion reactions are fairly in- dependent of the rotation about the trunnions, being primarily dependent only on the elevating gear reaction, the total resistance to recoil and the weight of the tipping parts. The additional forces induced by rotation about the trunnion can be treated as secondary forces. ~~~~~ The total trunnion reaction becomes, T=/V+Y* (Ibs) To determine the angular acceleration with a given elevating gear reaction E. We have, approx. P b e+Ks-Tr'ain\i t -Ej'(I t + I tc ) hence dt Assuming Tr 1 sin u t 1 E t I n J t +I tc Ks and Ej as constant, Ir'sin n +E since obviously Tr'sinu t +Ej must be greater than Ks. Further since t f P b dt (m+0.5 n)v where m = mass of projectile m = mass of powder charge v = velocity of projectile in bore (ft/sec) we have (m+0.5i)ve /Tr 'sinu t +Ej-gs ^ w 3 ~^ ~ (. ) t 873 e (m+0.5i)ve Tr ^ where u = travel up the bore, (ft) To allow for the reaction effect of the powder gases, we will assume the free angular displacement at the end of the powder period, given by (ui+2m) ve 9j = " Hence the angular velocity and * tc displacement at the end of the powder period become (mv + 4700i) ,Tr 'sina +Ej-Ks. - e - (_ - 1 - ) t e = (m*2m)ue _ Tr'simyEj-Ka t; where t = total powder period (sec) v = muzzle velocity of projectile (ft/sec) u = travel up the bore (ft) The remaining angular displacement is that due to a constant torque (Tr'sinu 1 +Ej-Ks) acting on a rotating mass with an initial angular velocity t . Hence (Tr 'sin u^Ej-Ks) (6 t - 9 t )= ^U t +I tc ) " and therefore, for the total angular displacement 6 t (m+2i)ue Tr 'sin^+Ej-Ks) t* ( I t + I tc^ w i tX I t + I tc it^tc 2 ^ 2(Tr'sinc t +Ej-Ks) mv+4700i) Tr'sinu^Ej-Ks where w^ =( e -( ) t. and t is computed by the methods of interior ballistics and T= /x*+Y* using a suitable value of E, we may compute from the above ex- pressions the total angular twist. GENERAL EQUATIONS:- ROTATION With rotation of OF THE TRUNNIONS ABOUT A the trunnions about FIXED AXIS OR A TRANSLATION an axis, the ele- OF THE TRUNNIONS. vating gear reaction is usually reversed and the magnitude of the reversed action on the 874 **& Fig. 876 elevating gear depends upon the product of the angular acceleration about this axis and the total moment of inertia about the trunnions of the tipping parts. Thus, in the rolling of a ship or in the jump of a field carriage where the angular ac- celeration upon the mount may be considerable, and with heavy tipping parts, a large reversed reaction is exerted on the elevating gear, which in turn modified the trunnion reactions. This same phenomena occurs in a double recoil system, or in a railway mount where the mount below the recoiling parts is accelerated up in an inclined plane or along the rails. Let us first consider, the angular motion induced in the tipping parts when the elevating gear reaction is nil. Assuming the trunnions to rotate about an axis 0, fig. (2) and the axis of the bore and center of gravity of the recoiling parts to be along the trunnion axis, then, The Kinetic reactions on the tipping parts, become (1) The trunnion reaction X and Y which impress the angular acceleration on the tipping parts. Due to the friction of the trunnions T = /X 2 +Y* has a moment about the trunnion axis: T r 1 sin u. (2) The tangential component of the ine,rtia force of the tipping parts = R and its moment about the g dt 2 trunnion, axis becomes, W t T R W_x r Let Tm= - sin(0+e) "hence R 1 - (Tm)= R 1 W g dt 8 g dt 876 (3) The centrifugal component of the inertia force of the tipping parts * w t dQ R ( ) and its monent about the g dt trunnion axis becomes dt g dt (4) The rotational inertia couple of tbe tipping parts j _. where w = tbe angular velocity about the trunnions, Itr = Inonient of inertia of the recoiling parts about tbe trunnions I tc = moment of inertia of the cradle about the trunnions (5) The weight of tbe tipping parts, its moment being W t (TG)cos0=W t W_x (- )cos0=W r x cos w t (6) Tbe complementary centrifugal inertia force due to tbe relative motion,of the recoiling parts 38 2m r * where x * tbe relative d t displacement of tbe recoiling parts. Its moment about tbe trunnion becomes, dx 2m r (x -x) * (7) Tbe powder reaction, and tbe relative inertia resistance due to tbe relative accelerstion of tbe recoiling parts. We are not concerned with these reactions, since their moment effect is nil, it being assumed that their line 877 of action passes through the trun- nion. We have, therefore for the moment equation about the trunnion, considering the kinetic equilibrium of the various inertia forces, Tr'sin a +(I tr +I tc ) - 2m r w - -W r Xcos0 d t d t Wu r d 2 Q r dQ R'X sin(0+e)+ R'X( ) cos(0+e)0 g dt* g dt If we assume R large, for an elementary dis- placement, R dQ may be considered rectangular, hence the term... w r dQ R'X(-r~)* cos(0+e) may be omitted. g dt Further R * R 1 appro*, R being the distance from axis to the trunnion. In experiments, conducted by the French at "Sevran-Livry" the term 2m r w -T was found to be negligible. Hence the equation of angular motion about the trunnion axis without an elevating gear interposed becomes, n T \dw W r ^aQ T r sina l **' 1 tr +1 tc'^fjr -W r Xcos0 RX T sin(0+e)0 since Trsina 1 and W r Xcosl are small for a large angular acceleration, we have, approximately, v dw *r From this equation we observe that immediate- ly upon the recoiling parts becoming out of battery, when the acceleration of the top carriage is backwards, as would occur in the jump of a field carriage, the upward rolling of a ship or in the recoil of the top carriage in a double recoil system or railway mount, we have an angular acceleration tending to cause a reversal or stress in the elevating gear. 878 ANGULAR ACCIL1RAT16N OP THI TIPPING PARTS. Invariable Elevating Gear Reaction Introduced. In this case, tbe angular acceleration of tbe tipping parts, is the same as the angular ac- celeration of the system about the fixed or in- stantaneous axis 0. To impress this angular ac- celeration on the tipping parts, as would occur in the jump of a field carriage, or in the upward rolling of a ship, tbe elevating gear reaction is lessened or completely reversed when tbe trunnions are located along tbe bore. Considering fig. (3) Let Pfc = the powder reaction on tbe breecb (Ibs) Q t and Q a = tbe front and rear clip reactions (Ibs) tan u = f = coefficient of guide friction m r and w r = mass and weight of recoiling parts (Ibs) 8 = total braking force (Ibs) X and Y = components of tbe trunnion reaction parallel and normal to tbe bore (Ibs) E = elevating gear reaction (Ibs) j = distance from trunnion axis to line of action of E (ft) 9 e = angle between E and tbe axis of tbe bore 7 r = relative velocity of recoiling parts in cradle (ft/seo) dQ = angular velocity impressed on tipping parts dt (rad/sec) I r - moment of inertia of recoiling parts about center of gravity of recoiling parts I tr moment of inertia of recoiling parts about trunnion axis. I tc = moment of inertia of the cradle about the trunnion axis. x o and y o = battery coordinates of the center of gravity of the cradle with respect to the 879 Fig. 3 880 truonioD axis, x and y t = battery coordinates of the center of gravity of the cradle with respect to the trunnion axis. d t b - distance from trunnion axis to line of action of B. r' = radius of the trunnion bearing, u = friction angle in the trunnion bearing. x y and x y = coordinates of the front and rear clip reactions with respect to the trunnions. BBACTIOM6 OH THE RECOILING PARTS. The reactions on the recoiling parts are: (1) The powder force P b (Ibs) (2) The reactions due to the con- straint of the cradle Q and Q 2 (Ibs) (3) The braking force exerted by the cradle B (Ibs) (4) The relative tangential inertia force due to the relative acceleration d v of the recoiling parts m r -^ (Ibs) (5) The relative complementary centrifugal force due to the com- bined angular and relative motion of the recoiling parts dQ 2m r v r (Ibs) d* (6) The tangential inertia force due t . rotation a"bout the axis m.R (Ibs) dt a (7) The centrifugal inertia force due to rotation about the axis 0- r R(^) 2 (Ibs) (8) The weight of the recoiling parts W r (Ibs) 881 (9) The angular couple resisting d*Q angular acceleration I r -j (ft.lbs) d * Tbe equations of notion for the recoiling parts, become, along x x 1 av r d*Q dQ p b -a r m r R -777- cos(e+0)-M r R(T-) sin(e+0)+W r sin0 since u = (1) along v v ' dQ ,/ dG N, r, d'Q r v_ - +m r R( --) z cos(e+0)-m r R - dt at at (2) for moments about the axis 0, we have, dv r r dt . dQ . dQ dQ . -ffi-R* 1_ +2m_v r [x x -x+Rsin(J+e)] dt 9 dt dt -W r [ (x -x)cos0+Rsin e- Now m r R 2 +I r =I or moment of inertia of recoil- ing parts about axis 0. Hence the above expression reduces to, (P b -m r ^)[Rcos(e+0)+s]+P b e b -ZW (Q t + Q 2 +B)-I or j^~ a t dQ +2m r r [x -x+Rsin(0+e)]-W r [^ -x)cos!+Rsin e-y o sin0] d t (3) BBACTIOHS OH THB CRAPLB; The reactions on the cradle are: U) Tbe reactions of the recoiling parts on the cradle Q 4 Q a and E. (2) The trunnion reaction T = /X 2 +Y 2 and having a moment about its center line Tr 1 sin u. 882 (3; The elevating gear reaction E (4) The tangential inertia force . R *I dt * da , (5) The centrifugal inertia ra( ) 2 c dt (6) Tbe weight of the cradle W c The equations of motion, become along the x x 1 axis (Q. +Q a )sin u +B+T c sin0-t-Ecos 9 d*Q .dQ cos(e+0)-m c R( d*Q dQ a -m c R cos(e+0)-m c R( ) sin(e+0)-X =0 (I 1 ) along the y y 1 axis Y+Esine -W cosD-CQ -Q )cos u-nn_R( ) a cos(e+0) (2 1 ) d t for moments about the axis 0, d*0 2M ( Q t * Q 2 *B)-XR cos(0+e)+YR sin(0+e )-m c R a - dt 2 d*Q -Iv > +Ecos6 e [R cos(0+e)-Jcos9 e ]+Esin9 e [Rsin(e+0)-J sin9 e )-W c (Rsin e-x c cos0+y c sin0)=0 Now, c R a +I c =I oc the moment of inertia about the axis of the cradle. and Ecos9 e tRcos(0+e)-Jcose e ]--Esine e [Rsin(0+e)-Jsin Q e ] =ER cose cos(0+e)+ER sin0 e sin(0+e)-EJ(co3 a e e +sin*e e ) =ERcos(0+e-9 e )-BJ=E[Rcos()+e-e e )-J] Hence the moment equation of the cradle about 0, reduces to ZM (Q +Q +B)-XRcos (0+e) + YR sin(0l+e)-I oc - dt* +E[Rcos(0+e-e e )-J]-W c (R3ine-x c cos0+y c sinJ)=0 (3 ' ) HgACTIONS ON THE TIPPIMQ PASTS. Since the tipping parts are balanced about the 883 trunnions in the battery position, we have, and *r x o~ w c x c* r x o~ m o x c =0 Adding (1) and (I 1 ), (2) and (2 1 ) and (3) and (3), we have d*Q dd t (H r +W c )sin0+Eeos8 e -X=0 (1") dQ jn - Y+Esine-(W r +W.. )cos0+2m r v r - + (m r +nu c r c r r j * * r c d*Q -( r -t-m c )R sin(0+e)=0 dv r Cl If M W V_l U [x +x+RCZJ+e)]+W r xcos0-(W r +W c )Rsin e-XRcos (0+e) +YR sin(0+e) + E [Rcos(0+e^-9 e )-J] =0 (3") Equations (1"), (2"), (3") are the general equations of a recoiling system, where the relative translation is along the axis of the bore and the trunnions have a rotation about some fixed axis 0. These equations may be simplified as fol- lows: W^ a W r +W c ana m t =iB r +m c where W t = the total weight of the tipping parts. mt = the total mass of the tipping parts Further I or I tr +m r R* Ioc* r tc*"c R * and I or +I oc =tI tr +I tc +B t Rt Rcos (0+e)+s*Rcos(0-e) approx. dv r d 2 Q dQ Xap w~ m r m t R t cos(0+e) + ( )* b dt dt* dt 884 d*Q dQ dQ [ - sin(0+e)-( )*cos (j0+e)]-2m p v r ' -Esin0 ft dt dt dt * Dd dQ dv r -(P b -a r Rcos(0+e-e e )-J sin e e -W p xcos0+H t Rsin e+XRcos(0+e)-Y in(0+e) 1 R) Substituting the values of X and Y in the equation of and simplifying, wa have, J (4) which is evidently the moment equation of the various kinetic reactions on the tipping parts about the trunnion as an axis. Since the term dQ 2m r v r (x -x) is always small, the elevating ^* dv gear reaction, reduces a to I tr + I tc ) (4') where I tr *I r + r [ (x o -x) 8 +yl I r = JDoment of inertia about center of gravity of recoiling parts. Hsnce I^ r is a variable depending upon the displace- ment in the recoil x, also Itc =I c* m c ^ x c +y a constant I c = moment of inertia about center of gravity of ths cradle. The equation (4) or (4 1 ) is of special im- portance in the study of the variation of the elevation gear reaction. The angular acceleration be detericined in the following discussion 886 on the jump of a carriage. In the case when s and e b = 0, that is when the center of gravity of the recoiling parts and trun- nion axis lie along the axis of the bore, we have -(Ti +T* ) - Thus the elevating gear reaction dt 8 is reversed and its monent about the trunnion imparts the required angular momentum in the tipping parts. We calculate the value of (-E) we must determine d 2 Q the maximum angular acceleration - . d t The condition that there will be no reversal of stress on the elevating gear on the jump of a field carriage, is that W r xcos0+(P b -m r )s+P b e b rc dv r Now roughly P b ~fl r "~ ~~~ ^ tne static resist- dt ance to recoil, hence for no reversal of stress on the elevating gear, In the battery position: Ks+P b e b ^ (i tr+ i tc Cut of battery position: >/ .d 2 Q cos =(It.r + It. From these equations we may determine the required distance from the center of gravity of the recoil parts to the trunnion axis, to prevent a reversal of stress on the elevating gear when the gun jumps as in a field carriage. RiCTILIKBAR ACCELERATION OF THE TIPPINS PASTS With a double recoil system, or in the case of a railway recoiling along the rails, the trunnions are accelerated to the rear due to the recoil re- 886 Fig. A- 887 action of the gun. Thus the tipping parts are sub- jected to a rectilinear acceleration to the rear and the elevating gear reactions is increased. Considering fig.( 4) we have the various re- actions as the recoiling parts and cradle as shown. BKACTIOHS OH THE BgCOILIHQ PARTS. The reactions on the recoiling parts, consist, (1) The powder force -- P b (Ibs) (2) The reactions due to the constraint of the cradle --- O t and Q f (Ibs) (3) The braking force exerted by the cradle B (Ibs) (4) The weight of the recoiling parts W r (Ibs) (5) The kinetic reaction of the recoil- ing parts due to the relative accel- eration dv_ r - (Ibs) (6) The kinetic reaction due to the ac- celeration +a) E= when the top car- J riage moves. when the top carriage is stationary, J Thus we have only a slightly additional load 890 m r brought on the elevating gear (Ibs) J This value however is somewhat compensated by the slightly decreased value of K x due to the fact that the relative velocity is somewhat less than the static velocity of recoil. RECAPITULATION. Reaction of Top Carriage on Tipping Parts: For tbe Trunnion Reactions dv r dv c =Pv-in r -- -- (m P +ni c )---rcos(0+a)+(W r --W c )sin0+Bcos9 e * dt dt dv c m+m) - sin(0+a)- sin 9 dt For the Elevating Gear Reaction dv r dv c in (0+a) J If we define K x =B+(Q t +Q 2 )sin u+W r sin0 then dv r dv c K x~ p b si "r [ - + - cos(0+a)] and W t =W r +W c total * weight of the tipping parts, M t =m r =m c Total mass of tipping parts. For the Trunnion Reactions dv c X=K x =m c - cos(0+a)+W t sin0--Ecos9 e dv c Y sW^cos^+mt- sin(0+a)-B sin9. at c For tbe Elevating Gear Reaction: dv c P b e b* K xyo +w r x cosef+m r [xsin(8f+a)+y o cos (0+a)] _ ___^_ __ dt E=^ - ' J ON THE JUMP OF A FIELD CARRIAGE Mounts are frequently de- signed for stability at a given minimum elevation and yet may be fired at a lower elevation. Con- 891 sideration, therefore, must be given to the inertia loadings and corresponding reactions induced by the jump of the carriage. ID the following discussion it will be assumed the total mount to rotate about its spade point. By the application of D 1 Alerabert 's principle we introduce the various inertia effects as kinetic reactions, the mutual reactions between the parts, of course having no effect on the kinetic equilibrium of the total system, gun cradle and carriage. From the acceleration diagram we have for the recoiling parts, (1) The relative acceleration along the axis of the bore -- dv r Ut/sec.) dt (2) The tangential acceleration of'tbe recoiling parts about the axis - B S= dt (3) The centripetal acceleration of the recoiling parts towards the axis wR (4) The acceleration due to the relative motion combined with the rotation of the recoiling parts 2w v r The accelerations in the remainder of the mount, the carriage proper, become (1) The tangential acceleration k c ""~~ dt (2) The centripetal acceleration w 2 L c KIHSTIC EQUILIBRIUM OP THE SYSTEM. (Gun and Carriage) Prom the principle of D'Alenbert, we have the external reactions in equilibrium with the various kinetic reactions induced by the angular rotation of the mount and the relative acceleration of the gun. The forces and kinetic reactions on the system 892 gun and carriage are : (1) The total powder reaction P^ (2) The weights of the recoiling parts and carriage W^ and W c (Ibs) (3) The tangential inertia force of the recoiling parts due to the angular acceleration about the spade point dw M R R (Ibs) (4) The centrifugal inertia force of the recoiling parts due to the angular velocity about the spade point M R R w 2 (Ibs) (5) The inertia resistance due to the relatire acceleration of the recoil- ing parts (Ibs) dv r r dt (6) The inertia resistance due to the combined rotation of the recoiling parts 2 m r wv r (Ibs) (7) The tangential inertia force of the carnage proper due to the angular acceleration about the spade point dw . . fflc c dt (8) The centrifugal inertia force due to the angular velocity about the spade point m o L c w* (Ibs) (9) The inertia couple about the center of gravity of the recoiling parts due to the angular acceleration of the system IT? TT (ft. Ibs) 893 (10) The inertia couple about the center of gravity of the car- riage proper due to the angular acceleration of the system *$*$ 'oJi (fulbs) - 0(1 ) n since 9 = -% +0+Q whence Q = angle turned in rotating about 0, we have ^Q ^ = = w for the angular dt dt velocity d a e d a Q dw -^ = = for the angular acceleration. dt* dt 2 dt Considering now the recoiling parts, above, we have d v t? d w p b~ a R m R d -mR*(x -x)-B-R t +W r sine(=0. dt dt R dw Simplifying we have, Pb^Rt-jT + d T^--w^c -x)]-B-R t +W r sin(?=0 (2) whence B=F v + Pb tne tota l braking reaction (Ibs) P y = the recuperator reaction (Ibs) P n - the total hydraulic resistance (Ibs) Now v * p b =p hs ~ whence v g = static recoil velocity (ft/sec) P ns = corresponding static hydraulic braking reaction (Ibs) 894 a dv R (j w V R We thus see that Pb~ IB RtT7- fd Tr tw * (x o~ x)33p hs ~T~* R t dt ut v F v -W R sin0 (3) From equation (1), we have dvp P b (d+e)-m R - d+2m R w v R (x o -x)-W R [ (x Q -x)cos0-d d _ _ dt dt " m R R*-m c L =I R +I c sinJ]-W c L c cos(0-B) If I s 3 the moment of inertia of the system about the axis 0, then I s m R R*+m c L c +I R + I c a variable since m R R*m R [d a +(x o -x ) a ] a function of x, hence dv R p b( d+e )~" m R~d"[ d+2m R w v R (x o -x)-W R [ (x o -x)cos(Z(-d dt 5 I s sin]-W.L r cos(6-B) - - (4a) Substituting the value in equation (3), we have dt dv R a dynamical equation in terms of and w. If dt now. we construct a table for the various intervals of time, we may compute V R , p , w and J* by the d t dt methods of a point by point procedure. APPROXIMATE CALCULATIONS FOR THE Prom equation JUMP OF A CARRIAGE (3) in the previous article, we have dv R dw V R P b -m R [ - +d. +v 2 (x -x)]-P hs -vF v +R dt dt v Cl tr The terms m d and m R w 2 (X Q -X) are usually small compared with dv R dvi m R , hence V R =V S approx dt and Pb~ m R~j~ = ^ ^^ e statlc resistance to recoil (approx ) 895 COMPONfA/TS OW JUMP /WffTM SOffCfS ON JUMP OF F/fLD Fig. 5 886 Substituting in equation (4a) of the previous article, and omitting the term 2 m r r v r (x Q -x) which is small, we have dw P b e b +Kd-W r [(x -X)cos 0-d sin0]-W L c cos(e-B) The moment effect of the weights, H p [(x -x)cos0-d sinJ)= W c L c cos (9-B)= w s L s~ B R Hence dw P b e b =Kd-W g L g +W R x cos ** " "V where W s = weight of entire system L s = horizontal distance from spade point to line of action, of W s I s = moment of inertia of total system about spade axis BARBETTE CHASSIS MOUNTS. In this type of mount, the top carriage and gun recoil up an inclined plane, and the recoil in general is not parallel to the bore. The characteristics of such mounts is that a component of the direct powder reaction is brought upon the mount and therefore the various parts are stressed considerably higher then with mounts recoil- ing in a cradle. During the powder period, we have an impulsive or percussion effect brought on to the mount, and the effect of finite forces as gravity and the braking force may be neglected. The gun together with the top carriage are considered in this type of mount as the recoil- ing parts. The gun has trunnions, and the trunnions are located at the center of gravity of the gun along the axis of the bore. Since there is no regular acceleration in the recoil, the reaction on the elevating gear is practically nil. Due to the weight and position of the center of gravity 897 ON fffCO/L/A/G Fig. 6 898 of the top carriage, the center of gravity of the recoiling parts is not located at the axis of the bore. During the powder pressure period, there- fore, we have a whipping action due to the powder pressure couple which increases the end roller re- action and the front clip reaction. The bottom carriage which supports the chassis for the top carriage is traversed on a roller base plate, the horizontal reaction being carried on the pintle bearing and the vertical reactions by the traversing rollers. This arrangement is typical of any Barbette emplacement. Let F b - the total powder reaction (Ibs) = angle of elevation of gun a = angle of inclination of chassis fflg and Wg = mass and weight of the gun (Ibs) DC and w c 3 mass and weight of the top carriage m r and w r - mass and weight of the recoiling parts 8 = the total braking reaction (Ibs) dt>= distance from trunnion to line of action of B (ft) Q A and 0^ = the front and rear roller reactions on the top carriage exerted by the chassis (Ibs) R t and R f = the front clip reaction and rear roller reaction exerted by the traversing base plate on the bottom carriage (Ibs) d x and d t = distance from trunnions to line of action of Q and Q respectively, n = friction angle of roller reactions. H - the horizontal reaction between the base plate and bottom carriage at the pintle bearing (Ibs) REACTIONS OtT THE RBCOILIHG PARTS GUN AMD TOP CARRIAGE T06STHBR We have for the motion of the recoiling parts, along the chassis:- 899 d*l P b cos(0+a)-W_sin a-B(Q +Q )sin u-m. = =0 (1) r dt normal to the chassis:- PhSin(0+a)+W r cos a-(Q t +Q t )cos a =0 (2) about the trunnions: d*l If we assume the braking constant throughout the recoil, we have, B+W r sina+(Q i +Q a )sin u = K and equation (1) becomes, P b cos(0+a)-m r - K =0 Integrating, we find, dl P b cos(0+a) K = v = / - dt -- t dt m m = Vf or the maximum free r r velocity of recoil for a recoiling mass m r , hence Kt t dt Integrating again, T K p 1. * / V f cos(0+a)dt- o 2m r KT 2 = E cos(0+a) - where E is the free recoil dis- 2ni r placement for a recoiling mass m. during the total powder period. During tfte re- g mainder of the recoil, we have J r v*=K(b-l t ) hence rr t f^ flpg -m r [V f cos(0+a) -- ]* *K[b-Ecos(0+a)+ ] m r 2m r Simplifying, mfV | cos (cr +a ) K = - where E and T 2[b-(E-V f T)cos(0+ a )] are obtained by the methods of Interior Ballistics. EFFECT Of CHASSIS ROLLER REACTIONS OH THE RKCOIL BRAKE. Assuming only the end roller reactions to come into play, we have, from eq. (1) and (2), 900 K-B-W r sin a tan u = = f P b sin(0+a)+W r cos a hence K -B-W r sina=f [P b sin(0+a)+W r cos a] where f coefficient of roller friction. After the powder period, K-B-W r sin a = f W* r cos a, therefore during the powder period, B 1 =K-W r sin8J-f [ P b sin(0+a)+W r cosa] in the recoil, B=K-W r sin0-fW r cos a , and the charge of required braking, becomes, B-B t fP b sin(0+a) BEACTION3 ON THE BOTTOM CARRIAGE. The reactions on the bottom carnage are:- (1) Q t and Q 2 reversed, the roller reactions on the chassis of the top carriage (Ibs) (2) V reversed, the braking reaction (Ibs) t- (3) The horizontal pintle bearing reaction n. (4) The weight of the bottom carriage Wtc- (5) The traversing roller and clip reactions R t and R 8 (Ibs) Then, resolving forces along and normal to the chassis, we have, (Q t +Q a )sin u +B-HCOS a+ (R ft -R t )sin a-W tc sin a =0 (I 1 ) (Q +Q )cos u+W t .cos a-Hsin a -(R -R )cos a =0 (2 1 ) a t wO a i and for moments about the trunnion, H dh-Bd^Q^-Q^-R^-R^-W,^ =0 (3 ) where x^ = the momentum of W tc about the trunnion. BXTSRNAL REACTIONS ON THE SYSTEM CONSIST- I NG OP 1 THS TOTAL MOUNT. Adding equations (1) and (I 1 ), we have, d a l F b cos(0+a)-W r sin a-m r - -Hcos a+(R a ~R t )sin a-W tc sin a (1") 901 Since P b cos(0+a)-m r pK and r +W tc =W s the total d t weight of the mount. Equation (I 11 ) reduces to, K-H s sin a - H cos a +(R a -R t )sin a = (1") Adding (2) and (2'), we have P b sin(0+a)+H r cos a+W tc cos a-Hsin a -(R 2 -R 1 )cos a (2") Adding (3) and (3 1 ), we have, - (y cos a+x sina)0 (3 B ) dt* Eliminating (R a ~R t ) from (1") and (2"), we have Kcos0-H+P b sin a sin(0+a) =0 (a) Eliminating H from (1") and (2"), (R 2 -R t )+Ksin a-P b sin(0+a)cos a-W s =0 (b) and equation (3") reduces to for moments about the trunnion, sina)=0 (c) From (a) and (b), we have, H=Kcos0+P b sina sin(0+a) R > -R 1 P b sin(!^+a)cos a+ s -K sin a. Substituting the value of H in (c) and combining with (b) we obtain R ( and R t respectively. PERCUSSION REACTIONS: The percussion reactions take place during the powder period and are reactions of a magnitude comparable with the powder forces. In an ordinary cradle recoil, the direct effect of the powder re- actions are practically eliminated by allowing the gun to recoil along the bore. In mounts of the chassis type, especially when the gun elevates, we have a large component of the powder reaction, which causes the chassis to offer a corresponding reaction. 902 OA/ Fig. 7 903 FERCUSStQH REACT/ON ON RECO/L/N6 PARTS PERCUSSION REACT/ ON ON TOP CARRIAGE *J t \ Fig. 8 904 In dealing with impulsive forces, the effect of continuous or finite forces is negligible com- pared with the percussion reactions. Hence in the following we will omit such forces as gravity, and the recoil brake reaction. PERCUSSION REACTIONS ON THE RECOILIN3 PAHTS: The percussion reactions are, (1) The powder force P b d a l (2) The inertia resistance I=ffi_ dt 2 (3) The resultant reaction of the chassis --- Q P^ acts along the bore, I acts parallel to the chassis and through the center of gravity of the recoiling parts, while Q. balances these reactions at their intersection, as shown in fig.(8) The force polygon of the percussions is abc, where a b //P^, bc//I and ca//Q. The direction of Q. is slightly inclined to the chassis due to the friction angle u. Further Q is the resultant of Q and the front and rear roller reactions. Now t 2 the resultant of Q and I, must intersect the resultant of P^ and Q 8 . Since P^ intersects at 2 at a, we have the direction of the resultant of Q, and I along ae . In the force polygon bd is drawn parallel to ae, and therefore cd is proportion- al to Ql while da is proportional to Q a . In the force polygon, we have, P b I Q t Q 2 vc . = = = hence I = P^, ab be cd da ab ""^* DYNAMICAL RELATIONS ON FIRING Small guns up to FROM AN AEROPLANE. a caliber of 75 have been successfully fired from large aeroplanes, 8/ 905 Larger calibers may be possible by the introduction of the muzzle brake, which thereby reduces the re- coil reaction. In this discussion, however, we will take the simple case of a gun without a muzzle brake. Let V Q = horizontal velocity of the plane before firing (ft/sec) V o V a velocity of the plane immediately after firing (ft/sec) V t V r = velocity of the gun at the end of the powder period (ft/sec) V^ v = muzzle velocity of projectile (absolute) (ft/sec) P b - powder reaction (Ibs) R = recoil reaction (Ibs) in r and w r = mass and weight of recoiling parts (Ibs) n g and * s ~ mass and weight of equivalent weight of aeroplane + weight of cradle and mount (Ibs) Assume the gun to be fired horizontally while the aeroplane flies horizontally: During the powder period, we have the mutual impulsive reaction between the gun and aeroplane = F b dt t For the gun, / * P b dt = m r (V o -V r ) (1) o the impulsive effect of the recoil reaction R being negligible. For the projectile and powder, we have, Ato f P b dt =(m+0.5m)(v-V ) during the travel up the bore. N 7 ~ V o P b dt = mU700-( ) during the powder expansion. 906 At / P b dt * I(v-V )+i4700 * mv+m4700 (approx) (2) Let us DON consider the effect of the recoil reaction R on the aeroplane and fixed part of the mount. On firing the aeroplane the aeroplane acts somewhat as an elastic beam, more or less supported by the air reactions at the ends. We may consider, the equivalent mass of the aeroplane and mount at- tached = 0.7 to 0.8 the actual mass of the plane and mount. We will denote m s as this equivalent mass. Then, for the motion of the aeroplane during the recoil period, we have, S (V -V ) R = S f Ubs) (3) t and for the motion of the recoiling parts during this same period, n r (V -V r ) R = * (IDs) (4) t since the recoiling parts must have the same velocity as the plane at the end of recoil. It is interesting to note the magnitude of the relation of the various velocities for a typical small mount. V = 100 miles/hour = 146.6 ft/sec. V -V r 30 ft/sec, roughly; V r = 116 ft. sec. roughly, V = between 116 and 146 ft/sec, say 130 ft/sec. Thus vie have a check in the velocity of the plane of several feet per second, the magnitude of which depends of course on the ballistics and relations of the various masses. Combining the previous equations, we have, r (V -V r )=mv+i 4700 (5) ' ( Wg + r )(V -V t )mv +1 4700 (7) That is, as we should expect from first principles, 907 the momentum imparted to the aeroplane backwards, equals the momentum imparted to the projectile and powder forwards. Let us now assume the recoil reaction con- stant, and let b equal the length of recoil. Now due to the superior motion of the aero- plane as compared with that of the gun, during the recoil the aeroplane does work on the gun, in bringing the velocity from the smaller value V r to the larger value V hence 2 The energy taken from the aeroplane, becomes 2 hence -R b = -| (V*-V) - -|(V -?) therefore, the recoil reaction, becomes _ v 8 + _ l r ^"o ""rvr. x m s* m r. ri , R %- [ ( -r * >- ( -T )v ; ' (Ibs) .hare V r =V o - ( 211i|225, ( {t/sec) aod v v o -(HIiZ22i) ((t/seo) m s +m r DISAPPBARISG AHP OTHBH TYPgS OP CARBIA6E8. TYPES OP DISAPPEARING Disappearing gun car- CAFRIAGES. riages, as evident by their terminology, are designed, so that in the recoil the gun is brought down below a parapet and disappears from the enemy's view. The gun is loaded in the lower position. By in- troducing a counterweight, the gun is brought by gravity to the firing position, the gun during the firing period only being above the parapet. Disappearing gun carriages may be broadly 908 classified in two general types:- (1) Revolving or rotating types, where the gun lever rotates about a fixed axis, as in the Monorieff. Howell and Krupp carriages. (2) Sliding carriage types, where the Cardon system of linkage is used, the gun lever being constrained to move at two of its points along guides practically at right angles, as in the Buffington Crozier models. APPROXIMATE THEORY OF THE The following as- ROTATING TYPE OF DISAPPEAR- sumptions are made and ING CARRIAGE. the validity of these assumptions will be considered more in detail later:- (1) The center of gravity of the gun will be assumed at the gun trunnioni (2) The angular displacement of the gun lever, during the powder period, will be assumed small and will therefore not effect the initial geometrical conditions greatly. (3) The inertia effect of the elevat- ing rods, will be assumed negligible as compared with that of the gun, lever, gun and counterweight. (4) The elevating arm, will be assumed approximately parallel to the axis of the gun lever and roughly equal to the upper half of the gun lever. (5) The angular movement of the gun itself during the powder period will be assumed very small. From assumptions (3), (4) and (5) we may neglect the reaction of the elevating arm during the powder action period, for the following reasons: (a) The tangential component of 909 the elevating arm reaction becomes zero due to assumption (3). (b) Condition (4) assumes the instantaneous center of the gun practically at infinity. Hence the angular velocity of the gun at the end of the powder period is negligible; the angular acceleration therefore may be assumed zero, and the normal reaction of the elevating arm becomes zero. In practice it is possible to obtain (1) com- pletely, and (2) and (3) are closely realized. The condition (4) may be met constructively at one elevation but is difficult to meet for all elevations, since the gun customarily is designed to recoil to the same loading angle. To reduce the reaction on the elevating arm it is customary to introduce a kick down buffer at the bottom end of the arm, and thus during the powder period a small minor reaction comparable with the buffer resistance is introduced between the elevating arm and gun. This reaction may be neglected as compared with the major reactions of the gun lever. Therefore, as a first approximation, however, we will neglect the reaction of the elevating arm, and assume the center of gravity of the gun located at the trunnions. Let Wg= weight of the gun (Ibs) W r = weight of the gun lever (Ibs) w cw = weight of the counterweight (Ibs) I r =W r kJ = moment of inertia of gun lever about fixed axis of rotation. I cw = W cw k w = moment of inertia of counter- weight about fixed axis of rotation. 910 REACTIONS ON THE. ROCKEIR AT GUN DURING POWDER PERIOD REACTIONS ON THE ROCKER AT GUN AFTER POWDER PERIOD w cw Fig. 9 911 T and N = tangential and normal trunnion re- action (Ibs) X and Y = horizontal and vertical reactions at axis of rotation of gun lever (Ibs) P * total powder reaction (Ibs) P ffl = maximum powder reaction (Ibs) = angle of elevation of gun 6^= initial angle of gun lever with respect vertical 0= final angle of gun lever with respect vertical r = radius of upper half of gun lever (ft) r 1 = radius to center of gravity of counter- weight (ft) R = reaction of oscillating cylinder brake d^= initial angle R makes with the normal to r' r o = distance from axis along r 1 to line of action of R. m = mass of projectile IE = mass of powder charge v = muzzle velocity (ft/sec) TfVf = total friction torque resisting rotation From fig.( 9). the gun axis makes an agle 0- 9j with the tangent of the path in the initial position of the gun. For the motion of tte gun lever, we have for moments about the fixed axis, + * cos d r o + Vf (1) and for the motion of t"he gun along the tangent to its initial path, Pcos(?- 6i)-T- -* r (2) g dt If s = the displacement along the arc of the gun trunnion V = the corresponding tangential velocity of the gun trunnions, 912 ds d6 d*s dV d*e * r ; = r - Hence, combining dt dt dta dt dt * the two equations, we have, w g *r *cw dV r o r f Pcos(0-6.j )=( + -+- r) +Rcosd; +T f (3) 1 g r 2 r a dt r r Evidently J+~""~ ma y be regarded as the so called equivalent translatory mass at the gun trunnions due to the rotational inertia effect of the gun lever and counterweight. Integrating equation (3), we have, r r f Rcosd dt T f dt Pcos(0-6 i ) r c r V = / dt - / - / g I r I cw Wg I r I cw Wg I r I cw ^P I -f-_^__ " I HM I ---- -,_S \ \ -- g r 2 r 2 g r 2 r 2 g r* r 2 Now both and d as well as the friction torque TfTf vary during the powder period but as the change is small, we are quite justified in assum- ing them constant. Further, since, Pdt=(m+0.5m)v (during the travel of the shot up the bore), we have ( m +0.5m)v - - dt = - - - - cos (0-i) or in terms Wrf 1 r i /% n- W/< 1 -l-riui 8 , r , cw (_5 + _j, cw \ of the free g r 2 r a g r 2 r 2 velocity of recoil, (m+0.5m)v V f cos(0-e.)= - - - - cos (f?-9i) W w r x r 1 cw. ( + + - ) g r 2 r 2 ' where Vj is the equivalent free velocity with a recoiling mass equal to "r l r x cw s ( + + - ) g r 2 r 2 ' Integrating again, we have X f eos(0-e i )=["* ' 5m i ]x' 008(0-6^ where x 1 = the g ^ r | cw absolute dis- & r placement of the projectile up the bore. Now 913 -*) x'cos(0-6 i )= u cosdy-QiJ-XfCosCCf-Si) hence, we have w g l r J cw + + - +m+0.5m g r 2 r 2 I r I cw now m+0.5m is small compared with + -T +- r bence g r 2 r 2 we may assume (m+0.5i)ucos(0-e i ) X f cos(gf-6 i ) = - - - - i (ft) The equations of recoil, become therefore ^2+T f )t r r r V=VfCos((?-9 i ) and I c , '"r 1 (Rcosd +T f )t* r A r (approx) (m+0.5m)v (m+0.5m) u where Vf= and X w r ^ J cw , , w g J r ^w. ( ++ r) ( +-T + r) + m+ 0.5 m g r 2 r 2 g r 2 r 2 We see the equations of recoil during the powder period are exactly similar to the previous recoil equation, the recoiling mass now including the inertia effect of the rotating elements. Hence the previous interior ballistic formulas are im- mediately applicable for the computation of the free recoil displacement E and the time of the powder period t e . For the maximum velocity of recoil, we have mv +47001 Vf m = " (ft/sec) and the max. ^- + :?L velocity of & r r constrained recoil along the path of the gun trunnion, becomes, 914 (Rcosd +T f )t c r r IS+T r r The corresponding maximum angular velocities and angular displacements, become, ,de. V " s. "* ( dI )m= T and e * " T The energy of recoil at the end of the powder period becomes, w A s i(T +T + i r 9 )n 9 H m a v r x cw TT m From the energy equation we may easily consider the remainder of the recoil. Since the brake and friction resistances are small compared with the powder reaction and the inertia resistance of the rotating parts, we may assume with sufficient accuracy that v m =v fm cos ^ ~ 9 i^ and s m 3 B cos(*-0 i ) We have, for the recoil energy at any angular displacement 6. f (Rcos d.r ) d 9 + / ' T f r f de+W cw r ' (cos Qj- cos 6) i 9 i - Wg r r(cos Qj - cos 9)= A m - A where W gr = weight of gun and that portion of the rocker, not including the counter weight reduced to an equivalent weight at the gun trunnion, that is T" r" = u Wg where r' = radius of 915 W r r r *g r =Wg+ ' r r = distance from axis to center of gravity of rocker. Since d varies with the angular displacement of the gun lever, from a layout we may readily evaluate the term 8 f (Rccs d r Q )de provided R is assumed constant i which is usually the case. Further since T^r does not vary greatly we may assume it constant. As a close approximation, u(W g +W r +W cw )rf* u =0.15 roughly radii bearing of axis of rotation of rocker r" = radius of trunnion. Further T*r f =T I r f +T fl r" f f f f f Hence 9* / T f r f de = T f r f (6 f - 6 i ) (ft/lbs) now A = r(I r + I cw + r a )w s 2 (rad/sec) REACTIONS OK THE CORDAN LINKAGE Reactions on the DISAPPEARING CARRIAGE DURING Gun: The center THE POWDER PERIOD. of gravity of the gun is assumed at the trunnion axis of the gun. The angular acceleration of the gun is assumed small and the reaction of the elevating arm on the gun is considered a secondary force, this being possible by a proper arrangement of the parts or by the introduction of a kick down buffer at the base of the elevating arm. The primary reactions on the gun consist: (1) The powder force along the axis of the bore = P b 916 f?ACT/OA/S ON T/if GUN . Fig. 10 917 (2) The trunnion reactions divided into horizontal and vertical com- ponents X and Y respectively. (3) The weight of the gun acting through the trunnion axis = Wg (4) The tangential inertia force along the path of the movement of the trunnions or normal to a line from instantaneous axis to the trunnion axis = d 2 s nirf *dt* (5) The centrifugal inertia force, normal to the path of the trunnion axis and proportional to the square of the angular velocity = d Q sftp* The secondary reactions on the gun are: (1) The elevation arm reaction on the gun comparable with the kick down buffer reaction at the base of the elevating arm. (2) The inertia couple due to the angular acceleration of the gun about the trunnion axis. . In the following analysis, we will neglect the effect of the secondary reactions. The forces on the gun neglecting the secondary forces are shown in fig . ( Since we assume the rotation negligible, we have the equations of motion, P b cos 0-mg cos B+mgl(- -) 2 sinB-X=0 T-m g ^| sinB-m g k(~) 2 cos B+W g -Y =0 b d 2 s d*6 where tan B = - tan 6 ; -= 1 -r approx. since a+b dt 1 * dt z 1 does not change greatly during the powder period. 918 ON TH GUN FACTIONS ON TflF SUD//V6 Fig. I 919 I = / (a 2 + 2ab)cos 2 6+b a Hence, the trunnion re- actions become, d 2 8 d6 IT rcosB+m,* 1 (~) 2 sinB at s at Y - Pvsinef-D-1 sin8-m c l(- -) 2 cosB+W tf 5 dt * dt BUCTIOMS OK THE BOCKKB. The reactions on the rocker, are: (1) The reaction of the gun on the rocker divided into components X and Y. (2) The reaction of tlie sliding car- riage on the rocker at the rocker trunnion, divided into components X 1 and Y 1 . (3) The reaction of the counterweight cross head at the wrist pin of ttie cross head, divided into components X" and Y". (4) The weight of the rocker at the center of gravity assumed at the rocker trunnion W r . (5) The rotational inertia couple due to the angular acceleration of the rocker = , 2 Ir d~t7 (6) The tangential inertia force of the rocker along d*x OX * m rI7jr actin fi through center of gravity of rocker. (7) Tbe centrifugal inertia force of the rocker normal to OX at The equations of motion of tte rocker, become, along OX - Y-Yi+Y"-m =Q x * A " 920 ~fri along OY about the instantaneous axis I, X(a+b)cos 9+Y b sin 0-X'a cos 9-Y" a sin 9 - m r d f x d 2 9 _ a cos e - l r _= o BBACTIOHS 01 THB SLIDIHG CARRIAGE AHD COOMTBR WEIGHT RB3PBCTIVSLY IN THB DIRECTION OF THBIB MOTIONS. Considering the sliding carriage, we have, d*x X 1 H-m c -=0 Where R is the hydraulic brake re- action en the carriage and for the counterweight, d y Y"-m cw w =0 dt 2 EQUATION OF MOTION OF THE SYSTEM DURING THE POWDER PRESSURE PERIOD. Substituting Vhe values of X',Y",X and Y in the moment equation about the instaneous axis of the rocker, we havs, Pb I ( a + b ) cos e cos 0+bsin 9]- m ,. [ (a+b ) 2 cos a 6+b 2 sin 2 9] * dt 2 + Wgb sin 9-[R*(m r +m c ) ]a cos 9 m cw a sin 9 * dt 2 dt -W cw a sin 9 - l f 1J[ = Now x = a sin 9 dx d6 = a cos 9 dt dt d 2 x d 2 9 He,, jpj- = a cos 9 a sin 9 (-^) 2 921 and y * a cos 9 dy d9 ~ - a sin 9 dt dt d*y d*9 de, 9 _ = - a Sln _ - a cos 6 (_) If we assume the positive direction of y upward, then Substituting these values in the arova equation we have the general dynamical equation of the disappearing carriage during the powder pressure in terms of a single coordinate variable e. The differential equation of motion, becomes, P b [(a+b)cos 9 cos(8+b sin 6 sin 0]-nig[ (a+b)*cos* 9+t> j 2 ft j j A sin 8 9] - 'W.jb sin 9-B a cos 9-(m r +m c )a 2 cos* 9 - dt* dt* d 2 9 do -m_ w a 2 sin 2 6 - +(m..--m..)a sin 9 cos 9( )* -m rw dt 2 dt a sin 9 cos 9(~) -W cw a sin 9 - I *0 dt dt* Combining terras, we have, Pb[(a+b)cos 9 cos0+b sin 9 sin0]- d ()**[ tan (0-Q ) co s(0-B)-sin (0-8)]* - It i ,dQ , 1 cos(0-B) d6. c( ) -cw 1 = - - ( - ) From the ac- dt C COS 8 (0-Q) dt celeration fU J.C e* fce*? rj**,",* -rj fc?'? t .. ceieranon diagram, we have along the x axis. d ' y g ^djO.a ,dQ, a r~z -- d rrr) cos0+a d sin0=a,,cos Q-c ( ) sinQ at* at at , N ., d ^dT slnjZJ ~ a d cos ^* a c sln +c ^!J ^ cos Q then, 2 dQ a c sindcosQ= -5 sinQ-dCrr) sinQcos0+a d sinQsin0+c (- ) ct t d L d t sin 2 Q a c sinQcosQ= cosQ-df--) sin0cosQ-ajcos0cosQ-cf~) dt 2 dt dt cos 2 Q ROTATING TYPE CARRIAGE: The maximum reactions REACTIONS ON TRUNNION on trunnions and main AND FIXED AXIS OF bearing (fixed axis of rocker) ROCKER. are at a maximum value at the maximum powder pres- sure, and therefore we only need to consider these values in determining the strength of parts. The powder reaction is mainly balanced by ths insrtia resistance offered by the gun and the revolving parts. The reaction exerted on the rocker at the trunnions, is that needed to overcome the angular inertia of the rocker and counterweight which in turn must be equal to the powder reaction increases the inertia resistance offered by the gun. There- fore the heavier the gun as compared with the re 930 volving parts, the smaller the effect of the powder reaction. The reaction of the main bearing is consider- ably augmented over that of the trunnion reactions due to the tangential inertia forces of the counter weight. The development of the Cordan linkage in which the rocker bearing is allowed to slide back on a top carriage has been largely to decrease the reaction at the main bearing when fixed as in the revolving type. At the maximum powder force, the recoil velocity of the gun is small and therefore the centrifugal force of the gun may be negleoted. The tangential component of the trunnion reaction, be- T=P jnax cos(0-e i ) + Wsin 9^ where m jjar* and for the normal component, N=P raax sin(0-6 j )+WgCos Therefore, we have T=p roax cos(ei-e i )(i -- -i - ]+w g *r *cw in** + - 1 r r N=P max sin(0-e i )+W g cos B i _ and for the resultant trunnion reaction S^ s / N 2 +T* The maximum bending moment in the rocker or gun lever occurs at a section adjacent to the center bearing of the rocker. This bending moment is due to the moment of the reaction of the gun at the trunnion minus the inertia moment of that part of the rocker above the section, which is practically one-half the mass of the rocker or gun lever. The moment of the inertia resistance of the rocker, becomes, - s t d 2 6 l . dt a i l r ds l r . - \- - = - I_ - = - r - where is the r dt 2 r r dt 2 r * equivalent mass of the gun lever referred to the 931 trunnions. The maximum bending moment at center section of the rocker or gun lever, becomes, t X r d*s, M =vT - - - - -)r or in terms of the maximum w * r at powder force I r A r x cw no ++ r r In addition the section is subjected to a compression, C O =N+-^ W r cos e i=P Bax sin(0-9 i ) + (Wg*^W r ) We will now consider the reaction at the fixed axis of the rocker or gun lever. Since the tangential inertia effect of the rocker practically balances, we will consider the reaction on the main center bearing as due only to the reaction of the gun at the trunnions and tne inertia of the counterweight. The tangential inertia resistance of the counter- weight, is de dt* F cw s lmq. - where q is the distance to any mass particle of the counterweight measured from the axis of rotation of the gun lever. If r cw = the distance to the center of gravity of the counterweight, then ImqM cw r CB hence d 2 6 , . a . r cw M cw FCW=MCW r c w Jj#*.wftrtQF*i>-J It is to be noted that the point of application of F cw is not at the center of gravity of the counterweight, but rather at the center of percussion of the counterweight with respect to the axis of rotation of the gun lever. If k is the distance from the axis of rotation to the center of per- cussion, (J2Q where Z - the /* + * A + a radius of gyration of the Counterweight with respect to the fixed axis, cw Z 2 therefore k = - Resolving" the resultant r c reactions at the fixed 932 axu of the gun lever into components normal and alop~ the axis of ths gun lever, we have, neglect- ing the centrifugal forces as small, Y=N+(* cw +W r )cos or substituting values for N, T, and F cw , r cw X=Pj nax cos(0-9 i )(H- - ~-^ - )+(W g +W cw +W r )sin 6 i M cw ^ Y * p mas and for the resultant we have, S Q From these equations, it is easy to see, that the reaction at the fixed axis is increased over that at the gun trunnions by the tangential inertia of the counterweight ^ P^cos (0-6 A r " " IM With a heavy counterweight, this term is larger and the bearing load at the fixed axis becomes very great with large guns. To reduce this re- action and consequent weight of members, etc., the Cordan linkage disappearing carriage developed by Buffington.Crozier and the Krupp linkage have been used for the larger gun mounts. Sub- tracting, we have, -^ sinQ- r-r^cos Q+d()*sin(0-Q) +a d cos (0-Q)t-c(- ) dt a dt 2 dt dt d*x g d 2 y g Substituting the values of - and - , we have dt 2 dt 2 (a+b)[cos 6 - sin 6 (^)]sinQ-b[ sine ll| +cos (li)t dt* dt dt z dt cosQ+d(~) a sin(0-Q) +a d cos(0-Q)+c $-)* =0 dt dt Expanding and simplifying, we find 933 d*0 d6 2 d a a[cos6 - - sin 6(--) ] sinQ-b[sin(6-Q) - at at dt 2 dt ,dQ. I 2 cosM0-B) ,d0 x a and I*) = ( ) dt c* cos*(0 Q) dt hence -:ni?f Jsap* sdj a d b[sin(e-Q) +co<^Q)(-j7)*]-a[cose S-t - sin dt at dt - cos(gf-Q) ()] sin Q * -- -*-t-h 'ft*' ... ^_pk t 55- V c ; -'l^ cos 8 (0-8)1 d6 , sin(0-Q)+ ,' ^. > ( ) c cos 2 (0-B)J dt f -*. ,- : : Combining the acceleration and velocity terms, we have [bsin(9-Q)-a cosSsind] d 2 6 <{a sin Qsin Q+b a , * + > cos(0-Q) dt* cos(0-Q) f *f^>d Jliii ^(iSJSo -- - ^ i) i* i cos(e-Q) - rtan(0-ft)cos(0-B)-sin(0-B)] sin(0-Q)> AX * d9 1^ cos 2 (0-B) d9 2 dt c cos 2 (0-B) dt Therefore the angular acceleration of the gun, be- comes, A a d d dt 2 ' dt 2 d d cos(0 - Q) (rad/sec. 2 ) 834 where A d b sin (9 - Q) - a cos 9 sin Q B d - a sin 9 sin Q+bcos(6-Q) [tan(0-Q) d cos(0-B)-sin(0-B)) a sin(J-Q) + cos 2 (KB) c cos*(0-Q) d Q For the acceleration a c = c -^ we eliminate a rf in dt* the equations:- dtt . cosH 3( ) a u dt< -a c cosQcos0 d*y g d0 a dQ 2 = TTT- sin0-D(-~) sin*0-c( ) cosQsinfl at dt dt -a c sinQsin(? Adding, we have ,d0 2 dQ a in0-d( ) 0-c( ) sin(0-Q)-a c cos(0-Q)=0 dt* at* dt dt hence * c 3 cos(0-Q) g g a a Substituting for , , , ., (-- ) and ( ) we obtain dt z dt 2 dt dt dt 2 dt dt* dt cos(0-Q) cos(0-Q) eos g (0-Q) dt 935 Combining, we have, r i /y u /*. /*\ 4bsin (6-0)+acos0sin9+- acos6eos0+bcos(e-0) d*e I d cos(0-Q) dt [tan(0-Q)cos(0-B)-sin(0-B)]*+ cos(0-Q) c cos a (0-Q)J dt d*Q Therefore since a c = c , the angular acceleration dt * of the gun lever, becomes, 2 d*Q 5 dt a i 160I.-3V d c cos(0-Q) where A., = a cos 9 cos + b cos (6-0) r = - sbsin(9-0) + a cos0sin9+- [ tan(0-B)-sin c cos* (0-0) RECAPITULATION 07 VELOCITIES AND ACCBLBRATIOHS IH A CORDAH LIHKAGB DISAPPEABIN6 GUN CARRIAGE: Let a+b = total length of gun lever a = distance from cross bead to top carriage trunnion b = distance from top carriage trunnion to gun trunnion d = distance from gun trunnion to elevating arm trunnion measured along gun. c = length of elevating arm. 3 = angle gun lever makes witn vertical = angle turned by gun Q = angle elevating arm makes with vertical M = angular velocity of gun lever dt 936 = angular velocity of gun at dQ angular velocity of elevating arm d9 = angular acceleration of gun lever d0 TTT z angular acceleration of gun d*Q 777- s angular acceleration of elevating arm dx g - = horizontal linear velocity of gun at trunnions - = vertical linear velocity of gun at trunnions a = horizontal acceleration of gun at trun- nions , s = vertical acceleration of gun at trunnions at dx = velocity of top carriage at dx - - * acceleration of top carriage dt* dy * velocity of counterweight and crosshead at dy TT7 = acceleration of counterweight and cross- u t bead Then, in terms of the angular velocity and acceleration of the gun lever, (a) The velocity and acceleration of top carriage, are dx d9 - - . e - (ft/iee) d*x d a 8 dfi 2 - = a cos 6 -- a sin Q( ) Cft/sec a ) dt a dt z dt 937 0>) The velocity and acceleration of tbe top carriage, are " ~ a sin 9 (ft/aec) at = - a sin 9 ~j - a cos 6 (2)' (ft/sec) (c) The velocity and acceleration of the gun, are ^ = (a+b)cos 9 ( 4r dt at - = (a+b)cos ^Ii 2 _ dy g at d * y - at de b sin 6 - at d9 s = b sin e T +b cos 6 (- ) at z dt i[tan(0-Q)cos(0-B)-sin(0-B)] d dt (ft/sec) (ft/sec) (ft/sec) (ft/sec) (rad/sec) df (rad/sec) where A = b sin(9-Q)-a cos 9 sin B = a sin 9 sin Q + b cos(9-Q) -- [tan(0-ft) cos(0-B) - sin(0-B)]in(e>-Q) + c cos 2 ( (d) The velocity and acceleration of tbe elevating arm, are dQ 1 cos(g-B) d9 = (rad/sec) c dt dt 2 c cos(CT-O) where A_ = a cos 9 cos + b cos (9-0) 12 B c = ~ "J b sin(8-0) + a cos $ sin 9 + [tan(0-Q) L c ..a 1* cos 2 (0-B)l cos(0-B)- sin(0-B)J + /* \ r c cos 2 (0-Q) J Coordinates of the system: Displacement of top carriage = x Displacement of counterweight = y Distance from instantaneous center of gun lever to gun trunnion: 1 = Aa+b) 2 cos 2 6+b 2 sin 2 6 COORDINATES OP THE CORDAN LINKAGE In estimating DISAPPEARING CARRIAGE. the work done by the various weights and resistances during the retard- ation period of the recoil it is necessary to com- pute the various displacements of the parts of the system in terms of the independent coordinate of the system. Prom the diagram, to determine VI and Q in terms of the angle 9 made by the gun lever with the vertical, we have x o = (a+b)sin 6+d cos 0-c sin Q y - b cos + d sin + c cos which may be written, d cos 0= x Q -(a+b) sin 6+c sin Q d sin 0=y o + b cos 6-c cos Q Squaring and adding, we have d*=[x o -(a+b )sin 9]" +2[x -(a+b)sin 9]c sin Q +(y o +b cos 9 )%2(y +bcos 9) c cos Q + c* This equation may be put in the form, 939 [x Q -(a+b)sin 9]sin Q+(y +bcos 9)cos Q [x -(a+b)sin 9) 2 +(y o +bcos 9) 2 2 hence m sin (A+Q) = J "S d a 7C 2 + [x o -(a+b)sin 9)%(y o +bcos9 )*> where m * /[x Q -(a+b )sin 9] 2 +(y o +bcos 9) a _. x ft -(a+b)sin 9 A = tan [- y o +bcos t From this equation we may solve for Q in terms of 9, and substituting in either equation below, cos = 3- [x -(a+b)sin 9+c sin a] a sin 0=5 lyo + k cos ~ c cos ^ we may then calculate the value of in terms of the independent variable. Further if, Displacement of top carriage - x Displacement of counterweight = y The distance from instantaneous center of gun lever to the gun trunnion, is 1 = /(a+b) a cos 2 9 +b a sin 2 6 8EACTION3 ON THE PARTS OF Considering the re- CORDAN LINKAGE. actions on the gun, it will be assumed that ths center of gravity is located at the gun trunnion. The gun is subjected to a translatory acceleration divided into horizontal and vertical components as well as an angular acceleration due to the reaction of the elevating arm. Let (1) PI, = the powder pressure along the axis of the bore (2) X and Y = the horizontal and vertical reactions at the gun trunnions. 940 (3) W s the weight of the gun acting through the gun trunnion (4) nu*-* and m d * ? = the inertia 8 dt a components along the horizontal and vertical axis (5) *gT~7 * the i ner *i a angular re- sistance (6) X"'and Y'"= the horizontal and vertical components exerted by the ele- vating arm on the gun Then for the motion of the gun, we have d a x g - - - x"'= c d t P b sin0-Y-m- dt y MI dco80-X'"d dt* For the elevating arm, we have (X 1 ' 'cosQ+Y 1 ' 'sinQ)c- dQ I c - = 5 dt 2 where I c = the moment of inertia about its fixed axis. Combining with the moment equation of the gun, we have cos " 1 * d~F c sin Q (Ibs) cd cos(0-Q) d 2 Q I c d sin0*Ig c cos Y"'= - (Ibs) cd cos (0-a) Next, to obtain the reactions X and Y we must consider the dynamical equations of the gun lever. By taking moments about the instantaneous center of the gun 941 lever, we eliminate the unknown normal reactions of the constraints of the carriage and counter- weight. Then for moments about the instantaneous center of the gun lever, we have X(a+b)cos 9+Y b sin 9-X'a cos 9~Y"a sin 8 - ra d'x d*9 d'x a cos 6 . I R = o where X'-R*m c R s the hydraulic brake reaction on the carriage "m c ". d a y Y"m cw +w cw m cw and w cw mass at dt* weight of the counterweight. Combining, we have the dynamical equation of the motion of the disappearing gun carriage during the powder pressure period, as follows: dQ d*0 dcosfl-I- csinQ c dt a Sdt a - - - ^] cd cos (0-Q) c cos cos 9 [P b sin 0-m c -- J b sin 9 dt 8 cd cos(fr-d) d a x d a y -(R+ra - ) a cos 9 - (m cw - + w cw ) a sin 9 dt dt d9 - ra B - a cos e - ID - * * dt* dt 2 For a solution of this equation we must substitute for the various accelerations their value in terms of a function of tne acceleration d*6 - . The hydraulic brake resistance R may readily dt * be obtained by considering the energy equation 942 of the linkage to its recoiled position. If Ag 3 kinetic energy of gun at end of powder period (ft/lbs) A c Kinetic energy of top carriage at end of powder period (ft/lbs) A e = kinetic energy of elevating gun at end of powder period (ft/lbs) A R * kinetic energy of gun lever at end of powder period (ft/lbs) A w ~ kinetic energy of counterweight at end of powder period Then for the kinetic energy of the gun, we have, if 1 - the distance to gun trunnion from the instantaneous center of gun movement, and k radius of gyration about center of gravity or trunnions of the gun. 2 ) d J dt (ft/lbs) For the kinetic energy of the elevating arm. -% BOO *$ } _ 1* eos*(0-B) d6 c 2 cos a (e-Q) dt For the kinetic energy of the gun lever, if k E * NOTE: If the path of the sliding carriage has an inclination to tha horizontal equal to angle d, then for the equation of tho gun lever, we have a 2 * X(a+b)coe O +Y(b sin O - a cos tan d ) - ( R + m c - ) dt 2 Substituting the values of X and Y as in the previous equations, we have the general dynaioal equation of Motion. 943 radius of gyration about the center of gravity of tbe gun lever, we have A R If tbe top carriage and sides are inclined plane making angle with tbe horizontal. ' 9(l+tan a V./TB. B . dt For tbe kinetic energy of the top carriage A c = r l "c a * cos8 e ^TT^ for horizontal plane and A e = 7 m e a a cos 2 9(l+tan* o)( )* for inclined plane dt For the kinetic energy of the counterweight and cross bead, t A w = I m w a2 sin * 9 ("cTt^ a When tbe sliding carriage rides an inclined plane the kinetic energy of the counterweight, becomes, 1 2 d 2 A w = -ffl w a a (sin 9+cos 9 tan <*) ( *-) From the principle of energy, where W B ~ work resisted by the recoil brake W CB = work resisted by the weight of the counterweight Wg = work done by the weight of the gun W e = work done by the weight of the elevating _ arm W R = work done by the weight of the gun lever W c = work done by the weight of the sliding carriage During the powder period, the sliding carriage moves a distance E and tbe gun lever angle increases from 6 to 9j . The length of recoil = L and the re- coilad position of the gun lever makes an angle 6 with the vertical. 944 Work, resisted by the recoil brake W B jf R=the brake resistance, then for the work of the recoil brake during the retardation period, we have W B R(L-E) (ft/lbs) where obviously L-Ea(sin 6 2 - 3 in and with an inclined plane sliding carriage, Work resisted by the weight of the counterweight * W cw "cw ~ *w y w "w * "eight of counterweight where y w = a(cos 9 t -cos 9 a ) and with an inclined plane sliding carriage y w = a(cos 6 t -cos 6 a )+L sin a Work due to the weight of ths gun = Wg Wg= "g^g "here yg = (a+b)(cos 9 t -cos & 2 ) and like- wise with an inclined plane sliding carriage yg=(a+b)(cos 6 t -cos 9 2 ) Work due to the weight of the sliding carriage and gun lever = W r +W,. , f Assuming the center of gravity of the gun lever at the gun lever trunnion, the center of gravity of the gun lever has the sane displacement as the sliding carriage. Hence W.+W *(w r +w c )y c where y e (L-E)sin a * (sin6 - 1 ** COS ** sin B I ). Hence when the plane is horizontal no work is done by the weights of the gun lever or sliding carriage. Work due to the weight of the elevating arm = W e where y e = d e (cos C^-cosQ,) d e = distance to center of gravity from fixed axis of elevating arm. = sin in-1 4 {d g -c+[x -(a+b)sin Q I ]' 2 /[x -(a+b)sin ej a +(y o +b cos 946 (a+b) sin 9 - tan" [- - - - -] yg+bcOsQj^ ' [{d a -c a +[x -(a+b)sin 8 l+(y o +bcos 6 Q a sin" 1 j - L 2/[x -(a+b)sin 9 a 1 +(y Q +bcos 6^ ) x n -(a+b )sin 6 - tan" 1 [-2 - g - ] y o +bcos 9 2 EQUIVALENT MASS OF CORDAN LINKAGE. During the powder period, it is convenient to express the dynamical equation of recoil in terms of the external moments or forces and the equivalent mass of the system tines the acceleration of the coordinate considered. The equivalent mass and corresponding reactions may be referred as a function of the angle made by the gun lever witb the vertical or as a function of the displacement of the slid- ing carriage. (1) Equivalent mass referred to angle "6" of gun lever witb vertical :- From the dynamical equation of recoil for the Cordan linkage previously derived, we have, for moments about the instantaneous center of the gun lever, Phla cos 9 cos(9-0)]-Ra cos 9H0 * a sin 9 m- [ (a+b )cos 9 - + 8 i n 9 ft] +* a dt dt cos 9 + m pw a sin 9 + I dt a dt dt c cos(0-Q) d a Q [bsin(9-Q)-a sin Q-cos 9] d 2 dT 2 * +I ^ d cos(Gf-O) dt a Neglecting the centrifugal components of the accelerations, as small, d*x a d6 d * x g . . d a 9 - - a cos 9 - ; - a =(a+b) cos 9 - dt dt 8 dt dt * 946 d a y d*9 d*y tf d*9 a sin 9 : * b sin 9 dt* dt* dt dt* d*Q 1 d a O TT * ,~ ^ t a c s e cos 0+ fa o s (9-^)]- r dt* c cos(Qf-Q) dt* Substituting, we have Pj,la cos 9cos0+bcos (9-0)]- Ra cos 9-ViL IB a sin 9 = -a sinQcos9]*l cos a (0-Q) \ (a*+2ab)cos* 9+b*] +m c a*cos* [acos 9 cos 0+b eos(9-0)] ^ [bsin(9-Q>- c a cos(Br-Q) g de _ d 9 Thus the equation is in the form of AP h -BR-CW.=D- dt* dt here A = a cos 9 cos + b cos ( 9 - (?) B = a cos 9 C = a sin 9 and for the equivalent mass "D" D * njg[(a a +2ab)cos* 9+b a ] +m c a*cos*9+m cw a 2 sin*9+I r [a cos 9cos0+b cos-(9-0)] a [bsin(9-0)-asinQ + I * I c*cos*(6-0) d 2 cos 2 (0-0) cos 9]* -^ For the solution, during the powder period, we express the powder re- action as a function of the time, then rlfl fl ^ h DC\TOn w A UO m U x W . A XX f > _./_. (_ )t , _ MS -(_ , t where Vf is the velocity of free recoil of the gun Integrating again, A BR+CW CW a _ _ u TP _ f \ +2 D ^ * ( - 20 ) where E is the displaceaent of the gun in free re- coil. From the solution of these equations, we obtain, 947 e t * e o* M g e ~ <~ - > T * "here T = powder interval BR+CW The angular displacement and the angular velocity of the gun at the end of the powder period. Sub- stituting these values in the energy equation, we have *B * A g*M* R + A c +A w +Wg+Jr e +i R +* C -W CW and there- fore can readily determine R the total braking resistance. (2) Equivalent mass referred to dis- placement of sliding carriage X;- In place of a movement and angular acceleration equation, we nay consider the inertia and the reactions of the system as re- duced to an equivalent translatory mass and force as a function of the displacement of the sliding carriage. Therefore reducing the motion of the system to one of translation along the path of the top carriage. By direct analysis, we have, if Pjj * powder reaction X and Y = components of gun trunnion reaction X I and Y' = components of top carriage on gun lever X" and Y" = components of reaction on gun lever at crosshead X' 11 and Y 1 ' ' = components of elevating arm reaction on gun mg = mass of gun m cw = mass of counterweight T. S = mass of elevating arm ro_ - mass of top carriage c r m B = mass of gun lever K e> Then, for moments about the instantaneous center 948 d*x X(a+b)cos 6+y b sin 6-Y" a sin 9-m R a cos 6-I r dt -X' a cos 6 = o dt Dividing through by a cos 9, we have )y tan e-Y-tan e -. a 2 R dt* a cos9 Since d 2 sin cd cos (0-G) d 2 cos cd cos ( -Q) ire have on substitution, P^( - cos(?+-tan 6 sin0)-W cw tan 6-X 1 as the equi- 3 8 valent force acting along the top carriage guides and _ d 2 Q J _ dC I r rr d cos0-I fl r re sin b t dt a a *dt 2 a cd cos a sin JtfTa c cos v a+b dt a gdt 2 b . d*y - - 9+ - tan a cd cos (fif-Q) ""dt* I r 4*9 + as the equivalent inertia resistance a cos9 dt offered by the mass of the total system reduced to the path along the top carriage guides. 949 THROTTLING CALCULATIONS WITH ADD WITHOOT A FILLING IM BPFF1B. 4.7 Gun Trailer Mount with U. S. Variable Recoil Valve. w = weight of projectile 45 Ibs. v = muzzle velocity 2400 ft/sec, v = 166.13 (in.) w - weight of ponder charge 11 Ibs. p b max * 34000 Ibs/sq.in. b = 36" (0 to 45) 4.7 (0 to 45) X = total resistance = 17806.9706 Ibs. * r * weight of recoiling parts = 7560 Ibs. S t 3 spring load * P v =S f 3 spring load at end of recoil 16140 Ibs F v j = S o = spring load at assembled height = W r * 1.3 = 9800 Ibs. 16140-9800 6340 St * S o " ~36 " "ST " 176 - U1 lb8 " increase of spring load per inch of recoil. * maximum angle of elevation * 45 W r sin a = weight component = 7550 * .70711.= 5338. Ibs.- 6805 B g = stuffing box friction * 2,25 * 100 * 225 Rg = guide friction = H r u cos u = coefficient of friction .15 R g = 7550x.70711x.15 -800.8021 Effective area of recoil piston = 9.337 (sq.in) 950 METHOD OF PLOTTING VELOCITY CURVE - VARIABLE RESISTANCE TO RECOIL. Kt, ' P V fo - ; (ft/sec) 2M (ft) When projectile leaves the muzzle. Kt. (ft/sec) 2M (ft) The maximum restrained recoil velocity and corresponding recoil. ob where V fm =V fo * (t m -t o )[l- _^l__^_r_ Mr 4M r (V f -V ) 'ob 6M r (V f -V ) ](t.-t ) (ft) and K(T-t ) t. - T - -= ( S ec) Kt V P -V f -- (ft/sec) M M Kt* E r >B -- (ft) 2M r > At the end of the powder period During the retardation period, / 2[K- \ (b+x-2E r )](b-x) / ^ and therefore cA 2[K- - (b-x)-2E r )](b-x) (sq.in) 13.2 - R t - W, 961 which gives the required throttling area with a variable resistance to recoil during the retard- ation period. S f -S x P h =K+W p sin0-R t -(S + ) for spring return re- cuperators. Equivalent throttling area : 4.7 A. A. Trailer, Model 1918 1_ 1 1 W2 \tf 2 U72 Area of one hole = .0113 sq. in. In battery - W xb = 20 holes = .226 sq.in. w| =.0510 W x =103 holes =1.1639sq.in. w_ =1.3546 t * t 4" Recoil - W* =81 holes =.9153 sq.in. W* = .8377 x f x a W =88 holes=.9944 sq.in. W^= .9888 8" Recoil -W x =86 holes =.8781 sq.in. WS * .9566 \ \ W Xj = 77 holes = .8701 sq.in.W| t = .7570 12" Recoil -W, =98 holes=1.1074 sq.in. W. = 1.2263 x a *a W x =67 holes * .7571 sq.in. W| = .5732 i x t 16" Recoil -W x =107 holes = 1.2091sq.in.W = 1.4619 W x =59 holes = .6667 sq.in. W = .4444 *i *i 20" Recoil-W_ =115 boles = 1.2995 sq.in.W* 1.6887 * 2 W x =50 holes = .5650 sq.in. WJ = .3192 24" Recoil -W, =125 holes= 1.4125 sq.in.W* = 1.9951 X 8 ** W x = 30 boles = .3390 sq.in. W = .1149 28" Recoil -W x =140 boles - 1.5820 sq.in.W* -2.5027 W X 2 =30 holes = .3390 sq.in. WJ 4 = .1149 32 M Recoil-W x =140 holes = 1.5820 sq.in. W$ =2.5027 t W x = 24 holes = .2712 sq.in. W| = .0735 36" Recoil -W y =152 boles=.1.7176 sq.in.W x = 2.9501 *2 2 W x =0 holes = sq.in. W* = 962 Equivalent throttling areas-4.7 A.A. Trailer, Mpdel 1918 -1 - y, . i , y -*- i e y In battery, i- + -=20.355, .049127, .221 W .1.3546 .0510 4" Recoil,^- = + - =2.204, .453720 .675 w e .9888 .8377 8-B.oo.il.ij. 7^570 +i 956e- 2.366 .422654 .650 12"Recoil,jL - -^ * j-^f 2.569, .389256. .623 6 168ecoil |j - -ijjj * -j-^ijj = 3.934 .340831 .683 20" R..011..JL. -ijg * T -i^ - 3.724 .2685S8, .818 11 1 24" Recoil, = + * 9.204 .108648 ,329 w e .1149 1.9951 28"Recoil,- * 4 = 9.102, .100865, .331 6 .1149 2.5027 32 "Recoil, = + =14.00, .071428, .267 w e .0735 2.5027 36" Recoil, Equivalent throttling area - 4.7 A. A. Trailer, Mgdel 1918 (Calculated) R = = W* = 175WJ 175R h K = 1.43, K a =2.045, A = 9.337 sq.in., A 3 =913. 994 .7 KA 8 = 1869-U7 863 i t 1" Recoil * 1869 - 117<16 - 60 ' - .800656, W...447 175x12072.42 A t 4" Recoil* V = 1869.117*19.330 176x11330.17 ____^* A t 12" Recoil=W| = 1869 ' 11?Xl6 ' 614 =.315279,W X = .561 175x9350.81 ^B^^^^_^B^2 A t 16" RecoilW| 1869.117x15.090 , t290878 ^ w , >531 175x8361.14 A t 20" Recoil=W = 1869>11?Xl3>245 =. 254184, W x * .504 175x7371.46 . 1869.117x11.569 A t 24" Recoil=W* = - =. 223998, W x = .473 175x6381.79 ^^vwft A t 28" Recoil -W 1869.117x9.395 = <174836 ^ w = >418 175x5392.11 9 A t 32" Recoil=W = 1869 ' 117x6 ' 604 3 . 105806, v .325 175x4402.43 A t 36" Recoil - Equivalent throttling area - With filling in buffer Aj, = 1.767 sq.in. A Q * .76 sq.in. A 9.337 sq.in., A- = .69 sq.in. ^- = -==+-=-^ +-^=1.73+2.10=3.83 w b .76 .69 .577 .476 K(A-A b ) 3 V Wg -^ a -, 1.43* (9.337-1.767) 3 xl9.33 At 4" Recoil, W| = , 176(113 30 - L33 -1.767 .19.88 W a "e 175x.261 2.04x433.76x373.64 1.43 (9.337xl.767) 3 xl8.03 At 8"Recoil,H| = 2.04x433.76x325.08 175(10340- ' * 1! - 160 ' " -* 00 A 12" 2.04x433.76x275.89 W = "" *" " ^^ *^^^m^m^m^* 1.76.5.51x275.89 - 160 ' " * _ it 16" Recoil, *. > 1.43'(9.337-1.767)-16.09' 176(8361 . 1.33 .1. 175x.261 a 2.04x433.76x227.7 175(8361 . 1 - 76>5 -"-^- 7 45.67 955 r 3_nV FI6. A FIG. B FIG.C FIG. D GUN LUGS PLATE I 966 . 201484_ , _ 138> 1454775 COH8TBUCTIVI DETAILS. GUN LUGS - Typical gun lugs are shown in PLATE I. Plate I, figures A, B, C and respectively. A gun lug properly speaking is an integral part of a gun, being an integral part of the bree ring. Fig. A, shows an arrangement used on the 75 m/m Futeaux Model 1897. Surrounding the lug is the piston rod yoke connecting the piston and gun lug, fig. H - Plate 3. The piston rod yoke and gun lug are connected by the key passing through both lug and yoke. Pig. B, shows a simple construction used on the 4.7" anti-aircraft mount, model 1917. Fig. C, shows the lug of the 155 m/ra G.P.F. gun. The two "holes in the lug are for the hydraulic and recuperator piston rods. Fig. D, is a lug for connecting the recoil sleigh n slide to the gun in the 155 m/m Schneider Howitzer. The sleigh and gun are further connected by a front clip, but the pull during the recoil however, is exerted through the lug, the front merely supporting the gun. Two sections are important in the design of a lug:- the section "ah" just above the rods, which carries mainly shear, and the section "mn" at the breech circumference which should be designed main- ly for bending. ARRANGEMENT OF GUIDES The recoiling parts AND CLIPS. are constrained to recoil ID the direction of the axis of the bore by the engagement of clips attached to the gun or recoiling mass, in suitable guides on 967 the cradle or recuperator forging. The reaction between the guides and clips balance the weight component of the recoiling parts normal to the bore and the turning moment, due to the pull of the various rods about the center of gravity of the recoiling parts. Due to the large turning moment caused by the pulls as compared with the weight component of the recoiling parts normal to the bore and more or less "play" between the guides and clips, the normal reactions exerted by the guides on the clips are more or less concentrated at the end contacts. The distribution of the bearing pressure, of course, depends upon the elasticity and play between the clip and guides, and there- fore, assumptions based on experience must be made as to the proper surfaces required. In older type mounts, we have a continuous clip on the gun, engaging in the guides of the cradle. Unless the gun clips are sufficiently long, we have a varying, (gradually decreasing distance), between the clip reactions assumed concentrated at the ends and thus the friction of the guides increases in the recoil. Due to heating of the guides firing unless sufficient play is allowed for, warping of the guides may cause a binding action between the clips and guides. Therefore, due to these considerations, (1) the increase in clip reaction towards the end of recoil, and (2) the difficulty of preventing warp- ing of the guides or clips and (3) the necessity of a long gun jacket, continuous gun clips have been nore or less discontinued in modern artillery. When gun clips are used we have combinations of three or more gun clips. When only three 966 clips are used it is possible to maintain practically only two clips in contact with the guides through- out the greater part of recoil. This is an advantage since any warping of the guides, etc., does not materially effect the operation of the recoil. With four or more gun clips, we have one or more inter- mediate clips, thus necessitating a more careful lining up of the gun clips and design of the guides to prevent warping unless considerable play is to be allowed. Referring to fig. (3) Plate we have an arrangement of three clips A, 8 and C, which recoil to an intermediate position A^B^C 1 , where the rear clip leaves the guide and the front clip enters the guide. If the clips are equally spaced as they should be, this intermediate position is one-half the length of recoil. In the final position the clips are in the position A" B" C" at the end of recoil. If "1" is the distance between clips, since A should not leave the guide until C enters the guide and at the end of recoil 6 must be still in contact with the guide, the length of guide should be: Win. length of guides = 2L = b, (3 gun clips) and therefore, b Distance between clips 1 = - 2 With three clips, during the first half of the recoil, the coordinates with respect to the center of gravity of the recoiling parts of the front and rear clips respectively, become those of B and A while during the latter half, they become those of C and B. With four clips we have an intermediate clips always in contact with the guides; hence a careful alignment is necessary with more or less to prevent any binding action of the middle clip and throw the greater part of the clip load on the extreme 969 front and rear clips respectively . Referring to fig. (3) Plate ( the clips A,B,C and D move from the battery position, to the midway intermediary position, that is when clip A leaves the guide and clip D just enters the guide. If "1" is the distance between the extreme clips in bat- tery, i.e. between A and C, or between B and when clips are equally spaced as they should be, we have 1 = b, that is the distance between clips equals the length of recoil. Further the minimum length of guide = - I - ~ b. With four clips, the coordinates of the front and rear clip reactions with respect to the center of gravity of the recoiling parts during the first half of recoil become those C and A respectively, while during the latter half they become those of 8 and respectively. Let us now consider the front and rear clip reactions between the guides and clips of the gun. The clip reactions, become, for the front Clip ' P for the rear clip, P bs + Bd b * r cos0(x t -ny x ) where P b = the max. total powder reaction on the breech (Ibs) e = the distance from the axis of the bore to the center of gravity of the recoil- ing parts (inches) B - the total braking pull excluding the guide friction (Ibs) dfc = the distance down from the center of gravity of the recoiling parts to the line of action of the center of pulls (inches) 960 W r = weight of recoiling parts (Ibs) x t and y t = coordinates of front clip reaction measured from the center of gravity of the recoiling parts. x a and y a = coordinates of rear clip reaction measured from the center of gravity of the recoiling parts. n = coefficient of guide friction = 0.15 approx = the angle of elevation. 1 = x t + x - the distance between clip re- actions . Since ny and ny are small as compared with x and x a respectively, we have for a close approx- imation, P b e+Bd b -W r cos0.x 2 Bd b -W r cos0 x 2 = - - (approx) , n 1-2 nd r 1 P b e+Bd b +W r cos0 x t Rd b +K r cos t (approx) l-2nd r where d r a mean distance from center of gravity of recoiling parts to guide. The guide friction, becomes, 2nBd b +nW r cos0(x -x. ) RI 3 n(Q +0.) - E (Ibs) l-2nd r The following table is useful in the layout arrangement for the gun clips and proper length of guides, as well as showing the change in clip reaction and guide friction for ths two combinations . 961 M M X X 1 1 X X ^w M o O o Ll Ll Ll t-, tit rt S a c c c c CV3 IN) + CO ( s q T ) Q 1 -o 1 \ Ml/ 'O^ 03 TJ jQ UOT^OTJjJ 3 p T n } T * ^ O J. CD CD C C * O3 X is cs M X O S O w Li o f s q T ) tf o t, uot^aeay djig jesg ja XXI * .> CD JQ N ' TJ CVJ CD X 9 M CO X ^Sl o 8 L! 5 o ( s qt ) 1 O Li uoj >o* aa <*T TO uoj,g CD I ft 03 CO psatnbsj .O sapinS jo q^JJuai *Ufh -O CO 1 CO s u o j 3. 3 p 3 j d | i o jeoj pus ^ u o j j u a 3 u ^ o q aouc^efQ ^) 1 03 J3 q i T o o 3 j jo sujta^ u T sdi-[o ussii^aq souF^sta XI 1 CM X( 1 CVJ sdTTO jo on CO sf 962 DESIGN AND STRENGTH OF In the design of gun GUN CLIPS AND GUIDES. clips and guides, the following points should be considered: (1) General considerations as to lay out, protection from dust, etc; (2) the arrange- ment of clips and guides as outlined in the previous paragraph; (3) the computation of the maximum clip reactions; (4) the design of the clip or guide for allowable bearing pressure; (5) the strength of the clip or guides at their various critical sections, to resist bending, direct stress and shear. (1) The location of guides in the direction normal to the axis of the bore should be based on the follow* ing considerations :- (a) From a cross section of the gun and recuperator forging, the best position of guides and gun clips can be located with consideration for minimum stress in gun clips. This requires that the guides be located as near the axis of the bore as possible. (b) For constructive reasons, it is good design to keep the various parts connected with the recoiling parts as near the axis of bore as possible. (c) The reactions of the guides, however, are quite independent of the position of the guides in a normal direction to the bore, but since the resisting section of the cradle or re- cuperator forging is very large 963 Fk5. F GUN CLIPS PLATE 2 964 Q< t>j -]gu u ^Bo 965 as compared with those of the gun clips; gun clips with long projections downward from the gun clip jacket due to guides too far below the axis of the bore are undesirable. Hence the location of the guides depends upon construction and fabrication features with due consideration to the strength of the gun clips. These features in general demand that the guides be located as close to the axis of the bore as possible. (2) For small guns, three clips equally spaced as described in the previous paragraph should be used. The front and rear clips should be bevelled off, so that smooth entrance may be made into the guides. Bronze liners either in the clips or guides should be used. For larger caliber guns, more clips should be used since the clip reactions and cor- responding friction are reduced. Considerable tolerance should be allowed but very careful alignment made in order to prevent possible binding . (3) The computation of clip reactions has been tabulated in a previous paragraph, for the common arrange- ment of either three or four equally spaced gun clips. (4) The bearing contact during the recoil between guides and clips, depends upon tolerance between the guides and clips as well as the elasticity of the material, and on the magnitude of the wear between 966 the clips and guides. Therefore, we see the distribution of bearing pressure and the length of contact is completely indeterminate. From practice, however, the following assumption will be made: (a) Length of gun clip 1 = 1.8d (in.) approx. where d = diam. of bore. (b) Constant length l'=1.5 d(in.) (c) Distribution of pressure assumed triangular. Therefore, if b 1 = contact width of clip and guide, (inches) we have for the maximum bearing pressure due to the clip reaction Q (Ibs) . Q 1 Ibs. per sq.in. b d Now the max. allowable bearing pressure steel on bronze, becomes, pg m = 600 to 800 Ibs. per sq*in. Hence b' = .0017 to .0022 - (inches) d The distance 1-1 ' should be the bevelled length of clip distributed on either end. With eccentric pulls the side thrust between clips and guides causes a bearing reaction Q 1 and if b" is the depth of guides in contact with clip, we have, b' = .0017 to .0022-=- (inches) d (5) The strength of gun clips depends upon the form or type of gun clip used. In fig. E, plate 2, we have the minimum bearing contact (w-x). The required thickness of the toe T is based on bending at section (a-b). Since the front clip re- action causes this bending, and the load is divided between two L. FIG. H 967 FIQ. K , j FIG. L. R5TOM ROD GUH Pt-ATE -4- 966 front clips on either side. We have, , (-x) /Q (w-x) 1.225 / 1 = 0.912 / (in) where f m = elastic limit of material used. STRENGTH OF RECOIL PISTON The greater part RODS. of recoil piston rods are subjected to ten- sion during the re- coil, and com- pression during the counter recoil due to the counter recoil buffer reaction. In a few types of recoil systems, we have compression in the rod during the recoil, an example being in the pneumatic cylinder of the 16" U. S. Railway mount. The critical diameter of 2 recoil piston rod is at the smallest section within the gun lug as shown in figures H, K and L, Plate 4. This diameter should be based on the recoil pull at maximum elevation and the inertia load at maximum acceleration This load is the same that occurs for the gun lug. Let P a the total fluid reaction + packing friction on piston and rod (Ibs) B = total braking (Ibs) Pb = total max. powder reaction on breech (Ibs) f m = allowable fibre stress of material used (Ibs/sq.in) w p = weight of rod and piston (Ibs) w r = weight of recoiling parts (Ibs) d = diameter of smallest free section at gun lug, 0.7854 f m For hollow piston rods, with a "filling in" or spear buffer chamber, we must consider a section the greatest distance from the piston but passing through the buffer for maximum inertia and minimum 969 thickness of the rod. Let w' = weight of piston* rod to section (Ibs) d ro = outside diam. of buffer rod (in) d r j = inside diam. of buffer chamber (in) Then using the previous symbols, we have, d ro ~ d ri = - usually d ri is fixed 0.785 f m in consideration of the buffer design, hence d ro is determined from the above formula. When piston rods are subjected to compression, during the counter recoil or with a pneumatic recuperator during the recoil, the rod should be treated as a column loaded and constrained at both ends. The maximum column load on the rod equals the maximum counter recoil buffer load, which may be roughly estimated on the basis of counter recoil stability at horizontal elevation. If Cg = constant of counter recoil stability = 0.85 to 0.9 W s = weight of total gun + carriage (Ibs) lg = distance from wheel contact to line of action of W s , recoiling parts in battery (in) b = height of center of gravity of recoil- ing parts above ground (in) Bj{ - counter recoil buffer reaction (Ibs) F vi = recuperator reaction in battery (Ibs) R'= approximate total friction (Ibs) = 0.3 W_ W 1 ' W 1 ' then BjJ +R'-F v i=C s -2-i from which BJ=F vi +C '--- - R 1 h h (Ibs) thus giving the maximum compression load on the rod. With pneumatic recuperators if the rod is under compression, the maximum compression is 970 x,- *- X, FIG. M kM-4H JLL FKa. M RECUPERATOR FORGlhSS Pt-ATE 5 971 liable to be either at the beginning or end of recoil. At the beginning we have the initial recuperator reaction + the inertia load of the rod, and at the end of recoil the maximum recuperator reaction. TRUNNIONS AND SUPPORTING In older mounts, the BRACKETS. trunnions were an integral part of the gun, the gun setting directly in the top carriage. With mounts using a recoil system between the gun and top carriage, the trunnions are usually bolted by a supporting bracket to the cradle, though when the recuperator becomes a guide support replacing the necessity of a cradle, the trunnions often are an integral part of the recuperator forging. Plate 4 shows recuperator forgings with trunnions an integral part of the forging, figures M and P, while fig. N shows a recuperator forging with a trunnion bracketed on. Plate 6 shows typical trunnions and their supporting brackets which are bolted to cradle . In fig. M, consideration only of the design of the trunnion itself is necessary, while in fig. P the strength of section m y should be considered as well. Section mn is subjected to bending and shear combined with direct stress. DESIGN Or TRUNN10KS: Let w = bearing length of trunnion Of 3 outside diam. of trunnion d* * inside diam. of trunnion at section "mn" f = max. fibre stress, - Ibs. per sq.in. ft, = allowable bearing pressure - Ibs. per sq. in. 972 Let w = width of section "ab" just above the rods w 1 = width of section "mn" at the contact of breech circumference and lug. dfc, = the distance down from "mn" to center of gravity of pulls d - depth of lugs T = longitudinal length of lug Pb - max. total powder pressure on breech n c = weight of recoiling parts attached to lug. w r = total weight of recoiling parts Then w B* r

Free movement of gun during powder expansion p ob fo x f'o = T~ g "7" +v fo t i n (ft) Total free movement of gun during powder pressure period E*X fo +X f , (ft) Total time of powder pressure period T = t t 991 STABILITY: TOTAL RESISTANCE TO BICOIL AT MAXIMUM AND MINIMUM ELEVATION, Weight of system (gun and carriage) W s (Ibs) Distance from spade point to line of action of YJ S (from preliminary layout) Height of trunnion from ground (assume) h t (ft) Horizontal distance from spade point to trunnion center (assume) l t (ft) Distance from center of gravity of recoiling parts to trunnion (assume) s (ft) Moment arm of resistance to recoil for angle of elevation d = b t cos0+s-l t sin0 (ft) Height to center of gravity of recoiling parts for horizontal recoil b - h+s (ft) APPHOIIMATI CALCULATION; (g and T not comp u t e d) Velocity of free recoil wv +47005 V f - I WJ 992 Travel up bore u (inches) Initial recoil constrained energy (approx) A r = - V* (ft/lbs) 2 8 where V r = 0.92 V f (approx) long recoil = 0.88 Vf short recoil Displacement of gun during powder period .w+0.5w. u , . E_ = ( ) (ft) w r 12 where a = 2.25 for long recoil = 2.22 for short recoil (1) Constant resistance throughout Recoil . Constant of horizontal stability Overturning moment Stabilizing moment (Usually assume 0.85) Kin. length of recoil con- sistent with stability at minimum elevation U1U p*9V.b*V/U i W s l s +W r E r cos0-/(N s l s +W r E r cos0) 2 -4W r cos0(W s l s E r - c 2 YT r cos At Elev. cos 0=1 and d = h (ft) Max. alienable recoil at horizontal elevation .035 V f /b (ft) Assumed length of horizontal recoil at min. elevation b b (ft) Total resistance to recoil at horizontal or minimum use A r for long recoil Assumed length of recoil at nax. elevation consistent Hith clearance b s (ft) Total resistance to recoil at nax. elevation (0 ffi = Use A p for short recoil 993 Variable Resistance to recoil Constant of horizontal stability C s Min. length of recoil con- sistent with stability at min. elevation 1 b Bf.)] (ft) at elev. cost=l and d=h 994 Max. allowable recoil at horizontal elevation b h -.035 V f /n~ (ft) "max Assumed length of recoil at horizontal or min. elevation b h (ft) Mean resistance to recoil during retardation period Stability slope W r cos0 JHI d C s (Ibs/ft) Mean resistance to recoil in battery K -K B + ~(b-E r ) (Ibs) Mean resistance to recoil out of battery k -K B - 5 (b-E r ) (Ibs) 2 Exact calculation E and T computed (See Interior Ballistics) . (1) Constant resistance to recoil. Constant of stability (assumed) C s = (C-0.85 usually) 9 A =W cos m i = 996 B*W r cosf min (V f T-E)-W s l s C * W s l s (V f THS)+ i - V f c c Min. length of recoil con- sistent with stability at mm. elevation -B+/B*-4AC /f (ft) Allowable recoil at horizontal elevation b h =.035 /h~ (ft) n min. Assumed length of recoil at minimum elevation b h (ft) Total resistance to recoil at min. elevation K h Max. elevation consistent with clearance b s (ft) Total resistance to recoil at max. elevation 996 (2) Variable resistance to recoil Constant of stability (assumed) C_ = W P cos*5 Stability slope ffl=C (Ibs/ft) S d Total resistance to recoil during powder period consistent with stability C 8 (* 8 l.-W r E cos 0) (lbs) COS A = m B , !! -2K-2BE K*mT 8 C =(2E 2 V f T)K + ~- B r 4ro* Min. length of recoil consistent with stability at min. elevation . =Biji*L (f t ) Allowable recoil at horizontal elevation b h = .035 V f /~h (ft) "max-. 997 Assumed length of recoil at minimum elevation b h (ft) Total resistance to recoil during powder period with assumed length of recoil at min. elevation m r Vf+ai(b-E) 2 K h 2fb h -E+V f T- - (b-E)] 2 ro Total resistance to recoil in out of battery position with assumed length of recoil at min. elev. Margin of stability at minimum elevation for the assumed long recoil in and out of battery respectively. Mean constant pressure on breech of gun 'be 1.12 Max. overturning force in battery (stability limit) l-2nd r n = 0.15 to 0.25 1 = k , 3 clips = b, 4 clips Ci d r = mean distance to guide friction from bore. 998 Max. overturning force out of battery *l~ W f b h cos k{ -2-2 F. !] Margin of stability in battery K-R (Ibs) Margin of stability out of battery k'=k (Ibs) Estimated Jump of Carriage at Horizontal elevation, Distance from spade to center of gravity of W g d s (ft) = Time of recoil (approx.) "r v f t - (sec) = Ang. vel. about spade at end of tine t g(K n d-W g l s )t i (rad/sec) Tine to nax . lift of carriage from end of time t d| t * - w (sec) Total angular displacement about spade to max. lift. i 6 = - w (t +t ) (rad) 999 Lift of wheel from ground S w = I w e =l 8 6 (appro*) (ft) where l w = distance from spade to wheel base (ft) Potential energy at nax. lift E 8 =W S 1 8 6 (ft/lbs) VABYIKe TBK RECOIL OB BLKVATIOMt In general assume the length of recoil at horizontal recoil constant from )j to 0} degrees, (usually from to 20 elevation), then, decrease the recoil proportionally with the elevation, or consistent with clearance. Length of intermediate recoil (ft) from A^ to m b b b = - (H0 n ) + b s degrees. b b ri i b h = long recoil b 8 = short recoil Resistance to recoil: from 0j to 0j Variable K n = 2[b h -E+V f T- - ~(b-E)] o - Lj ID ft ] Constant K h ** b-E+V f T from 5 2 to ^ assumed constant = K = -^ exact b-E+V f T 1000 0.47 Moment arm of resistance to recoil about spade d = h t cos0+S-l t sin# (ft) * a a. > C.' -H i-l W 4* M o * V) 41 n 10 I/I H M p> 1i H in i-i 6 o ll +> P -O 4* v w Q) *rt (V i h h O .a . _ rH o *** 9 "& H 6 H *J rH 4) a i/i > ft 00 a 00 a o a JB -H <0 * * CO > 4> (0 C 4 (O O (0 * V o o O -rl H E *! P. -* . b h C s W r cos0 i K h i Usually J * from # . . * to 20 jj b h C s WrCCS 01 K Q d i 01 b h Kh ^ From 20 to max. elev . b K d L i. * * 1001 Initial recuperator reaction (appro*. ) (Max. elev. =0 m ) F vi = 1.3 W f (sin O m +0.3cose( in )(lbs) 5623.8 Min. Bean recuperator reaction (Max. elev. m ) F VB) =2W r rsin0 m -0.3(l-cos0 m )+0.3W r l (Ibs) 7418 Min. allowable ratio of com- pression P^ _ 1.5(1.665F vm -F vi ) r,,^ fm* M 1.79 Max. allowable ratio of com- pression (stability limit) "max VI VI = 9. Max. allowable ratio of com- pression (heating limit) m = 2 to 2.5 Mean temperature in recuperator (assumed) !, = 20 to 30 C (centigrade) Max. temperature due to com- pression where k = 1.3 with floating piston = 1.1 air contact with oil 1002 Ratio of compression used. M Max. allowable air pressure Pafa = 200 to 250 Final air pressure p a f (Ibs/sq.in) Initial air oressure Paf P ai = " (Ibs/sq.in) ID Initial recuperator air e P, volume required: vi , m . V_ = bj,( - ) (cu.in) Pai i m k - b n * length of horizontal recoil (inches ) Effective area of recuperator piston F vi A., = (Ibs/sq.in) Pai Length of air column in terns of recoil stroke j = 0.8 to 1.2 usually Actual length of air column 1 * ' j b n (in) 1003 Air cyl. cross section A a Ratio of - = Effect. area of recuperator A v A a 1, m* r = = -( - ) =2.8 A v j J - 1 Area of cross section of air cylinder A a = r A v (sq.in) = Max. allowable fibre stress in the recuperator piston rod f_ (Ibs/sq.in) = a t f D = - to - elastic limit usually Required area of cross section of recuperator piston rod = 1.2 -~ (sq.in) Required diam.of recuperator piston rod (in) 0.7864 Assumed diam. of recuperator piston rod d v (in) Area of cross section of re- cuperator piston rod a = 0.7864 d Required area of recuperator cylinder A -A +a v (sq.in) 1004 Required diao. of recuperator cylinder vo 'vo 0.785 (in) Assumed diam. of recuperator cylinder d vo (in. ) Area of recuperator cylinder A (sq.in) Effective area of recuperator piston A v = A vo -a v (sq.in) Initial recuperator pressure P. ai F vi = (Ibs/sq.in) A v Final recuperator pressure Paf = m Pai (Ibs/sq.in) Initial air volume (exact) i ''" ' k=1.3 to 1.1 Length of air col uran(exact ) 1 - - (in) 1006 RECOIL BRAKE LAYOUT. Max .hydraulic pull (at max. elev.) - F (Ibs) Max. hydraulic pull(at elev.) P hc =K h -0.3W r -F vi (Ibs) Max. allowable brake pressure ph max. (Ibs/sq .in) Ph max = 400 to 500 (lbs/sq. in) Required effective area of recoil piston P hm A = - (sq.in) Ph max Reciprocal of contraction factor of orifice assumed C Win. recoil throttling area C A V r Bin max where V r = 0.9 V f (approx.) Hydraulic brake pressure (at elev.) p bo pho = -7- (Ibs/sq.in) Max. recoil throttling area (at elev.) C A V r W h max = - where V r = 0.92 V f (approx.) The battery stabilizing moment of counter recoil W s L,j = 150 to 250 L a (inch Ibs.) 1006 L * distance from wheel base to W s L & - distance from spade to wheel contact Max. buffer reaction of counter recoil "S L B DIMIB8IOR8 07 HOLLOW PISTON BODS. Max. allowable buffer pressure p b ' m (Ibs/sq.in) Assume from 1600 to 2500 (Ibs/sq.in) Area of buffer chamber A . = JLL (sq.in) b Required inside diam.of piston rod / *b 0.7834 Area of inside cross section of rod Filloux recoil system A b s 3 w bn (sq.in) Required inside diam. of piston rod Filloux recoil system 0.7834 Assumed inside dian. of piston rod d (in) 1007 Max. allowable fibre stress brake piston rods f m (Ibs/sq.in 1 ) ^ to J elastic limit (Ibs/sq.in) Outside diam. of piston rod based on max. allowable tension /df +1.6 (in) Outside diam. of piston rod based on max. hoop compression .. Assumed outside diam. of rod d o (in) Area of total cross section of rod a r 0.7854 d (sq.in) - (2) Dimensions of Solid piston rods Max. allowable fibre stress f B (Ibs/sq.in) 3 1 - to j elastic limit (Ibs/sq.in) Required area of piston rods p bm al 1.3 (sq.in) r. Corresponding diam. rod 1008 Assumed dism. of rod d r (in) Area of rod a r = 0.7854 dj (sq.in) AREA OF DIAM. OF RECOIL OYL1BDIB Required area of recoil cylinder AJ=A'+a r (sq.in) Corresponding diam. of recoil cylinder is Assumed diam. of recoil cylinder d (in) Area of recoil cylinder A. = 0.7864 d* (sq.in) Effective area of recoil piston A = A r -a r (sq.in) = Max. pressure in recoil cylinder P hm max PRINCIPLE RBACTION8 AND STRB8SEE THROUGH- MOUNT. Total resistance to recoil at max. elevation i m r v r K, = (Ibs) 1009 Initial recuperator reaction F vi =1.3W r (sin0 m +0.3cos0 m ) (Ibs) = Max. hydraulic pull (approx.) max. elev. FROM RECUPERATOR AND RECOIL BRAKE LAYOUT DETERKINE: Distance from axis of bore to line of action of P 1 n d h (in) Distance from axis of bore to line of action of d v (in) Distance from axis of bore to line of action of resultant braking B Distance between guide friction 1 (in) = 1 = for 3 clips (in) 2 1 = b j, for 4 clips (in) Coordinates along bore of front and rear clip reactions with respect to center of gravity of recoiling parts x (in) x t (in) 1010 Distance from axil of bore to line of action of mean guide friction (from layout) d r (in) Total braking at max. elevation U t -x 1 K.+W p (sin0-ncosfl * 1 (Ibg) I where n * 0.1 to 0.2 Recuperator piston friction F vi Rpv -04 n D vo T^- w p Assume W. width of packing (in) Recuperator stuffing box friction F vi R - .04 d y - W 8 (Ibs) Ay Assume w g width of packing (in) Total recuperator packing friction R (e*p) v(lb > Hydraulic piston friction K. R ph - .04 * D -jp ir p (Its) Hydraulic stuffing box friction R ib - ,04n d p w, (Iba) Total hydraulic packing friction 1011 Total hydraulic pull (max. elev.) Total hydraulic reaction (max. elcv.) Max. hydraulic pressure p hm p hm - (Ibs/sq.in) A Max. recuperator reaction F vf m P vi (Ibs) where m * ratio of compression UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. -?5 6 1361 FfcBi RECD Form L9-25wi-8,'46 (9852) 444 THE MBRARY TIP 6Uo TJ.S. dept. - IJbrary Sign Of systems UC SOUTHERN REGIONAL LIBRARY FACILITY UF Ubrary ST' SEP '73