®l?f 1. B. Bill ICibrarg TL670 This book is due on the date indicated below and is subject to an overdue fine as posted at the circulation desk. EXCEPTION: Date due will be earlier if this item is RECALLED. wR'«% AIRPLANE DESIGN AND CONSTRUCTION AIRPLANE DESIGN AND CONSTRUCTION BY OTTORINO POMILIO CONSULTING AERONAUTICAL ENGINEER FOR THE POMILIO BROTHERS CORPORATION First Edition Fifth Impression McGRAW-HILL BOOK COMPANY, Inc. NEW YORK: 370 SEVENTH AVENUE LONDON: 6 & 8 BOUVERIE ST., E. C. 4 1919 Copyright, 1919, by the McGraw-Hill Book Company, Inc. PRINTED IN THE UNITED STATES OF AMERICA THE MAPLE PRESS COMPANY, YORK, PA. ilJitir nnis (!^)rbilU Brigljl 65924 INTRODUCTION By far the major part of experimental work in aero- dynamics has been conducted in Europe rather than in America, where the feat of flying in a heavier than air ma- chine was first accomplished. This book presents in greater detail than has hitherto been attempted in this country the application of aerodynamic research conducted abroad to practical airplane design. The airplane industry is now shifting from the design and construction of military types of craft to that of pleasure and commercial types. The publication of this book at this time is, therefore, opportune, and it should go far toward replacing by scientific procedure many of the "cut and try" methods now used. Employment of the data presented should enable designers to save both time and expense. The arrangement, presentation of subject matter, and explanation of the derivation of working formulae, together with the assumptions upon which they are based, and consequently their limitations, are such that the book lends itself to use as a text in technical schools and colleges. The dedication of this volume to Wilbur and Orville Wright is at once appropriate and significant; appropriate, in that it is a tangible expression of the keen appreciation of the author for the great work of these two brothers; and significant, in that it is a return, in the form of a rational analysis of many of the problems relating to airplane design and operation, on the part of the product of an older civili- zation to the product of the new, as a sort of recompense for the daring, courage and inventive genius which made human flight possible. J. S. Macgregor. New York, 1919. CONTENTS Introduction Paoe . vii Chapter I. II. III. IV. V. VI. VII. VIII. IX. X. XL PART I Structure of the Airplane The Wings 1 The Control Surfaces 19 The Fuselage 37 The Landing Gear 44 The Engine 51 The Propeller 72 PART II The Airplane in Flight Elements of Aerodynamics 87 The Glide 102 Flying with Power On 115 Stability and Maneuverability 134 Flying in the Wind 151 PART III The Efficiency of the Airplane XII. Problems of Efficiency 161 XIII. The Speed 167 XIV. The Chmbing 188 XV. Great Loads and Long Flights 204 PART IV Design of the Airplane XVI. Materials 221 XVII. Planning the Project 261 XVIII. Static Analysis of Main Planes and Control Surfaces .... 276 XIX. Static Analysis of Fuselage, Landing Gear and Propeller . . . 324 XX. Determination of the Flying Characteristics 358 XXI. Sand Tests— Weighing— Flight Tests 379 Index 401 ACKNOWLEDGMENT The author desires to express his sincere thanks to Mrs. Lester Morton Savell for her valuable assistance in matters pertaining to English and to Mr. Garibaldi Joseph Piccione for his intelligent assistance in drawing the diagrams. O.P. AIRPLANE DESIGN AND CONSTRUCTION PART I STRUCTURE OF THE AIRPLANE CHAPTER I THE WINGS While for birds, and in general for all animals of the air, wings serve to insure both sustentation and propulsion, those of the airplane are used solely to provide the means of sustaining the machine in the air. The phenomenon of sustentation is easily explained. A body moving through the air produces, because of its mo- tion, a disturbance of the atmosphere which is more or less pronounced and complex in character. In the final analy- sis, this disturbance is reduced to the formation of zones of positive and negative pressures. The resultant of these pressures may then be classified into its three components : 1. Vertical or sustaining force, called Lift, 2. Horizontal component parallel and opposite the line of flight, called Drag, and 3. Horizontal component perpendicular to the line of flight, called Lateral Drift. The vertical component may be positive or negative. An example of the negative component is found in the elevator used for the climbing maneuver of an airplane, as will be shown later. 1 Library N„ C. State College 2 AIRPLANE DESIGN AND CONSTRUCTION The horizontal component parallel to the line of flight, is always negative; i.e., it tends to retard the motion of the body. "Conservation of energy"^ is the principle underlying this phenomenon. The horizontal component perpendicular to the line of flight is called the force of "drift," because it tends to make the body drift from the Une of flight. This compo- nent, generally not existing in normal flight, is of great importance in the directional maneuvers of airplanes. For a body having a plane of symmetry and moving through space so that the line of flight is contained in that plane, the force of drift is zero and the only components acting are the lift and the drag. Observations made of birds' wings and results based upon the experiences of experimenters in aeronautics, have demonstrated the possibility of devising surfaces of such form that by properly moving them through the air they create reactions, of which the vertical component has a far greater magnitude than the horizontal. Thus, a surface capable of developing high values of lift with small values of drag is called a wing. In actual practice, as will be shown further on in a more detailed study of aerodynamical principles (Chapter 7), the value of the ratio j^ — varies from 15 to 23. This means that wings may be built, which, for every 23 lb. of load carried, offer a resistance to motion of but I lb. It is natural, then, that designers direct all efforts toward in- creasing the y^ — ratio, which is used to define the efficiency of the wing. Three factors influence such efficiency: the profile of the wing section, the ratio of the wing span to its depth or chord (called the Aspect Ratio), and 1 This principle states that energy can be neither created nor destroyed. If the horizontal component were positive, perpetual motion would ensue, since it would be necessary only to furnish the initial force to set the body in motion. The body would then continue in its path without further applica- tion of energy. THE WINGS 3 the relative position of the wings (in multiplane ma- chines). The profile of a wing section is its major section at right angles to the span of the wing. Because of the simplicity of modern construction, wings are generally built with Leadinq/ ^Edse ^ Back. Bo+1-or Fig. 1. a constant section throughout the span. In the early days of aeronautics, however, many types of wings were built with a variable wing section, but the aerodynamical advantages derived from their use were never sufficient to compensate for the complicated construction required. In the profile of a wing, there are the following distinct elements (Fig. 1) : leading edge, back, bottom and trailing edge. The proper use of these elements makes it possible to obtain the highest values of the y^ ratio, as well as to vary the Lift coefficient according to the load to be carried per square foot of wing surface. Line of Fl.gh+. The angle between the wing chord and the line of flight, called the angle of incidence of the wing (Fig. 2), may vary between greater or smaller limits. As a result, the distri- bution and value of the positive and negative pressures will vary, and give different values of Lift, Drag and t^ The laws of variation of these factors are rather complicated and cannot be expressed by means of formulae. It is pos- sible, however, to express them by mea.ns of curves as 4 AIRPLANE DESIGN AND CONSTRUCTION illustrated in Figs. 3 and 4. These illustrate the laws of variation for the values of the Lift, Drag and yz — - coeffi- cients for two types of aerofoils, which, although having the same lengths of chord, differ in other elements. It is now necessary to introduce a new factor, namely, the speed or velocity of translation of the wing. All aerodynamical phenomena, when considered with respect to speed, follow the general law that the intensity of the phenomenon increases not in proportion to the speed, but to the square of the speed. This is accounted for by the fact that for redoubled speed not only is the velocity of impact of air molecules against the body moving in the air redoubled, but so also is the number of molecules that are struck by the body. Consequently it is seen that the intensity of the phenomenon is quadrupled. Assuming a wing with an area of A square feet, the fol- lowing general equations may be written : (1) L = X X .4 X V D = 8X A XV'~ where L = total Lift for area A in pounds D = total Drag for area A in pounds V = speed of translation in miles per hour (m.p.h.). In practice it is convenient to refer the coefficients X and 8 to the velocity of 100 m.p.h., whence the equation (1) becomes L = X X A X (4)^ IS A = I sq. ft., and V = 100 m.p.h., then Li = X (2) that is, X is the load in pounds carried by a wing with an area of 1 sq. ft. and moving at a velocity of 100 m.p.h., THE WINGS 8 X 1.75 35 » 1^ / y k. ' 1.50 30 V J,.- J Z ^ ' ? ^ 7 ^ -I (^ ^ t - \ al I.E5.Z5 _ J - \ ^ t \ 1 i^ 7 \ / \ t - ^, -t ^^ ^4 S ^ ,^ 1.0020 _ / ^ ,^ .^k^ t ^^'^\ f ^ ^.^ i ^^ ^^ o.75:i5 i ^yi ^. '^ ^%^ 050riO S IE ^^ ^y ^: X y^^ :-^**t -^^-^ = = 3: — "^ JL ^'^ 0.2505 _ X / J2 ^T ^- y ±.'': :: : 25 12.5 7.5 3-2-1 1 2 34 56 7 &9 Degrees Fig. 3. 5 X 2.25 35 ~ ~' — ~ "- ... ■ ~ — — ~ "~ '~ ^ ~ ~" ~" -f J f ' / 2.00 30 ' / ' > y > ^ 1.75 25 i' ^ J < /. y / ,^ ' / T.5020 y p" / / y J / ^ / / \ 1.2515 fJ- / / / \ \ / / \ V s /* / / V / y N 1.0010 ^ . / / ^ ->^ ^ ^ _ - _ _ _ __ _'_ ^:^ -- 7 r = -— - - — — - ^ - - - — -- - — ' f 0.75 5 i ' / f J 1 OC -^ 25 22.5 20 17.5 12.5 2.5 -3 -2 -1 234 5^739 degrees. FlQ. 4. 6 AIRPLANE DESIGN AND CONSTRUCTION and 8 the head resistance in pounds for a wing with an area of 1 sq. ft. and moving at a velocity of 100 m.p.h. Knowing X and 5, by using equation (2) the values of L and D may be found for any area or any speed. Also, the ratio - is equal to y: which is obtained by dividing the L U equation by the D equation. Now, the coefncicnts X and b may assume an entire series of varied values by changing the angle of incidence of the wings. Figs. 3 and 4 show the laws of variation of X, 5 and - for tw^o different types of wings to which we will refer as wing No. 1 and wing No. 2. An examination of the diagrams is instructive because it shows how it is possible to build wings which may have totally different values of Lift, the speed being the same for both wings. For example, at an angle of incidence of 3°, wing No. 1 gives X = 11.8, while wing No. 2 gives X = 17.6; in other w^ords, with equal speeds, wing No. 2 carries a load 49 per cent, greater than wing No. 1. The laws of variation of X and 5 depend upon the several elements of the wing, namely, the leading edge, top, bottom and trailing edge. Let us consider separately the function of each of these elements: Actually, the function of the leading edge is to penetrate the air and to deviate it into two streams, one w^hich will pass along the top and the other which will pass along the bottom of the wing. In order to obtain a good efficiency it is necessary that this penetration be made with as little disturbance as possible, in order to prevent eddies. Eddies give rise to considerable head resistance and are therefore great consumers of energy. For that reason, the leading edge should be designed with the same criterions as those adopted in the design of turbine blades. Figs. 5 and 6 show the phenomenon schematically. Due to inertia, the air deviated above the wing tends to continue in its THE WINGS rectilinear path, thus producing a negative pressure or vacuum on top of the wing. This negative pressure exerts a centripetal force on the air molecules, tending to deflect Fig. -Leading edge of good efficiency. their path downward so as to flow along the top curvature of the wing. A dynamic equilibrium is thereby established between the negative pressure and the centrifugal force of Fig. G. — Leading edge of poor efficiency. the various molecules (Fig. 7). It is obvious, then, that the top curvature has a pronounced influence not only upon the intensity of the vacuum, but also on the law of negative pressure distribution along its entire length. POSITIVE PRESSURE. Fig. 7. The air deviated below the wing tends instead, also due to inertia, to condense, thus producing a positive pressure which forces the air molecules to follow the concavity of 8 AIRPLANE DESIGN AND CONSTRUCTION the bottom curvature. Because of this change in the direc- tion of velocities, a centrifugal force is developed which is in dynamic equihbrium with the positive pressure produced TFig. 7). Curves showing the laws of distribution of the positive and negative pressures are given in Fig. 8. The resultant -6 /^ N. - 5 / \ k - 4 / \ - 3 / / Aver age Ne ga+ive Pre&sv re. N ^ -^-2 / 4.9 Lb. 3erSc| Ft \ ^ / . / i 5 1 1 5 Z ? •S 3 .3 5 4 45 •f 1 ) \ Aver age P D&i+iv z Pres sure. y + 2 \ 1.1 Lb.per Sq.Ft. ^ + 3 \ - y / V^ ^ Fig. 8. of these pressures represents the value -r- It will be noted that the portion of the sustentation due to the vacuum above is much greater than that due to the positive pressure below. In the case under consideration, it is 2.9 times greater, and equal to 74 per cent, of the total Lift, There- fore, the study of the top curvature must be given more careful consideration than that of the bottom curvature, as a wing is not at all defined by the bottom curvature alone. In practice, the means adopted to raise the value of X is THE WINGS 9 to increase both the convexity of the top and the concavity of the bottom of the wing, thereby increasing the intensi- ties of the negative and positive pressures. The traiUng edge also has its bearing on the efficiency. Its shape must be such as to straighten out the air stream- liness when the air leaves the wing, affecting a smooth, gradual decrease in the negative and positive pressures Fig. 9. — Trailing edge of good efficiency. until their difference becomes zero. In this manner, the formation of a wake or eddies behind the wing, with the resulting losses of energy, is avoided (Figs. 9 and 10). In brief, for good wing efficiency, it is primarily necessary for the leading and trailing edges to be of a design which will avoid the formation of eddies, and in order to obtain a higher value of the Lift coefficient X the top and bottom curvatures must be increased. Fig. 10. — Trailing edge of poor efficiency. From the foregoing it is easy to understand the impor- S tance of the ratio ^ ; that is, the relation between the span S and the chord C of a wing. Considering the front view of a wdng surface. Fig. 1 1 , which represents a section parallel to the leading edge, and shows the mean negative and positive pressure curves for the top and bottom of the wing, it will be seen that while in 10 AIRPLANE DESIGN AND CONSTRUCTION the central part the curves are represented by hnes parallel to the wing, at the wing tips A and B, they suffer serious disruption, for at the end of the wing a short circuit between the compression and the depression occurs. This is due to the air under pressure rushing toward the vacuum zone, thus establishing an air flux (the so-called marginal losses) , with the result that at the wing tips the average pressure curves come together, and the Lift is decreased consider- ably, thus lowering the value X of the wing. It is therefore Nega+i ve Pressure Posi + ive Pressure necessary to reduce the importance of this phenomenon to a minimum, this being done by increasing the ratio of the span to the chord ( ts ) • Assume, as it is sometimes done in practice, that the disruption in the average curves due to marginal losses extends for a distance AC and BD, equal to the chord of the wing; and also that the diagram is modified according to a linear law. This is equivalent to assuming a decrease in the Lift measured by the triangles AA'C, AA"C", BB'D' and BB" D" . The same result is obtained as though the average X remained constant and the lifting surface were re- duced by the amount c-, which means that the total surface would be reduced by sXc — c'^. If the product sXc is kept constant by increasing s and diminishing c correspondingly, the importance of the term c is greatly decreased. The loss is c~ c expressed by -^v,— = -' that is, by the inverse of the ratio S /\ c s THE WINGS 11 So it is seen that by increasing the ratio -> the span chord average value of the coeffi- cient of Lift is increased, and it is therefore advantageous to build wings of large spread. In practice, however, there is a limit beyond which this ad- vantage becomes a minimum, and there are also static and structural problems to be con- sidered which limit the value of the ratio In modern Fig. 12. machines, this value varies from 5 to 12, and even more. In biplanes, triplanes and multiplanes, another very im- portant problem is presented ; that of the mutual interference of each plane upon the others. In view of the close arrange- ment of the surfaces necessitated by structural considerations, and the high values of their negative and positive pressures of air, a conflic- tion of air flow is formed over the entire wing surface, with the result that the value of the Lift coefficient is lowered. Figs. 12 and 13 illus- trate this phenomenon for a biplane and triplane respectively. In the case of the biplane, the following effects ensue : 1. Decrease in vacuum on top of lower plane, and 2. Decrease in positive pressures on bottom of upper plane. Fig. 13.-Triplane system. J^ ^^^ ^^^^ o£ ^j^^ triplane, the losses are still greater, due to 12 AIRPLANE DESIGN AND CONSTRUCTION 1. Decrease in vacuum on top of bottom plane, 2. Decrease in positive pressures on bottom of inter- mediate plane, 3. Decrease in vacuum on top of intermediate plane, and 4. Decrease in positive pressures on bottom of upper plane. It is thus seen how undesirable, from an aerodynamical point of view, the triplane really is. At the present time, however, the triplane is not a common type of airplane, so the discussion here will be limited to the biplane. Another important ratio in aeronautics is the unit load on the wings, or the number of pounds carried per square foot of wing surface. Theoretically this value may vary between wide limits; for example, for wing No. 2 set at an angle of 6° and moving at a speed of 150 miles an hour, the ratio is 51 lb. per sq. ft. In practice, however, that value has never been reached. Special racing airplanes have been built whose unit loads were as high as 13 lb. per sq. ft., but the principal disadvantages of such high unit loads are the resulting high gliding and landing speeds, and an appreciable loss in maneuverability. For this reason designers strive to confine the unit load between the limits of 6 and 8 lb. per sq. ft. Consider a biplane with a chord and gap each of 6 ft. with a unit load equal to 8 lb. per sq. ft. Keeping in mind what has been previously stated (Fig. 8), it can be as- sumed that the values of positive and negative pressures (vacuum) found at the top and bottom of both wings would be equal to 2 lb. per sq. ft. and 6 lb. per sq. ft. respectively, provided, of course, that the two wing surfaces had no effect on each other. Now, if a difference in pressure of 8 lb. per sq. ft. is produced between two points in the air at a distance of 6 ft. from each other, the air under pressure rushing violently to fill up the vacuum will result in a veri- table cyclone in the intervening space. When a wing is in motion, condensed and rarefied conditions of the air are being constantly produced, so that THE r/INGS 13 .15 35 _ ___ ~r n^ i ■ ^ X 1 ^ 1 50 ?0 J_ > -L y- 1 1 / 1 ?5 25 1 1 III / 1 ■ 1 Ml/ 1 1 1 y'^ 1 1 "T 1 1 1 / 1 1 1 1 1 / 1 1 on ?n 1 1 ! 1 1 J/ ' '■"" ^0 1 \ \ ^ 1 >^' oJ /-^ 1 ^ JiS -IV 4- «-■■ / ' / 07^ 15 1 ^ ' N>. / 1 ^^ ■ 4I Z _L 5^ ± ^^ 1 / ± ^^^^ ±^^ i ^ «^ ^-^ L y^ ^^^^ -t 0.50 10 — L_L-^ Lyj^_^::_^^__::s __i_ ::::::::::::#:i::;?:::::::::: ::s-: nz ^^ ^. zL^y nor' c 1 />n 0.25 b -^ ^-'^ ^^^ ^"^ ? 2 25 22.5 17.5 12,5 15 -3-2-1 1 234 5e7&9 Degrees. Fig. 14. 8 A, 2V, S 25 22.5 ■ .Iw- ^^ ' 1.50 30 . / f / / / 1.25 25 . i /■ Ub .a, / '<" / s . b / 1 00 20 - s. / 1 N / s Jf - ) /V, ^' 1 J > 0.75 15 : V ^ /' s ,^ s ^ i 1 <> , X SI / 1 y^ " , 0.50 10 - 1 11^ >y s -- — _ - - — 5^ y _ _ _ _ - -— r — ■ - - V — - - - - f y 0.25 5 . 1 y "j ^ f f 1 : L - 20 17.5 12.5 75 3 -2 -I 1 2 3 4 5 6 7 6 .9 Degrees. Fig. 15. 14 AIRPLANE DESIGN AND CONSTRUCTION a certain dynamic equilibrium ensues. In order to study the phenomenon more closely, a few brief computations will be made. Again consider the type of wing curve whose characteris- tics are given in Fig. 3, and assume that it is to be adopted for a biplane. In such a case, the curves in Fig. 3 are no longer applicable and new curves must be determined experimentally, since the aerodynamical behavior of the wing shown in Fig. 3 will change for every one of the three following conditions : 1. Acting alone, as for a monoplane, 2. Serving as the upper plane of a biplane structure, and 3. Serving as the lower plane of a biplane structure. Fig. 14 gives the characteristics for wing No. 1 serving as a lower plane. Considered as an upper plane, the aerody- namical curve is practically the same as that in Fig. 3. Fig. 15 gives the characteristics of a complete biplane whose upper and lower planes are similar. Compare now a monoplane having a wing surface of 200 sq. ft., possessing the type of wing mentioned above, with a biplane also having the same wing section, and whose planes are each 100 sq. ft. in area. Assume each machine to carry a load of 1500 lb. at a speed of 100 miles per hour. The problem then is to find the values of the angles of inci- dence and the thrust efforts required to overcome the Drag. From the equation X XA X' ^ ,100 smce L = 1500 lb. and A = 200 sq. ft., then X-^^^^ -75 '^ - -20-0 - ^'^ which value of X gives, for the monoplane (Fig. 3), i = r 5 = 0.415 D = 0.415 X 200 = 83 lb. THE WINGS 15 and for the biplane (Fig. 15), i = r 45' 8 = 0.450 D = 0.450 X 200 = 90 lb. In the case of the biplane -^ is seen to be 8 per cent, smaller than in the case of the monoplane. The thrust required is 8 per cent, greater, therefore 8 per cent, more H.P. is re- quired to move the wing surfaces of this biplane than that necessary to move a similar wing in the monoplane structure. However, the final deduction must not be made that a bi- plane requires 8 per cent, more power than the monoplane of equal area. The power absorbed by the wing system is really only about 25 per cent, of the total H.P. required by the machine, so that the total loss due to the employment of a biplane structure is 8 per cent, of 25 per cent., or 2 per cent. Of late, the biplane structure has almost entirely sup- planted that of the monoplane, due largely to the great superiority, from a structural point of view, offered by a cellular structure over a linear type. For lifting surfaces of equal areas, the biplane takes up much less ground space and is much lighter than the monoplane. Regarding the former, the pr — ^ ratio being the same, the span of the biplane is only 0.71 that required by the monoplane. As to weight, it is to be noted that a wing structure usually consists of two or more main beams called wing spars, running parallel to the span. Wing ribs, constructed to form the outline of the wing section, are fitted to the spars. The junction of the wings to the body or fuselage of a machine is made by means of the spars, which are the main stress-resisting members of the wing. The spars of monoplane wings are fixed or hinged to the fuselage and braced by steel cable rigging (Fig. 16). In the biplane, instead, the corresponding spars of both upper and 16 AIRPLANE DESIGN AND CONSTRUCTION lower planes are held together by struts and cross bracing, forming a truss (Fig. 17). For those familiar with the principles of structures it is easy to see the great superiority of the biplane structure over the monoplane structure in stiffness and lightness, and the impossibility of monoplane structure in large machines because of its excessive weight. Fig. 1G. Wing structure is becoming more and more uniform for all types of airplanes. As already pointed out, the frame consists of two or more spars on which the ribs are fitted (Fig. 18). A leading edge made of wood connects the front extremities of the ribs, while for the trailing edge a steel wire or wood strip is used. The spars are also held together by wooden or steel tube struts and steel wire cross bracing, the function of which is to stiffen the wing horizontally. The rib is usually built up with a thin veneer web, to which strengthening flanges are glued and nailed or screwed (Fig. 19). The spars are usually of an I, channel, or box section for lightness (Fig. 20). The vertical struts between the upper and lower wings of a biplane may be either of wood or steel tubing (Fig. 21). THE WINGS 17 In either case, they must have a streamHne section in order to reduce to a minimum their head resistance. Wood struts are often hollowed to obtain lightness. Many ,' In+vrmedia-te Rihs i-o insure a good Curva+ure •for the Leading Bdge. ^ Forward Spar. Angle Slru-h BoxSecf/on--"^ End Rib. EndFrH-mg-for to the Fuselage-'' ^ In+ermedialv "V'Sed-ion Pib... ngTip Interior yfing Trussing Strut Rear Spar Fig. is. Interior Sleel tVire Cross SraC'nc SECTION A-D (ENLARGED) Fig. 19. I Fig. 20. different systems of connecting the struts and cables to the spars are used, and some of the many possible methods are shown in Fig. 22. The wing skeleton is covered with linen fabric, attached by sewing it to the ribs, and tacking or sewing it to the 18 AIRPLANE DESIGN AND CONSTRUCTION leading and trailing edges. It is then given an application of spec makes al varnish, called ''dope," which stretches it and t air tight. The surface is then finished with bright -B(ENLAR6ECi) Fig. 21. Fig. 22. waterproof varnish, which leaves the fabric smooth so as to reduce frictional losses to a minimum, thereby detract- ing as little as possible from the efficiency. CHAPTER II THE CONTROL SURFACES In studying the directional maneuvers of an airplane, reference must be made to its center of gravity (C.G.) and to its three principal axes. Two of the axes are contained in the plane of symmetry of the machine while the third is normal to this plane. One of the two axes in the plane is parallel to the line of flight while the other is perpendicular to it. By a known principle of mechanics, every rotation of the machine about its C.G. may be considered as the resultant of three distinct rotations, one about each of the three principal axes. On the other hand, if three systems of control are used, each capable of producing a rotation of the airplane about one of its principal axes, any rotation of the machine about its C.G. can be brought about or prevented. The principal axis perpendicular to the plane of sym- metry, is called the pitching axis. Rotations about that axis are called pitching movements. The devices used to bring about, or prevent a pitching movement are called devices of longitudinal stability. The axis perpendicular to the line of flight, in the plane of symmetry is called the axis of direction of flight. The devices which cause or prevent movements about that axis are called devices of directional stability. The axis parallel to the line of flight is called the rolling axis, and the devices causing or preventing rolling move- ments are called devices of lateral stabilit3^ There are usually two surfaces which control longitudinal stability, one fixed, called the stabilizer or tail plane, and the other movable, called the elevator. The stabilizer or tail plane is a relatively small surface fixed at the rear end of the fuselage. Its function is, first 19 20 AIRPLANE DESIGN AND CONSTRUCTION of all, to offset or even completely invert the phenomenon of the inherent instability of curved wings, and secondly, to act as a damper on longitudinal or pitching movements. The stabilizer may be of various shapes and sections. It may be either lifting or non-lifting, but it must always satisfy the basic condition that its unit loading per sq. ft. be lower than that of the principal wing surface. Under this condition only, will it act as a stabilizer; otherwise it would add to the instability of the wings. As to the proper dimensions of the stabilizer, they depend on various factors such as the weight of the airplane, its longitudinal moment or inertia, its speed, and the distance the stabilizer is set from the center of gravity of the machine. Moreover, the proportions of the stabilizer with respect to the other parts of the airplane are also dependent on another factor: the tj^pe of airplane. For small, swift combat machines which require a high degree of maneuvera- bility, the stabilizer will require relatively less surface than that required for large, heavily loaded machines, such as those used for bombing operations and requiring a much lower degree of maneuverability. The framework or skeleton of the stabilizer is generally of wood or steel tubing. In general its angle of incidence may be adjusted either on the ground or while in flight. However, that incidence must never be greater than the angle used for the main wing surfaces. Its value is gen- erally 1° to 4° less than that of the wings. The elevator or movable surface is hinged to the rear edge of the stabilizer, and it may be raised or lowered while in flight. In normal flight the elevator is set parallel to the air flow so that there is no air reaction on its faces. If it is swung upward or downward the air will strike it, producing a reaction whose direction is upward or downward respec- tively, thus tending to set the machine for climbing or descending. The size of the elevator also depends on the weight, moment of inertia, speed of the machine, and on its dis- THE CONTROL SURFACES 21 tance from the center of gravity of the machine; also the type of airplane and the service for which it is intended must be given consideration. However, for quick and responsive machines the elevator must be proportionally larger than Fig. 23. for slow machines endowed with a greater degree of stabil- ity. In other words, the two proportions vary inversely as those of the stabiUzers. However, this will be more easily understood upon considering the functions of 22 AIRPLANE DESIGN AND CONSTRUCTION the two devices which are in a certain sense, completely opposite. The function of the stabilizer is to insure longitudinal stability, just as its name implies. The elevators function instead, is to disturb the equilibrium of the machine in order to bring about a change in the normal flying. An outline of a type of stabiUzer and elevator system is given in Fig. 23. A closer study may now be made of the function of these two parts of longitudinal stability. First of all, examina- tion will be made of the mechanism by which the stabilizer, when properly set, exercises its stabilizing property. When, in an airplane, the incidence of the wing is changed with respect to the air, through which it is progressing, the air reaction will not only vary in intensity but also in loca- tion. If the new reaction is such as to antagonize the deviation, the airplane is said to be stable; otherwise it is said to be unstable. Wings having curved profiles, when acting alone, are un- stable. Laboratory experiments have shown that for a wing with a curved profile, the reaction moves forward as the incidence is increased, and vice versa; thus the reaction moves in such a way as to aggravate the disturbance. The point of intersection of the air reaction on the wing chord is called the center of pressure of the wing (Fig. 24) . The location of the center of pressure is usually indicated by the ratio -• The curves for X and for as functions of the c c angle of incidence for a given wing section, are shown in Fig. 25. By applying the data from these curves to a wing of 5 ft. chord and 40 ft. span, supposing the normal speed to be 100 m.p.h. and the normal angle of flight 2°, the wing loading will be L = 7.5 X 200 = 1500 lb. and it will be in equilibrium if the center of gravity of the load falls at a distance of 40 per cent, of the chord, or 2 ft. from the leading edge. Suppose now that the inci- THE CONTROL SURFACES 23 dence is increased from 2° to 4°, then the sustaining force becomes L = 10 X 200 - 2000 lb. 17.5 15.0 12.5 10.0 7.5 5.0 2.5 > / I - * ■> ^ *c > ^ v^ ■» ^ ■''^ •" / 1 ^ w ^ ' ^^s y^ '^s /'^ ^s / "^^ - ^Z ^k_ tL ^^ ^^ ^-' / ^^^^ ^^ 7 ^ /^ -?/ / ^'^ 7 y / / - ( \j / A- 2C. c 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 -3-2-10 1234567 69 Degrees. Fig. 25. and it will be applied at 37 per cent, of the chord, or 1.85 ft. from the leading edge; this result will then produce around the center of gravity, a moment of 2000 X 0.15 = 300 ft. lb. 24 AIRPLANE DESIGN AND CONSTRUCTION and such moment will tend to make the machine nose up; that is, it will tend to further increase the angle of inci- dence of the wing. Following the same line of reasoning for a case of decrease in the angle of incidence, it will be found in that case that a moment is originated tending to make the machine nose down. Therefore, the wing in question is unstable. A practical case will now be considered, where a stabil- izer is set behind this wing, and constituted of a surface of 15 sq. ft. (2 X 7.5) set in such a manner as to present an angle of —2° with the line of flight w-hen the wdng in front presents an angle of + 2°. In normal flight there is 1 . The sustaining force of the main w ing, equal to L, = 7.5 X 200 = 1500 lb. 2. The center of pressure of the main wing located at 0.40 X 5 = 2 ft. from the leading edge, 3. The sustaining force of the elevator equal to L, = 2.3 X 15 = 33.5 lb., and 4. The center of pressure of the elevator located at 0.44 X 2' = 0.88 ft. from its leading edge. Suppose now that the incidence of the machine is in- creased so that the angle of incidence of the front wing changes from +2° to +5°, then there is 1. The sustaining force of the main wing equal to L, = 11.30 X 200 = 2260 lb. 2. The center of pressure of the main wing located at 0.355 X 5' = 1.78 ft., 3. The sustaining force of the elevator equal to L, = 6.05 X 15 = 91 lb., and 4. The center of pressure of the elevator located at 0.410 X 2' = 0.82 ft. from its leading edge. With these values, the total resultant of the forces acting in each case is obtained, and it is found that while in nor- mal flight, the moment of total resultant about the e.g. of the machine is equal to zero; when the incidence is increased Library N. C. State College THE CONTROL SURFACES 25 to 5°, that moment becomes equal to 2351 X (2.71' - 2.50') = 493 ft. lb. tending to make the machine nose down; that is, tends to prevent the deviation and therefore is a stabilizing moment (Fig. 26). In analogous manner it can be shown that if the incidence of the machine is decreased, a moment tending to prevent \J'l5Sq.Ff Ls'33.5/tS. iL^^9//t>s. Fig. 26. that deviation is developed. It is obvious, then, that if the airplane were provided with only a stabilizer and with no elevator, it would fly at only one certain angle of inci- dence, since any change in this angle would develop a stabil- izing moment tending to restore the machine to its original angle. Thus the exact function of the elevator is to pro- duce moments which will balance the stabilizing moments 26 AIRPLANE DESIGN AND CONSTRUCTION due to the stabilizer. This will allow the machine to assume a complete series of angles of incidence, enabling it to maneuver for climbing or descending. There are also usually two parts controlling directional stability; one fixed surface called the fin or vertical stabilizer, and one movable surface called the rudder. Consider, for example, an airplane in normal flight; that is. with its line of flight coincident with the rolling axis (Fig. 27). In this case there is no force of drift, but if for some reason the line of flight is no longer coincident with the rolling axis, a force of drift is developed (Fig. 28), whose point of application is called center of drift. If this center is found to lie behind the center of gravity, the machine tends to set itself against the wind; that is, it becomes endowed with directional stability. If, instead, the center of drift should fall before the center of gravity, normal flight would be impossible, as the machine tends to THE CONTROL SURFACES 27 turn sharply about at the least deviation from its normal course. In practice, since the center of gravity of an air- plane is found very close to the front end of the machine, the condition of directional stability is easily attained by the use of a small vertical surface of drift which is set at the extreme rear of the fuselage. This surface is called the fin or vertical stabilizer. There is, however, a type of airplane called the Canard Fig. 28. type, in which the main wing surface is the one in the rear, (and consequently the e.g. falls entirely in the rear) and in which the problem of directional stability presents considerable difficulty. This type of airplane, however, is not used at the present time. A machine provided with only a fin would possess good directional stability, but for that very reason it would be impossible for the airplane to change its course. For that reason it is necessary to have a rudder; a vertical movable 28 AIRPLANE DESIGN AND CONSTRUCTION surface, which, when properly deviated, will produce a balancing moment to overcome the stabilizing moment of the fin, thus permitting a change in the course of the drift. The phenomenon may now be studied more in detail. Let us suppose that the directing rudder is deviated at an angle; this deviation will then provoke on the rudder a reaction D' (Fig. 29), which will have about the center of gravity a moment D'Xd'; as a result, the airplane will Fig. 29. rotate about the axis of direction and the line of flight will no longer coincide with the rolling axis; that is, when the airplane starts to drift in its course, a drifting force D" is originated, which tends to stabilize, and when D" X d" = D' X d' , equilibrium will be obtained. Obviously, then, the line of flight will no longer be rectiUnear, since the two forces D" and D' are unequal, and if transported to the center of gravity they will give a resultant D = D" — D' other than zero. The equilibrium will be obtained only if the line of THE CONTROL SURFACES 29 flight becomes curvilinear; in fact, a centrifugal force 4> is then developed which will be in equilibrium with the re- sultant force of drift D. Then equilibrium will be obtained when ^ = D; as W V $ = ^ X — g r where W is the weight of the airplane, g the acceleration due to gravity, V the velocity of the airplane and r the radius of curvature of the line of flight, therefore a r from which is obtained W V W V g ^ D g ^ D" -D' From this equation it will be seen that to obtain remarkable maneuverability in turning, the difference D" — D' must have a large value. Or, since D' d" it is necessary that the center of drift, although being in the rear of the center of gravity, must be not too far behind it, and it is necessary that the rudder be located at a consider- able distance from the center of gravity. In other words, for good maneuverability, an excessive directional stability must not exist. The foregoing applies to what is called a flat turn without banking, which is analogous to that of a ship. The airplane, however, offers the great advantage of being able to incline itself laterally which greatly facilitates turning, as will be shown when reference is made to the devices for transversal stability. In summarizing the foregoing, it is seen that in addition to the fixed surfaces, stabilizer and fin, whose functions are to insure longitudinal and directional stability, airplanes are provided with movable surfaces, elevator and rudder, which are intended to produce moments to oppose the stabilizing moments of the fixed devices. It will now be 30 AIRPLANE DESIGN AND CONSTRUCTION better understood that excessive stability is contrary to good maneuverability. In like manner, for transversal stability, there are two classes of devices opposite in their functions. Some are used to insure stability while others serve to produce moments capable of neutralizing the stabilizing moments. Let us consider an airplane in normal flight, and suppose that a gust of wind causes the machine to become inclined laterally by an angle a. The weight W and the air reaction L will have a resultant D„ which will tend to make the f V r \ .,__- ^ b4 1 ::>^ w Fio. 30. machine drift (Fig. 30) ; this drifting movement will produce a lateral air reaction — Z>„ acting in the direction opposite to Z)„. The resultant of the lateral wind forces acting on the machine is — Z)„. If this reaction is such as to make with the force D^ a couple tending to restore the machine to its original position, the machine is said to be transver- sally stable; this is the case shown in Fig, 30. If —D„ has the same axis as Z)„, the au-plane is said to have an indif- ferent transversal stability. If, finally, —D^ and D^ form a couple tending to aggravate the inclination of the machine, the latter is said to be transversally unstable. Consequently, in order to have an airplane laterally stable, conditions must be such that the lateral reaction — Z)„ together with the force D„ form a stabilizing couple; that is, the point of application of the force — Z)„ must be THE CONTROL SURFACES 31 situated above the point of application of force Z)„, which is the center of gravity. However, the couple of lateral stability must not have an excessive value, as it would decrease the maneuverability to such an extent as to make the machine dangerous to handle, as will now be explained. It has been explained before how a turning action may be obtained by merely maneuvering the rudder, and how ^=-D^=Lsinoc = Wtanoc this cannot be actually done in practice since there is a possibility of the machine banking while turning. Now, when the airplane "banks," the forces L and W will admit a lateral resultant D^ which tends to deviate laterally the line of flight. A centrifugal force is thereby developed, tending to balance the force Z)„ and equilibrium will obtain when = Z)„ (Fig. 31); that is, when 32 AIRPLANE DESIGN AND CONSTRUCTION where r is the radius of curvature of the Hne of flight; therefore D„ -- W g r vhich will give r = w v^ g^D~. isD„ = IF tan a, we obt; ain r = 'x 72 tan This equation shows that the turn can be so much sharper as the speed is decreased, and the angle a of the bank is increased. This explains why pilots desiring to turn sharply, make a steep bank and at the same time nose the machine upward in order to lose speed. Now the angle of bank may be obtained in two ways; by operating the rudder or by using the ailerons which are the controls for lateral stability. In using the rudder, it has been observed that the machine assumes an angle of drift. If the force of drift D = D" - D' (Fig. 29) passes through the center of gravity, a flat turn without banking will result. If force D passes below the center of gravity, the airplane will incline itself so as to produce a resultant Z)„ of L and W, in a direction opposite to force D. Then the total force of drift is equal to Z) — D„. This case is of no practical interest, since it corresponds to the case of lateral instability, which is to be avoided. If, instead, force D passes above the center of gravity, then the angle of bank a is such that D^ is of the same direction as D. Therefore, the total force of drift is D + D„. Now if force Z)„ had its point of application too far above the center of gravity, the result would be that with a slight movement of the rudder, a strong overturning moment would develop which would give the machine a dangerous angle of bank. Therefore it is evident that an excessive stabilizing moment must be avoided. THE CONTROL SURFACES 33 The ailerons are two small movable surfaces located at the wing ends (Fig. 32) . Let us now observe what happens when they are operated. The ailerons are hinged along the axes AA' and BB', and are controlled in such a manner that when one swings upward the other swings downward. With this inverse movement, the equity of the sustaining force on both the FiQ. 32. right and left wings, is broken. Thus a couple is brought into play which tends to rotate the machine about the rolling axis. Since it is possible to operate the ailerons in either direction, the pilot can bank his machine to the right or to the left. Supposing that the pilot operates the ailerons so that the machine banks to the right; let a be the angle of bank; then, a force D^ is produced, which, in a laterally stable machine will tend to oppose the banking movement caused 34 AIRPLANE DESIGN AND CONSTRUCTION by the ailerons. The rapidity of turning, and consequently the mobility of the machine, will increase in proportion as the rapidity of the banking movement increases. Now, all other conditions being similar, the rapidity with which the machine banks is proportional to the difference of the couple due to the actions of the ailerons, and the couple due to the force of drift a; if the value of the latter is very large (that Fig. 33. is, if D„ is applied very far above the center of gravity) the maneuver will be slow. Therefore for good mobility of the airplane, the force Z)„ must not be too far above the center of gravity. The foregoing considerations show the close interdepend- ency existing between the problems of directional stability and those of transversal stability. It is practically possible Fig. 34. to control directional stability by means of the lateral con- trols, and vice versa. For example, birds possess no means of control for directional stability alone, but use the motion of their wings for changing the direction of their flight. To raise the force Z)„ with respect to the center of gravity, we may either install fins above the rolling axis, or, better still, give the wings an upward inclination from the center THE CONTROL SURFACES 35 to the tip of the wing, the so-called dihedral angle (Fig. 33). The effect of this regulation is that when the machine takes an angle of drift, the wing on the side toward which the machine drifts, assumes an angle of incidence greater than the inci- dence of the opposite wing, thereby developing a lateral couple which is favorable to stability. The framework of the ailerons is usually of wood, steel tubing or pressed steel members. An outline of wood ailer- ons is given in Fig. 34. |3i Concluding, to be relatively safe and controllable at the same time, an airplane must be provided with devices which will produce stabilizing couples for every deviation from the position of equilibriimi: but these couples must not be 36 AIRPLANE DESIGN AND CONSTRUCTION of excessive magnitude, for the machine would then be too slow in its maneuvers, and consequently dangerous in many cases. These stabilizing couples must be of the same magnitude as the couples which can be produced by the controlling devices. In this manner the pilot always has control of the machine and it will answer readily and effectively to his will. The control system of maneuvering by the pilot usu- ally consists of a rudder-bar operated by the feet, and a hand-controlled vertical stick (called the ''joy stick") piv- UNBM,ANCED RUDDER BALANCED RUDDER I A' Fig. 37. oted on a universal joint, moved forward and backward to lower and raise the elevator, and from left to right to move the ailerons (Figs. 35 and 36). Balanced rudders are found on some of the high-powered machines, as they reduce, to a slight degree, the muscular effort of the pilot. The effort required to move a control surface depends on the distance h (Fig. 37) between the center of pressure C and the axis AB of rotation. If axis AB is moved to A'B', the value of h is reduced to h', and therefore the required effort for the maneuver is decreased. CHAPTER III THE FUSELAGE The fuselage or body of an airplane is the structure usu- ally containing the engine, fuel tanks, crew and the useful load. The wings, landing gear, rudder and elevator are all attached to the fuselage. The fuselage may assume any one of various shapes, depending on the service for which the machine is designed, the type of engine, the load, etc. In general, however, the fuselage must be designed so as to have, as nearly as possible, the shape of a solid offering a minimum head resistance. In the discussion on wings, it was observed that the air reaction acting on them is gen- erally considered in its two components of Lift and Drag. For a fuselage moving along a path parallel to its axis, the Lift component is zero, or nearly so; the Drag component is predominant, and must be reduced to a minimum in order to minimize the power necessary to move the fuselage through the air. Let S indicate the major section of the fuselage, and V the velocity of the airplane. Laboratory experiments have shown that head resistance is proportional to S and V^. Assuming our base speed as 100 m.p.h. for a given fuselage, then R-^ KXSX (^f (1) therefore, if >S = 1 and F = 100, then R ^ K. Thus the coefficient A' is the head resistance per square foot of the major section of the fuselage, when V = 100 m.p.h. This is called the coefficient of penetration of the fuselage. The lower K is, the more suitable will be the fuselage, as the corresponding necessary power will be decreased. Equation (1) shows two ways of decreasing the necessary power; (a) By reducing the major section of the fuselage to a minimum, and (6) by lowering the value of coefficient K as much as possible. 37 38 AIRPLANE DESIGN AND CONSTRUCTION In order to solve problem (a) it is necessary first to adapt the section of the fuselage to that of the engine. The Fr;. 38. fuselage may be of circular, square, rectangular, triangular, etc., section, so designed that its major section follows the form of the major section of the engine. In the second place, it is good practice, when other reasons do not prevent it, to arrange the various masses constituting the load (fuel, pilot, pas- sengers, etc.) one behind the other, so as to keep the transversal dimension as small as possible. To decrease the coefficient of head resistance, the shape of the fuselage must be carefully designed, especially the form of the bow and of the stern. Analogous to that of the wings, the phenomenon of head resistance of the fuselage is due to the resultant of two positive and negative pressure zones, developing on the forward and rear ends respectively (Fig. 38). Whatever be the means employed to reduce the importance of those zones, the value of K will be lowered, thus improving the penetration of the fuselage. To improve the bow, it must be given a shape which will as nearly as possible approach that of the nose of a dirigible. This is easily affected with engines whose contours are circular, but the problem presents greater difficulties with vertical types of engines, or V types without reduction gear. Sometimes a bullet-nosed cowling is fitted over the propeller hub, fixed to and rotating with Fig. 39. THE FUSELAGE 39 the propeller. Its form is then continued in the front end of the fuselage contour, its lines gradually easing off to meet those of the fuselage (Fig, 39). To improve the stern of the fuselage it must be given a strong ratio of elongation, and the shaping with the rest of the machine must be smoothly accomplished. A special advantage is offered by the reverse curve of the sides; in fact, in this case, a deviation in the air is originated in the zone of reverse curving (Fig. 40) tending to decrease the pressure, and consequently increasing the efficiency. Fig. 40 The value of coefficient K varies from 7 (for the usual types of fuselage) to 2.8 (for perfect dirigible shapes). It is interesting to compare such values with the coefficient of head resistance of a flat disc 1 sq. ft. in area, which is equal to 30. To move the above disc at a speed of 100 m.p.h. we must overcome a resistance of 30 lb., while in the case of the fuselage of equal section, but having a perfect streamline shape, we must overcome a resistance of only 2.8 lb., or less than one-tenth the head resistance of the disc. Practically, a well-shaped fuselage has a coefficient of about 6, so if its major section is, for instance, 12 sq. ft., the resistance to be overcome at a speed of 150 m.p.h. is 6 X 12 X (J-^V = 162 lbs. which will theoretically absorb about 66 H.P. Fuselages may be divided into three principal classes, depending on the type of construction used: (a) Truss structure type, (6) Veneer type, and (c) Monocoque type. 40 AIRPLANE DESIGN AND CONSTRUCTION Mo+orSupporj-s. ^ Motor Supportinq Beams. Tran&verse Stru-h. Fig. 41. Veneer Panel Sfrffening Wood Cross-Bracing. ^ Moior Supports, Motor Supportinj-Beams. Fig. 42. THE FUSELAGE 41 The truss type generally consists of 4 longitudinal longer- ons, held together by means of small vertical and horizontal struts and steel wire cross bracing (Fig. 41). The whole frame is covered in the forward part with veneer and alumi- num and in the rear with fabric. The longerons are gen- erally of wood, and the small struts are often of wood, although sometimes they are made of steel tubing. Fuselages built of veneer are similar to the truss type as they also have 4 longitudinal longerons, but the latter, instead of being assembled with struts and bracing, are held in place by means of veneer panels glued and attached by nails or screws. By the use of veneer, which firmly holds the longerons in place along their entire length, the section of the longerons can be reduced (Fig. 42). The monocoque type has no longerons, the fuselage being formed of a continuous rigid shell. In order to insure the necessary rigidity, the transverse section of the mono- coque is either circular or elliptical. The material gener- ally used for this type is wood cut into very thin strips, glued together in three or more layers so that the grain of one ply runs in a different direction than the adjacent plies. This type of construction has not come into general use because of the time and labor required in comparison with the other two types, although it is highly successful from an aerodynamical point of view. Whatever the construction of the fuselage be, the distribu- tion of the component parts to be contained in it does not vary substantially. For example, in a two-seater biplane (Fig. 43) , at the forward end we find the engine with its radia- tor and propeller; the oil tank is located under the engine, and directly behind the engine are the gasoline tanks, located in a position corresponding to the center of gravity of the machine. It is important that the tanks be so located, as the fuel is a load which is consumed during flight, and if it were located away from the center of gravity, the constant decrease in its weight during flight would disturb the balance of the machine. 42 AIRPLANE DESIGN AND CONSTRUCTION THE FUSELAGE 43 Directly behind the tanks is the pilot's seat, and behind the pilot is the observer. Fig. 43 shows the positions of the machine-guns, cameras, etc. The stabilizing longitudinal surfaces and the directional surfaces are at the rear end of the fuselage. The wings, which support the entire weight of the fuselage during flight, are attached to that part on w^hich the center of gravity of the machine will fall. Under the fuselage is placed the landing gear. Its proper position with respect to the center of gravity of the machine will be dealt with later on. CHAPTER IV THE LANDING GEAR The purpose of the landing gear is to permit the airplane to take off and land without the aid of special launching apparatus. The two principal types of landing gears are the land and marine types. There is a third, which might be called the intermediate type, the amphibious, which consists of both wheels and pontoons, enabling a machine to land or ''take Fig. 44. off" from ground or water. This discussion will be devoted solely to wheeled landing gears, the study of which pertains especially to the outlines of the present volume. The ''take off" and landing, especially the latter, are the most delicate maneuvers to accomplish in flying. Even though a large and perfectly levelled field is avail- able, the pilot when landing must modify the line of flight until it is tangent to the ground (Fig. 44) ; only by doing this will the kinetic force of the airplane result parallel to the ground, and only then will there be no vertical com- ponents capable of producing shocks. 44 THE LANDING GEAR 45 In actual practice, however, the maneuvers develop in a rather different manner. First, the fields are never perfectly level, and secondly, the line of flight is not always exactly parallel to the ground when the machine comes in contact with the ground. The landing gear must therefore be equipped with shock absorbers capable of absorbing the force due to the impact. The system of forces acting on an airplane in flight is generally referred to its center of gravity, but for an air- = Tofal Liffoffhe Ylmga and Horizontal Tail Planes. T- Propeller Thrust. ^ G= Reaction of 6round. Fig. 45. plane moving on the ground, the entire system of the acting forces must be referred to the axis of the landing wheels. Such forces are (Fig. 45), T = propeller thrust, W = weight of airplane, L = total lift of wing surfaces, = total head resistance of airplane, = inertia force, = friction of the landing wheels, and = reaction of the ground. The moments of these forces about the axis of the landing gear may be divided into four groups: 1. Forces whose moments are zero (the reaction of the ground, G), 46 AIRPLANE DESIGN AND CONSTRUCTION 2. Forces whose moments will tend to make the machine sommersault (forces T and F), 3. Forces whose moments tend to prevent sommersault- ing (forces W and R), and 4. Forces whose moments may aid or prevent sommer- saulting (forces L and 7) . In group 4, the moment of the force L may be changed in direction at the pilot's will, by maneuvering the ele- vator; force I prevents sommersaulting when the machine accelerates in taking off, and aids sommersaulting in landing when the machine retards its motion. In practice it is possible to vary the value of these mo- ments by changing the position of the landing gear, placing it forward or backw^ard. By placing the landing gear forward, the moment due to the weight of the machine is particularly increased, and it may be carried to a limit where this moment becomes so excessive that it cannot be counterbalanced by moments of opposite sign. Then the airplane wdll not "take off," for it cannot put itself into the line of flight. By placing the landing gear backward, the moment due to the weight is decreased, and this may be done until the moment is zero, and it can even become negative; then the machine could not move on the ground without sommer- saulting. Consequently it is necessary to locate the land- ing gear so that the tendency to sommersault will be de- creased and the "take off " be not too difficult. In practice this is brought about by having an angle of from 14° to 16° between the line joining the center of gravity of the machine to the axis of the wheels, and a vertical line pass- ing through the center of gravity. Let us examine the stresses to which a landing gear is subjected upon touching the ground. Assume, in this case, an abnormal landing; that is, a landing with a shock. (In fact, in the case of a perfect landing, the reaction of the ground on the wheels is equal to the difference between the weight W and the sustaining force L, and assumes a maximum value when L = 0; that is, when the machine is THE LANDING GEAR 47 standing.) In the case of a hard shock, due either to the encounter of some obstacle on the ground, or to the fact that the Hne of flight has not been straightened out, the kinetic energy of the machine must be considered. That kinetic energy is equal to 2 g where g is the acceleration due to gravity, and V the velocity of the au'plane with respect to the ground. The foregoing is the amount of kinetic energy stored up in the airplane. Naturally, it would be impossible to adopt devices capable of absorbing all the kinetic energy thus developed, as the weight of such devices would make their use pro- hibitive. Experience has proven that it is sufficient to pro- vide shock absorbers capable of absorbing from 0.5 per cent, to 1 per cent, of the total kinetic energy. Then the maximum kinetic energy to be absorbed in landing an air- plane of weight TT^ and velocity V, is equal to W 0.0025 to 0.0050 X — X V^ g For example, for an airplane weighing 2000 lb., moving at a velocity of 100 m.p.h. (146 ft. per sec), assuming 0.004, it will be necessary that the landing gear be capable of absorbing a maximum amount of energy equal to 0.004 X ^ X 146^ = 5300 ft .-lb. The parts of the landing gear intended to absorb the kinetic energy of an airplane in landing, are the tires and shock absorbers. Fig. 46 gives the work diagrams for a wheel. The wheel is capable of absorbing 900 ft. -lb. with a deformation of 0.25 ft. Fig. 47 gives the diagram of the work referred to per cent, elongation for a certain type of elastic cord. The work absorbed by n ft. of elastic cord under a per cent, elongation of x is equal to the product er YPJTT times the area of the diagram corresponding to x per 48 AIRPLANE DESIGN AND CONSTRUCTION cent, elongation. Supposing, for instance, to have a shock ab- sorbing system 32 ft. long, allowing an elongation of 150 per 8000 7000 6000 5000 tfj 4000 3000 2000 1000 ~ . - - -. 1 , 1 1 - - - - - - ---jjrft^ TZOO/b.A / ' 1 1 : _: : _ '^ j '- ±" t \/ I /t J |_ -T- \/^X^tu.°i^-Ai^-A ' 41 i 411 ?i ILLL _j_ T'. ^-- % :q:::: z± . : - : 0.5 010 0.15 0.20 025 Fig. 46. cent. ; the work that it can absorb is equal as shown in the diagram to 1800 ft.-lb. As this gives a total of 2700 ft.-U)., 150 ~ ~ ~ ~ ~ ~" ~ ~ ~ ~ ~" "" ~ ~" — ~ ~ ~ - ~ r ~ ~ ~ ~ - ~ ~ ~ no 150 110 110 ■ ^ " ■* ^ - ~ "■ ~ ■ " ■ ■■ ^ '°1 ^ ' '' , / 70 60 50 / / / i so f / / c ^ ' ? ° 10< 10 5 07 08 09 IC 01 w 1 «r 2C )02 2 a 30 3 3 !03 3E D3( « 10* 1" ) Fig. 47. two wheels and two shock absorbers of such type will be sufficient for the airplane in question. THE LANDING GEAR 49 Rubber cord shock absorbers, which perform work by their elongation, have proven to be the lightest and most Fig. 49. Fig. 50. practical. Experiments have been made with other types, such as the steel spring, hydraulic and pneumatic, but the 50 AIRPLANE DESIGN AND CONSTRUCTION results have shown these types to possess but Uttle merit. Fig. 48 illustrates an example of elastic cord binding. Fig. 49 shows the outline of a landing gear. Up to this point, our discussion has been only on the vertical component of the kinetic energy. Consideration must also be given the horizontal component, whose only effect is to make the machine run on the ground for a cer- tain distance. \Mien the available landing space is limited, the machine must be slowed down by means of some brak- ing device, in order to shorten the distance the machine has to roll on the ground. Friction on the wheels, head resistance and the drag all have a braking effect, but it often happens that these retarding forces are not sufficient. The practice therefore prevails of providing the tail skid with a hook, which, as it digs into the ground, exerts on the machine an energetic braking action (Fig. 50). On some machines, a short arm, with a small plow blade at its lower end, is attached to the middle of the landing gear axle, which can be caused to dig into the ground and pro- duce a braking effect. Similar to the landing gear, the tail skid is also provided with a small elastic cord shock absorber to absorb the kinetic energy of the shock. On certain airplanes, use is made of aerodynamical brakes consisting of special surfaces which normally are set in the line of flight, and consequently offering no passive resist- ances, but when landing they can be maneuvered so as to be disposed perpendicularly to the line of motion, producing an energetic braking force. CHAPTER V THE ENGINE The engine will be dealt with only from the airplane designers point of view. For all the problems peculiar to the technique of the subject, special texts can be referred to. There are various types of aviation engines — with rotary or fixed cylinders, air cooled or water cooled, and of ver- tical, V, and radial types of cylinder disposition. Whatever the type under consideration, there exist certain funda- mental characteristics which enable one to judge the engine from the point of view of its use on the airplane. Such characteristics may be grouped as follows: 1. Weight of engine per horsepower, 2. Oil and gasoline consumption per horsepower per hour, 3. Ratio between the major section of the engine and the number of horsepower developed, 4. Position of the center of gravity of the engine with respect to the propeller axis, and 5. Number of revolutions per minute of the propeller shaft. In order to judge the light weight of an engine, it is not sufficient to know only its weight and horsepower; it is also essential to know it specific fuel consumption. If we call E the weight of the engine, P its power, C the total fuel consumption per hour (gasoline and oil), and x the number of hours of flight required of the airplane, then the smaller the value of the following equation, the lighter will be the motor: y = P + X X p (1) For a given engine, equation (1) gives the linear relation between y and x, which can be translated into a simple, 51 52 AIRPLANE DESIGN AND CONSTRUCTION graphic, representation. Let us consider two engines, A and B, having the following characteristics: Table 1 Engine E lbs. r H.P. '4 lbs. per H.P. c Iba. 1 lbs. per H.P. A B 600 750 300 300 2 2.5 180 144 1 0.6 0.48 For engine A, equation (1) will give 7/ = 2 + 0.6a-. For engine B, equation (1) will give y = 2.5 + 0.48x. y X .r^ ^ \^~ A e^ -r^ A ^ y y 01 23456789 10 ;x Hours Fig. 51. Translating these equations into diagrams (Fig. 51), we see that engine A is lighter than engine B, for flights up to 4 hours 10 minutes beyond which point, B is the lighter. If a; = 10 hours, then y. = 8 lb. y. = 7.3 that is, B has an advantage of 0.7 lb. per H.P. ; since P = 300 H.P. the total advantage is 270 lb. THE ENGINE 53 Practically, for engines of the same general types, the C value of the specific consumption c = p> varies around the same values. In that case, only the weight per horsepower, e = p is of interest. In fact, that ratio is so important that it may often be convenient to adopt an engine of lower power in comparison with another of high power, for the sole reason that for the latter the above ratio is higher. Let us suppose that we wish to build an airplane of given horizontal and climbing speed characteristics, capable of carrying fuel for a flight of three hours and a useful load of 600 lb. (pilot, observer, arms, ammunition, devices, etc.). Fixing the flying characteristics is equivalent to fixing the maximum weight per horsepower, of the machine with its complete load. In fact, we shall see further on in discussing W the efficiency of the airplane, that the lower the ratio p- between the total weight W, and the power P of the motor, the better will be the flying characteristics of the machine. W Supposing for example, that -p = 10 lb. Analyzing the weight W, we find it to be the sum of the following components : TF^ = weight of airplane without engine group and accessories, Wp = weight of the complete engine group, Wc = weight of oil and gasoline, Wu = useful load. We can then write W = W^ + Wp + Wc + Wu Generally W^ = % W; Wp = eP. In this case (assuming 4 hours of flight) Wc = 4CP, where C is the specific con- sumption per horsepower which can be assumed to be equal to 0.55; this gives Wq = 2.2P; furthermore Wv = 600 lb. 54 AIRPLANE DESIGN AND CONSTRUCTION We shall then have W = }i W + eP + 2.2P + 600 W that is, since -p must be equal to 10 P _ 600 4.46 - e and consequently W = ^^ — 0.446 - O.le In Fig. 52 these relations have been translated into curves, and it is seen that there are innumerable couples of values e, P, which satisfy the conditions necessary for the construc- tion of the airplane under consideration. Let us examine the extreme values f or e = 2 lb. per H.P. and e = 3 lb. per H.P. We see that if e = 2; P = 246 H.P. and W = 2460 lb. if e = 3; P = 416 H.P. and W = 4160 lb. From these it is obvious then, that although using an engine of 70 per cent, more power, the same result is ob- tained, plus the disadvantage of having an airplane whose surface (and consequently the required floor space), is 70 per cent, greater. However, in practice it often happens that an engine of higher power than another, not only does not possess higher weight per horsepower, but on the contrary, has a lower weight per horsepower. It is only necessary to note the importance of this matter. Another important consideration is the bulk of the en- gine. Of two engines having the same power, but different major sections, we naturally prefer the engine of lesser major section, because it permits the construction of fusel- ages offering less head resistance. An example will make the point clearer. Supposing we have two engines, each of 300 H.P., whose characteristics with the exception of their bulk, are absolutely similar. Suppose that one of THE ENGINE 55 these engines has a major section of 6 sq. ft., and the other of 9 sq. ft., the head resistance of the fuselage of the second engine is 50 per cent, greater than that of the first. Let 400 y y y \^ ^ \ .^ y ^ l^ 300 ^ ^ ^ ^ <-- ^ "^ .-^ ZOO 100 e 00 4 46- s . _J 5000 4000 3000 2000 1000 2.25 2.50 175 us assume that the power developed is used up in the fol- lowing manner : 30 per cent, for the resistance of the wing surface, 40 per cent, for the resistance of the fuselage, and 30 per cent, for the resistance of all the other parts. The result is that with the second engine, a machine can be constructed whose head resistance will be 20 per cent. 56 AIRPLANE DESIGN AND CONSTRUCTION greater, thereby losing about 7 per cent, of the speed, due to the relations between the various head resistances and the speeds, as we shall see in the discussion on the efficiency of the airplane. The position of the center of gravity with respect to the propeller axis, has a great importance in regard to the installation of an engine in the airplane. An ideal engine should have its center of gravity below, or at the most, coin- cident with the line of thrust. This last condition is true for all rotary and radial engines. Instead, for engines with vertical or V types of cylinders, the center of gravity is generally found above the line of thrust, unless the pro- peller axis is raised by using a transmission gear. In speaking of the problems of balancing, we shall see the great importance of the position of the center of gravity of the machine with respect to the axis of traction, and the con- venience there may be in certain cases, of employing a trans- mission gear in order to realize more favorable conditions. Furthermore, the transmission gear from the engine shaft to the propeller shaft, may in some cases prove very con- venient in making the propeller turn at a speed conducive to good efficiency. In the following chapter we shall see that the propeller efficiency depends on the ratio between the speed of the airplane and the peripheral speed of the propeller; since the peripheral speed depends on the number of revolutions, this factor consequently becomes of vast importance for the efficiency.. Let us see now which criterions are to be followed in installing an engine in an airplane, and let us discuss briefly, the principal accessory installations such as the gasoline and oil systems, and the water circulation for cooling. As has been pointed out before, in the type of machine most generally used today, the tractor biplane — the engine is installed in the forward end of the fuselage — on properly designed supports, usually of wood, to which it is firmly bolted. The supports, in turn, are supported on transverse fuselage bridging and are anchored with steel wires which take up the propeller thrust (Fig. 53). THE ENGINE 57 58 AIRPLANE DESIGN AND CONSTRUCTION The oil tank is generally situated under the engine, so as to reduce to a minimum the piping system. There are two pipe lines — one leading from the bottom of the tank and which is used for the suction, the other, for the return and leading into the top of the tank (Fig. 54). The oil tank is usually made of copper or leaded steel sheets; it generally weighs from 10 per cent, to 12 per cent, as much as the oil it contains. It is easy to place all the oil in one tank, as the oil con- sumption per horsepower is about ^{qq of the gasoline Oil Feed and Refui'n Pump -=-^ Return Oil Pump Filter Return Pipe Fig. 54. consumption, but it is a difficult matter to contain all the required gasoline in a single tank, especially for powerful engines. Therefore, multiple tanks are used. As the gaso- line must be sent to the carburetor which is generally located above the tanks, it is necessary to resort to artifices to insure the feeding. The principal artifices are a. Air pump pressure feed, h. Gasoline pump feed. The general scheme of the pressure feed is shown in Fig. 55. The motor M, carries a special pump P which compresses the air in tank T; the gasoline flowing through cock 1, goes to carburetor C Cock 1 enables the opening or closing of the flow between tank T and the carburetor. Further- THE ENGINE 59 more, it allows or stops a flow between the carburetor and a small auxiliary safety tank t, situated above the level of the carburetor, so that the gasoline may flow to the carbu- FiG. 55. — Gasoline pressure feed sj^stem. retor by gravity; the gasohne in this tank is used in case the feed from the main tank should cease to operate. Fi- I I I I I I p r^i r^^ f^i tTi r]°i f^ ^ Hei'ghi of Fall of Gasoline nally, cock 1 also enables a flow between the main tank T and the auxiliary tank t, in order that the latter may be replenished. The scheme of circulation is completed by a 60 AIRPLANE DESIGN AND CONSTRUCTION hand pump p, which serves to produce pressure in the tank before starting the engine; cock 2 establishes a flow between tank T and either or both of the pumps P and p, or excludes them both. Fig. 56 shows the scheme of circulation by using the gaso- line pump feed. The gasoline in the main tank T flows to a pump G, which sends it to the carburetor. Cock i permits or stops a flow between tank T and the carburetor, or between tank t and the carburetor, or between T and /. Pump G may be operated by a special small propeller or by the engine. In the schemes of Figs. 55 and 56, an example of only one main tank is shown. If there are two or more tanks the conception of the schemes remains the same, the cocks only changing so as to allow simultaneous or single func- tioning of each of the tanks. Gasoline pump feed is much more convenient than pres- sure feed because it is more reliable. It does not use com- pressed air, is less tiresome for the pilot, as it requires of him only the maneuver of opening or closing a cock, and finally, because the tanks can be much lighter as they do not have to withstand the air pressure. As a matter of interest, a tank operating under pressure weighs from 14 per cent, to 18 per cent, as much as the gasoline it contains, while a tank operating without pres- sure weighs from 10 per cent, to 13 per cent. We shall note finally, that it is necessary to install proper metallic filters or strainers in the gasoline feed system, in order to prevent impurities existing in the gasoline, from clogging up the carburetor jets. The piping systems for gasoline and oil are made of copper. The joints are usually of rubber. As to the diam- eter of the piping system, it must be comparatively large for the oil, in order to avoid obstruction due to congealing. For the gasoline, the diameter must be such that the speed of gasoline flow does not exceed 1 ft. to 1.5 ft. per second; thus for instance, supposing an engine to consume 24 gallons an hour (that is, 0.00666 gallon a second) the THE ENGINE 61 inside diameter of the gasoline pipe must be from ^{q in. to % in. It is often necessary to resort to special radiators to cool the oil. On the contrary, in order to avoid freezing, in winter, it is necessary to insulate the tank with felt. The water circulation exists only in water-cooled engines. Fig. 57 shows the principle of the water-cooling system. The engine is provided with a water pump P, which pumps the water into the cylinder jackets; after it has been ^ ! Ill —Water-cooling system. warmed by contact with the cylinders, it flows to the radi- ator R, which lowers its temperature. Finally, from the radiator, the water flows back to the pump, and the circuit is completed. The gasoline consumption of the engines varies from 0.45 to 0.55 lb. per H.P. per hour. Assuming an average of 0.5 lb. per H.P., and since the heat of the combustion of gasoline is about 18,600 B.t.u. per lb., then for 1 H.P. per hour, 9300 B.t.u. are necessary. Now, the thermal equivalent of 1 H.P. per hour is 2550 B.t.u., therefore only 2550 qoXTv = 27.5 per cent, of the heat of combustion of the 62 AIRPLANE DESIGN AND CONSTRUCTION gasoline is utilized in useful work; the rest, 72.5 per cent, or 6550 B.t.u. are to be eliminated through exhaust gases or through the cooling water. The B.t.u. taken up by the exhaust, compared with those taken up by the cooling water, vary not only for each engine, but even for each type of exhaust system. On the average, we can assume the water to absorb about 30 per cent, of the B.t.u., or about 2800 B.t.u. for every horsepower per hour; the quantity of B.t.u. to be absorbed by the cooling water of an engine of power P, is consequently equal to 2800P B.t.u. This quantity of heat must naturally be given up to the air, and the radiator is used for that purpose. From the standpoint of its application to the airplane, the radiator must possess tw^o fundamental qualities, which are : First, it must be as light as possible, and Second, It must absorb the minimum power to move it through the air. Since the weight also involves a loss of power, suppose that, as w^e have indicated, the flying characteristics depend on the weight per horsepower, we may then say that the lower the percentage of power absorbed the more efficient wdll be the radiator. It is possible to determine experi- mentally the coefficients which classify a given type of radiator according to its efficiency, with respect to its application to the airplane. Before all, it must be remembered that a radiator is nothing more than a reservoir in which the water circulates in such a way as to expose a large wall surface to the air which passes conveniently through it. There are two main types of radiators : the water tube type, and the air tube or honeycomb type. In the first, the water passes through a great number of small tubes, disposed parallel to, and at some distance from each other; the air passes through the gaps between the tubes. In the air tube radiators (also called honeycomb radiators because of their resemblance to the cells of a beehive), the water circulates through the interstices between the tubes, while the air flows through the tubes. For the present great flying speeds, the latter THE ENGINE 63 type of radiator has proven much more suitable, and therefore is more generally used. To compare two types of honeycomb radiators, we will take into consideration a cubic foot of radiator, and study its weight, water capacity, cooling surface, head resistance, and cooling coefficient. The first three are geometrical elements which can be defined without uncertainties. The head resistance depends not only on the speed of the airplane, but also on its position in the machine, and frontal area. Finally, the cooling coefficient beside depending on the type of radiator, depends on the velocity of water flow and air flow, and the initial temperatures of the air and water. As one can see, there are many factors which would be difficult to condense into one single formula. We must therefore content ourselves with studying separately, the influence of each of the above factors. In the following table are given the values of the weight Wji, water capacity W-^, and radiating surface S per cubic foot, of radiator for certain types of radiators; also let us call a the ratio between the weight of 1 cu. ft. of radiator including the water, and its radiating surface. Table 2 Type of radiator Circular tubes with hexagonal sides Square tubes .... Square tubes Hexagonal tubes. Weight Wr lb. per cu. ft. 34.8 38.9 42.8 29.7 Water capacity Ww lb. per cu. ft. 20.5 9.3 8.8 12.9 Total weight W lb. per cu. ft. 55.3 48.2 51.6 42.6 Radiating surface 2 sq. ft. per cu. ft. 97.8 188.5 161.5 132 Total weight of radiating surface lbs. per sq. ft. 0.4660 0.2653 0.3095 0.3227 The power absorbed by the head resistance of 1 cu. ft. of the radiator, may assume the following expression: where S is the frontal area of the radiator, and V is the speed of the machine in feet per second. 64 AIRPLANE DESIGN AND CONSTRUCTION Let us call d the de])th of the radiator core; S X d = 1 1 d' OT S = J ; thus the preceding expression becomes fixlxV (1) a The coefficient /3 varies not only with the different types of radiators, but with the same radiator, depending on whether it is placed in the front of the fuselage, or whether it is completely surrounded by free air. Equation (1) shows that to decrease the head resistance it is convenient to augment the depth of the radiator d. This increase, however, is Umited by the fact that it is advisable to keep at a maximum the difference in the water and air temperatures; then if the depth of the radia- tor tubes is greatly increased, the air is excessively heated, thus decreasing the difference in temperature between it and the water. For this reason the depth d may become greater as the air flow v through the tubes is increased in velocity. The following is a practical formula that may be used in determining d : d = SXlXVv (2) where I is the diameter of the tubes in feet, and v the velocity of the air flow^ through the tubes in feet per second. The quantity of heat radiated by 1 cu. ft. of radiator, not only depends on the type, but on the difference between the temperature t^ of the water, and t^ of the air, on the velocity of water flow, on the velocity v of air flow through the tubes, and on the radiating surface S per cubic foot of the given radiator. Assuming the velocity of water flow to be constant, the quantity of B.t.u. may be expressed by yX (L - L) X vi: (3) where y is the cooling coefficient, varying with the type of radiator. Now, if the engine has power P, the radiator must take THE ENGINE 65 care of 2800P calories. Therefore the volume C of the radiator must be such that C X y X {ty, - ta) X V X ^ = 2800P or, ^ 2800P ,4s yX{L-ta)XvX^ The weight of the radiator will be C X W, and the power absorbed by its head resistance will be CXBX\XV' ^^^^ ^'' (5) d SXlXVv If we call ^ the ratio =^ » the power required to carry D Drag C X TFlb. will be (in ft. lbs.), C X W X ^ X 7 Therefore the total power absorbed by the cooling system will be ^ CX^XV ^ ^ ^^r x^XV SXlXVv D and by equation (4) P, = P X , ^^" X \^+ W X |X F yX{L-ta)XvX^ V^l\/v D We can further simphfy the preceding expression. First of all we will note that v (the velocity of air flow inside of the tubes), is proportional to the speed of the airplane; we can then write V = 8 X V The temperature L is usually taken at 176°F. (80°C.) ; it is not convenient to increase it, as the airplane must be able to fly at considerable altitude, where due to the atmos- pheric depression, the boiling point of water is lowered. For the air temperature ta, we must take the maximum annual value of the region in which the machine is to fly; in cold seasons, the cooling capacity of the radiator becomes 66 AIRPLANE DESIGN AND CONSTRUCTION excessive, and therefore, special devices are resorted to, for cutting off part, or all of the radiator. In very warm climates, we may take for example <„ = 104°, then the result is L - ta = 176° - 104° = 72°F. As to the dimension I (the diameter of the tube through which the air passes), experiments have shown that to diminish TF, and increase 2, I must be kept around 0.396 in. = 0.033 ft. Finally, the ratio ^ for a good wing, varies p around 15. Then letting p = ratio ~^> where P/^ is the power absorbed by the radiator, and P the total power, W equation (5), remembering that -;^ = a, by the proper re- ductions, becomes "^ 7 X 25'^ y X d where the coefficients have the following significance: Pr p = -p = percentage of power absorbed by the radiator, W a = -:^ = weight of radiator per square foot of radiating surface, /3 = coefficient of head resistance, 7 = cooling coefficient of the radiator, 8 = y: = coefficient of velocity reduction inside the tubes, with respect to the speed of the airplane, and 2 = radiating surface per cubic foot of radiator. C Similarly, if we call c = p the volume of radiator re- quired per horsepower, and simplifying as before, equation (4) gives '^■' X 1 (7) 7X2-5 V THE ENGINE 67 The two equations (6) and (7), allow one to solve the problem of determining the volume of the radiator and the power absorbed. For a given type of radiator, a, jS, b, and 2 are constants, then one can write 149/3 _ ^. 583 X a _ ^. 38.9 ^ . TXSXa^^ ' 7X5 ' y X 8 X and therefore equations (6) and (7) become, respectively, f "^7 (8) I p = 5 X 7^^ + C Naturally, such relations can be used within the present limits of airplane speeds (80 m.p.h. to 160 m.p.h.). They state that the volume of the radiator is inversely propor- tional to the speed, and the power required is proportional to the M power of the speed. Before leaving the discussion on radiators, we will briefly discuss the systems of reducing the cooling capacity. There are two general methods; to decrease the speed of water circulation, or to decrease the speed of air circulation. The second is preferable, and is today more generally adopted. It is effected by providing the front face of the radiator with shutters which can be more or less closed until the air passage is completely obstructed. Mufflers have not as yet been extensively adopted for aviation engines, principally because they entail a direct loss of power amounting to from 6 per cent, to 10 per cent.; and because of their bulk and weight. Ordinary exhaust tubes are used, exhausting singly for each cylinder, or joined together, the point being, to convey the gases away from those parts of the machine that might be damaged by them. Before concluding this chapter, it is desirable to note the functioning of the engine at high altitudes. Modern air- planes have attained heights up to 25,000 ft.; battleplanes carry out their mission at heights varying from 10,000 to 20,000 ft., therefore it is necessary to study the actions of the engine at such altitudes. 68 AIRPLANE DESIGN AND CONSTRUCTION Since the density of the air decreases as one rises above the ground, according to a logarithmic law, let H be the height in feet, at some point in the atmosphere above sea level, and fi the ratio between the density at height H, and that at ground level; then H = 60,720 log ^ Fig. 58 shows the diagram for /i as a function of H, con- structed on the basis of the preceding formula. wouy ~ ~ ~ ~ ~" ■" ~ ~ — ~ ~ ~ ~ ~ 27000 - - / - > H- 60 720 loq-h - r ?I000 ' / ^ . 13000 / " - / / ^ 15000 / / c / X. 12000 / ^ ' 9000 ,/ / / 6000 / / 3000 / / y Ll Iz L \_ L \_ L L _ L L _ _ _ _ P3 Fig. 58. In practice, however, it happens that the temperature of the air also decreases as one rises above the ground. Then at a given height H, the density /x with respect to the ground level, is greater than the value given by the above formula. In the following discussion, which is primarily qualitative in nature, we will not take into account this decrease in tem- perature, in order not to complicate the treatment of the subject. With this foreword, let us remember that the moving THE ENGINE 69 power P, is equal to the product of the angular velocity co by the engine torque M. P = o^XM At height H, the engine torque M is proportional to the q.70o 0.500 0.100 5000 lOOOO 15O0O 20000 25000 H .n Feet FiG. 59. mass of oxygen burned in one unit of time, or to the density of the air. Therefore P = ^ilf^eo = ^xFoX- (1) Wo where Po = oioMo = power at sea level. It is obvious then, that as the machine climbs, the power of the engine decreases. 70 AIRPLANE DESIGN AND CONSTRUCTION In Fig. 59, a diagram is given for the reduction in per- P centage ^ of the power, corresjionding to the increase of //. JL o In one of the following chapters will be shown the in- fluence that the decrease in the air density exerts on the power required for the sustentation of the machine. It will be readily perceived, that if a machine is to climb 25,000 ft., it must be able to maintain itself in the air with 0.251 of the power of the engine; in other words, it must carry an engine which will develop „ ^.^ = = 4 times the U.zol minimum power strictly necessary for its sustentation. In practice, these are the actual means chosen by designers to attain high altitudes. That is, the machines are equipped with engines of such excess power, as to be sufficient to maintain flight even after the strong reduction of power mentioned above. Such a method is evidently irrational, since at ground level the airplane employs a useless excess of power, while at high altitudes it is overloaded with a weight of engine entirely out of proportion to the power actually developed. To eliminate this loss of efficiency, tw^o solutions present themselves. One provisional solution (but of inestimable value in augmenting the efficiency of engines as they are actually conceived and constructed) consists of providing the engine with an air compressor which will feed the car- buretor. In this way, the mass of gas mixture taken in by the engine at each admission stroke, is greater than the amount which would be sucked in from the atmosphere directly, and as a result, the engine torque is increased. Two types of compressors have thus far been experi- mented with; the turbo compressor designed by Rateau (France), actioned by means of the exhaust gases, and the centrifugal multiple compressor designed by Prof. Anastasi (Italy), actioned by the engine shaft. The latter type, for example, with an increase in weight of less than 10 per cent., allows a complete recuperation of the power at 13,000 ft., or it recuperates 50 per cent, of the THE ENGINE 71 power. Since it absorbs 10 per cent, of the power in operation, the actual power recuperated is 40 per cent. These compressors have not yet been adopted for practi- cal use, because of reasons inherent to the operation of the propeller, which will be seen in the following chapter. The second solution (the one toward which engine technique must inevitably direct itself in order to open a way for further progress), consists in predisposing the engines so that the compression of air at high altitudes may be effected without the aid of external compressors. CHAPTER VI THE PROPELLER The propeller is the aerial propulsor universally adopted in aviation. Its scope is to produce and maintain a force of traction capable of overcoming the various head resistances of the wings and other parts of the airplane. Calling T the propeller traction in pounds, and V the velocity of the airplane in feet per second, the product T X V measures the useful work in foot pounds per second accomplished by the propeller. If Po is the power of the engine in horsepower, the propeller efficiency is expressed by TV ^ " 550 X Po ^^'' Every effort must of course be used in making the pro- peller efficiency as high as possible. In fact, equation (1) may also be written as P_ TV 550 X p which means that having assumed a given speed and a given head resistance, the power required for flight will be so much greater as the value of p is smaller. Suppose for example that T = 500 lb. and V = 200 ft. per sec, then for PI = 0.70 Pi = 260 H.P. for p2 = 0.80 P. = 227 H.P. and P2 is 13 per cent, less than Pi. A propeller is defined by a few geometric elements, and by its operating characteristics. The geometric elements of a propeller are the number of blades, the diameter, the pitch, the maximum width of the blades and their profile. 72 THE PROPELLER 73 Propellers are built with 2, 3, and 4 blades. The type most commonly used is the 2-blade propeller, especially when quick-firing guns with synchronized devices for firing through the propeller, are mounted on the airplane. On machines that have their propellers in front, the problem of firing directly forward is solved by equipping the machine guns with special automatic devices operated by the engine Fig. 60. (devices called synchronizers), which release the projectiles at the instant the propeller blades have passed in front of the machine gun muzzle; in other words, the projectile is fired through the plane of rotation of the propeller when the blade has rotated by an angle a (Fig. 60). Angle a is not fixed, but varies with the number of revolutions of the propeller, which is easily understood if one considers that the velocity of the projectile remains constant, while the angular velocity of the propeller varies. Thus, as the number of revolutions change, there is a dispersion of pro- jectiles; these fall in a sector 8, which is called the angle of dispersion of the synchronizer (Fig. 61). Now, if this angle is greater than 90°, as it often happens, it is impossible to use 4-bladed propellers, altho in certain cases, 4-bladed 74 AIRPLANE DESIGN AND CONSTRUCTION propellers may be convenient for reasons of efficiency, as will be observed further on. The diameter of the propeller depends exclusively^ upon the power the propeller has to absorb, and upon its number of revolutions. The pitch of the propeller, from an aerodynamical point of view, should be defined as "the distance by which the propeller must advance for every revolution in order that the traction be zero.^^ In practice, however, the pitch is measured by the tangent of the angle of inclination of the propeller blade with respect to its plane of rotation; if d is the angle for a cross section AB of the propeller, at a distance r from Fig. 62. the axis XX (Fig. 62), the pitch of the propeller at that section will be p = 2-Kr tang 6 Practically, propellers are made with either a constant pitch for all sections, or a more or less variable one. Figs. 63 and 64 illustrate respectively, two examples of propellers, one with constant pitch, the other with variable pitch. The width of the blade is not important as to its absolute value, but is important with respect to the diameter. Since the propeller blade may be considered as a small wing moving along an helicoidal path, it is evident that to increase the efficiency, it is convenient to reduce the width of the blades to a minimum with respect to the diameter. However, it is not possible to reduce the blade width below a certain limit, for reasons of construction and resistance of THE PROPELLER 75 the propeller. Practically, it oscillates from 8 to 10 per cent, of the diameter. The profile of a propeller, although varying from section to section, characterizes the type of the propeller. It bears a great influence on the characteristics of a propeller. Fig. G3. All propellers having the same type of profile, are said to belong to the same family. Numerous laboratory experiments on propellers, by Colonel Dorand, have demonstrated that there exist cer- tain well-determined relations between the elements of Fig. 64. propellers that are of the same family and geometrically similar, so that once the coefficients of these relations are known, it is easily possible to obtain all the data for the design of the propeller. Let D = the diameter of the propeller in feet, p = the pitch of the propeller in feet, Po = the power absorbed by the propeller on the ground, 76 AIRPLANE DESIGN AND CONSTRUCTION N = number of revolutions per second, V = the speed of the machine in feet per second, and p = the efficiency of the propeller, than the relations binding the preceding parameters are irnD (1) (2) (3) Equation (2) states that the coefficient a of equation (T V IS not a constant, but depends on the ratio ■nD Let us examine the graphical interpretation of this ratio. V Since -kiiD is the peripheral speed of the blade tip, — ^ measures the angle d that the path of the blade tip makes with the plane of rotation of the propeller (Fig. 65). Now, the angle of incidence i of the blade with respect to its path, is measured exactly by the difference B — d'; as 6 is fixed, i varies with the variation of 6' ; this explains why as tan- y gent d' = — ^ varies, the power absorbed by the propeller varies, and consequently coefficient a varies. This also explains equation (3), which shows that the propeller V efficiency is dependent upon - — ^; in fact, as in the case of a wing, the efficiency of a propeller blade varies with the variation of the angle of incidence i. THE PROPELLER 77 Returning to equation (1), and assuming a given value for a, for instance, a = 3 X 10"^ then that equation becomes P„ = 3 X 10-« nW and states 1. For a propeller of a given diameter, the power required to rotate it, increases as the 3d power of n. In Fig. 66 a curve is drawn illustrating that law, assuming D = 10 ft.; the curve is a cubic parabola. 1000 900 &00 TOO • &00 500 400 300 200 r 1 / r y , ' ' ' 'o' i^-a' ' ' / D=^ lOFt.. > / > / / > / J j^ ' 1 V ' tf' ?'^ -* / / X X ^ ^ ^ Ml J " 15 20 25 n in R.p.s. Fig. 66. 2. For a given number of revolutions, the power required to rotate a propeller, increases as the 5th power of the diameter. In Fig. 67 the curve is drawn illustrating that law for n = 25 r.p.s. = 1500 R.P.M. It is a parabola of the 5th degree. 3. Assuming the power, the diameter to be given to the propeller is inversely proportional to the ^i power of the number of revolutions. The curve for that law is drawn in Fig. 68. It is an hyperbola. 78 AIRPLANE DESIGN AND CONSTRUCTION 400 --- HH P= 3x10 ^2SxD i - 300 :::::::::::: :::|::; = ::::::::::;^(::: 200 --- lililB 100 ' " :::::::::::-r:::::::::4 : ::::::;:i_;::::::::::::: < n 1:-- = :^ : fflfflwiiiiiiiiiiiiiiiiy OI234567&9 10 Diameter ,Ft. Fig. 67. tu 1 1 n 1 .-a \ P= 300 Up. 300=3xlO'^An^^D^ \ \ 1 k 20 1 " . N , « , s _ _ _ __ ._ __ _ _ _ _ __ _ _ !, ^ _ __ - - - -- -- -^ - - - - -- - - - - ^ - - - - 10 5 10 5 20 25 n R.p.s Fio. 68. THE PROPELLER 79 Equation (2), which gives a as a function of — ^' is of interest only inasmuch as it is necessary to know the value of a for equation (1). Therefore, we shall not pause in examination of it. riR P 0.8 - p = nT ^ ;^^1-s ^^ ^' ' O K ^^ ^^ ClA. ^'^ "^^ 0.3 . / 0? ^ - 0.2 ^y 0! ^< - n ^ 0.05 0.10 015 020 0.227025 O30 0.35 V Fig. 69. l.V ~ ~ "■ ~ ~ ~^ ~ ~ " ~ ' p J _, _ ^ ■^ N -\ ^ ^ 1 if* 1 ^ 1 ' ^ •■ 0/ ^ /* ^ "^ , 7 ' /' - ^^ j ,,^ *\ n2' J .. _J _ U -J _]_ .. .. 0.05 0.15 020 V TtnO Fig. 70. 0.25 0.30 035 On the contrary, it is of maximum interest to examine equation (3), which gives the efficiency of the propeller. Let us consider all geometrically similar propellers of the same family, having diameter D and pitch p, so that ^ 80 AIRPLANE DESIGN AND CONSTRUCTION = 0.8; Fig. 69 gives the diagram p = /of — v^) for such propellers. The diagram shows that p increases and reaches a maximum value pmax = 0.71 corresponding to V the value — ^ = 0.227. Let us now consider a group of propellers also of similar P profile, but having j^ = 1.0, and let us draw the efficiency diagram (Fig. 70). This will be similar to the preceding one in shape, but will reach a value p„,ax = 0.77 corre- V spending to a value of - — ^ = 0.275. P If this experiment is repeated for various values of ^s' it will be observed that the maximum efficiency obtainable from a propeller of certain profile, varies with the variation of that ratio; it is easy to construct a diagram giving all the P values of pmax as functions of j^- Such a diagram shows that a propeller of a certain type, gives its maximum effi- P ciency when yc = 1.20. Naturally this condition does not suffice, as the propeller must rotate at a number of revolu- V tions n, such that the ratio — f^ will be the one at which the propeller actually attains the maximum efficiency. p Fig. 71 gives the values of ^> a, and p, as functions of V — j::, for the best propellers actually existing. The use of these diagrams requires a knowledge of all the aerodynamical characteristics of the machine for which the propeller is intended. However, even a partial study of them is very interesting for the results that can be attained. p V First, we see that for ^ = 1.18 and — ^ = 0.32, the maximum efficiency p reaches a value of 82 per cent. Obviously that is very high, especially when the great THE PROPELLER 81 simplicity of the aerial propeller is considered. But un- fortunately, it often occurs in practice, that this value of efficiency cannot be attained because there are certain 7x10-^ 6x10 5x10' 3x10 J ' S Oaaaat m il ll ll lll Il III I MI I I I I I Iim i mnmTTTTTTTTTTm i MM 014 0.16 0.18 0.20 0.22 0.24 0.26 0.2& Q30 032 V TtnD Fig. 71. parameters which it is impossible to vary. An example will illustrate this point. Let us assume that we have at our disposition an engine 82 AIRPLANE DESIGN AND CONSTRUCTION developing 300 H.P., while its shaft makes 25 r.p.s., and let us assume that we wish to adopt such an engine on two different machines, one to carry heavy loads and conse- quently slow, the other intended for high speeds. Let the speed of the first machine be 125 ft. per sec, and that of the second 200 ft. per sec. We shall then determine the most suitable propeller for each machine. For the first machine, as n = 25, and V = 125, the ex- pression —^ becomes equal to k • We must choose a value of D, such that together with the value of a corre- sponding to-^-, (Fig. 71), it will satisfy the equation 300 = anW" or, for n = 25 a X D' ^ 0.0192 Now the corresponding values of a and D satisfying those equations are a = ^^1.4 X 10-" and D = 10.7; in fact, for this value of D, ^ = 3.14X25^X107 = '^^•^^^' *^ '''^''^ corresponds a = 1.4 X 10"^; the corresponding value of p is ~0.62, that is, our propeller will have an efficiency of 62 per cent. ; its pitch will be 0.46 X 10.7 =5.0 ft. For the second machine instead — n = 25, and V = «^^ .t. . ^ !_ 200 2.55 200-the expression ,;^ becomes 3-^^^25^^^ = -^ and a X D^ = 0.0192; the two values satisfying the desired conditions are V 200 ^ = •UOr25-X8r6 = 0-296; « = 4.1 X 10-, and corresponding to these values p = 0.79. The pitch results equal to 9.3 ft. We can see then, that the propeller for the second 79 machine, has an efficiency of 79 per cent. ; that is ..^ = '^1.27 more than that of the first machine. It would be THE PROPELLER 83 possible to improve the propeller efficiency of the first machine by using a reduction gear to decrease the number of revolutions of the propeller. In this case, it would even be possible by properly selecting a reduction gear, to attain the maximum efficiency of 82 per cent. But this would require the construction of a propeller of such diameter, that it could not be installed on the machine. Consequently we shall suppose a fixed maximum diameter of 14 ft. Then it is necessary to find a value of V n, such that value a corresponding to ^ gives a X n^ X D" = 300. That value is n = 12.4 r.p.s., Y for which — ^ = 0.23 and p = 0.72. We see then that m 0.72 this case, the reduction gear has gained p. ^o ^ ^-^^ °^ ^^ per cent, of the power, which may mean 16 per cent, of the total load; and if w^e bear in mind that the useful load is generally about H of the total weight, we see that a gain of 16 per cent, on the total load, represents a gain of about 50 per cent, on the useful load; this abundantly covers the additional load due to the reduction gear. From the preceding, we see that in order to obtain good efficiency, modern engines whose number of revolutions are very high, must be provided with a reduction gear when they are to be applied to slow machines. On the contrary, for very fast machines, the propeller may be directly connected, even if the number of revolutions of the shaft is very high. Concluding we can say, that it is not sufficient for a propeller to be well designed in order to give good efficiency, but it is necessary that it be used under those conditions of speed V and number of revolutions n, for which it will give good efficiency. Until now we have studied the functioning of the propel- ler in the atmospheric conditions at sea level. Let us see what happens when it operates at high altitudes. The equation of the power then becomes 'P = aXaXn'XD' 84 AIRPLANE DESIGN AND CONSTRUCTION where /x is the ratio between the density at the height under consideration and that on the ground (see Chapter 5). This means that the power required to rotate the propeller decreases as the propeller rises through the air, in direct proportion to the ratio of the densities. As to the number of revolutions, the preceding equation gives P Theoretically, the power of the engine varies proportion- ally to yu, that is so that theoretically we should have 3 ^ JJ^Po ^ Po and this would mean that the number of revolutions of the propeller would be the same at any height as on the ground. Practically, however, the motive power decreases a Uttle more rapidly than proportionally to m (see Chapter 5) , and consequently the number of revolutions slowly decreases as the propeller rises in the air. If instead, by using a compressor or other device, the power of the engine were kept constant and equal to Po, then the number of revolutions would increase inversely as \/;u. So for instance, at 14,500 ft., where m = 0.5 the n n revolutions would be y^_ = tt^^ = 1.26 n. A propeller making 1500 revolutions on the ground, would make 1900 revolutions at a height. This, then, is one of the principal difficulties that have until now opposed the introduction of compressors for practical use. In fact, as it is unsafe that an engine designed for 1500 revolutions make 1900, it would practically be necessary for the propeller to brake the engine on the ground, so as not to allow a number of revolutions greater than 1500 X 0.79 = 1180. In this way, however, the engine on the ground could not develop THE PROPELLER 85 all its power, and therefore the characteristics of the machine would be considerably decreased. To eliminate such an inconvenience, there should be the solution of adopting propellers whose pitch could be vari- able in flight, at the will of the pilot; thus the pilot would be enabled to vary the coefficient of the formula P = aXn^ X D^ and consequently could contain the value of n within proper limits. Today, the problem of the variable pro- peller has not yet been satisfactorily solved; but tentatives are being made which point to positive results. The materials used in the construction of propellers, the stresses to which they are subjected, and the mode of designing them, will be dealt with in Part IV of this book. PART II THE AEROPLANE IN FLIGHT CHAPTER VII ELEMENTS OF AERODYNAMICS Aerodynamics studies the laws governing the reactions of the air on bodies moving through it. Very few of these laws can be established on a basis of theoretical considerations. These can only give indications in general; the research for coefficients, which are definitely those of interest in the study of the airplane, cannot be completed except in the experimental field. Direction Perpendicular' to Line ofFligh-f-and Coniafned in ihe Yerhical Plane. Direci-ion Perpend icularfo the VerHcal Plane Containing the f^^.^ LineofFnghh For these reasons, we shall consider aerodynamics as an "Applied Mechanics" and we shall rapidly study the experimental elements in so far as they have a direct application to the airplane. Let us consider any body moving through the air at a speed V, and let us represent the body by its center of gravity G (Fig. 72). Due to the disturbance in the air, positive and negative pressure zones will be produced on the various surfaces of the body, and in general, the resultant 87 88 AIRPLANE DESIGN AND CONSTRUCTION R of these pressures, may have any direction whatever. Let us resolve that resultant into three directions perpen- dicular to one another, the first in the sense of the Hne of flight, the second perpendicular to the line of flight and lying in the vertical plane passing through the center of gravity, and the third perpendicular to that plane. These components R^, R^, and R's, shall be called respectively : R^, the Lift component, Rg, the Drag component, R\, the Drift component, RaR Fig. 73. If we wished to make a complete study of the motion of the body in the air it would be necessary to know the values, of 7?x, R5, and R\, for all the infinite number of orientations that the body could assume with respect to its line of path; practically, the most laborious research work of this kind would be of scant interest in the study of the motion of the airplane. Let us first note that the airplane admits a plane of sym- metry, and that its line of path is, in general, contained in that plane of symmetry; in such a case, the component R\ = 0. This is why the study of components R^ and Rs is made by assuming the line of path contained in the plane of symmetry, and referring the values to the angle i that the line of path makes with any straight line contained in the plane of symmetry and fixed with the machine. In general, this reference is made to the wing chord (Fig. ELEMENTS OF AERODYNAMICS 89 73), and i is called the angle of incidence; as to the force of drift, usually the study of its law of variation is made by keeping constant the angle i between the chord and the projection of the line of path on the plane of symmetry, and varying only the angle 5 between the line of path and the plane of symmetry (Fig. 74) ; the angle 8 is called the angle of drift. Summarizing, the study of components R^, R^, and R's, is usually made in the following manner: 1. To study i?x and Rs, considering them as functions of the angle of incidence i. 2. To study R's by considering it as a function of the angle of drift 5. For the study of the air reactions on a body moving through the air, the aerodynamical laboratory is the most important means at the disposal of the aeronautical engineer. The equipment of an aerodynamical laboratory consists of a special tube system of more or less vast proportions, 90 AIRPLANE DESIGN AND CONSTRUCTION inside of which the air is made to circulate by means of special fans (Fig. 75). The small models to be tested are Fig. 75. suspended in the air current, and are connected to instru- ments which permit the determination of the reactions OD .Z^ Fig. 76. provoked upon them by the air. The section in which the models are tested is generally the smallest of the tube sys- ELEMENTS OF AERODYNAMICS 91 tern, and a room is constructed corresponding to it, from which the tests may be observed. The speed of the air current may easily be varied by varying the number of revolutions of the fan. The velocity of the current may be measured by various systems, more or less analogous. We shall describe the Pitot tube, which is also used on airplanes as a speed indi- cator. The Pitot tube (Fig. 76), consists of two concentric tubes, the one, internal tube a opening forward against the wind, the other external tube h, closed on the forward end but having small circular holes. These tubes are con- nected to a differential manometer. The pressure trans- dV- mitted by tube a is equal to P + ~2~'^ ^^^ pressure trans- mitted by tube b is equal to P; thus, the differential man- ometer will indicate a pressure h in feet of air, equal to that is ^ 2g ^ consequently -f y__^M as ^ = 32.2, the result will be ^-^-ll d represents the specific weight of the air. The preceding formula consequently gives us the means of graduating the manometer so that by using the Pitot tube it will read air speed directly. With this foreword, let us note that experiments have demonstrated that the reaction of the air R, on a body moving through the air, and therefore also its components Rx, Rs and R's, may be expressed by means of the formula R = a "^ X A X V- U 92 AIRPLANE DESIGN AND CONSTRUCTION where a = coefficient depending on the angle of incidence or the angle of drift, d = the specific weight of the air, g = is the acceleration due to gravity (which at the latitude of 45° = 32.2), A = the major section of model tested (and defined as will be seen presently), and V = the speed. As a matter of convenience we shall give the coefficients assuming that the specific weight of the air is the one cor- responding to the pressure of one atmosphere (33.9 ft. of water), and to the temperature of 59°F. Furthermore the coefficients will be referred to the speed of 100 m.p.h. Then the preceding formula can be written R = K X A X im)'' (^^ and knowing K, it gives the reaction of the air on a body similar to the model to which K refers, but whose section is equal to A sq. ft., and the speed to V m.p.h. It is of interest to know the value of coefficient K, when the pressure and the temperature of the air are no more 1 atmosphere and 59° F., but have respectively any value h whatsoever (in feet of water), and t° (degrees F.). The value of the new coefficient Kht is then evidently given by ^'' ~ ^ ^ 33:9 ^ TeoM^l^ This equation will be of interest in the study of flight at high altitudes. Interpreted with respect to the speed, formula (1) states that the reaction of the air on a body moving through it, is proportional to the square of the speed of translation. This is true only within certain limits. In fact, we shall soon see that in some cases coefficient K, determined by equation (1), changes with the variation of the speed, although the angle of incidence remains constant. ELEMENTS OF AERODYNAMICS 93 From the aerodynamical point of view, the section of the parts which compose an airplane may be grouped in three main classes which are: 1. Surfaces in which the Lift component predominates, 2. Surfaces in which the Drag component predominates, and 3. Surfaces in which the Drift component predominates. The first are essentially intended for sustentation. Among them, the elevator is also to be considered, of which the aerodynamical study is analogous to that of the wings. The second, surfaces in which the component of head resistance exists almost solely, are the major sections of all those parts, as the fuselage, landing gear, rigging, etc., which although not being intended for sustentation, form essential parts of the airplane. Finally, the last surfaces are those in which the air reaction equals zero until the line of path is contained in the plane of symmetry of the airplane, but manifests itself as soon as the airplane drifts. In Chapter I, we have spoken diffusely enough of the criterions followed for the aerodynamic study of a wing. Consequently, we shall repeat briefly what has already been said. Let us consider a wing which displaces itself along a line of path which makes an angle ^ with its chord ; a certain reaction will be borne upon it which may be examined in its two components R^^ and R^ respectively perpendicular and opposite to the line of path, and which shall be called Lift and Drag, indicating them respectively by the symbols L and D. We may then write. ^ = ^ ^ ^ ^ {mj Vioo Where the coefficients X and 8 are functions of the angle 94 AIRPLANE DESIGN AND CONSTRUCTION of incidence, and define a type of wing, and .1 is the total surface of wing. The wing efficiency is given by L _ X D 5 and measures the number of pounds the wing can sustain for each pound of head resistance. In Chapter I, we have given the diagrams for X, 8 and -> as functions of i for two types of wings; consequently, it is unnecessary to record further examples. For a complete aerodynamical study of a wing, it is necessary to determine in addition to the diagrams of X, 5 and -' as functions of i, also the diagram of the ratio X ^ as a function of i, which defines the position at the center of pressure (see Chapter II). Knowledge of the law of X variation of ^ as a function of i, is necessary to enable the study of the balance of the airplane. In the reports on aerodynamical experiments conducted in various laboratories, American, English, Italian, etc., the reader will find a vast amount of experimental material which will assist him in forming an idea of the influence borne on the coefficients X and 5, not only by the shape and relative dimensions of the wings, as for instance the ,. span , thickness of the wing , ^ , ratios -T- — J — TTv -' — and — i i Frr • ' but also chord of the wmg chord of the wmg by the relative positions of the wings with respect to each other; such as multiplane machines with superimposed wings, with wings in tandem, etc. In the study of coefficients of resisting surfaces, in gen- eral, solely the knowledge of the component Rs is of interest; the sustaining component R^ is equal to zero, or is of a negligible value as compared with that of R^. We then have Rs = K X A X Vioo/ ELEMENTS OF AERODYNAMICS 95 where i^ is a function of i, and A measures the surface of the major section of the form under observation, taken K\V\\\\VVVV\\^V^ CD L^!Z3 perpendicular to the axis of symmetry of the body, or to the axis parallel to the normal line of path. 96 AIRPLANE DESIGN AND CONSTRUCTION In general, the head resistance is usually determined for only one value of i, that is, for the value corresponding to normal flight. In fact, it should be noted that an airplane normally varies its angle of incidence within very narrow hmits, from 0° to 10°; now, while for wings such variations of incidence bring variations of enormous importance in the values of L as well as in those of D, the variation of coefficient K for the resistance surfaces is relatively small. Consequently, in laboratories, only one value is found. Nevertheless, exception is made for the wires and cables, which are set on the airplanes at a most variable inclina- tion, and therefore it is interesting to know coefficient K for all the angles of incidence. A table is given below compiled on the basis of Eiffel's experiments, which gives the value of K for the following forms (Fig. 77), and for a speed of 90 feet per second: A = Half sphere with concavity facing the wind, B = Plain disc perpendicular to the wind, C = Half sphere with convexity facing the wind, D = Sphere, H = Cylinder with ends having plain faces, with axis parallel to the wind, / = Cyhnder with spherical ends, with axis parallel to the wind, E = Cylinder with axis perpendicular to the wind, F = Airplane strut — fineness ratio }4, G = Airplane strut — fineness ratio }i, L = Airplane fuselage with radiator in front, M = Dirigible shape, N = Airplane wheel without fabric, and = Wheel covered with fabric. Iable 3 A B C D E F G H I L " JV 43.5 28.6 7.8^4.1 8.7 15.6 3.5 22.6 6.1 8 8 28 14 ELEMENTS OF AERODYNAMICS 97 In the above table, one is immediately impressed by the very low value for the dirigible form. Its resistance is about 10 times less than that of the plain disc. The preceding table contains values corresponding to a speed of 90 ft. per sec. If the law of proportion to the square of the speed were exact, these values would also be available for other speeds. On the contrary, at different speeds these wu ■ 50 - ^ 40 i^^SO 20 10 ■^ ^v.. 10 20 30 40 50 60 70 &0 90 100 110 Speec^ Ft. per Sec. Fig. 78. values vary. An example will better illustrate this point. In Fig. 78 diagrams are given of the variation of K for the forms A and D, and for the speed of from 13 to 100 ft. per sec. (Eiffel's experiments). We see that coef- ficient K of form A, increases with the speed, while that of D decreases. These anomalies can be explained by admit- ting that the various speeds vary the vortexes which are formed behind the bodies in question, therefore varying the distribution of the positive and negative pressure zones, and consequently the coefficients of head resistance. 98 AIRPLANE DESIGN AND CONSTRUCTION Figs. 79 and 80 give the diagrams of the coefficient K, for the wires and cables (Eiffel); for the wires, coefficient K first decreases, then increases; for the cables instead, the value of K shows an opposite tendency. Finally, 30 -^ ___ - ^^ZO 10 n 10 20 30 40 50 60 70 30 90 100 110 Speed Ft. per Sec. Fig. 79. 40 30 10 20 30 40 50 60 70 30 20 100 110 Speed Ft. per Sec. Fig. 80. Fig. 81 gives the diagram showing how coefficient K varies for the wires and cables when their angle of incidence varies from 0° to 90°. In studying the airplane, it is more interesting to know the total head resistance than that of the various parts; ELEMENTS OF AERODYNAMICS 99 if we call Ai, Ai, . . . and A^ the major sections of the various parts constituting the airplane and which produce a head resistance, (fuselage, landing gear, wheels, struts, wires, radiators, bombs, etc.), and Ki, K^, . . . and X„, 1.0 0.& ,0.6 0.4 02 00 .^ J _/ y 7 t 7 r 1 t t J t 7 7 15 30 45 eO 75 90 Angle of Incidence in Degrees Fig. 81. the respective coefficients of head resistance, the total head resistance R^ of the airplane will be fi. = if.A.Q^ + AV4. (4)> . . . X»A„Q' = (ifiAi + KiA2 + . . . A'^„)(jQjjj = „ (jqqJ 100 AIRPLANE DESIGN AND CONSTRUCTION where a = KiAi + K2A2 + . . . A'„.4„ and is called the total coefficient of head resistance of the airplane. As to the study of the drift surfaces, it is accomplished by taking into consideration only the drift component, and not the component of head resistance, as the latter is negligible with respect to the former. Furthermore, in this study it is interesting to know the center of drift at various angles of drift, in order to determine the moments of drift and their efficaciousness for directional stability. When the line of path lies out of the plane of symmetry, all the parts of the airplane can be considered as drift surfaces. Nevertheless, the most important are the fusel- age, the fin, and the rudder. From the point of view of drift forces, the fuselages without fins and without rudders, may be unstable; that is, the center of drift may be situated before the center of gravity in such a way as to accentuate the path in drift when this has been produced for any reason whatsoever. For what we have already briefly said in speaking of the rudder and elevator, and for what we shall say more dif- fusely in discussing the problems of stability, it is opportune to know both of the coordinates of the center of drift, which define its position on the surface of drift. Finally, we shall make brief mention of the aerodynamical tests of the propeller. Let us suppose that we have a propeller model rotating in the air current of an experimental tunnel. By measur- ing the thrust T of the propeller, its number of revolutions n, the power P absorbed by the propeller, and the velocity V of the wind, it is possible to draw the diagrams of T, P, and the efficiency p. Numerous experiments by Colonel Dorand have led to the establis-hing of the following relations ; T = a' n 2£)4 P = a n' .£)5 TV a V p ^^ P a X nD ELEMENTS OF AERODYNAMICS 101 where D is the diameter of the propeller, and a and a are numerical coefficients which vary with the variation of V —j^ This ratio is proportional to the other V _ velocity of translation ■jrnD peripheral velocity which defines the angle of incidence of the line of path with respect to the propeller blade. V Knowing the values of a and a as functions of — ^' it is possible to obtain those of T, P, and p, thereby possess- ing the data for the calculation of the propeller. CHAPTER VIII THE GLIDE Let us consider an airplane of weight W, of sustaining surface A, and of which the diagrams for X, 8 and the total head resistance o-, are known. Let us suppose that the machine descends through the air with the engine shut off; that is gliding. Suppose the pilot keeps the elevator fixed in a certain position maintaining the ailerons and the rudder at zero. Then if R Wcose the airplane is well balanced, it will follow a sloping line of path d (Fig. 82), which will make a well-determined angle of incidence i, with the wing; in fact, if this angle should vary, some restoring couples (see Chapter II), tending to keep the machine at incidence i, would be produced. Let us study the existing relations among the parameters W, A, X, 8, (X, d and V. When the machine has reached its normal gliding speed (that is, V = constant), the forces 102 THE GLIDE 103 acting on it are reduced only to the weight W, and the total air reaction R. By a known theorem of mechanics, all the forces acting on a body in uniform rectilinear motion, balance each other; that is, in this case force R is equal and of opposite direction to W; that is, R + W ^ Let us consider the two components R^ and R^ of R (on the line of the path and perpendicular to the line of path). The preceding equation can then be divided into two others R, + W sin 0-0 (1) R^ + W cos = (2) Let us express the components Rx and R^ as function of X, 5, 0- and 7, Remembering what we have said in the preceding chapters, R^ = 10-' XAF2 Where R^ is expressed in lb., A in sq. ft., V in m.p.h. and X is a coefficient which depends upon the angle of incidence and of which the law of variation must be found experimentally. As to i?5 its expression results from the sum of two terms, V one due to the wings 5 X A X ( jr.r. j and the other due to parasite resistances a of the form ' (loo)' Thus we shall have R,= 10-UAV^+10-' represents J~ for V = 100 m.p.h. Equation (6) enables us to state the following general principles : 4. Other conditions being equal, the angle of glide 9 is inversely proportional to the ratio -j that is, to the efficiency of the wing. 5. Other conditions being equal, the angle of glide d is directly proportional to the ratio -j between the coeffi- cient of parasite resistance and the surface of the wings. This ratio is also usually called coefficient of fineness. 6. The angle 6 of volplaning is independent from the weight of the airplane. This weight doesn't influence but the speed. In other words, by increasing the load, the gliding speed will increase but the angle of descent will not change. THE GLIDE 105 With this premise we propose, following a method sug- gested by Eiffel, to draw a special logarithmic diagram which will enable us to study all the relations existing among the variable parameters of gliding. 1.75 35 1.50 30 125 25 1.00 20 0.75 15 0.50 10 0.25 5 1 1 1 ■ 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 - cs 7??s -- ■ / ''■•' 7 ■^K > 20 ^ ' ' *« Z ^ s^ i -^ -.^ 175 rrrr / N l^ U i ^/ j^ p^ 1 /y i^ t / \^^ 1 ^ ^^ ^^ '^ t ^t*^ \ i ^^^ s i s ^^ - ^^ t >^^ ^«Nl?5 :::::::::::: ;:^^^^-:::i -'" -^f---^^ i ^^ >^ 10 J ?d T/ lt± 75 ^3-2-10 1 2 3 4 5 6 7 8 9 Degrees. Fig. 83. Let us go back to formulas (3) and (5) and write them in the following form - TF sin 10-4 (3^ + ^) w ^ = V [10-^- (5A + To, the angle 6 with the horizontal line changes sign; that is, the line of path ascends. First of all let us study horizontal flight. Then, as = equation (1) and (2) become T = 10-'' {8A + < 8 I FLYING WITH POWER ON 119 Let us consider then any point whatever of this curve for instance the point A ; the abscissa OX of this point is OX = log — p^ 550P Now log Tr3 "^ log 55OP1 — 3 log V; thus we can con- sider OX as the algebraic sum of segment log 550Pi, and segment —3 log V. Analogously the ordinate of point A is W OY = log ^2 W and as log ^"2 = log TF — 2 log V we can consider OF as the algebraic sum of the two segments log W and —2 log V. Thus, in order to pass from the origin to point A of the diagram, it is sufficient to add the segment log 550 Pi and —3 log V along the axes OX and log W and —2 log V along the axes OY. Since evidently these segments can be added in any order whatever, we can take first log 550Pi parallel to the axes of abscissa, then —3 log V also parallel to the axes of abscissa, then — 2 log V parallel to the axes of ordinates and finally log TF parallel to the axes of ordinates. Now it is evident that the two segments —3 log V and — 2 log V corresponding to V, can be replaced by a single oblique segment whose inclination is 2 on 3 and w^hose length is — \/2^ + 3^ . log V. Thus we can pass from the origin to point A by drawing three segments, one parallel to the axes OX, the second parallel to an axes of an inclination of 2 on 3 and the third parallel to the axes OY which segments measure in the respective scales Pi, V and W. The condition necessary and sufficient in order that a system of values of Pi, V and W, may be realized with the given airplane is evidently that the three corresponding segments, summed geometrically starting from the origin, end on the diagram. 120 AIRPLANE DESIGN AND CONSTRUCTION The units of measure selected for drawing the diagram of Fig. 88 are the following: Pi in H.P. V in m.p.h. W in lb. In order to determine the relation between the scales of A and A and the scales of Pi, V and W, it is necessary to fix the origin of the scale of V; we shall suppose to assume as origin V = 100 m.p.h. Then for V = 100 m.p.h., the coordinates A and A measure also W and P; in fact for the particular value V = 100 the segment to be laid off parallel to the scale of V becomes zero and so we go from the origin to the diagram through the sum of the only two seg- ments W and P. Let us consider then the point A whose coordinates are A = 0.3 and A = 0.0463 Corresponding to these points we shall have which gives W = 3000 lb. Pi = 84.2 H.P. Thus the scales of W and Pi are determined. In order to determine the scale of V we proceed as follows: Let us give to W and Pi two values whatever, for instance W = 3000 and Pi = 200 H.P. Applying the usual construction we shall lay off OB = 3000, BC = 200 in the respective scales; from point C we draw a parallel to the scale V to meet the diagram in point D. We shall have in CD the corresponding speed. Now for point D A = 0.153. Consequently, as we have 0.153 = ?^ we will have V = 140 m.p.h, FLYING WITH POWER ON 121 that is, the segment CD laid off in 0"D' gives the scale of V. The scales being known it is easy to study the way the airplane acts, that is, it is possible to find for each value of the speed the value of the power necessary to fly. In Fig. 88 we have disposed the scales so as to facilitate the readings; that is we have made the origin 0" of the scale of V coincide with the intersection of this scale and a Une O'X' parallel to the axis OX and passing through the value W = 2700 which is the weight of the airplane; and we have furthermore repeated on O'X' the scale of power. Then, in order to have two corresponding values of P and V we draw from any point whatever E on the scale of the speed, the parallel to OX up to F, point of intersection with the diagram; we draw then FF' parallel to the scale of the speed and we have in F' on O'X' the value of the power Pi corresponding to a speed E. The examination of the diagram enables us to make some interesting observations. Let us draw first the tangent t to the diagram which is parallel to scale V; this tangent will cut the axis O'X' in a point corresponding to a power of 58 H.P. ; this is the minimum power at which the airplane can sustain itself and the corresponding speed Fmin is 72.3 m.p.h. An airplane having an engine capable of giving no more than this power, could hardly sustain itself; it would be, as one says, tangent, and could only fly horizontally or de- scend, but could by no means follow an ascending line of flight. For all the values of speed greater or lower than the above value, the necessary power for flying increases. The phenomenon of power increasing for the decreasing speed may seem strange ; even more so, if the comparison is made with all other means of locomotion, for which the necessary power for motion is so much greater as the speed of motion increases. But we must reflect that in the airplane, the power necessary for motion is partly absorbed in overcom- ing the passive resistances, partially in order to insure sustentation ; this dynamical sustentation admits a maxi- 122 AIRPLANE DESIGN AND CONSTRUCTION mum efficiency corresponding to a given value of speed, below which, consequently, the efficiency itself decreases. Practically, the speed Fn,in corresponds to the minimum value which the speed of the airplane can assume. It is quite true that theoretically the speed of the airplane can still decrease, but the further decrease is of no interest, as it requires increase of power which makes the sustentation more difficult, and therefore the flight more dangerous. When the speed increases to values greater than Fmin; the power necessary for sustentation rapidly increases. The maximum value the airplane speed can assume, evidently depends upon the maximum value of useful power the propeller can furnish. Let P2 be the power of the engine, and p the propeller efficiency; the useful power furnished by the propeller is evidently pP2- To study flying with the engine running, it is necessary to draw the diagram pP2 as a function of V, in order to be able to compare for each value of V, the power pP2 available for that speed, and the power necessary for flying, also at that speed. Therefore, it is necessary to know the following diagrams : (1) P,=f (n) (2) a = f I ^^ ] , which gives the value of coefficient a of the formula Pp = anW^, corresponding to the power absorbed by the propeller, and The first of the three diagrams must be determined in the engine testing room, and the other two in the aerody- namical laboratory. When they are known, the determi- nation of values pP2 as a function of V becomes possible by using a method also proposed by Eifell, and which is interesting to expose diffusely. FLYING WITH POWER ON 123 Let US consider the equation Pp = anW> or n^D' As we have seen in chapter 6, a = •^("n)' ^-herefore n' D' '' \nD/ Now, instead of drawing the diagram by taking the values V Pp . . of — T< as abscissae, and those of -^t^^, as ordinates on uni- nD n^D-" form scales, let us take these values, respectively, as abscissae and as ordinates, on paper with logarithmic graduation (Fig. 89). P / V \ Let us now consider a point on the curve ^ w^ = /( --^ ) ; for instance, point A. The abscissa of this point is OX = V V log —j^; but log -j< = log V — log n — log D, conse- quently we can consider OX as the algebraical sum of the following three, log V, — log n, and — log D. Analogously, p the ordinate 07 of point A, is 07 = log -^£^' and we can write 07 = log Pp — 3 log n — 5 log D, considering 07 as the algebraic sum of the following, log P, — 3 log n and — 5 log D. Then, in order to pass from the origin 0, to point A of the diagram, it is sufficient to add log V, — log n and — log D following axis OX, and log Pp, — 3 log n and — 5 log D following axis OY. Since evidently these segments can be added in any order whatever, we can first take log V, then — log n parallel to axis OX, and — 3 log n parallel to the axis of the ordinates, then again — log D parallel to the axis of the abscissae, and — 5 log D parallel to the axis of the ordinates, and finally log Pp. Now it is evident that the two segments — log n and — 3 log n corresponding to n, can be replaced by a sin- gle oblique segment with an inclination of 3 on 1 and having 124 AIRPLANE DESIGN AND CONSTRUCTION a length proportional to log n. Analogously, the two segments — log D and — 5 log D corresponding to D, SCALE D A50 i 4"-"o ~ "Tvi^^ , 1/ s / ^ '^J/ ZL II n — F'T"" z ■300 aoxio^ — — — = = - /Tf'^ S •^ 4 A f '^^W -= - — = - - /- f^V"- - u ■it" -zsoLaxio"^ .150 9 as = E E E E — ---^ =: z. ...^. :_.:^ — — — — ATI f- Z--SS 5" , 1 _ n t - '-- X, ^ '/ 1 ~ 1 * l7 tZc j 17 1 t j / j ] / ^/^M." 1 '.T / s 1 t-t^ 1 1 i ] r ~1 ~r ::r: 1 1 mtttr 1 ^ m 0.6 -100 0.7 -90 Pp HP06 •80 ■70 0.5 •60 •50 E m 1 E _ t. — H-^ ^ ^ ,.i — '— — -^n : -::;?!'- — ^ ?! ^ — , "^r V ~i ^^ I > '^ Hj -jo. „ 1 ji:'^ 1 f j / y / / r 7 / / / / L / , -.11 I 4C -^ 4x10'^ 5x10'^ 6x10 50 60 70 80 ■^ 7x10^ &xl0'^9xl0"^0xl0'^ 12x10 90 100 150 -3 14x10-' 200 V.m.p.h. Fig. 89. can be replaced by a single oblique segment with inclina- tion of 5 on 1 and having a length proportional to log D. We can definitely pass from origin to point A of the diagram, by drawing four segments parallel respectively to FLYING WITH POWER ON 125 axis ox, to an axis of inclination 3 on 1, to an axis of inclina- tion 5 on 1, and to axis OY, and which measure V, n, D, and Pp, in their respective scales. The condition necessary and sufficient for a system of values of V, n, D and Pp to be realizable with a propeller corresponding to the diagram, is evidently that the four corresponding segments (added geometrically starting from the origin) terminate on the diagram. The units of measure selected for drawing the diagram of Fig. 89 are: V, in miles per hour n, in revolutions per minute D, in feet and PpinH.P. In order to determine the relation between the scales V P of — ^ and -T^ and those of V, Pp, n, and D, it is neces- sary to fix the origin of the scales of n and D. Let us suppose that the origin of the scale n be 1800 r.p.m. and that of scale D be 7.5 ft. Then for n = 1800 and D = 7.5 the V P coordinates — ^ and ~jfjb evidently also measure V and Pp-, in fact for these particular values, the segments to be laid off parallel to the scales n and D, become zero, and so we go from origin to the diagram by means of the sum of only the two segments V and Pp. Then, considering for instance the speed V = 100 m.p.h., it must be marked on the axis OX at the point where ^ = ., onn v^ t g = 0.0074. ^ nD 1800 X 7.5 In this way the scale of V is determined. Corresponding to F = 100 m.p.h. we have (see diagram p Fig. 89) ^^ = 2.46 X lO-^^. thus, making n = 1800 and D = 7.5 we shall have Pp = 340 H.P. ; marking the value of p P = 340 in correspondence to -yg^ = 2.46 X lO-^^ deter- mines the scale of powers Pp. In order to find the scale of D, make n equal to 1800, for which the segment n is equal to zero. 126 AIRPLANE DESIGN AND CONSTRUCTION Now, by giving V and Pp any two values whatever (for instance V = 100 m.p.h. and Pj, = 100 H.P.) by means of the usual construction a segment BC is determined, which measures the diameter D on the scale of D. The p value of D results from the value ,J!,.' which is read on the diagram at point C; in our case, this value is 2.22 X lO"*'^ and consequently, as Pp = 100 and n = 1800, we shall have 1800« X D' which gives D = 6 ft. Thus, by taking to the scale of D, starting from origin 0' (which is supposed to correspond to D = 7.50 ft.), a segment O'D' = BC, and marking the value 6 ft. on the point D', the scale of D is obtained. Finally, to find the scale of n, it is sufficient to make D = 7.5, V = 100 m.p.h. and Pp = 100, and by repeating analogous construction we find that the segment BC p corresponding to C is ^^ = 2.06; then for Pp = 100 and D = 7.5 the result is n = 1270. Then, by taking to the scale of n, starting from origin 0" (which by hypothesis is equal to n = 1800), a segment 0"D" = BC, and marking the value 1270 r.p.m. on the point D", the scale of n is defined. Analogously, we can also draw the diagram p = f[~Tj)' on the logarithmic paper, by selecting the same units of measure (Fig. 89), Let us suppose that we know the diagram Pi = f (n), (Fig. 90), which is easily determined in the engine testing room; we can then draw that diagram by means of the scale n, and the scale of the power shown in Fig. 89 (Fig. 91). Disposing of the three diagrams n'D'' ' \nDJ = '© P2 = / in) FLYING WITH POWER ON 127 drawn on logarithmic paper, it is easy to find the values pPi corresponding to the values of V. In fact let us draw in Fig. 91, starting from the origin of the scale of n, a- segment equal to diameter D of the propeller adopted, measuring D to the logarithmic scale of Fig. 89, in magnitude and direction. We shall have ?oo, ^ ^y^ y ^^ y y /^ /_ -^^,4 J^ AU ^T 1 7 / t t 7 f J 1400 l&OO n Fig. 90. point F'; then draw the horizontal line Y'x. Supposing that Fig. 91 be drawn on transparent paper, let us take it to the diagram of Fig. 89, making Vx coincide with axis OX, and the point Y' with any value Y whatever, of the speed. Fig. 92 shows how the operation is accomplished, suppos- ing Y' to be made coincident with Y = 100 m.p.h. and supposing Z) = 9.0 feet. The point of intersection A between the curves Pj, and 128 AIRPLANE DESIGN AND CONSTRUCTION -500 ^450 -400 -350 -300 -250 -200 -150 ■100 FLYING WITH POWER ON 129 :500 450 -400 350 ■300 250 ■eoo 150 •100 ^50 i Pp. Q9 08 07 0.6 0.5 r/ A / 1 / > aV^/ /I V \ mo w'l mo X \J'<00 Jpoo ci mo fa / ' I I L. 40 50 60 TO 50 90 100 Fig. 92. 150 200 130 AIRPLANE DESIGN AND CONSTRUCTION P-i determines the values of Po, p and rt corresponding to an even speed. ^ We can then determine for each value of V, the corre- sponding value Pn, and we can obtain the values p X P2 corresponding to those of V in Fig. 88. This has been done in Fig. 93. Comparing, in this figure, the values of pPi and Pi corresponding to the various speeds, we see that pP^ = Pi for V = 160 m.p.h. ; this value represents the maximum speed that the airplane under consideration can attain; in fact for higher values of V, a greater power to the one effectively developed by the engine at that speed, would be required. For all the speed values lower than the maximum value V = 160 m.p.h. the disposable power on the propeller shaft is greater than the minimum power necessary for horizontal flight; the excess of power measured by the difference be- tween the values pP2 and Pi, as they are read on the loga- rithmic scales, can be used for climbing. The climbing speed V is easily found when the weight TT^ of the machine is known. In fact in order to raise a weight W at a speed v, a power of vXW lb. ft. = ^^ X V X W H.P. is necessary; we now dispose of a power pPi — Pi, consequently the climbing speed is given by "^^ - ^' = 550 ^ " >^ '^ that is, t' = ^ X (pp, - Pi: The climbing speed is thus proportional to the difference pP2 — Pi; it will be maximum corresponding to the maxi- mum value of pP2 — Pi; in our example, this maximum is found for V = 95 and corresponding to it i; = 33 ft. per sec. ' In fact, point A determines a pair of values of V and n, which are com- patible either to the diagram of the power absorbed by the propeller, or to the diagram of the power developed by the engine. FLYING WITH POWER ON X 131 Tx: — ■ :.::: ^ o _ o X ^ — - - s g ■•"% " - ' o_ \ '--' ^ ' 5 ! ;:: ;:;i;::|;;;:;:;::::::::i - S - H^'^M^i ^jfJ 1 1 Tin °1 "= . .Jk-^-J^iJ a Ml ^ V — -s --- \ s \ o ^ ^-r "1 hsX-S V <2' + <,\ ^ K 1 "h^UZ \ ■ 'i >N :::::::::::::is o 'i -R ^ CJ s S ^vJi : :::::v::t:::::::::::::::: .^S t:::::::::: = . ^. ::=-=-- S - :: i;;::::::ft:!;=E=;nln^ H^l 1 k - ^h|||||||iI44tI '^^ ■''S |::i^:--::iiE=;EEEn-- 1 1 nJ m^^ +11 1 ' -^ ai:i::s-::::::::-: : : ::::'.::\,(^ _ trj^cc ■■•■ + ■• Ml' TV 1 t ! 1 ■ ^ 1 ' 1 l^o V6' L :_:::::::::::s±-: g ^ Bi#^^#sp 'rnti^^ |§HbJ4i,itfcHW.H :T1iinT I^IiI^^Ieeee -V ^--- e^^-°cu r:;;:;::::::::$z:i-^=^ ,^. :::::::::::::: : $ s;-g -\ - V . 1 " " V "" _ s is;:::: ::: : "^ " A •?> -- - . JV"^ - - 7\^ : :^s::::::___ _ q \ ^ - - - -,f ly i ..iinig-.- — I]: : :::::::v%Ti::: :- g##^S==^======t===== 1 1 1 1 1 1 N| 1 1 1 1 1 II 1 1 ■ :-:::::::::::\ OL . \>^ - - a \ ::._____ s n yi^D. . y-'. V| " Itl o - --- -fs;;; g 1 1 _^_^ o ° <: iSigsi i I § = 132 AIRPLANE DESIGN AND CONSTRUCTION The ratio y gives the value sin d which defines the angle d, as being the angle which the ascending line of path makes with the horizontal line (Fig. 94) . We then have V = V sin e This equation shows that the maximum v corresponds to the maximum value of V sin d, and not to the maximum value of sin 6; that is, it may happen that by increasing the angle d, the climbing speed will be decreasing instead of increasing. Fig. 94. In Fig. 95 we have drawn, for the already discussed ex- ample, diagrams of v and sin d as functions of V. We see that V is maximum for sin 6 = 0.35; for the value sin d = 0.425, which represents the maximum of sin 6, we have V = 29, which is less than the preceding value. We also see that in climbing, the speed of the airplane is less than that of ths airplane in horizontal flight, supposing that the engine is run at full power. The maneuver that must be accomplished by the pilot in order to increase or decrease the climbing speed, consists in the variation of the angle of incidence of the airplane, by moving the elevator. In fact, as we have already seen, W = 10-'\AV'" Fixing the angle of incidence fixes the value of X, and consequently that of V necessary for sustentation; the air- plane then automatically puts itself in the climbing line of path, to which velocity V corresponds. But the pilot has another means for maneuvering for height; that is, the variation of the engine power by ad- justing the fuel supply. In fact, let us suppose that the pilot reduces the power pPi] then the difference pP2 — Pi, will decrease, consequently decreasing V and sin e. If the FLYING WITH POWER ON 133 pilot reduces the engine power to a point where pPi — Pi = 0; the result will be z' = and sin = 0. We see then the possibility, by throttling the engine, of flying at a whole 110 120 130 V M.p.h. Fig. 95. 140 150 160 170 series of speeds, varying from a minimum value, which depends essentially upon the characteristics of the airplane, to a maximum value which depends not only upon the airplane, but also upon the engine and propeller. CHAPTER X STABILITY AND MANEUVERABILITY Let us consider a body in equilibrium, either static or dynamic; and let us suppose that we displace it a trifle from the position of equihbrium already mentioned; if the system of forces applied to the body is such as to restore it to the original position of equilibrium, it is said that the body is in a state of stable equilibrium. In this way we naturally disregard the consideration of forces which have provoked the break of equilibrium. From this analogy, some have defined the stability of the airplane as the "tendency to react on each break of equilibrium without the intervention of the -pilot.'' Several constructors have attempted to solve the problem of stability of the airplane by using solely the above criterions as a basis. In reality in considering the stability of the airplane, the disturbing forces which provoke the break of a state of equilibrium, cannot be disregarded. These forces are most variable, especially in rough air, and are such as to often substantially modify the resistance of the original acting forces. The knowledge of them and of their laws of variation is practically impossible; therefore there is no solid basis upon which to build a general theory of stability. Nevertheless, by limiting oneself to the flight in smooth air, it is possible to study the general conditions to which an airplane must accede in order to have a more or less great intrinsic stability. Let us consider an airplane in normal rectilinear hori- zontal flight having a speed V. The forces to which the airplane is subjected are: its weight TT^, the propeller thrust T, and the total air reaction R. 134 STABILITY AND MANEUVERABILITY 135 These forces are in equilibrium; that is, they meet in one point and their resultant is zero (Fig. 96). The axis of thrust T generally passes through the center of gravity. Then R also passes through the center of gravity. Supposing now that the orientation of the airplane with respect to its line of path is varied abruptly, leaving all the control surfaces neutral ; the air reaction R will change not only in magnitude, but also in position. The varia- tion in magnitude has the only effect of elevating or low- ering the line of path of the airplane; instead, the varia- tion in position introduces a couple about the center of -^ gravity, which tends to make the airplane turn. If this turning has the effect of re- establishing the original posi- tion, the airplane is stable. If, however, it has the effect of increasing the displacement, the airplane is unstable. For simplicity, the displacements about the three prin- cipal axes of inertia, the pitching axis, the rolling axis, and the directional axis (see Chapter II), are usually considered separately. For the pitching movement, it is interesting only to know the different positions of the total resultant R cor- responding to the various values of the angle of incidence. In Fig. 97 a group of straight lines corresponding to the vari- ous positions of the resultant R with the variation of the angle of incidence, have been drawn only as a qualitative example. If we suppose that the normal incidence of flight of the airplane is 3°, the center of gravity (because of what has been said before), must be found on the resultant R^^. Let us consider the two positions Gi and Gi. If the center of gravity falls on Gi the machine is un- FiG. 96. 136 AIRPLANE DESIGN AND CONSTRUCTION stable; in fact for angles greater than 3° the resultant is displaced so as to have a tendency to further increase the incidence and vice versa. If, instead, the center of gravity- falls in G2, the airplane, as demonstrated in analogous considerations, is stable. ^0" Fig. 97. In general, the position of the center of gravity can be displaced within very restricted limits, more so if we wish to let the axis of thrust pass near it. On the other hand, it is not possible to raise the wing surfaces much with respect to the center of gravity, because the raising would produce a partial raising of the center of gravity, and also because of constructional restrictions. Then, in order to obtain a good stability, the adoption of STABILITY AND MANEUVERABILITY 137 stabilizers is usually resorted to, which (as we have seen in Chapter II) are supplementary wing surfaces, generally situated behind the principal wing surfaces and making an angle of incidence smaller than that of the principal wing surface. The effect of stabilizers is to raise the zone in which the meeting points of the various resultants are, thus facilitating the placing of the center of gravity within the zone of stability. Naturally it is necessary that the intrinsic stability be not excessive, in order that the man- euvers be not too difficult or even impossible. The preceding is applied to cases in which the axis of thrust passes through the center of gravity. It is also neces- sary to consider the case, which may happen in practice, in which the axis of thrust does not pass through the center of gravity. Then, in order to have equilibrium, it is necessary that the moment of the thrust about the center of gravity T X ^, be equal and opposite to the moment i2 X r of the air reaction (Fig. 98). Let us see which are the conditions for stability. To examine this, it is necessary to consider the meta- centric curve, that is, the enveloping curve of all the resultants (Fig. 99). Starting from a point 0, let us take a group of segments parallel and equal to the various resultants Ri 138 AIRPLANE DESIGN AND CONSTRUCriON corresponding to the normal value of the speed. Let us consider one of the resultants, for instance Ri. At point A, where Ri is tangent to the metacentric curve a, let us draw oa parallel to h, which is tangent to curve /3 at 5 the extreme end of R^. We wish to demonstrate that the straight line oa is a locus of points such that if the center of gravity falls on it, and the equilibrium exists for a value of the angle of incidence, this equilibrium will exist for all the other values of incidence Fig. 99. (understanding the speed to be constant). In other words, we wish to demonstrate that oa is a locus of the points corre- sponding to the indifferent equilibrium, and consequently it divides the stability zone from the instability zone. Let us suppose that the center of gravity falls at G on oa, and that the incidence varies from the value i (for which we have the equilibrium) to a value infinitely near i'. If we demonstrate that the moment of R'i about G is equal to the moment of Ri, the equilibrium will be demonstrated to be indifferent. Starting from C point of the intersection of Ri and R'i, let us take two segments CD and CD' equal to the value Ri and R'i respectively. The joining line DD' is parallel to BB' ; now when i' differs infinitely little from i, BE' becomes tangent to the curve /8 at point B; conse- quently, DD' becomes parallel to tangent 6; that is, also to straight line ao. Now point C, if i' differs infinitely STABILITY AND MANEUVERABILITY 139 little from i, is coincident with A (and consequently the segments GC with GA) then the two triangles GCD' and GCD (which measure the moment of Ri and R'i with respect to G), become equal, as they have common bases and have vertices situated on a line parallel to the bases: that is, the equihbrium is indifferent. To find which are the zones of stabilitj^ and instability, it suffices to suppose for a moment that the center of gravity falls on the intersection of the propeller axis and the resultant Ri, then the center of gravity will be on Ri', and since A is on the line oa, it will be a point of indifferent equilibrium, consequently dividing the line Ri into two half lines corresponding to the zones of stability and instability. From what has already been said, it will be easy to establish the half line which corresponds to the stability, and thus the entire zone of stability will be defined. The calculation of the magnitude of the moments of stabiUty, is not so difficult when the metacentric curve and the values i?i for a given speed are known. The foregoing was based upon the supposition that the machine would maintain its speed constant, even though varying its orientation with respect to the line of path. Practically, it happens that the speed varies to a certain extent; then a new unknown factor is introduced, which can alter the values of the restoring couple. Nevertheless, it should be noted that these variations of speed are never instantaneous. In referring to the elevator, in Chapter II, we have seen that its function is to produce some positive and negative couples capable of opposing the stabilizing couples, and consequently permitting the machine to fly with different values of the angle of incidence. All other conditions being the same (moment of inertia of the machine, braking moments, etc.), the mobility of a machine in the longitudi- nal sense, depends upon the ratio between the value of the stabilizing moments and that of the moments it is possible to produce by maneuvering the elevator. A machine with great stability is not very maneuverable. On the other hand, 140 AIRPLANE DESIGN AND CONSTRUCTION a machine of great maneuverability can become dangerous, as it requires the continuous attention of the pilot. An ideal machine should, at the pilot's will, be able to change the relative values of its stability and maneuvera- bility ; this should be easy by adopting a device to vary the ratios of the controlling levers of the elevator. In this way, the other advantage would also be obtained of being able to decrease or increase the sensibility of the controls as the speed increases or decreases. Furthermore, we could resort to having strong stabilizing couples prevail normally in the machine, it being possible at the same time to imme- diately obtain great maneuverability in cases where it became necessary. As to lateral stability, it can be defined as the tendency of the machine to deviate so that the resultant of the forces of mass (weight, and forces of inertia) comes into the plane of symmetry of the airplane. When, for any accidental cause whatever, an airplane inclines itself laterally, the various applied forces are no longer in equilibrium, but have a resultant, which is not contained in the plane of symmetry. Then the line of path is no longer contained in the plane of symmetry and the airplane drifts. On account of this fact, the total air reaction on the airplane is no longer contained in the plane of symmetry, but there is a drift component, the line of action of which can pass through, above or below the center of gravity. In the first case, the moment due to the drift force about the center of gravity is zero, consequently, if the pilot does not intervene by maneuvering the ailerons, the machine will gradually place itself in the course of drift, in which it will maintain itself. In the other two cases, the drift com- ponent will have a moment difTerent from zero, and which will be stabilizing if the axis of the drift force passes above the center of gravity; it will instead, be an overturning moment if this axis passes below the center of gravity. To obtain a good lateral stability, it is necessary that the axis of the drift component meet the plane of symmetry of STABILITY AND MANEUVERABILITY 141 the machine at a point above the horizontal Une contained in the plane of symmetry and passing through the center of gravity; that point is called the center of drift; thus to obtain a good transversal stability it is necessary that the center of drift fall above the horizontal line drawn through the center of gravity (Fig. 100). This result can be obtained by lowering the center of gravity, or by adopting a vertical fin situated above the center of gravity, or, as it is generally done, by giving the wings a transversal inclination usually called ''dihedral". Naturally what has been said of longi- tudinal stability, regarding the convenience of not having IfCenhr of Drift falls on fhislone. the Machine is Lahratly Stable. If Cen-kr of Drift falls on this Zone the Machine is Laterally Unskibte Fig. 100. it excessive, so as not to decrease the maneuverability too much, can be applied to lateral stability. Let us finally consider the problems pertaining to directional stability. The condition necessary for an airplane to have good stability of direction is, by a series of considerations analogous to the preceding one, that the center of drift fall behind the vertical line drawn through the center of gravity (Fig. 101). This is obtained by adopting a rear fins. By adding Figs. 100 and 101, we have Fig. 102 which shows that the center of drift must fall in the upper right quadrant. Summarizing, we may say that it is possible to build machines which, in calm air, are provided with a great in- trinsic stability; that is, having a tendency to react every time the line of path tends to change its orientation rela- tively to the machine. It is necessary, however, that this 142 AIRPLANE DESIGN AND CONSTRUCTION tendency be not excessive, in order not to decrease the maneuverability which becomes an essential quality in rough air, or when acrobatics are being accomplished. If Center of Dri-ff falls on ihis Zone fhe Machine has ._ Directional Instabilitij. If Center of Drift fa lis on this Zone the Machine has Directional StabilitLj. Fig. 101. Thus far we have considered the flight with the engine running. Let us now suppose that the engine is shut off. Then the propeller thrust becomes equal to zero. Let us Zone within nhich the Center pf Drift must '/ in Order that the Machine be Trans versa II tj and Directionallu Siable Fig. 102. first consider the case in which the axis of thrust passes through the center of gravity. In this case, the disappearance of the thrust will not bring any immediate disturbance in the longitudinal equilibrium of the airplane. But the equihbrium between STABILITY AND MANEUVERABILITY 143 144 AIRPLANE DESIGN AND CONSTRUCTION weight, thrust, and air reaction, will be broken, and the component of head resistance, being no longer balanced by the propeller thrust, will act as a brake, thereby reducing the speed of the airplane ; as a consequence, the reduction of speed brings a decrease in the sustaining force; thus equi- librium between the component of sustentation of the air reaction and the weight is broken, and the line of path becomes descendent; that is, an increase of the angle of incidence is caused; a stabilizing couple is then produced, tending to restore the angle of incidence to its normal value; that is, tending to adjust the machine for the descent. The normal speed of the airplane then tends to restore itself; the inclination of the line of path and the speed will increase until they reach such values that the air reaction becomes equal and of opposite direction to the weight of the airplane (Fig. 103). Practically, it will happen that this position (due to the fact that the impulse impressed on the airplane by the stabilizing couple makes it go beyond the new position of equilibrium) is not reached until after a certain number of oscillations. Let us note that the glid- ing speed in this case is smaller than the speed in normal flight; in fact in normal flight, the air reaction must balance W and T, and is consequently equal to y/ W^ + T^; in gliding instead, it is equal to W; that is, calling R' and R" respectively, the air reaction in normal flight and in gliding flight, R^ ^ yw + T^ ^ ITVTl R" W V "^ W^ and calling V and V" the respective speeds, we will have -= K- W r., V^" \^ ^' '«' ux; Y" \R" \"^IF' When the axis of thrust does not pass through the center of gravity, as the engine is shut off, a moment is produced equal and of opposite direction to the moment of the thrust with respect to the center of gravity. Thus if the axis STABILITY AND MANEUVERABILITY 145 of thrust passes above the center of gravity, the moment developed will tend to make the airplane nose up. If instead, it passes below the center of gravity, the moment developed will tend to make the airplane nose down. If the airplane is provided with intrinsic stability, a gliding course will be established, with an angle of incidence different from that in normal flight, and which will be greater in case the axis of thrust passes above the center of gravity, and smaller in the opposite case. The speed of gliding in the first case, will be smaller, and greater in the second case than the speed obtainable when the axis of thrust passes through the center of gravity. Naturally, the pilot intervening by maneuvering the control surfaces can provoke a complete series of equilibrium, and thus, of paths of descent. We have seen that when a stabilizing couple is intro- duced, the airplane does not immediately regain its original equilibrium, but attains it by going through a certain number of oscillations of which the magnitude is directly proportional to the stabilizing couple; in calm air, the oscilla- tions diminish by degrees, more or less rapidly according to the importance of the dampening couples of the machine. In rough air, instead, sudden gusts of wind may be en- countered which tend to increase the amplitude of the oscillations, thus putting the machine in a position to pro- voke a definite brake of the equilibrium, and consequently to fall. That is why the pilot must have complete con- trol of the machine; that is, machines must be provided with great maneuverability in order that it may be possible, at the pilot's will, to counteract the disturbing couple, as well as to dampen the oscillations. In other words, if the controls are energetic enough, the maneuvers accompUshed by the pilot can counteract the periodic movements, thereby greatly decreasing the pitching and rolling movements. In order to accomplish acrobatic maneuvers such as turning on the wing, looping, spinning, etc., it is neces- sary to dispose of the very energetic controls, not so much to start the maneuvers themselves, as to rapidly regain the 146 AIRPLANE DESIGN AND CONSTRUCTION normal position of equilibrium if for any reason whatever the necessity arises. Let us consider an airplane provided with intrinsic auto- matic stability, as being left in the air with a dead engine and insufficient speed for its sustentation. The airplane will be subjected to two forces, weight and air reaction, which do not balance each other, as the air reaction can have any direction whatever according to the orientation of the airplane and the relative direction of the line of path. Let us consider two components of the air reaction, the vertical component and the horizontal component. The vertical component partly balances the weight; the differ- ence between the weight and this component measures the forces of vertical acceleration to which the airplane is subjected. The horizontal component, instead, can only be ba^.anced by a horizontal component of acceleration; in other words, it acts as a centripetal force, and tends to make the airplane follow a circular line of path of such radius that the centrifugal force which is thereby de- veloped, may establish the equilibrium. Thus, an air- plane left to itself, falls in a spiral line of path, which is called spinning. Let us suppose, now, that the pilot does not maneuver the controls; then, if the machine is pro- vided with intrinsic stability, it will tend to orient itself in such a way as to have the line of path situated in its plane of symmetry and making an angle of incidence with the wing surface equal to the angle for which the longitudinal equilibrium is obtained. That is, the machine will tend to leave the spiral fall, and put itself in the normal gliding line of path. Naturally in order that this may happen, a certain time, and, w^hat is more important, a certain vertical space, are necessary. The disposable ver- tical space may happen to be insufficient to enable the machine to come out of its course in falling; in that case a crash will result. We see then what a great convenience the pilot has in being able to dispose of the energetic controls which can STABILITY AND MANEUVERABILITY 147 be properly used to decrease the space necessary for restoring the normal equilibrium. Summarizing, we can mention the following general criterions regarding the intrinsic stability of a machine : 1. It is necessary that the airplane be provided with in- trinsic stability in calm air, in order that it react auto- matically to small normal breaks in equilibrium, without requiring an excessive nervous strain from the pilot; 2. This stability must not be excessive in order that the maneuvers be not too slow or impossible; and 3. It is necessary that the maneuvering devices be such as to give the pilot control of the machine at all times. Before concluding the "chapter it may not be amiss to say a few words about mechanical stabilizers. Their scope is to take the place of the pilot by operating the ordi- nary maneuvering devices through the medium of proper servo-motors. Naturally, apparatuses of this kind, cannot replace the pilot in all maneuvers; it is sufficient only to mention the landing maneuver to be convinced of the enormous difficulty offered by a mechanical apparatus intended to guide such a maneuver. Essentially, their use should be limited to that of replacing the pilot in normal flight, thereby decreasing his nervous fatigue, especially during adverse atmospheric conditions. We can then say at once that a mechanical stabilizer is but an apparatus sensible to the changes in equilibrium which is desired to be avoided, or sensible to the causes which produce them, and capable of operating, as a conse- quence of its sensibility, a servo-motor, which in turn maneuvers the controls. We can group the various types of mechanical stabilizers, up to date, into three categories: 1. Anemometric, 2. Clinometric, and 3. Inertia stabilizer. There are also apparatus of compound type, but their parts can always be referred to one of the three preceding categories. 148 AIRPLANE DESIGN AND CONSTRUCTION 1. The anemometric stabilizers are, principally, speed stabilizers. They are, in fact, sensible to the variations of the relative speed of the airplane with respect to the air, and consequently tend to keep that speed constant. Schematically an anemometric stabilizer consists of a small surface A (Fig. 104), which can go forward or back- ward under the action of the air thrust R, and under the reaction of a spring S. The air thrust R, is proportional to the square of the speed. When the relative speed is equal to the normal one, a certain position of equilibrium is ob- tained; if the speed increases, R increases and the small disk goes backward so as to further compress the spring. If, instead, the speed decreases, R will decrease, and the Fig. 104. small disk will go forward under the spring reaction. Through rod S, these movements control a proper servo- motor which maneuvers the elevator so as to put the air- plane into a climbing path when the speed increases, and into a descending path when the speed decreases. Such functioning is logical when the increase or decrease of the relative speed depends upon the airplane, for instance, because of an increase or decrease of the motive power. The maneuver however, is no longer logical if the increase of relative speed depends upon an impetuous gust of wind which strikes the airplane from the bow; in fact, this man- euver would aggravate the effect of the gust, as it would cause the airplane to offer it a greater hold. STABILITY AND MANEUVERABILITY 149 Thus we see that an anemometric stabilizer, used by itself, can give, as it is usually said, counter-indications, which lead to false maneuvers. In consideration of this, the Doutre stabilizer, which is until now, one of the most successful of its kind ever built, is provided with certain small masses sensible to the inertia forces, and of which the scope is to block the small anemo- metric blade when the increase of relative speed is due to a gust of wind. 2. Several types of clinometric stabilizers have been pro- posed; the mercury level, the pendulum, the gyroscope, etc. The common fault of these stabilizers is that they are sensible to the forces of inertia. The best clinometric stabilizer that has been built, and which is to-day considered the best in existence, is the Sperry stabilizer. It consists of four gyroscopes, coupled so as to insure the perfect conservation of a horizontal plane, and to eliminate the effect of forces of inertia, including the centrifugal force. The relative movements of the airplane with respect to the gyroscope system, control the servo-motor, which in turn actions the elevator and the horizontal stabilizing surfaces. A special lever, inserted between the servo- motor and the gyroscope, enables the pilot to fix his machine for climbing or descending; then the gyroscope insures the wanted inclination of the line of path. There is a small anemometric blade which fixes the air- plane for the descent when the relative speed decreases. A special pedal enables the detachment of the stabilizer and the control of the airplane in a normal way. 3. The inertia stabilizers are, in general, made of small masses which are utilized for the control of servo-motors; and which, under the action of the inertia forces and reacting springs, undergo relative displacement. In general, the disturbing cause, whatever it may be, can be reduced, with respect to the effects produced by it, to a force applied at the center of gravity, and to a couple. 150 AIRPLANE DESIGN AND CONSTRUCTION The force admits three components parallel to three prin- cipa axes, and consequently originates three accelerations (longitudinal, transversal, and vertical). The couple can be resolved into three component couples, which originate three angular accelerations, having as axis the same principal axis of inertia. A complete inertia stabilizer should be provided with three linear accelerometers and three angular accelerometers, which would measure the six aforesaid components. CHAPTER XI FLYING IN THE WIND Let us first of all consider the case of a wind which is constant in du-ection as well as in speed. Such wind has no influence upon the stabihty of the air- plane, but influences solely its speed relative to the ground. Let V be the speed proper of the airplane, and W the speed of the wind; in flight the airplane can be considered as a body suspended in a current of water, of which the /^ / ■ \ 1 1 ._u^_ b i /I / 1 ^^ .--'' Fig. 105. speed U, with respect to the ground, becomes equal to the resultant of the two speeds V and W (Fig. 105). We see then, that the existence of a wind W changes speed V not only in dimension but also in direction. Furthermore, if from a point A we wish to reach another point B, and co is the angle which the wind direction makes with the Une of path AB, it is necessary to make the air- plane fly not in direction AB, but in a direction AO making an angle 8 with AB such that the resulting speed U is in 151 152 AIRPLANE DESIGN AND CONSTRUCTION the direction AB. By a known geometrical theorem, we have U = VF2 -\-W' - 2UW cos (180° - 5 - a,) and • . "^^ • sm 5 = ^ sm w A simple diagram is given in Fig. 106, which enables the calculation of angle 5, when the speeds V and W, and the angle to which the wind makes with the line of flight to be covered, is known. This diagram is constituted of concentric circles, whose radius represents the speed of the wind, and of a series of radii, of which the angles with respect to the line OA give the angles to between the line of path and the wind. Let us find the angle 5 of drift, at which the airplane must fly, for example, with a 30 m.p.h. wind making 90° with the line of path (the drift angle of the trajectory must not be confused with the angle of drift of the airplane with respect to the trajectory, of which we have discussed in the chap- ter on stability). Let us take point B the intersection of the circle of radius 30 with the line BO which makes 90° with OA ; making B the center, and speed V of the air- plane the radius, which we shall suppose equal to 100 m.p.h., we shall have point C which determines U and 8 ; in fact OC equals U, and angle BCO equals 6. In our case U = 95.5 m.p.h., and sin 8 = 0.3. The speed of the wind varies within wide limits, and can rise to 110 miles per hour, or more; naturally it then be- comes a violent storm. A wind of from 7 to 8 miles an hour is scarcely percepti- ble by a person standing still. A wind of from 13 to 14 miles, moves the leaves on the trees; at 20 miles it moves the small branches on the trees and is strong enough to cause a flag to wave. At 35 miles the wind already gathers strength and moves the large branches; at 80 miles, light obstacles such as tiles, slate, etc., are carried away; the big storms, as we have already mentioned, even reach a speed of FLYING IN THE WIND 153 110 miles an hour. As airplanes have actually reached speeds greater than 110 m.p.h. (even 160 m.p.h.), it would be possible to fly and even choose direction from point to point in violent wind storms. But the landing maneuver, consequently, becomes very dangerous. At least during the present stage of constructive technique, it is wise not to fly in a wind exceeding 50 to 60 m.p.h. After all, such winds are the highest that are normally had, the stronger ones being exceptional and localized. On the contrary, 154 AIRPLANE DESIGN AND CONSTRUCTION for the aims of an organization, for instance, for aerial mail service, it would be useless to take winds higher than 30 to 40 m.p.h. into consideration. If we call M the distance to be covered in miles, V the speed of the airplane in m.p.h., and W the maximum speed in m.p.h. of the wind to be expected, the travelling time in hours, when the wind is contrary, will be L = M M V -W X -^ 200 50 100 150 200 V M.p.h Fig. 107. When the wind is zero the travelling time will be M consequently L = loX V 1 - w FLYING IX THE WIND 155 Supposing that we admit, for instance in mail service, a maximum wind of 35 m.p.h., a diagram can easily be drawn which for every value of speed V, will give the value 100 ^ which measures the percent increase in the travel- to ling time (Fig. 107). This diagram shows that the travelling time tends to become infinite when V approaches the value of 35 m.p.h. L To For each value of V lower than 35 m.p.h. the value 100 is negative; that is, the airplane having such a speed, and flying against a wind of 35 m.p.h. would, of course, retrocede. As V increases above the value 35, the term 100 f decreases; for V = 100 we have for instance 100 f = 154 per cent.; for V =130, 100 " = 137 per to ''o cent., etc. We see then, because of contrary wind, that the per cent increase in the travelling time, is inversely proportional to the speed. Before beginning a discussion on the effect of the wind upon the stability of the airplane, it is well to guard against an error which may be made w^hen the speed of an airplane is measured by the method of crossing back and forth between two parallel sights. Let AA' and BB' be the two parallel sights (Fig. 108). Let us suppose that a wind of speed W is blowing parallel to the line joining the parallel sights. Let ti be the time spent by the airplane in covering the distance D in the direction of A A' to BB', and ^2 the 156 AIRPLANE DESIGN AND CONSTRUCTION time spent to cover the distance in the opposite direction. It would be an error to calculate the speed of the airplane by dividing the space 2D by the sum ti + U. In fact the speed in going from A A' to BB' is equal to D '' u and in going the other way V -W = ^ By adding the two above equations : member to member, we have 27 = 5 + ^ U to that is J. ,JD , D\ Now this expression has a value absolutely different from 2D the other - — r—r- For example : supposing D = 2 miles, ti tl -f- 02 = 0.015 hours, and ^2 = 0.023 hours, we will have while 2D 4 h + U 0.015 + 0.023 105 m.p.h. When the speed of the wind is constant in magnitude and direction, the airplane in flight does not resent any effect as to its stability. But the case of uniform wind is rare, especially when its speed is high. The ampli- tude of the variation of normal winds can be considered proportionally to their average speed. Some observations made in England have given either above or below 23 per cent, as the average oscillations; and either more or less than 33 per cent, as the maximum oscillation. In certain cases, however, there can be brusque or sudden variations of even greater amplitude. FLYING IN THE WIND 157 Furthermore, the wind can vary from instant to instant also in direction, especially when close to broken ground. In fact, near broken ground, the agitated atmosphere pro- duces the same phenomena of waves, suctions, and vortices, which are produced when sea waves break on the rocks. If the airplane should have a mass equal to zero, it would instantaneously follow the speed variations of the air in which it is located; that is, there would be a complete dragging effect. As airplanes have a con- siderable mass they consequently follow the disturbance only partially. It is then necessary to consider beside the partial dragging effect, also the relative action of the wind on the airplane, action which depends upon the temporary variation of the relative speed in magnitude as well as in direction. The reaction of the air upon the airplane takes a different value than the normal reaction, and the effect is that at the center of gravity of the airplane a force and a couple (and conse- quently a movement of translation and of rotation), are produced. We have seen that in normal flight the sustaining com- ponent L of the air reaction, balances the weight. That is, we have L = m-'\AV^ If the relative speed V varies in magnitude and direction, the second term of the preceding equation will become 10~^X^A V^, and in general we will have lO-^V X Ax V"' J 10-'\AV- Consequently we shall have first of all, an excess or deficiency in sustentation and then the airplane will take either a climbing or descending curvilinear path, and will undergo such an acceleration that the corresponding forces of inertia will balance the variation of sustentation. 158 AIRPLANE DESIGN AND CONSTKL'CTWN If, for instance, the sustentation suddenly decreases, the Une of path will bend downward. In such a case, all the masses composing the airplane, including the pilot, will undergo an acceleration g' contrary to the acceleration due to gravity g. If m is the mass of the pilot, his apparent weight will no longer be mg but m(g — g') ; if it were that g'>g, the relative weight of the pilot with respect to the airplane would become negative, and tend to throw the pilot out of the airplane. Thence comes the necessity of pilots and passengers strapping themselves to their seats. Let us suppose that an airplane having a speed F undergoes a frontal shock of a gust increasing in intensity from W to W + aT^; if the mass of the airplane is big enough, the relative speed (at least at the first instant), will pass from the value V to that of V + ATF; the value of the air reaction which was proportional to V^ will become proportional to {V + AT7)2; the percentual variation of reaction on the wing surface will then be (F + AWy- - V _ 2 X V X A W + (aW) 72 ~ Y2 = 2f Wf that is, it will be inversely proportional to the speed of the airplane. Great speeds consequently are convenient not only for reducing the influence of the wind on the length of time for a given space to be covered, but also in order to become more independent of the influence of the wind gusts. Let us now consider a variation in the direction of the wind. Let us first suppose that this variation modifies only the angle of incidence i; then the value X will change. For a given variation Ai of i, the percent variation of X will be inversely proportional to the angle i of normal flight. From this point of view, it would be convenient to fly with high angles of incidence; this, however, is not possible, for reasons which shall be presented later. FLYING IN THE WIND 159 Let US now suppose that the gust be such as to make the direction of the relative wind depart from the plane of symmetry; there will then be an angle of drift. A force of drift will be produced, and if the airplane is stable in calm air, a couple will be produced tending to put the airplane against the wind and to bank it on the side opposite to that from which the gust comes. Naturally it is necessary that these phenomena be not too accentuated in order not to make the flight difficult and dangerous with the wind across. We find here the confirmation of the statement that stabiliz- ing couples be not excessive. PART III THE EFFICIENCY OF THE AEROPLANE CHAPTER XII PROBLEMS OF EFFICIENCY Factors of Efficiency and Total Efficiency The efficiency of a machine is measured by the ratio be- tween the work expended in making it function and the useful work it is capable of furnishing. For a series of machines and mechanisms which successively transform work, the whole efficiency (that is, the ratio between the energy furnished to the first machine or mechanism and the useful energy given by the last machine or mechanism), is equal to the product of the partial efficiencies of the successive transformations. To be able to effect the calculation of efficiency in an airplane, it is necessary to consider two principal groups of apparatus: the engine-propeller group and the sustenta- tion group. There is no doubt of the significance of the engine-propeller group efficiency; it is the ratio between the useful power given by the propeller and the total power supplied to it by the engine. The sustentation group comprises the wings, the controlling surfaces, the fuselage, the landing gear, etc.; that is, the mass of apparatus which forms the actual airplane. For the sustentation group, the efficiency, as it was pre- viously defined, has no significance, because neither sup- plied energy nor returned energy is found in it. The function of the sustentation group is to insure the lifting of the airplane weight, with a head resistance notably less than the weight itself The ratio between the lifted weight 161 162 AIRPLANE DESIGN AND CONSTRUCTION and the head resistance is usually taken as the measure of the efficiency of the sustentation group. The hfted load of an airplane is given by the expression L = 10-" XA72 and the head resistance is equal to the sum of two terms; one referring to the wing surface, the other to the parasite resistances : D - 10-' (5.4 +<^--^ 550p' which can also be written Iw 147,,^ p max. Since -^ is the load per sq. ft. of the wing surface, and ,^5 is the weight per horsepower of the airplane, v^^^. and p' being known, ^max- is easily calculated. In the preceding example we have for instance W P' IL. = 10- €l = 7.3; i/ = 33; p' = 0.695 A yy consequently Qm... = 0.117 CHAPTER XIII THE SPEED In ordinary means of locomotion, speed is usually con- sidered as a luxury, but in the airplane, it represents an essential necessity, for the whole phenomenon of sustenta- tion is based upon the relative speed of the wing surfaces with respect to the surrounding air. The future of the airplane, as to its application in every- day life, stands essentially upon its possibility of reaching average commercial speeds far superior to those of the most rapid means of transportation. When the airplane is in flight, high speeds present dangers incommensurably smaller than those which threaten a train or a motor car running at high speed. On the con- trary we have seen that the faster an airplane is, the better it fights against the wind. It is quite true that high speeds present real dangers when landing, but modern speedy airplanes are designed so as to permit a strong reduction in speed when they must return to earth. Let us remember that the two general equations of the flight of an airplane are: W = 10-^ \AV^ (1) 550p X P2 = 1.47 10-^ {6A + ^"^^ p'"max- corrcspoud to the three Y' Y" Y'" values — n < -^ < — n (^ig- HO). tuD irnD irnD Now, if with a given machine we wish to have the maxi- mum horizontal speed, it is convenient to select the pro- peller of such pitch and diameter so as to give the maximum efficiency at that speed. In formula (3), the propeller efficiency is seen to be to the % power; this means that for each 1 per cent, of increase of the efficiency, the speed increases only by }yi per cent. The increase of the motive power P2 is another means of increasing the speed; alsoP2 is seen to be to the % power and consequently at first glance, we may think that for a per- centual increase of P2 the same may be applied as that which has been said for a per cent, increase of p. Practically though, to increase P2 means adopting an engine of higher power, consequently of greater weight and difTerent incumbrance. Thus the change of P2 is reflected upon the terms 6 , A and c. It is not possible to translate into a formula the relation which exists between P2, 5, A and / / / 131 / > r-^ / / ion 0.74 076 P Fig. 112. 0.7S 0.50 113, 114, 115, and IIG, always adopting the preceding values for the constant terms. All the foregoing presupposes the air density constant and equal to the normal density; that is, to the one corre- THE SPEED 173 spending to the pressure of 33.9 ft. of water and to the tem- perature of 59°F. 137r 136 135 134 > 133 132 131 I3Q , / / / / / / y / / / ^, / ,i/ ^ ,^ / / / / / / / / / / / / / 350 360 370 380 PeHp. 390 400 Now as it is known tlie density of the air decreases as we rise in the atmosphere (see Chapter V), following a logarith- mic law given by the equation ^^^ - = 00720 log 1 R 60720 log ^"X^^^"^*' (1) 174 AIRPLANE DESIGN AND CONSTRUCTION Where H is the height in feet, Po . p- is the ratio between the pressure at sea level and the pressure at height H; t" is the Fahrenheit temperature at sea level, and fi is the ratio between the density at height H and the normal density defined above. 140 133 136 134 132 I3a ■ V \ \ s V \ \ \ N 1 \ \ ^\ \ k^^ f^:^ 1 1 s s?^ ^^N \ 1 \ \ ! 1 i\ s 1 N \ \ S — — \ s. -N 0,40 0.44 046 0.52 u Fig. 114. 0.5^ 0.60 Equation (1) can be translated into linear diagrams by using a paper graduated with a logarithmic scale on the ordinates, and with a uniform scale on the abscissae, giving to t" successively various values. In Fig. 117 these lines are drawn for t" = 0°, 20°, 40°, 59° and 80°F. By using these diagrams, the density corresponding to a given height for a given value of the temperature at ground level, is easily found. THE SPEED 175 Then let us again take up the examination of the formula for speed F = 155 X '''^''' 136 ■= 134 a. E 132 128 ^ T^ \ ^ V \ V > \ 4^X \^ \ V N S V \ 5^ j^ ^ \ 200 250 300 A 3q Pt-. Fig. 115. and let us place in evidence the influence of the variation of the density on various parameters which appear in it. The efficiency p is a function of ~^: now this ratio is nU influenced by the variation of the density, since V and n vary; then also p varies with a variation of fj.. We have already spoken of the influence of the density 176 AIRPLANE DESIGN AND CONSTRUCTION on the motive power in Chapter V, where we saw that the ratio between the power at height H and that at ground level is equal to m- 136 135 134 133 132 J3I 130 \, N s \ w \ \ \ \, s s. \ J \ VQ \ ?. \> > S, s S, \ s \ k. \ \ \ \, \, V \, \ V, s — 150 160 ITO 180 Fig. IIG. 190 ZOQ The useful power pP2 given by the engine propeller group is thus a function of the air density; therefore the diagram P-P2 = f{V) changes completely with a variation of /x. In Chapter IX we saw how to draw that diagram when the density is normal; that is, ^ = 1. Let us now consider THE SPEED 177 the case of /^ < 1. The ratio -~r-. = a is not only a function V of -^, but also of /x; and precisely that ratio is proportional 25O0O H=e0120hgjj:=e07?0hg(^^ f;/^) \ \ \Mv" \ NK\\ S^r" 20000 ^^ ^^ S^S ^Si^^ VC^s.*^V - 15000 _^^XA ^s^t 5;£^s >^^3po lOOOO "vi^Vv ly V\\ ^saK\\ sT^^^ ^o^^^ 5000 S\^ vn __.^_.VAa '^5\ ^ M\ n ■: :::::::: mmr 0.4 0.5 0.6 0.7 0.8 09 1.0 1.10 1.20 Fig. 117. to li.. Consequently for each value of /x a diagram n'D' needs to be drawn. In Fig. 118 such diagrams have been drawn on a logarithmic scale for the propeller family to which Fig. 89 of Chapter IX refers, and for the values y. = 178 AIRPLANE DESIGX AND CONSTRUCTION 1.0, 0.55, 0.41, 0.25, corresponding, for a temperature of 59° R, to the heights of 0, 16,000, 24,000 and 28,000 ft. The diagram which gives the motive power A as function of the number of revolutions is also to be decreased propor- r500 •350 i5x:a' -300 -200 0.9x10^ ~^ r TTi T rn T ! n I I nTTTTT TTTT TTT' ] ■ ; M 1 1 1 1 ! 1 '?• .ffJL J nil! 1 1 11'' \l M 1 1 1 1 , 1 M j n^DS ^{nD/~^. _|_ -J — h — — — =^ ^— ^ t n\ -^ T -j- 1 1 1 1 t i >«il:'"'': ' i ±_ : ::::::! 1^' 1 1 T 1 s 1 1 - _ ^ — — 1 ^ \ 1 =^=t=H=^X ^ i::i -- 1 "S-^ i ' //.=C!55rr -^ „ — ; — \ — 1 — 1 — ^*-^w^ \ ^ : , 1 1 1 1 ''T^^ lit! ;:{;■ s. \. ^ 1 ! 1 1 s s i 1 1 ! 1 1 1 \ \ • 1 ' 1 1 u '.: "":"^ \ — _ri -^-'^_ .35 _ ^ i ^ 1 ^E i : 1 ,1 = £>^ '^ <; i 1 E -^ -07 ,y^ V. V < ' ' I 1 ' 1 /^ ' : . \ \ " 1 ' ' ' 1 i ' 1 05 { ! 1 1 111 1(1 1 1 : 1 ; ' 1 1 1 , , , 1 i [ ill 111 ' i 1 ' 1 1 1 III nil I'll hi ! 1 L tf \ / 4x10'^ 5x 0'^ 6x10"^ 7x 10"^ 8 00-^9 xlO-^l 3x10" 12 UO' I4x Cf^^ 30 100 iiliiiiliiiil 200 J_l V m . p. h . Fig. 118. tional to /x. In Fig. 119 we have taken up again the diagram of Fig. 91 of Chapter IX, drawing it for the preced- ing values /i. Then by the known construction, we can draw the dia- grams pPi = /(F) for the preceding values of m (Fig. 120). THE SPEED 179 •550 -500 ■450 -400 ■350 -300 ■250 -200 -150 iOO 50 / 1 w / fm 1 // // I / 1 / / 1 / f 1 / X 'i / mo 1 / 1 / 1 / 1 -7 / / / / / / 1 1 7 ^ / ^ / \ 1 1 jj ij '80oJ -- "!::^^^ _. :::: \ \ rr k P ^ s s _s S^ 'n^ S, \ \ 3v. •x\ ■'?>:^'"'E -. ^»^s;:^ XI ^N. '^ ^<. \ ^^T '^1 — "^Ttffi'rr r/ A r>\ ^\ 1 -■ ^ s: ::-^'^ ^i_ "^^^i z^-"^- - M \ f 1 " '^ n^'^ t H« xT^^^ ^: 1 -Pm-- -^^v^^^-ffn- T 1 S 'i \ 1 T'"**^ *y A J\<4- ^ t ^ ' ^ ^1 <• S i '„ ^ 4- Nt-^l^r^ nJ IV-^^ s : :: ^\±:.\:.. ^, s\ ^ _o _ - ^ ^^__.. — ±" sJ^ - § o ----t"- i!^,,|--- 8 •= THE SPEED 181 In order to make evident the influence of the decrease of the air density on the parameter proper of the airplane, or in other words on the power Pi necessary to flying, let us take up again the general equation of flight W = 10~'\AV-' 550Pi = 1.47 X 10-4 (U + X'^^ > '?>A vs ^^^^ N ^ \\ ' VX V 1 • -M^" s::;it-:::;:::: f^mnw^ ' ^^1 \\\\\\\\\m i\^- ::::± -f-"-'- r^ r-'-ffF^= — U^, ^^jtitlt-'^ t""-± ^ -t - _i V ---f ji ^iiaiiILU...J>| - 1,%^ """ — " T 1^1 1 " ^ t ' ^ "" " o. 1 — ^ 1::::::::::::::^%^:::::::::::::: r^: i T m\\\m TTt " ++4 ||i tj- M::::::::::ii ::::::::::5t '-'-^ ±ii±::::::::::ii :::::i::>^A "'^" f ''" f-flH ' 1 ' LiiiJd — — r^— ::::::±:::::^ ^.j — ' ' 1 ' ' } ' j 1 - ^Hn- - 1 : d t3^ N - ^^— k 1 'i — M 1 h H- k^M'-'i ^ i::: : : — -j ■ : nyX'" t 1 ^'S^rv" 1 It """t%'^~ I " <^ \ S-- - '''' 1 Ju o^ t-'T--| M,% \ " Ci (5 186 AIRPLANE DESIGN AND CONSTRUCTION pairs of corresponding diagram, which give Pi and pPz. This has been done in Fig. 123, in which has been drawn :::it % C^ - m J hi! Ni ' ,^ 11': — r^*— W ! ! ' ' In i^r e^-T- rV x^ H ;! : 1 [ + - o. n o. LpL.-i--. ^^-^iry---^^ \ fn^ ^-- il M^-'*.^^ ^ £. ^^5 — s- cL = 5± Ss^ttt — i f^ ^ ::::^::z::|:te dt 3"5 ;::::^Ei5J# ^?-^ — g- --—"z^nM t4_-1- II rrrr t— o iir^:-^^: . - -tT mm - .:-{4.ili! ;;:i:pi I : i ; only part of the diagrams containing the intersections which define the maximum speeds. We see how these THE SPEED 187 speeds vary, as they increase and how flight becomes pos- sible even at 28,000 ft. and for greater altitudes. For our example we find that the speed at 28,000 ft. is equal to 265 m.p.h., while at sea level it was 160 m.p.h. Thus we also find that the number of revolutions of the propeller at 28,000 ft. is of 2450 r.p.m. against 1500 r.p.m. at sea level. Let us note first of all that in practice it would not be possible to run the engine at 2450 r.p.m. without risking or breaking it to pieces, if the engine is designed for a maxi- mum speed of say 1800 r.p.m. In second place we shall note that it would be practically impossible to build an engine or a special device such as to keep the same power at any height whatsoever. The utmost we can suppose is that the power is kept con- stant for instance up to 12,000 ft., after which it will natu- rally begin to decrease again. In order to make a more likely hypothesis, we shall suppose that the power is kept constant up to 12,000 ft. and then decreases following the usual law of proportionality. Based on this hypothesis we have drawn the diagram of Fig. 124 for the values M = 1.00; 0.64; 0.55; 0.41; 0.35 We see then that as we raise, the speed increases but much less than in the preceding case; furthermore after 12,000 ft. the speed remains about constant. If we could build propellers with diameter and pitch variable in flight, the operation of the engine-propeller group would be greatly improved and a great step would be made toward the solution of the aviation engine for high altitudes, because the problem of propeller is one of the most serious obstacles to be overcome for the study of the devices which make it possible to feed the engine with air at normal pressure at least up to a certain altitude. CHAPTER XIV THE CLIMBING In Chapter IX we have seen that the chmbing speed can be easily calculated as a function of V, when the power p X P2 furnished by the propeller and the power Pi neces- sary for the sustentation of the airplane at that speed, are known; and we have seen that the climbing speed v (ex- pressed in feet per second), is given by pP2 -Pi V = 550 W L^ R.f(V) Pa-f(V) Fig. 125. Practically, the maximum value v^^^_ of the climbing speed, obtained when the difference pPo — Pi is maximum, is of interest to us ipP2 — Pi) max. = 550 W (la) Thus if we wish to increase the climbing speed it is neces- sary to make the value {pP2 — -fOmax. the maximum possible. Let us suppose that the power P2 be given ; then first of all it is necessary that the airplane be built so that the mini- mum value of Pi be the lowest possible ; in the second place it is necessary that the propeller be selected so as to give 188 THE CLIMBING 189 the maximum efficiency, not at the maximum speed of the airplane, but at lower speeds, in order to increase the difTerence pPo —Pi. Fig. 125 shows how this can be accomplished; the diagrams p and p" correspond to two propellers having different ratio v/D. While the propeller p is better for speed than p", the propeller which corresponds to the lower value of p/P is decidedly better for climbing. Thus, practically, it is possible to adopt an entire series of propellers on a machine, to each one of which corresponds two special values for the maximum horizontal and climbing speeds. Naturally the selection of the propeller will be made according to whether preference is given to the horizontal speed or to the climbing speed. In order to study in full details, the climbing of an air- plane in the atmosphere, it is necessary to study the influ- ence the decrease of the air density has upon the climbing speed. Let us, as before, call ju the ratio between the air density at height H, and at sea level. At sea level m = 1 and the maximum climbing speed is the one given by formula (la). As the airplane rises, the value /x decreases and then formula (1) should be written ^max. = /(m) Referring to what has been said in the preceding Chap- ter when the characteristics of the airplane for ;u = 1 are known, it is easy to draw for different values of p., the curves pP, = /(F) and Pi =/(F) In Fig. 120 of the preceding chapter we have drawn these curves for the example of Chapter IX, and for values of p. = 1.0, 0.55, 0.41 For convenience, these curves are reproduced in Fig. 126. Comparing the pairs of curves corresponding to the same value of p., it is easy to plot the diagram which gives 190 AIIWLANE DESIGN AND CONSTRUCTION j< k H I -■ ----- o ^ w O 11: -lis i 4- --- K :::: :\ o :::::::: o r:::::::::::::;:::; \ :: :: ": \ 5! \ ' , 1' 1 o\ 1 5 v-7 . "" 8 ^ 1 mm — S-- ^^N ::E:E:EE 1 — ^-_ ^ i — "1 |j4t p:Mii|; 1. f M Ml — \ - -s .s ^::::5-::r=:::::: t-- ' \,~~^" oX ""^"" _S- ^5-- — 1 ^ s:x:-i.'^^""^s 1 X^ i^ ^ S""^v■"^--h^. -^^ -r/ 1 ? _ ^l:::":.^: .,__S__ -"^TT'" g: ^:::±:::::::: -H =^r=-=^?^--r^:| 1 ... - i---^-\,<5--- .... — ^%\ll- — 9 \M o , ^ o- ■ V^f^^ — 5 ffi "o PK.ii 7^ 8 2=1 THE CLIMBING 191 the climbing speed at the various heights. In Fig. 127 we have drawn this diagram, taking ?'n,ax. as abscissae and H as ordinates. It is interesting to draw the diagram t = f{H) ', £0 30 -if(max)-ft.perseo. Fig. 127. giving the time spent by the airplane in reaching a certain height H. To construct this diagram it is necessary first of all to draw the diagram of the equation ^ = f(H), Fig. 128a, which is easily obtained, from V = f{H) 192 AIRPLANE DESIGN AND CONSTRUCTION ~ ~ ~ 1 ) to= 59°F. !/ / ; / / [ ' / 1 / j / 1 / 1 / ^ ^ ^ * 1 V AH - - - -^ — — ^- L ■9. 1 6000 12000 l&OOO £4000 H(F+.) a lO 1000 7 r 7 t y t 7 r Tp a» r -^ /_ ^ y -.^^ ^^ ^ ^^ |1^ _^^ ** -e-''''' '' 6000 12000 laoOO 24000 H(Ft) b FlQ. 128. THE CLIMBING 193 By integrating- = f{H) we have t = f{H), (Fig. 128 b). In fact the elementary area of the diagram - = f{H) is equal to but consequently and -XdH V dH X dH = dt n X dH = t that is, the integration of diagram = f{H) gives t. In Fig. 128 a, b, we have drawn the scales of H ior t = 59°. Since by increasing H the value v tends toward zero, that of - tends toward <» , and consequently that of t also tends toward 0° . That is to say, when the airplane reaches a certain height, it no longer rises. It is said then, that the airplane has reached its ceiling. In actual practice the time of climbing is measured by means of a registering barograph. In Fig. 129 an example of a barographic chart has been given. This chart gives directly the diagram H = m that is, it gives the times on the abscissae and the heights on the ordinates. Since to reach its ceiling, the airplane would take an infinitely long time, practically the ceiling is usually defined as the height at which the ascending speed becomes less than 100 ft. per minute. It is advisable to stop a little longer in studying the influence the various elements of the airplane have upon the ceiling. 194 AIRPLANE DESIGN AND CONSTRUCTION THE CLIMBING 195 Let US again consider the formula . = 550 X ^,,^ W and let us place in evidence the influence of ^ on the differ- ence pP^ — Pi. Supposing that we adopt a propeller best for climbing; that is, one which gives the maximum efficiency correspond- ing to the maximum ascending speed, we can, with sufficient practical approximation, assume p constant; then, since P2 varies proportionally to m, the useful power available, can be represented by MP-P2 As for Pi, 55OP1 = 1.47 X 10-4 {8 A + (t)V' but W = IO-UA72 thus eliminating V from the two preceding equations \^A^ Now 5, 0-, and X are proportional to m, therefore Pi = 267 X 10-3 {8A + (t) X md X are proportional to m, Pi = 267 X 10-3(m5A + m^) = 267 X 10-3 i (5A -\- a) — ^^ and we can then write .= f [..A-267X10-3-l=(M+.)^-] Since the ceiling is reached when y = 0, it will correspond to value m', w^hich makes the second term of the preceding equation equal to zero. 267X10-3 ,, , , TF?^ That is m'pP2 ^j=^ {8A + c) -I^ = , 267^^ X 10-^ ,, , .,,W 196 AIRPLANE DESIGN AND CONSTRUCTION Remembering that H = 60,720 log ^^ the maximum vaUie //max. of ceiUiig will be that is //max. = 60,720 X log i //„„. = 60,720 log ^-^-^™,_^ (1) We can then enunciate the following general principles: 1. Every increase of p, Po, and X.4 increases the ceiling of the airplane and vice versa. 2. Every decrease of 8A, 1.2 1.4 1.5 5A-t-(r A Fig. 133. 1:7 Let us furthermore remember that the head resistance Rg and sustaining force R^ are expressed by Rs = 10-' {^ A +a)V^ R. = lo-^x^y^ THE CLIMBING 201 and consequently Rl \A 33000 32000 31000 30000 o 29000 E 2: 28000 27000 26000 25000 6.0 6.5 7.0 75 &.0 8.5 9.0 W A Fig. 134. Now, in a well-constructed airplane, the minimum value of — is between 0.15 and 0.18. Assuming 0.15, we shall have 5A + u^^ Wx shall have a-i > ai, and vice versa; that is, if the useful load of the first machine is greater than that of the second, the weight of its structure will instead be less. Now the weight of the structures, if the airplanes are studied with the same criterions and calculated with the same method, evidently characterize the solidity of the machine; and in that case the airplane having a lesser weight of structure, also has a smaller factor of safety, and if this is under the given limits, it ma}^ become dangerous to use it. Therefore, it is undesirable to increase the value oi u = ,fr by diminishing the solidity of a machine. It may also happen that two machines having different weights of structure, can have the same factor of safety, and in that case, the machine having less weight of structure is better calculated and designed than the other. The effort of the designer must therefore be to find the maximum possible value of coefficient u, assigning a given value to the GREAT LOADS AND LONG FLIGHTS 207 factor of safety and seeking the materials, the forms and the dispositions of various parts which permit obtaining this coefficient with the minimum quantity of material, that is, with minimum weight. In modern airplanes, the coeffi- cient u varies from the minimum value 0.3 (which we have for the fastest machines, as for instance the military scouts) , to the value of 0.45 for slow machines. The low value of u for the fastest machines depends upon two causes: 1. The factor of safety, necessary for very fast machines, must be greater than that necessary for the slow ones, there- ., 1 .,.., :::::::::::::::::::::::::::::::::::; ^^^:: ::::::::::::::::::::: ;,^|5t_.,- ::::: :: _::;::::::::::^^: :i i li : :::::::::^s^ ::: -r H ' ' r iSvL : _ _ .^y'..- . r- [ r 1- -\ L _ ■ J^ HI l' 1 J yf^ ' ^TiN ' ' ::::: y-.z- -\- -- i":-- - ] r -+ : \Jr\ \ W '1 J<1 Li Ml H 1 J J Fig. 137. fore the value of coefficient a in the fast machines is greater than in the slow ones, with a consequent reduction of the value u. 2. A fast machine having the same power, must be lighter than a slow machine (see the formula of total efficiency). That is to say, the importance of coefficient e increases, and therefore u diminishes. (6) In Chapter XII, we studied coefficient r and saw that T it was a function of Y . Let us now study ratio ^ ^"^^ ^^^ in it the maximum value to be put in the formula of useful load. 208 AIRPLANE DESIGN AND CONSTRUCTION Fig. 137 shows the diagram r = f{V) already given in Fig. 109 of that chapter. The diagram refers to a par- ticular example; its development, however, enables making some considerations of general character. From origin let us draw any secant whatever to the diagram. This, in general, will be cut in two points A' and A"; let us call r' and r" the values of efficiency and V and V" the values of speeds corresponding to these points. Then evidently r r , y, = yT, = tan a T Since we seek the maximum value of y> in order to have two values r^ and Vo such that their ratios will be the maximum possible, it will suffice to draw tangent t from origin to point Ao of the diagram, To . = tana^ax. Therefore infinite pairs of speeds V and V" exist, re- spectively greater and smaller than Vo, which individual- T ize equal values of ratio y ] naturally one would choose only the values of speed V , which are greater. Practically it is not possible to adopt the maximum To t: fore scarcely sustain itself; it is then necessary to choose a lower value of y- and corresponding to a speed V\>Vo. The value y^ must be inversely proportional to the height to be reached. In fact the equation WV r = 0.00267 -^ r . W states that ^ is proportional to ^-- Now as the maxi- W mum height //max. is a function of p-> consequently it is also jt 2 T a function of y' GREAT LOADS AND LONG FLIGHTS 209 (c) We treat finally the problems which relate to the increase of power P^. The increase of motive power has the natm-al consequence of immediately increasing the dimensions of the airplane. The question naturally arises, ''up to what limit is it possible to increase the dimensions of the airplane?" First of all it is necessary to confute a reasoning false in its premises and therefore in its conclusions, sustained by some technical men, to demonstrate the impossibility of an indefinite increase in the dimensions of the airplane. The reasoning is the following: Consider a family of airplanes geometrically similar, having the same coefficient of safety. In order that this be so, it is necessary that they have a W similar value for the unit load of the sustaining surface -^ , and for the speed, as it can be easily demonstrated by virtue of noted principles in the science of constructions. Let us furthermore suppose that the airplanes have the same total efficiency r. Then, as WV r = 0.00267 %^ and as r and V are constant, W will be proportional to P2 ; that is the total weight of the airplane with a full load will be proportional to the power of the engine W = vPo The weight of structure a X TF of airplanes geometric- ally similar, is proportional to the cube of the linear dimen- sions, which is equivalent to the cube of the square root of the sustaining surface; then aW = a'A^^ W but -T- = constant, therefore A is proportional to W and consequently we may write aW = a"W^ 210 AIRPLANE DESIGN AND CONSTRUCTION that is a = a"W^^ Since the weight of the motor group e X W m propor- tional to the power Po, e XW = e' X P2 but that is Then as we will have i>. = E e X W ^^-W V e = constant u + a -[- e = 1 u = I - e - a'WW and this formula states that the value of coefficient u di- minishes step by step as W increases, that is, as the dimen- sions of the machine increase step by step, until coefficient u becomes zero for that value of W which satisfies the equation I - e - a" ^yw = that is Thus the useful load becomes zero and the airplane would barely be capable of raising its own dead weight and the engine. So for example supposing e = 0.25 a" = 0.004 we shall have Ci W = ( ~^) = 35,000 lb. Now all the preceding reasoning has no practical founda- tion, because it is based on a false premise, that is, that the airplanes be geometrically similar. In fact, it is not at all GREAT LOADS AND LONG FLIGHTS 211 necessary that it be so; on the contrary, the preceding reasoning demonstrates that to enlarge an airplane in geo- metrical ratio would be an error. Nature has solved the problem of flying in various ways. For example, from the bee to the dragon fly, from the fly to the butterfly, from the sparrow to the eagle, we find wing structures entirely different in order to obtain the maximum strength and elasticity with the minimum weight. It may be protested that flying animals have weights far lower than those of airplanes; but if we recall, that along- side of insects weighing one ten thousandth of a pound, there are birds weighing 15 lb., we will understand that if nature has been able to solve the problem of flyiug within such vast limits, it should not be difficult for man, owing to his means of actual technical knowledge, to create new structures and new dispositions of masses such as to make possible the construction of airplanes with dimensions far greater than the present averaue machines. For example, one of the criterions which should be followed in large aeronautical constructions is that of dis- tributiDg the masses. The wing surface of an airplane in flight must be considered as a beam subject to stresses uniformly distributed represented by the air reaction, and to concentrated forces represented by the various weights. Now by distributing the masses respectively on the wing surface, we obtain the same effect as for instance in a girder or bridge when we increase the supports; that is, there will be the possibility of obtaining the same factor of safety by greatly diminishing the dead weight of the structure. Another criterion which will probably prevail in large aeronautical constructions, is the disposition of the wing surfaces in tandem, in such a way as to avert the excessive wing spans. The multiplane dispositions also offer another very vast field of research. As we see, the scientist has numerous openings for the solution; so it is permissible to assume that with the in- 212 AIRPLANE DESIGN AND CONSTRUCTION crease of the airplane dimensions not only may it be possi- ble to maintain constant the coefficient of proportionality u but even to make it smaller. Thus with the increase of power we shall be able to notably increase the useful load. Concluding, we may say that the increase of useful load can be obtained in three ways : (a) Perfecting the constructive technique of the airplane and of the engine, that is reducing the percentage of dead weights in order to increase u, (b) Perfecting the aerodynamical technique of the ma- chine, reducing the percentage of passive resistance and increasing the wing efficiency and the propeller efficiency, T so as to increase the value of ratio y corresponding to the normal speed V, and (c) Finally, increasing the motive power. Let us now pass to the problem of increasing the cruising radius. Let us call ^S^ax. the maximum distance an airplane can cover, and let us propose to find a formula which shows the elements having influence upon ^S^ax. The total weight W of the airplane is not maintained constant during the ffight because of the gasoline and oil consumption; it varies from its maximum initial value Wi to a final value Wf, which is equal to the difference between Wi and the total quantity of gasoline and oil consumed. Let us consider the variable weight W at the instant t, and let us call dW its variation in time dt. If P is the power of the engine and c its specific con- sumption (pounds of gasoline and oil per horsepower), the consumption in time dt will be cPdt and since that consumption is exactly equal to the decrease of weight in the time dt, we shall have dW = - cPdt (1) From the formula of total efficiency we have WV P = 0.00267 -^^— GREAT LOADS AND LONG FLIGHTS 213 then substituting that value in (1) dW = - 0.00267cTF - dt r and since y ^dS dt ^ = - 0.00267c ^ W r and integrating 'cdS Jf=- 0.00267/^ The value of c, specific consumption of the engine, can, with sufficient approximation, be considered constant for the entire duration of the voyage. Regarding r, we have already seen that it is a function of F; we shall now see that it is also a function of W. In fact, let us suppose that we have assigned a certain value Fi to 7; then the total efficiency will be W W r = 0.00267 7i -^ = const X y Supposing now that W is made variable; it would also vary P, following a law which cannot be expressed by a certain simple mathematical equation; it will then also vary ratio W -p and consequently r. Practically, however, it is convenient, by regulating the motive power and therefore the speed, to make value r about constant and equal to the maximum possible value. We can also consider an average constant value for r. Thus the preceding integration becomes very simple. In fact, as TF = W^ for 5 = 0, and TF = Wf for S = S^,,_, we shall have, loge Wf=- 0.00267 ^ 5„,ax. + log. IF. r that is r ^ , Wi 214 AIRPLANE DESIGN AND CONSTRUCTION and iiitroduciiifi; the decimal logariihiu instead of the Napierian r W >S„.,x = 8(55 X ^ X log ^ (1) Smax = 865-^ '°9 F" C=0.45 3600 3200 2S00 2400 ^ 2000 E 1600 IS) 1200 &00 400 1 / / / / 1 , ^/ f / / / / / / // ^ / f / k / / / ,/ / h ¥ y / / / / / / / / / ' / '/ f ^y / / j '/, // / V K/ / ^A / / / y /// // / / / r> y // // / / y y/ /. / y y' // / ^ /^ 7A^ y ^ V/ y y 1.0 1.1 1,2 1.3 1.4 1.5 1,6 1.7 I.& 1.9 2.0 Wf Fig. 138. The cruising radius therefore depends upon three factors: 1. Upon the total aerodynamical efficiency This de- pendency is linear; that is to say, an increase of say 10 per GREAT LOADS AND LONG FLIGHTS 215 cent, of aerodynamical efficiency, equally increases the maximum distance which can be covered by 10 per cent. 2. Upon the specific consumption of the engine. That dependency is inverse; thus, for example, if for we could seoo 3200 2600 2400 ^ 2000 1600 1200 .&00 AOO Smax = 865- log Wf 0=0.54 / / / ' // / / / / / / / / // V / / y ^ f / \/ y / // y A /I 1 // / / y / , / / / < / f / V A r / ,/ / y. // / / ^ '^Z / / y /' , '/// / / / / / ^ ^' Y/ y / y^ ^ //// V/ ''/ /' W/. // x ^ ^ v/^ '^ y^ ^ r.o 1.2 1.3 1.4 1.5 1.& 1.7 1.8 1.9 2.0 Wf reduce the specific consumption to half, the radius of action would be doubled. 3. Upon the ratio between the total weight of the airplane and this weight diminished by the quantity of gasohne and 216 AIRPLANE DESIGN AND CONSTRUCTION oil the airplane can carry. That ratio depends essentially upon the construction of the airplane; that is, upon the ratio between the dead weights and the useful load. Smax-= 865 -§- log -^ C=0.60 3600 .1 1.2 1.3 1.4 15 1.6 1.7 l.a 1.9 2.0 Fm. 140. We see, consequently, that the essential difference between the formula of the useful load and that of the cruising radius is in the fact that in the latter the total specific con- sumption of the engine, an element which did not even ORE AT LOADS AND LONG FLIGHTS 217 appear in the other formula, intervenes and has a great importance. From that point of view, almost all modern aviation engines leave much to be desired; their low weight per horsepower (2 lb. per H.P. and even less), is obtained at a loss of efficiency; in fact they are enormously strained in their functioning and consequently their thermal efficiency is lowered. The total consumption per horsepower in gasoline and oil, for modern engines is about 0.56 to 0.60 per H.P. hour; while gasoline engines have been constructed (for dirigibles) , which only consume 0.47 lb. per H.P. hour. A decrease from 0.60 to 0.48 would lead, by what we have seen above, to an increase in the cruising radius of 25 per cent. Starting from formula (1) we have constructed the dia- grams of Figs. 138, 139 and 140 which give the values of Wi ^m&x. as a function of ^ for the different values of r and c. In Fig. 138 it has been supposed that c ^ 0.48 lb. per H.P. hour, in Fig. 139 c = 0.54 and in Fig. 140 c = 0.60. The diagrams have a normal scale on the ordinates and a logarithmic scale on the abscissae. The use of the diagrams is most simple, and permits rapidly of finding the cruising radius of an airplane when Wi r, c and ~, are known. W f Before closing this chapter, it is interesting to examine as table resuming the characteristics of the best types of military airplanes adopted in the recent war, for scouting, reconnaissance, day bombardment, and for night bombard- ment. In Table 6 the following elements are found: Wi = weight of the airplane with full load. Wf = weight of the empty machine with crew and instruments necessary for navigation. W ■ ^ = ratio between initial weight and final weight. W f We shall suppose therefore that all the useful load, com- prising military loads, consists of gasoline and oil. 218 AIRPLANE DESK.'N AND CONSTRUCTION P = iiuixiinuin i)o\vor of the engine. Wi -p- = weight per horsepower. Wi —7^ = load per unit of the wing surface. ^max. = the maximum horizontal speed of the airplane. ^max. = the maximum ascending speed averaged from ground level to 10,000 ft. W X V . P' = — — "^-" is the power absorbed in horsepower 0.75 X 550 ^ to obtain the ascending speed v^^^^, supposing a propeller efficiency equal to 0.75. \p p/ V = 7jnax.-\/— p — is the horizontal speed of the airplane for which we have the maximum ascending speed ' max.* r - 0.00267 ^^' ^^""^ -^ is the total efficiency cor- responding to 7„ax.- Wi XV . r' = 0.00267 ~ py- is the total efficiency corre- sponding to v. S and S' = the maximum distances covered in miles Wi Wi corresponding to F^ax. f, tj^* and V, r', ^ respectively, supposing c = 0.60. Of „ = the gain in distance covered, flying at speed V instead of V. 85 P W'i = 375 X r' X ™y7- is the total weight the air- plane can lift at speed V, supposing an allowance of excess power of 15 per cent. W'f = Wf + H {W'i - Wi) is the new value of the final weight, supposing that >^ of the gain in weight is necessary to reinforce the airplane so as to have the same factor of safety. W'i =1^,- = the new ratio between the new initial weight GREAT LOADS AND LONG FLIGHTS 219 ^ C-l 1 cc!^ C<1 --1 r-i —1 -S ^'s g S cj 00 lO t^ to gK^l^-ii O CO ^ ^ •-I . ^ § § hl^aK^ii rt 00 00 00 •'•'Is-, s ^ g in fe:|^ w" m ^5 5 g§ ?5 i o o fe:a s § ^ 00 to o ^^1^ "^ .-< l-H ^ 1-1 m « o ^■s " "^ . ® CO o 5^ s -1 (N ■* ffli g 5 ^ s M •* cc M o M n m IM fl ^ 2 S S S r^ s ^ft,^ ■Das J3d -^j •xBtaa S ^ o O M^' Is. (N IN 00 Sh5S.^4i 00 t- t- 00 > OS 5 C3 !0 ^ Q S ^ 0) fe •s SI 1 °5 t 1 lit H H H-^ 220 AIRI'LAXE DESIGN AND CONSTRUCTION and the new final weight. — J-' = the new load per unit of wing surface. -p-^ = the new load per horsepower. S" = the maximum distance covered corresponding to :^ and to r . The examination of Table 7 enables making the following deductions: 1. Whatever be the type of machine it is convenient to fly at a reduced speed T"', because in that way the cruising radius increases. 2. All war airplanes are utiUzed very little as to useful load and consequently as to cruising radius. As column o// ^, shows, they could have a radius of action far superior if their enormous excess of power could be renounced. The gain is naturally stronger for the more light, quick air- planes, as for instance the scout machines, than for the heavier types. PART IV DESIGN OF THE AIRPLANE CHAPTER XVI MATERIALS The materials used in the construction of an airplane are most varied. The more or less suitable quality of material for aviation can be estimated by the knowledge of three elements: specific weight, ultimate strength and modulus of elasticity. Knowing these elements it is possible to calculate the coefficients _ ultimate strength in pounds per squa re inch ^ ~" specific weight in pounds per cubic inch and modulus of elasticity in pounds per square inch specific weight in pounds per cubic inch A, The coefficients Ai and A 2 are not plain numbers, but have a Unear dimension, and a very simple physical inter- pretation can be given to them; that is. Ax measures the length in inches which, for instance, a wire of constant section of a certain material should have in order to break under the action of its own weight; A 2 instead, measures the length in inches which a wire (also of constant section) of the material should have in order that its weight be capable of producing an elongation of 100 per cent. The higher the coefficients Ai and A 2, the more suitable is a material for aviation. It may be that two materials have equal coefficients A 1 and A 2, but different specific weights. In that case the 221 222 AIRPLANE DESIGN AND CONSTRUCTION material of lower specific weight is preferable when there are no restrictions as to space; instead, preference will be given to the material of higher specific weight when space is limited. This because of structural reasons, or in order to decrease head resistance. In all of the following tables whenever possible, we shall give the values of specific weight and coefficients Ai and A,. We shall briefly review the principal materials, grouping them into the following broad categories: A. Iron, steel and their manufactured products. B. Various metals. C. Wood and veneers. D. Various materials (fabrics, rubbers, glues, varnishes, etc.). A. IRON, STEEL AND THEIR COMMON FORM AS USED IN AVIATION Iron and steel are employed in various forms and for various uses; for forged or stamped pieces, in rolled form for bolts, in sheets for fittings, plates, joints, in tinned or leaded sheets for tanks, etc. Fio. 141. In Table 7 are shown the best characteristics required •of a given steel according to the use for which it is intended. Steel wires and cables are of enormous use in the con- struction of the airplane. Tables 8 and 9 give respectively tables of standardized wires and cables. MATERIALS 223 o c OS %1 22 o o O O R8 RS : II U : CO r-t CO -4 ■o o t^ OS o d o o n o_o_ o o lO 1-0 sggs o o o o lis U5 lO o o d d siii o o o o d d lO iO lO o •O lO lO o O O .-H o 2 I « " » sec o J2 m S t t M g ^ m 03 C3 o3 o ffl o " " 5 I I saai3jo^ joj 224 AIRPLANE DESIGN AND CONSTRUCTION Table 8.- -SizEs, Wekjhts and Physical Properties of Steel Wire English Units Diameter.! l^^% ft.-lb. Torsion test* Bend test Breaking strength Tensile strength Number of turns (minimum) Number of bends (minimum) Pounds (minimum) Lb. per sq. in. (minimum) 6 0.162 7.010 16 5 4500 219,000 7 0.144 5.560 19 6 3700 229,000 8 0.129 4.400 21 8 3000 233,000 9 0.114 3.500 23 9 2500 244,000 10 0.102 2.770 26 11 2000 245,000 11 0.091 2.200 30 14 1620 249,000 12 0.081 1.744 33 17 1300 252,000 13 0.072 1.383 37 21 1040 255,000 14 0.064 1.097 42 25 830 258,000 15 0.057 870 47 29 660 259,000 16 0.051 0.690 53 34 540 264,000 17 0.045 0.547 60 42 425 267,000 18 0.040 0.434 67 52 340 270,000 19 0.036 0.344 75 70 280 275,000 20 0.032 0.273 84 85 225 280,000 21 0.028 0.216 96 105 175 284,000 Metric Units American Diameter, Weight per 100 m., kg. Torsion test* Bend test Breaking strength Tensile strength wire gage Number of turns (minimum) Number of bends (minimum) Kilograms (minimum) Kg. per sq. mm. (minimum) 6 4.115 10.440 16 5 2041.0 154.0 7 3.665 8.280 19 6 1678.0 161.1 8 3.264 6.550 21 8 1361.0 163.8 9 2.906 5.210 23 9 1134.0 171.6 10 2.588 4.120 26 11 907.0 172.2 11 2.305 3.280 30 14 735.0 175.0 12 2.053 2.597 33 17 590.0 177.2 13 1.828 2.060 37 21 472.0 179.4 14 1.628 1.635 42 25 376.5 181.5 15 1.450 1.295 47 29 299.4 182.1 16 1.291 1.028 53 34 244.9 185.6 17 1.150 0.814 60 42 192.8 187.7 18 1.024 0.646 67 52 154.2 189.8 19 0.912 0.512 75 70 127.0 193.4 20 0.813 0.406 84 85 102.1 196.8 21 0.724 0.322 96 105 79.4 199.6 'The minimum number of complete turns which a wire must withstand may be computed from the formula: 2.7 _ 68^6 diameter in inches dia. in millimeters MATERIALS 225 T.^BLE 9. — Weights, Sizes and Strength of 7 X 19 Flexible Cable English units Metric units Diameter, in. Approxi- mate weight, lb. per 100 ft. Breaking strength, lb. (minimum) Diameter, mm. Approxi- mate weight, kg. per 100 m. Breaking strength kg. (nainimum) 0.375 (%; 26.45 14,400 9.525 39.36 6,532 0.344 (IK2) 22.53 12,500 8.731 33.53 5,670 0.312 (Ke) 17.71 9,800 7.938 26,35 4,445 0.281 ir32) 14.56 8,000 7.144 21.67 3,629 0.250 (K) 12.00 7,000 6.350 17.86 3,175 0.218 (%2) 9.50 5,600 5.556 14.14 2,540 0.187(^6) 6.47 4,200 4.763 9.63 1,905 0.156 (M2) 4.44 2,800 3.969 6.61 1,270 0.125 (Vs) 2.88 2,000 3.175 4.29 907 The formation of cables is shown in Fig. 141. The cable is made of 7 strands of 19 wires each; the figure shows how these strands are formed. The smaller diameters are extra- flexible so that they can be used as control wires as they well adapt themselves in pulleys. Recently, steel streamline wires have been introduced to replace cables, in order to obtain a better air penetration. Fig. 142 shows the section of one nf su^h wires. Their use Fig. 142. has not yet greatly broadened, especially because their manufacture has until now not become generalized. It is foreseen though, that the system will rapidly become popular. We shall now take up the attachments of wires and cables. The attachment most commonly used for wu-es, is the so-called "eye" (Fig. 143). It is an easy attachment to make, but it reduces, however, the total resistance of the wire by 20 to 30 per cent, depending on the diameter of the wire. 226 AIRPLANE DESIGN AND CONSTRUCTION Wires with larger threaded ends (called ''tie rods") (Fig. 144), are becoming of general use. A very good attachment can be obtained by covering the bent wire with brass wire and soldering the whole with tin (Fig. 145); in this way an attachment is obtained which gives Fi.;. H;J Fig. 144. Fig. 145. Fig. 146. 100 per cent, of the wire resistance. The soldering is made with tin in order to avoid the annealing of the wire. The best attachment of cables is made by so-called spUcing after bending it around a thimble (Fig. 146), which is made either of stamped sheets or of aluminum (Fig. 147). MATERIALS 227 Steel is also much used in tube form, either seamless, cold rolled, or welded. Table 10 gives the characteristics of the steel of various tube types. ^ 4M Fig. 148. Tables 11 and 12 give the standard measurements of round tubes with the values of weight in pounds per foot and the values of the polar moment of inertia in in.^ 228 AIRPLANE DESIGN AND CONSTRUCTION Steel tubing having a special profile formed so as to give a minimum head resistance is also greatly used for inter- plane struts as well as for all other parts which must necessarily be exposed to the relative wind. The best profile (that is, the profile which unites the best requisites of mechanical resistance, lightness and air penetration) is given in Fig. 148 which also shows how it is drawn, and gives the formulae for obtaining the peri- meter, the area, and the moments of inertia I:, and /„ about the two principal axes as function of the smaller diameter d and thickness t. Tables 13 and 14 give all the above mentioned values, and furthermore the weight per linear foot for the more commonly used dimensions. A greatly used fitting in aeronautical construction is the turnhuckle, which is designed to give the necessary tension to strengthening or stiffening wires and cables. A turnbuckle is made of a central barrel into which two shanks are screwed with inverse thread; the shanks have either eye or fork ends* thus we have three classes of turnbuckles : Double eye end turnbuckle (Fig. 149a) Eye and fork end turnbuckle (Fig. 1496) Double fork end turnbuckle (Fig. 149c) MATERIALS 229 o - U5 O O lO lO lO o o o o o o o o o |§3 £-5 s iC o o o ssss d d d d 09 O s i o o o p M lo in lo d d d d o o o o o i« lo ic in m -< — ' C^ IN M ddd d d X| B 0) a as ' o o S 8 «^i^i iS.g'.E " S S S I t; -2 filh 230 AIRPLANE DESIGN AND CONSTRUCTION li 5 M 15 s o «? O r~ o> ^ ^ rt 05 ^ o 0! . o e^ ■* t^ o -ll UO 3 m ^ rr ^ -V g o —1 a t •^ •o S i, Ilia ^ ^ nn o> o> Ui Si ^ 6 o •-1 M w N d o o o o o O o o o o o o o O n ^ N 00 « CO (N ^ O) O M ^ 00 lO CO 05 o f ■ i c^ 1 d d d d d d d d d d d d d d d d H eg 1, s s g ;?: ^ •* t:; S 00 ?^ s S f? z o IM •>!< lU d o o o o o O o o o o o o o o o o 5 c 00 i o to i t^ c^ ™ M o X s lo IN o S o ?n o 6 S rt c^ i. d o o o o o o O o o o o o o '-' -■ 66666666 d 6 S 00 S 2 00 s m 0 CO 03 SJM V* \,ec ^(N v» v-ji sjjo \Hi «\ B\ l-\ - J\ r^ «S -JS «\ M\ l.\ ,^ rt^„rtrt,-i,-,^C^(M X 1 232 AIRPLANE DESIGN AND CONSTRUCTION 25 f! II ^ .£ o o o o O IM t->. 00 lO iiiii 0437 0608 0808 1032 CM CO »0 O O »0 CO 1-H CM CM OS CM CO X r-. o o o o o o c o o o o o o o o o o o g (N (N OC OS lllgl t^ OS CO CO IC CO o »c o O O CO OS -^ o o o o o o o o ^ O O -H ^ ^ CM CM CM CO 1-^ 00134 00170 00213 00239 iC p. O CO t- o o o o o o o c o 0465 0617 0788 0890 1032 o o c o o o o o o o o o o o o o o o Id 5^' 0560 0710 0890 1245 (N CO Tf lO t^ t^ OS CO CO ffi S CO ?i CO o o o o o o o o o o o »^ ^ ^ CM CO CO Tt* Oi -^ O Oi rH (N CO TJH (N CO I- CM 00 00 O CO -^ O Tt* t- O CO o o o o o o o o ^ o O ^ ^ ^ ^ ^ CM CM is (M t^ 00 b- SgSii »C CO o o ■<* CD l^ CO 00 CM CD rH o o o o o o o o o O O O ^ r-* O ^ ^ CM ^ CM CM CO CO CO < a iiii l-H ,-H tH CM CO .-H CM CO ■* mil o o c o o o o o o O O O O o o o o o S S 5§2gg oo o ^ o o iC o o o o »o c o o »0 lO CO 05 CO Tf* Id t> l> lO OS 00 CM CD 00 '-I lO O §§ii OS CD CO oo "^ rH CM CO CO ■* fl c ssss S 05 s s s CM CM iM CM sssss 0 CO l> C30 10 (N 10 »C 00 IC t^ CD 00 -H 00 CO CO t^ 00 CI - a 0.0803 0.1065 0.1364 0.1542 0.1793 0.2045 d d d d d CO CO ^ ^ 5 § 6 6 6 6 6 6 "b ^^g^SS §Si2§g§ S^g^t^g CO '^ lO «0 1> 00 t^ "H CO -* CO 10 i> th 10 ^ ^ ^ C^ — 1 <35 t^ 01 10 00 CO t^ <-< (N (N CO CO Tji CO -* CD (N 00 00 (N t^ CO CO CO CJ CO CO ■«* -"^^ "5 I' S52§§?S g?2?5§S§ 05 10 r^ 00 ^ ^ t^ iiliis _2 (M C<) 05 05 05 05 000000 CO CO CO CD CO CO 00 00 06 00 00 06 05 05 05 C5 05 05 1> t^ t^ l> 1> I> s s ^ s s § 05 05 05 Cft 05 OJ s s ^ s s s ^ ^ ^ S S ^ a a 05 10 CO -* 05 »c § ^ d 065 083 094 109 125 134 ||§§gg 000000 c 000000 B B ic ic ■* 10 'H 00 t^ t^ 10 lO .-< ^ (N (N (M CO --1 C^ (N IM CO CO C S aSS « M a-- a ■S A -< II s W5 h3 Per ce) Elong tion (min. In 2" 50.8 m : : " « 2 g 2 o >o ^ •So as 3S CO s'e J, . . . 1-1 o .2 1^ 88 5 H o o t- ^- r- m t^ O Tt< 'J- to -H C ° IT a hi d d xj lo 00 w ^ a .f — lo ■S.S 1 ws^ l.a lllllll o rj tj -." r-" (N (N lo o a r-00- H 3 ■" (M Ol lO U5 •!( t^ : -< ^ '. 6. 6 ■ Z Z 1 s : a a B •*- J .2 ■ 3 ; S £ : iJ ^ § ss >^ '^ "^ S^ >^ >; h5 g a o 6 ^ ^ ^ g Izz-s « " ■^ § S E S S E 2 IS S^S 3 1 1 E S E E £ si d 3 3 3 3 f, S ^ -- -= — 5 V . " e. < i < < -< 1 236 AIRPLANE DESIGN AND CONSTRUCTION Table 16. — Properties o Strength Values at 15 Per Cent. Mo Specific gravity based on volume and weight when oven dry Specific weight at 15 per cent. moisture Static Common and botanical names Fiber stress at elastic limit Modulus of rupture Aver- age Mini- mum per- mitted Lb. per cu. ft. Lb. per sq. in. Lb. per sq. in. (/6) Ash (commercial white) (Fraxinus Ameri- cana; Fraxinus LanceoLata; Fraxinus Quad- 0.62 0.5,3 0.40 0.66 0.67 0.53 0.43 0.66 0.53 0.81 0.54 0.50 0.66 0.72 0.42 0.56 0.56 0.48 0.36 0.60 0.61 0.48 0.39 0.60 0.48 0.73 0.50 0.46 0.60 0.65 0.38 0.52 40 35 25 41 43 35 28 44 34 50 36 34 42 46 28 38 7700 5800 4700 7400 8400 7300 4500 6700 6700 8900 7000 7100 8100 6700 4800 7900 12700 10500 7200 12600 13500 10600 7000 12500 10400 16300 10000 10400 12900 12000 7500 11900 Ash (black) (Fraxinus Nigra) Bass wood {Tillia Americana) Birch (Betula Lutea, Lenta) Cherry (black) (Prunus Serotina) Gum (red) {Liquidnmhar Styraciflua) Hickory (true hickories) {Higoria Glabra, Mahogany (true) (Swietenia Mahagoni) Mahogany (African) (Khayn Senegalensis) . Oak (commercial white) (Quercus Alba Mac- Poplar (yellow) '(Liriodendriim Tulipifera) . . timbers for aviation uses; they must be free from disease, homogeneous, without knots and burly grain, and above all they must be thoroughly dry. Artificial seasoning does not decrease the physical qualities of wood, but, on the con- trary, it improves them if such seasoning is conducted at a temperature not above 100°r. and is done with proper precautions. It is very important, especially for the long pieces, as for instance the beams, that the fiber be parallel to the axis of the piece, otherwise the resistance is decreased. Furthermore, it is important to select by numerous laboratory tests the quality of the wood to be used, because between one stock of wood and another, great differences may usually be found. As an example of the importance which the value of the density of wood has upon the major or minor convenience of its use in the manufacture of a certain part, let us suppose that we design the section of a wing beam which has to MATERIALS 237 p Various Hard Woods isture, for Use in Airplane Design bending Compress- ion parallel to grain max. crushing strength Compress- ion perpen- dicular to grain fiber stress at elastic limit Shearing strength parallel to grain Hardness side load required to imbed 0.444 in. ball to one-half its diameter /6 /6 fb Modulus of elasticity Work to maxi- mum load E S.W. S.W. 1000 lb. In. lb. Lb. per Lb. per Lb. per persq. per cu. sq. in. sq. in. sq. in. Lb. in. (E) '- (/c) (/O 1500 14.2 6000 1300 1750 1150 0.472 0.138 317.5 37500 1400 14.1 4900 800 1350 740 0.467|0. 1281300. 40000 1300 6.4 38C0 400 880 340 0.503|0.122'288. 52000 1500 13.3 5900 1100 1700 1060 0.468 0.135 307.4 36585 1800 17.6 6600 1060 1620 1070 0.489 0.120 314 41860 1400 12.0 5800 700 1500 830 0.548 0.141302.8 40000 1200 7.3 3800 400 800 380 0.527 0.114 250.0 42855 1400 19.3 5800 1200 1650 1200 0.464 0.132 284.1 31818 1400 11.0 4900 700 1500 650 0.471 0.144 305.9 41777 J 1 1900 28.0 7300 1800 1800 0.448 0.110 326.0 38000 1300 9.1 5500 1000 1420 "860 0.5500.1421277.8 36111 1400 10.3 5100 900 1270 730 0.489,0.102 305.9 41777 1600 12.9 6500 1200 1990 1200 0.504 0.155 307.1 38095 1400 12.7 5900 1300 1760 1270 0.490 0.147 260.9 30435 1300 6.2 4100 400 900 370 0.546 0.120 267.9 46430 1500 13.1 6100 1000 1300 950 0.513 0.110 313.2 39474 resist, for example, to a bending moment of 20,000 Ib.-inch; and let us suppose that the maximum space which it is possible to occupy with this section is that of a rectangle having a base equal to 2.2" and a height equal to 2.8". We shall make a comparison between the use of spruce and the use of douglas fir, for which the value of coefficient Ai is about the same. Table 17 gives a modu- lus of rupture of 7900 lb. per sq. in. for the spruce with a weight per cu. ft. of 27 lb.; that is, 0.0156 lb. per cu. inch. Since the maximum bending moment is equal to 20,000 Ib.- inch, the section modulus of the section will equal „. 20,000 ^ Ko • u^ ^^^ = -7-;9oo = '-^^ ^"^^ For fir, instead, we shall have W, = 20,000 9,700 with a density of 0.0197 lb. cu. in = 2.06 in.--" 238 AIRPLANE DESIGN AND CONSTRUCTION Table 17.— Properties Strength Values at 15 Per Cent. M Specific gravity based on volume and weight when oven dry Specific weight at 15 per cent. moisture Static Common and botanical names Fiber stress at elastic limit 1 Modulus of rupture Mini- Aver- mum age per- mitted Lb. per cu. ft. Lb. per sq. in. Lb. per Cedar (incense) {Libocedrua Decurrens) Cedar (Port Orford) (Chamaecyparis Law- soniana) Cedar (western red) (Thuja Plicata) Cedar (white northern) {Thuja Occidentalis) Fir (Douglas) 0.36 0.47 0.34 0.32 0.52 0.39 0.45 0.39 0.41 0.47 0.32 0.42 0.31 0.29 0.47 0.36 0.40 0.36 o;42 26 31 23 22 34 27 29 27 27 31 4900 6200 4200 4200 0800 5300 5100 5100 5100 5100 7100 10300 6400 5800 9700 7400 7800 7400 7900 8800 Pine (western white) {Pinus Monticola) Pine (white) (Pinus fitrobiis) .... Spruce (red, white, Sitka) (Picea Rubens; Canaden^i.t Sitrhevxix) Cypress (bald) (T-ixodium Distirhitm) Let us call x the thickness of the flange (Fig. 150a). Making the thickness of the web equal to 0.8x, the section modulus and the area of the section will be respectively W = H [2.2" X 2.8''2 - (2.2 - 0.8a;)(3 - 2xy] cu. in. A = 2.2'' X 2.8 - (2.2 - 0.8a:) X (3 - 2x) sq. in. For spruce W = W 2.53 in.= from which we have X = 0.9" A = 4.37 sq. For fir we shall have analogously X = 0.65" A = 3.29 sq. in. Consequently, the spruce beam will weigh 4.37 X 0.0156 = 0.069 lb. per in. of length, while the fir beam will weigh 3.29 X 0.0197 = 0.064 lb. Supposing then for instance, that the total length of the beams be 150 ft., i.e., 1800 in., the weight of the spruce beams would be 1800 X 0.069 = 124 lb., while the weight of the fir beams would be 1800 X MATERIALS 239 OF Vahious Conifers oisture, or Use in Airplane Desifin bending Modulus of elasticity Work to maxi- mum load ion parallel to grain max. crushing strength Compress- ion perpen- dicular to grain fiber stress at elastic limit Shearing strength parallel to grain Hardness side load required to imbed 0.444 in. ball to one-half its diameter S.W. Ai = E S.W. 1000 lb. In. lb. Lb. per Lb. per sq. in. Lb. per per sq. per cu. sq. in. sq. in. Lb. in. {E) in. (/O (A) 1000 6.0 4300 600 850 430 0.606 0.120 273.1 38464 1 1700 9.7 5300 700 1160 580 0.51310. 112:332. 3 54840 1000 5.5 4000 400 790 300 0.625:0.123:278.3 43478 750 5.1 3400 350 800 300 0.586 0.138 263.6 34091 1780 7.2 6000 750 1020 580 0.619 0.104 285.3 52353 1100 5.0 4300 540 950 410 0,58110.128 274.1 40740 1400 6.9 4800 480 670 360 0.615 0.086 269.0 48276 1200 6.1 4500 530 850 380 0.608 0.115 274.1 44444 1 1 1300 7.4 4300 500 920 430 0.5440.117292.648148 1300 6.8 5400 670 940 460 0.012 0.107,284.0 41936 0.064 = 115 lb.; that is, a gain of 9 lb., more than 7 per cent., would be obtained. If we use elm, which has the same coefficient A i as the preceding woods, but a resistance of 12,500 lb. per sq. in. and a weight per cu. in. of 0.0255 lb. we would have (Fig. 1596) X = 0.48" A = 2.44 sq. in. with a weight per inch of 2.44 X 0.0255 = 0.062 and for 1800 in., a weight of 112 lb.; that is, a gain of about 10 per cent, over the spruce. Let us now examine an inverse case, a case in which the piece is loaded only to compression and no limit fixed upon the space allowed its section; this for instance is the case of fuselage longerons. Then the product E X I (elastic mod- ulus X moment of inertia) , is of interest for the resistance of the piece. Let us suppose that the longeron has a square section of side X. We then have 1 . / = 12 240 AIRPLANE DESIGN AND CONSTRUCTION Supposing that we have two kinds of wood of modulus El and Eo and specific weight Wi and Wo respectively; and suppose that coefficient A2 be the same for both kinds, that is DOUGLAS FIR ELM Fig. 150. Let US call /i and 1 2 the moments of inertia which the section must have respectively, according as to whether it is made of one or the other quality of wood. If we wish the piece to have the same resistance in both cases then Eili = E2I2 MATERIALS 241 that is from which W, xi' = W2 X2' (1) The weights per Hnear inch evidently will be in both cases TFi X xi^ and W2 X X2' and their ratio w will be Wi X xi^ w = But from (1) consequently W2 X x^ TTi X X,'- ^ xl Wi X X22 xi^ that is, the piece having the greater section will weigh less, therefore it is convenient to use the material of smaller specific weight. Let us now consider the veneers, which have become of very great importance in the construction of airplanes. Wood is not, of course, homogeneous in all directions, as for instance, a metal from the foundry would be; its struc- ture is of longitudinal fibers so that its mechanical qualities change radically according to whether the direction of the fiber or the direction perpendicular to the fiber is considered. Thus, for instance, the resistance to tension parallel to the fiber can be as much as 20 times that perpendicular to the fiber, and the elastic modulus can be from 15 to 20 times higher. Yice versa, for shear stresses we have the reverse phenomenon; that is, the resistance to shearing in a direc- tion perpendicular to the fiber is much greater than in a parallel direction to the fiber. Now the aim in using veneer is exactly to obtain a material which is nearly homogeneous in two directions, parallel and perpendicular to the fiber. Veneer is made by glueing together three or a greater odd number of thin pUes of wood, disposed so that the fibers 242 AIRPLANE DESIGN AND CONSTRUCTION cross each other (Fig. 151). It is necessary that the num- ber of plies be odd and that the external phes or faces have the same thickness and be of the same quality of wood, so that they may all be influenced in the same way by humidity, that is, giving perfect symmetrical deformations, thus avoiding the deformation of the veneer as a whole. It is advisable to control the humidity of the plies during the manufacturing process, so that the finished panels may have from 10 to 15 per cent, of humidity. If we wish to have the greatest possible homogeneity in both directions, it is advisable to increase the number of plies to the ut- most, decreasing their thickness; this also makes the joining more easy by means of screws or nails, because the veneer offers a much better hold. Considerations analogous to those given for the density of wood, lead to the conclusion that, wishing to attain a better resistance in bending, it is preferable to use plies of low density for the core. In fact, the weight being the same, the thickness of the panels will be inversely proportional to the density; but the moment of inertia, and consequently the resistance to column loads are pro- portional to the cube of the thickness; we see, therefore, the great advantage of having the core made of light thick material. Light material would also be convenient for the faces, but they must also satisfy the condition of not being too soft, in order to withstand the wear due to external causes. In Tables 18 and 19 we have gathered some of the tests MATERIALS 243 IM -S. 1^' pi CO lO t^ b- §§ ogooooooooo a: co^ co^ c^^ i> CO C5_ co_ (M_ CO o lo ai co t^ Tt o o o o o O t^ (M 00 t^ lO Ci 0-- »o o im" .-T rt" r-T ci o o o o CD CO (N 05 I> Oi .-H CO -§1 CD CD CD CD CO -* c^ ^ TjH 00 (-^ io~ ccT CO (m' lo 00 t--' o o o c; CO t> O t^ 05 0-. (M X' GO t^ 0-. '-H(N(MC0OC0r-»t^00C0t---*05CDC0t^ OiOOOOQOCOOSrtOOOQOC-.OCCOOO ■^iCiTfiTti'^Tt'CDiOiCiO OOOOOOOOOOOOOOC « 5t) 03 C O ell on 73 eS >o^^ 3 3 3 iCQOOOOOQWHOOO 244 AIRPLANE DESIGN AND CONSTRUCTION g_-l £.5 C-. (N iC T*< C5 t^ 10 05 00 IC 00 lO ■2 &S& °? C5'He«5-^cOi-i-«noo(M.-icr. rtOOco-<** 00 -w (S^S 2 V So 11 0< •3.S £•- S 5S S S g S S 2 S ^ ^ ^ S § ?5 ^ S := g !• M i^ "1. "^ ^„ "^ °^ '^^ '^ ""1 '^ '^ *^_ "?. "1. '^_ '^ ^ £2 h S Jj3 ""If SSS :§ -.S-^^^^:^^^^^ « D.° aS-2 li oooooooooooooooo J3 o^ g t> £•= M ° ^- ^^ t^ ^- ^ ^- "i ^- ^- °°^ ^^ °l ^^ ^^ "^^ i^ 00 (2^2 TjT -t^jT »o |^f lo" cc lo CD CO -^^ cc TtT co lO 21 ti oooooooooooooooo C<>OOt^t^O00O00CC-<^C5t^OC0»C (22 05 00 O CO O CC --I — ■* t^ O CO t^^ O^ 0_ 05_ H 3^ CO CO C> lO O CO 00 O' U5 CCT lO" t^" •*" lO" CO CO tMi I.S oooooooooooooooo 00CD'*O^Tt<'M^t^--r-(MiO00Tft^ .2 aa ^„ oodcoococr. Tfcoocoi^c: lococoi^ c 5 .^12 it ^^ ^ im' c" OO" OO" !>"'-<" of H - £ •«? S3 8^ c^t^C5t^t^'*a>oeo»C(N'*t^"*(N'-i « 2 S « daioJco-*iocoioco-*»o-*Tf ^J-J any ( any ( any ( (soft) (suga omme omme vhite) (yelk i 1^ § ^^g^-^^3-^53 § s a 2 1 £ a oi 5 C 1 . c 1 ^1 c 0. c "c ^ "•1 p: C a: ''I "5 1 MA TERIALS 245 Table 20. — Haskelite Designkng Table Not Sanded Haskelite Research Laboratories- FOH Three-ply Panels -Report No. 109 Nominal thickness of panel Faces Thickness and kind of wood Thickness and kind of wood ViQ in. Spanish cedar. . Spanish cedar . . Spanish cedar. . Spanish cedar. . IVIex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Maple Maple Maple Maple Birch Birch Birch Birch Vio in. Spanish cedar . . Spanish cedar . . . Spanish cedar. . . Spanish cedar. . . Spanish cedar. . . Spanish cedar. . . Mex. mahogany. Mex. mahogany. Mex. mahogany. Mex. mahogany. Mex. mahogany. Mex. mahogany. Maple Maple Maple Maple Maple Maple Birch Birch Birch Birch Birch Birch '4o m. Spanish cedar. . . Mex. mahoganj- Maple Birch Spanish cedar. . . Mex. mahogany. Maple Birch Spanish cedar , . . Mex. mahogany. Maple Birch Spanish cedar . . . Mex. mahogany. Maple Birch Approxi- mate weight Lb. per 100 sq. ft. -Approximate strength, lb. per in. of width Along I Along face I core grain j grain >20 in- Bassvvood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Spanish cedar. . . Poplar Mex. mahogany. Maple Birch Basswood Spanish cedar . . . Poplar Mex. mahogany. Maple Pirch Basswood Spanish cedar . . . Poplar Mex. mahogany. Maple Birch 21 22 24 25 22 23 20 26 27 28 31 31 28 28 31 31 26 27 27 28 33 33 27 28 28 29 34 35 32 33 33 34 39 40 33 33 34 35 40 40 400 400 400 400 550 550 550 550 600 600 600 600 820 820 820 820 200 270 300 410 200 270 300 410 200 270 300 410 200 270 300 410 400 500 400 400 400 650 400 550 400 600 400 820 550 500 550 400 550 650 550 550 550 600 550 820 600 500 600 400 600 650 600 550 600 600 600 820 820 500 820 400 820 650 820 550 820 600 820 820 246 AIRPLANE DESIGN AND CONSTRUCTION Table 21. — Haskelite Designing Table for Three-ply Panels — Not Sanded Haskelite Research Laboratories — Report No. 109 Nominal thickness of panel 0.121 in. Thickness and kind of wood Ms in. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Mex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Maple Maple Maple Maple Maple Maple Birch Birch Birch Birch Birch Birch Core Approxi- mate weight Approximate strength, lb. per in. of width Thickness and kind of wood Ho in. Basswood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Spanish cedar. . Poplar Mex. mahogany Maple Birch Basswood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Spanish cedar . . Poplar Mex. mahogany Maple Birch Lb. per 100 sq. ft. Along face grain 31 570 32 570 32 570 33 570 38 570 1 38 570 ! 33 790 33 790 34 790 35 790 40 790 40 790 40 860 41 860 41 860 42 860 47 860 48 860 40 1180 41 1180 42 1180 42 1180 47 1180 48 1180 Along core grain 500 400 650 550 600 820 500 400 650 550 600 820 500 400 650 550 600 820 500 400 650 550 600 820 made at the ''Forest Product Laboratory;" the veneers to which these tests refer were all three pUes of the same thickness and the grain of successive pUes was at right angles. All material was rotary cut. Perkins' glue was used throughout. Eight thicknesses of plies, from %o" to %" were tested. In Tables 20 to 29 are quoted the characteristics of three-ply panels of the Haskehte Mfg. Corp., Grand Rapids, Michigan. MATERIALS 247 Table 22. — Haskelite Designing Table for Three-ply Panels — Not Sanded Haskehte Research Laboratories — Report No. 109 Nominal thickness Faces 1 Core Approxi- mate weight Approximate strength, lb. per in. of width of panel Thickness and kind of wood Thickness and kind of wood Lb. per 100 sq. ft. Along face grain Along core grain 3^8 in. Spanish cedar. . . . Spanish cedar Spanish cedar .... Spanish cedar .... Spanish cedar .... Spanish cedar Mex. mahogany . . Mex. mahogany. . Mex. mahogany . . Mex. mahogany . . Mex. mahogany . . Mex. mahogany . . He in. Basswood Spanish cedar Poplar 33 34 35 36 42 43 35 36 37 38 44 45 42 43 44 45 51 52 43 44 44 46 52 52 570 570 570 570 570 570 790 790 790 790 790 790 860 860 860 860 860 860 1180 1180 1180 1180 1180 1180 620 500 810 Mex. mahogany.. 690 750 Birch 1030 Basswood Spanish cedar Poplar 620 500 810 Mex. mahogany . . Maple 690 750 0.133 in. Birch 1030 Basswood Spanish cedar. . . . Poplar 620 Maple Maple Maple Maple 500 810 Mex. mahogany.. Maple Birch 690 750 Maple Birch 1030 Basswood Spanish cedar Poplar 620 Birch Birch 500 810 Birch Birch Mex. mahogany . . Maple 690 750 Birch Birch . . . 1030 One of the best veneers for aviation is one obtained with spruce plies; this is easily understood if we consider the low density of spruce. D. VARIOUS MATERULS (a) Fabrics. — Fabrics used for covering airplane wings are generally of Unen or cotton, though sometimes of silk. The fabric is characterized by its resistance to tension and 248 MHI'l.ASK DESICX AS I) COS STRICT ION Table 23. — Haskelite Designing Table for Three-ply Panels — Not Sandeu Haskolito Research Laboratories — Report No. 109 Cor Nominal thickness of panel Approxi- Approximate mate strength, lb. per weight Thickness and kind of wood Thickness and kind of wood Lb. per 100 sq. ft Along face grain Along core grain 154 in. Ks in- Spanish cedar . . . . Spanish cedar. . . . Spanish cedar. . . . Spanish cedar . . . . Spani.sli cedar. . . Spanish cedar. . . . Spanish cedar . . . . Mex. mahogany. Mex. mahogany . Mex. mahogany . Mex. mahogany . Mex. mahogany. Mex. mahogany. Mex. mahogany . Maple Maple Maple Maple Maple Maple 712 in- Basswood Redwood Spanisli cedar . . . Poplar Mex. mahogany. Maple Birch Basswood Redwood Spanish cedar. . . Poplar Mex. mahogany. Maple Birch Basswood Redwood Spanish cedar . . . Poplar Mex. mahogany. Maple Maple j Birch Birch Basswood Birch Redwood Birch I Spanish cedar. . Birch I Poplar Birch Mex. mahogany Birch Maple Birch Birch 38 38 39 40 41 50 50 40 40 41 42 43 51 52 47 47 48 49 50 58 59 47 47 49 49 51 59 GO 570 570 570 570 570 570 570 790 790 790 790 790 790 790 860 860 860 860 860 860 860 1180 1180 1180 1180 1180 1180 1180 830 710 670 1080 920 1000 1380 830 710 670 1080 920 1000 1380 830 710 670 1080 920 1000 1380 830 710 670 1080 920 1000 1380 MATERIAL,^ 249 Table 24. — Haskeltte Designing Table koh Three-ply Not Sanded Haskelite Research Laboratories — Report No. 109 Panels- Nominal thickness of panel Faces Thickness and kind of wood Thickness and kind of wood Approxi- mate weight Lb. per . 100 sq. ft. Approximate strength, lb. per in. of width Along face grain Along core grain 0.183 in. }4o in. Spanish cedar. . . . Spanish cedar . . . . Spanish cedar .... Spanish cedar .... Spanish cedar .... Spanish cedar .... Spanish cedar .... Mex. mahogany. . Mex. mahogany. . Mex. mahogany . . Mex. mahogany. . Mex. mahogany. . Mex. mahogany . . Mex. mahogany. . Maple Maple • Maple Maple Maple Maple Maple Birch Birch Birch Birch Birch Birch Birch Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch 800 800 800 800 800 800 800 1100 1100 1100 1100 1100 1100 1100 1200 1200 1200 1200 1200 1200 1200 1650 1650 1650 1650 1650 1650 1650 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 250 AIRPLANE DESIGN AND CONSTRUCTION Table 25. — Haskelite Desigxing Table fok Three-ply Not Sanded Haskelite Research Laboratories — Report No. 109 Panels — Nominal thickness of panel Core Thickness and kind of wood Thickness and kind of wood Approxi- mate weight Lb. per 100 sq. ft. Approximate strength, lb. per in. of width Along face grain Along core grain 0.208 in. Vie in. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Max. mahogany. . Mex. mahogany. , Max. mahogany . . Mex. mahogany . . Max. mahogany . . Mex. mahogany . . Mex. mahogany. . Maple Maple Maple Maple Maple Maple Maple 3- {2 in. Basswood Redwood Spanish cedar .... Poplar Mex. mahogany. . Maple Birch Basswood Redwood Spanish cedar. . . . Poplar Mex. mahogany. . Maple Birch Basswood Redwood Spanish cedar. . . . Poplar Mex. mahogany. . Maple Birch Birch ' Basswood , Birch i Redwood Birch i Spanish cedar . . Birch Poplar. Birch Birch Birch Mex. mahogany. Maple Birch 1000 1000 1000 1000 1000 1000 1000 1370 1370 1370 1370 1370 1370 1370 1500 1500 1500 1500 1500 1500 1500 2060 2060 2060 2060 2060 2060 2060 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 MATERIALS 251 Table 26. — Haskelite Designing Table for Three-ply Panels- NoT Sanded Haskelite Research Laboratories — Report No. 109 Nominal thickness Faces Core Approxi- mate weight Approximate strength, lb. per in. of width of panel Thickness and kind of wood Thickness and kind of wood Lb. per 100 sq. ft. Along Along face core grain grain He in. Spanish cedar Spanish cedar .... Spanish cedar. . . . Spanish cedar. . . . Spanish cedar Spanish cedar. . . . Spanish cedar .... Mex. mahogany . . Mex. mahogany . . Mex. mahogany . . Mex. mahogany . . Mex. mahogany . . Mex. mahogany . . Mex. mahogany . . Maple }s in. Basswood Redwood Spanish cedar Poplar 58 58 61 62 64 76 77 62 62 64 65 67 79 80 74 74 76 77 79 92 93 75 75 77 78 80 93 94 1000 1000 1000 1000 1000 1000 1000 1370 1370 1370 1370 1370 1370 1370 1500 1500 1250 1060 1000 1620 Mex. mahogany . . Maple 1370 1500 Birch 2060 Basswood Redwood Spanish cedar Poplar 1250 1060 1000 1620 Mex. mahogany . . Maple 1370 1500 0.250 in. Birch 2060 Basswood Redwood Spanish cedar. . . . Poplar Mex. mahogany . . Maple 1250 Maple in^n Maple Maple Maple Maple Maple Birch 1500 1000 1500 : 1620 1500 1370 1500 i-'inn Birch 1500 2060 2060 2060 2060 2060 2060 2060 2060 Basswood Redwood Spanish cedar .... Poplar 1250 Birch 1060 Birch 1000 Birch 1620 Birch Mex. mahogany.. Maple 1370 Birch Birch 1500 Birch 2060 252 AIIiPLANE DESIGN AND CONSTRICTION Table 27. — Haskelite Designing Table for Thkee-ply Panels — Not Sanded Haskelite Research Laboratorios — Report No. 109 Nominal thickness of panel Faces Thickness and kind of wood Core Thickness and kind of wood Approximate strength, lb. per in. of width Along face grain Along core grain 0.250 in. H2 in. Spanish cedar. Spanish cedar. S|)anish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Mex. mahogany. . Mox. mahogany . Mex. mahogany. Mex. mahogany . Mex. mahogany . Mex. mahogany . Mex. mahogany . Maple Maple Maple Maple Maple Maple Maple Birch Birch Birch Birch Birch Birch Birch H2 in. Basswood Redwood Spanish cedar. . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar. . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar. . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch 1330 1330 1330 1330 1330 1330 1330 1830 1830 1830 1830 1830 1830 1830 2000 2000 2000 2000 2000 2000 2000 2750 2750 2750 2750 2750 2750 2750 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 830 710 670 1080 920 1000 1370 830 710 670 lOSO 920 1000 1370 MATERIALS 253 Table 28. — Haskelite Designing Table p-or Three-ply Panels — Not Sanded Haskelite Research Laboratories — Report No. 109 Nominal thickness P„e. Core Approxi- mate weight Approximate strength, lb. per in. of width of panel Thickness and kind of wood Thickness and kind of wood Lb. per 100 sq. ft. Along face grain Along core grain H. in. Spanish cedar .... Spanish cedar. . . . Spanish cedar. . . . Spanish cedar .... Spanish cedar Spanish cedar .... Spanish cedar .... Mex. mahogany. . Mex. mahogany . . Mex. mahogany . . Mex. mahogany. . Mex. mahogany . . Mex. mahogany. . Mex. mahogany. . H in. Basswood Redwood Spanish cedar Poplar. . . . 68 68 70 71 73 86 87 72 72 74 75 77 90 91 89 89 91 92 94 106 107 90 90 92 93 95 108 109 1330 1330 1330 1330 1330 1330 1330 1830 1830 1830 1830 1830 1830 1830 2000 2000 2000 2000 2000 2000 2000 2750 2750 2750 2750 2750 2750 2750 i 1250 1060 1000 1620 Mex. mahogany. . Maple 1370 1500 Birch . 2060 Basswood Redwood Spanish cedar. . . . Poplar . . . 1250 1060 1000 1620 Mex. mahogany . . Maple 1370 1500 Birch 2060 0.291 in. Basswood Redwood Spanish cedar .... Poplar Mex. mahogany. . Maple Birch Basswood Redwood Spanish cedar. . . . Poplar Mex. mahogany. . Maple Birch 1250 1060 Maple Maple 1000 1620 Maple Maple 1370 1500 Maple. 2060 Birch 1250 Birch Birch 1060 1000 Birch Birch 1620 1370 Birch 1 Birch 1500 20(0 254 AIRPLANE DESIGN AND CONSTRUCTION Table 29. — IIaskkliti: Designing Table for Three-ply Panels — Not Sanded Haskelite Research Laboratories — Rejxtrt No. 109 Nominal thickness of panel Faces Thickness and kind of wood Thickness and kind of wood Appi Approximate strength, lb. per weight in. of width Lb. per 100 sq. ft. Along face grain Along core grain 0.375 in. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Spanish cedar. Mex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Mex. mahogany Maple Maple Maple Maple Maple Maple Maple Birch Birch Birch Birch Birch Birch Birch H iri. Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar. . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch Basswood Redwood Spanish cedar . . Poplar Mex. mahogany Maple Birch 87 87 89 90 92 104 105 93 93 95 96 98 111 112 118 118 120 121 123 136 137 120 120 122 123 125 138 139 2000 2000 2000 2000 2000 2000 2000 2750 2750 2750 2750 2750 2750 2750 3000 3000 3000 3000 3000 3000 3000 4120 4120 4120 4120 4120 4120 4120 1250 1060 1000 1620 1370 1500 2060 1250 1060 1000 1620 1370 1500 2060 1250 1060 1000 1620 1370 1500 2060 1250 1060 1000 1620 1370 1500 2060 MATERIALS 255 >C * iC CD lO CD 00 05 t^ M fC t>. e<3 lO IN CC »0 CO -^ CD J J J J J J C^ Cq C^ C^ CM C^ IC o CO d J 7 00 00 00 CO O 00 CO o 00 t^ (N 00 t^ t^ (M IC O IM lO 00 (M CC 00 t^ £1 Tf I> O 00 (M t^ lO CO O O t^ 05 J J J J J J ^ |j ^ 2 ^ ^ o o f f 00 00 H^ o 00 CD O 00 CD O 00 t^ (N 00 t^ t>. C0 lO 00 CM -* -* (M CO O CD CO CI 00 t^ ,-1 ,-H CM CO rH rH S"2 o o o CO r^ o -H OO CO o o o o o o 00 CM ■* rt< CM O O CO CO CM O O ^ ^ CM CO CM CM CM CM ■* lO CM CM I I , ' I ^ ■"TT 1 X'-, ^ ' ' ^£^ > ' ' \ ^j-<^ ^h'V'^ "• " > . .\. ^ 100 ^. , ; ' , ' ^ '■~:~.p^ Vt ■' ■ ., .,v^. . ^ - • ; ^ ~ » V ^ .' [„ ^"^ > ^~y _ Diameter 1 ^ ^ = 0531 in \ 50 ■• y 1 .^„J.^-,, CJ., ,-J- lifO 1 l_ -i-/ iu^SR xir -N^ fi "" " '^- [~ li^J ^ 50 200 250 300 Loading, Pounds. Fig. 154. wrapping. The work can be easily calculated by measuring the shaded areas in Figs. 153 and 154. Naturally to do this it is necessary to translate the per cent, scale of elonga- tion into inches, which is easy when the weight per yard is knowm. For 150 per cent, of elongation the work absorbed by 1 lb. of elastic cord without initial tension and without cotton wrapping is 1280 Ib.-in.; while that absorbed by elastic cord with 127 per cent, of initial elongation is equal to 20,200 Ib.-in.; that is, in the second case a work about 16 times greater can be absorbed with the same weight. This shows the great convenience in using elastic cords with a high initial tension. MATERIALS 259 (c) Varnishes. — ^Varnishes used for airplane fabrics are divided into two classes: stretching varnishes (called "dope"), and finishing varnishes. The former are intended to give the necessary tension to the cloth and to make it waterproof, increasing at the same time its resistance. The finishing varnishes which are applied over the stretching varnishes have the scope of protecting these latter from atmospheric disturbances, and of smoothing the wing surfaces so as to diminish the resist- ance due to friction in the air. The stretching varnishes are generally constituted of a solution of cellulose acetate in volatile solvents without chlorine compounds. The cellulose acetate is usually con- tained in the proportion of 6 to 10 per cent. The solvents mixtures must be such as not to alter the fabrics and not to endanger the health of men who apply the varnish. The use of gums must be absolutely excluded because they conceal the eventual defects of the cellulose film. A good stretching varnish must render the cloth absolutely oil proof, and will increase the weight of the fabric by 30 per cent, and its resistance by 20 to 30 per cent. Finally it should be noted that it is essential for the var- nish to increase the inflammability of the fabric as little as possible; precisely for this reason the cellulose nitrate varnish is used very seldom, notwithstanding its much lower cost when compared with cellulose acetate. In general for linen and cotton fabrics three to four coats of stretching varnish are sufficient; for silk instead, it is preferable to give a greater number of coats, starting with a solution of 2 to 3 per cent, of acetate and using more concentrated solutions afterward. The finishing varnishes are used on fabric which have already been coated with the stretching varnishes. These have as base linseed oil with an addition of gum, the whole being dissolved in turpentine. A good finishing varnish must be completely dry in less than 24 hours, presenting a brilUant surface after the drying, 260 AIRPLANE DESKIN AND CONSTRUCTION resistant to crumpling, and able to withstand a wash with a solution of laundry soap. {d) Glues. — Glues are greatly used both in manufac- turing propellers and veneers. Beside having a resistance to shearing superior to that of wood, a good glue must also resist humidity and heat. There are glues which are applied hot (140°F.), and those which are applied cold. A good glue should have an average resistance to shearing of 2400 lb. per sq. in. CHAPTER XVII PLANNING THE PROJECT When an airplane is to be designed, there are certain imposed elements on the basis of which it is necessary to conduct the study of the other various elements of the design in order to obtain the best possible characteristics. Airplanes can be divided into two main classes: war air- planes and mercantile airplanes. In the former, those qualities are essentially desired which increase their war efficiency, as for instance: high speed, great climbing power, more or less great cruising radius, possibility of carrying given military loads (arms, muni- tions, bombs, etc.), good visibihty, facility in installing armament, etc. For mercantile airplanes, on the contrary, while the speed has the same great importance a high climbing power is not an essential condition; but the possibility of transport- ing heavy useful loads and great quantities of gasoline and oil, in order to effectuate long journeys without stops, assumes a capital importance. WTiatever type is to be designed, the general criterions do not vary. Usually the designer can select the type of engine from a more or less vast series; often though, the type of motor is imposed and that naturally limits the fields of possibility. Rather than exposing the abstract criterions, it is more interesting to develop summarily in this and the following chapters, the general outline of a project of a given type of airplane, making general remarks which are applicable to each design as it appears. In order to fix this idea, let us suppose that we wish to study a fast airplane to be used for sport races. 261 262 AIRPLANE DESIGN AND CONSTRUCTION The future aviation races will certainly be marked by imposed limits, which may serve to stimulate the designers of airplanes as well as of engines towards the increase of efficiency and the research of all those factors which make flight safer. For instance, for machines intended for races the ultimate factor of safety, the minimum speed, the maximum hourly consumption of the engine, etc., can be imposed. The problem which presents itself to the designer may be the following : to construct an airplane having the maximum possible speed and also embodying the following qualities: 1. A coefficient of ultimate resistance equal to 9. 2. Capable of sustentation at the minimum speed of 75 m.p.h. 3. Capable of carrying a total useful load of 180 lb. (pilot and accessories), beside the gasoUne and oil necessary for three hours flight. 4. An engine of which the total consumption in oil and gasoHne does not surpass 180 lb. per hour when running at full power. Let us call W the total weight in pounds of the airplane at full load, A its sustaining surface in sq. ft., W^ the useful load in pounds, P the power of the motor in horsepower, and C the total specific consumption of the engine in oil and gasoline. Remembering that in normal flight W = 10-'\AV' since the condition is imposed that the airplane sustain itself for V = 75 m.p.h., we must have W -J- < 0.56 Xmax. that is, the load per square foot of wing surface will have to equal ^%qq of the maximum value Xn,ax. which it is possible to obtain with the aerofoil under consideration. The total useful load will equal W^ = 180 + 3cF PLANNING THE PROJECT 263 Let US call Wp the weight of the motor including the propeller, W„ the weight of the radiator and water, W^ the weight of the airplane. Then W = W^ + W, + W, + W^ (1) Calling p the weight of the engine propeller group per horsepower we will have Wp = pP The weight of the radiator and water, by what we have said in Chapter V, can be assumed proportional to the power of the engine and inversely proportional to the speed. As to the weight of the airplane, for airplanes of a certain well-studied type and having a given ultimate factor of safety, it can be considered proportional to the total weight; we can therefore write W^ = aW Then (1) can be written W = ISO -\-ScP + pP + by + aW that is ^ = ^ + r^„(3c4-P+|) (2) The machine we must design is of a type analogous to the single-seater fighter. Consequently in the outhne of the project we can use the coefficients corresponding to that type. For these, the value of a is about 0.34; also, expressing V in m.p.h. we can take b = 45. Remembering the imposed condition that cP must not exceed 180 lb., we will have to select an engine having the minimum specific consumption r, in order to have the maximum value of P; at the same time the weight p per horsepower must be as small as possible. 264 AIRPLANE DESKIX A\D CONSTRUCTION Let US suppose that four types of engines of the following characteristics are at our disposal: Table 31 /' II. l» II). per II. P. C I 250 2.3 0.54 575 135 II 300 2.2 0.53 660 159 III 350 2.1 0.56 735 196 IV 400 2.0 50 800 23G It is clearly visible that engines No. Ill and No. IV should without doubt be discarded since their hourly con- sumption is greater than the already imposed, 180 lbs. Of the other engines the more convenient is undoubtedly type II for which the value of p is lower. Then formula (2), making a = 0.34, P = 300, c = 0.53, p = 2.2, b = 45, becomes 20^ (3) TT^ 1992 + To determine W as a first approximation, let us re member that the formula of total efficiency gives WV 0.00248 (4) and that for a machine of great speed we can take r = 2.8; then making P = 300 we have J_ ^ J)^0248 ^ V 840 and substituting in (3) W (1 - 0.06) = 1992 that is, W = 2130 Then V = 159 m.p.h. Consequently we can claim; in the first approximation, that the principal characteristics of our airplane will be W = 2130 lbs. F_. = 159 m.p.h. F„.i„. = 75 m.p.h. P = 300 H.P. PLANNING THE PROJECT 265 Let us now determine the sustaining surface. We have seen that we must have W - ^ < 0.56 Xn,ax. where X^^x. is the maximum value it is practical to obtain. S X l./O ^& 1 1 1 / t t ^ ^« — kIu Z a ^ / 1 ^s. ^ ^ y ^^£ ^< nc:n ,n ^ ^^^ a1 ^S r ^^ 4 ^^ no^ c fc-i ^'i ^ 1- . oi ::^ n,5 2 Degree© Fig. 155. T5 From the aerofoils at our disposition, let us select one which, while permitting the realization of the above con- dition, at the same time gives a good efficiency at maximum Let us suppose that we choose the aerofoil having the characteristics given in the diagram of Fig. 155. Then as X^ax = 14.4, we must have 266 AIRPLANE DESIGN AND CONSTRUCTION ■i^iair -J Uf L9S9 ^ WOS/9 N. VO ^ 5i ^ ^ w c ^ s 1 § 1 1 - oS OQ t: s § ^1^ ^» 55 l\ X /'^ \\/ i ^ ^ v X ^^ J^\ m S s § A: ^ ^,^''^^^0^ i§ '♦^ (^ / t \ / s o «A ci CM N )\ oa 1 § / \ ^ ^ / \ s § / \ § ? / ^ L -^ ^ ^ r-; •^ (0 K vt> ^ 1 PLANNING THE PROJECT 267 W For -^ = 8 and W = 2130 lb. A = 265 sq. ft. Let us select a type of biplane wing surface adopting a chord of 65". The scheme will be that shown in Fig. 156. We can then compile the approximate table of weights, considering the following groups: 1. Useful Load Pilot 180 lb. Gasoline and oil 477 lb. Instruments 11 lb. Total 668 lb. 2. Engine Propeller Group Dry engine and propeller 660 lb. Exhaust pipes 6 lb. Water in the engine 30 lb. Radiator and water 125 lb. Total 821 lb. 3. Wing Truss Spars 100 1b. Ribs 26 lb. Horizontal struts and diagonal bracings 20 lb. Fittings and bolts 30 lb. Fabric and varnish 25 lb. Vertical struts 40 lb. Main diagonal bracing 35 lb. Total 2761b. 4. Fuselage Body of fuselage 155 lb. Seat, control stick, and foot bar 25 lb. Gasoline tanks and distributing system 40 lb. Oil tanks and distributing system 6 lb. Cowl and finishing 25 lb. Total 251 lb. 5. Landing Gear Wheels 32 lb. Axle and spindle 25 lb. Struts 15 1b. Cables 4 lb. Total 76 lb. 268 AIRPLANE DESIGN AND CONSTRUCTION 6. Controls and Tail Group Ailerons 12 lb. Fin 2 1b. Rudder 6 lb. Stabilizer 8 lb. Elevator 10 lb. Total 38 lb. We can then compile the following approximate table: Table 32 Donoininatioii Weight in lb. Per cent . of total weight 668 31.0 821 38.5 276 13.0 251 12.0 76 3.5 38 2.0 2130 100.0 1. Useful load 2. Engine propeller group. 3. Wing truss 4. Fuselage 5. Landing gear 6. Controls Total. A schematic side view of the machine is then drawn in order to find the center of gravity as a first approximation. In determining the length of the airplane, or better, the distance of tail system from the center of gravity, we have a certain margin, since it is possible to easily increase or decrease the areas of the stabilizing and control surfaces. For machines of types analogous to those which we are studying, the ratio between the wing span and length usu- ally varies from 0.60 to 0.70. Since we have assumed the wing span equal to 26.6 ft., we shall make the length equal to 18 ft.; that is, we shall adopt the ratio 0.678. The side view (Fig. 157) shows the various masses, with the excep- tion of the wings and landing gear; these are separately drawn in Figs. 158 and 159. Then with the usual methods of graphic statics we determine separately the center of gravity of the fuselage (with all the loads), of the wing truss, and of the landing gear. It is then easy to combine the three drawings so that the following conditions be satisfied: 1. That the center of gravity of the whole machine be on PLANNING THE PROJECT 269 270 AIRPLANE DESIGN AND CONSTRUCTION Pounds. Fig. 158. PLANNING THE PROJECT 271 Pounds FiQ. 159. 272 AIRPLAXE DESICN AND CONSTRUCTION •spunoj PLANNING THE PROJECT 273 the vertical line passing through the center of pressure of the wings. 2. That the axis of the landing gear be on a straight line passing through the center of gravity and inclined forward by 14°; that is, by about 25 per cent. The superimposing has been made in Fig. 160. The ideal condition of equilibrium is that the center of gravity, thus found, not only must be on the vertical line passing through the center of pressure, but must also be on the axis of thrust; if it falls above the axis of thrust it is advisable that its distance from it be not greater than 4 or 5 inches at the maximum ; if instead it falls below the axis of thrust, we have a greater margin as the conditions of stability improve. This shall be seen in Chapter XXI. In our case, it falls 2.5 in. above the propeller axis. The center of gravity having been approximately de- termined we can draw the general outline (Figs. 161, 162 and 163). It is then necessary to calculate the dimensions of the stabiUzer, fin, rudder, and elevator. To do this, it would be essential to know the principal moments of inertia of the airplane. The graphic determination of these moments is certainly possible but it is a long and laborious task be- cause of the great quantity and shape of masses which compose the airplane. Practically a sufficient approximation is reached by con- sidering the weight W instead of the moment of inertia. Then calling M the static moment of any control surface whatever about the center of gravity (that is, the product of its surface by the distance of its center of thrust from the center of gravity) we shall have M = aX 72. Value a can be assumed constant for machines of the same type. Then, having determined a based on machines which have notably well chosen control surfaces, it is easy to determine M. Value a in our case can be taken equal 274 AIRPLANE DESIGN AND CONSTRUCTION Fig. 161. Fig. 162. ¥ia. 163. PLANNING THE PROJECT 275 to 3900 for the ailerons, 2100 for the elevator, and 2500 for the rudder, taking as the units of measure pounds for W and feet per second for V. Then it is possible to compile the following table where a and M have the above significance, I is the lever arm in feet and *S is the surface of the rudder elevator and ailerons in square feet. The velocity V has been taken equal to 150 M.P.H., i.e. 220 ft./sec. Table 33 Controls " M (cu. ft.) I (feet) S (sq. ft.) 3900 4100 2400 172 178 105 8.3 14.8 15.6 21.0 Elevator 12 Rudder 6.7 CHAPTER XVIII STATIC ANALYSIS OF MAIN PLANES AND CONTROL SURFACES Owing to the broadness of the discussion we shall limit ourselves to summarily resume the principal methods used in analyzing the various parts, referring to the ordi- nary treaties on mechanics and resistance of materials for a more thorough discussion. In this chapter the static analysis of the wing truss and of the control surfaces is given. ELEVATION Fig. 164. JO 601n Scale C7f Length* Fig. 164 shows that the structure to be calculated is com- posed of four spars, two top and two bottom ones, con- nected to one another by means of vertical and horizontal trussings. For convenience the analysis of the vertical trussings is usually made separately from the analysis of the horizontal ones, and upon these calculations the analysis of the main beams can be made. 276 MAIN PLANES AND CONTROL SURFACES 277 First of all it is necessary to determine the system of the acting forces. An airplane in flight is subjected to three kinds of forces: the weight, the air reaction and the pro- peller thrust. The weight is balanced by the sustaining component L, of the air reaction; the propeller thrust is balanced by the drag-component D. The weight and the propeller thrust are forces which for analytical purposes can be considered as applied to the center of gravity of the airplane. The components L and D instead, are uniformly distributed on the wing surface. Practically, the ratio ^ assumes as many Fig. 165. different values as there are angles of incidence. The maxi- mum value, which is assumed in computations, is, usually, y- = 0.25. Thus it will be sufficient to study the distribu- L/ tion of L, because, when this is known the horizontal stresses can immediately be calculated. Let us suppose that the aerofoil be that of Fig. 165 and that the relative position of the spars be that indicated in this figure. The first step is to determine the load per linear inch of the wing. Fig. 164 shows that the linear wing development of the upper wing is 320.48 inches while that of the lower wing is 288.58 inches. We know that the two wings of a biplane do not carry equally because of the fact that they exert a disturbing influence on each other; in general the lower wing carries less than the upper one; usually in practice the load per unit length of lower wing is assumed equal to 0.9 of that 278 AIRPLANE DESIGN AND CONSTRUCTION of the upper wing. Then evidently the load per linear inch of the upper wing is given by 2130 320.48 + 0.9 X 288:58 ^ ^■^'^' ^^' P^' ^'''^ and for the lower wing it is given by 0.9 X 3.66 = 3.29 lb. per inch From these linear loads we must deduct the weight per linear inch of the wing truss, because this weight, being 029 I =18.85 In. ^S§^ ZiSS - 0.43 L ^M ^ Fig. 166. applied in a directly opposite direction to the air reaction, decreases the value of the reaction. In our case the figured weight of the wdng truss is 276 lb.; thus the weight per linear inch to be subtracted from the preceding values will be 0.45 lb. per linear inch. We shall then have ultimately: Upper wing loading 3.21 lb. per linear inch Lower wing loading 2.84 lb. per linear inch Knowing these loads, it is possible to calculate the dis- tribution of loading upon the front spars and upon the rear spars. For this it is necessary to know the law of variation ' of the center of thrust. MAIN PLANES AND CONTROL SURFACES 279 It is easily understood that when the center of thrust is displaced forward, the load of the front spar increases, and that of the rear spar decreases; and that the contrary happens when the center of thrust is displaced backward. We shall suppose that in our case the center of thrust has a displacement varying from 29 per cent, to 37 per cent, of the wing cord (Fig. 166). In the first case the front spar will support 0.62 of the total load and the rear spar will support 0.38; in the second case these loads will be respectively 0.43 and 0.57 of the total load. Thus the normal loads per linear inch of the four spars can be summarized as follows: Front spar upper wing 1.98 lb. per inch Rear spar upper wing 1.82 lb. per inch Front spar lower wing 1.75 lb. per inch Rear spar lower wing 1.62 lb. per inch Practically it is convenient to make the calculations using the breaking load instead of the normal load; in fact there are certain stresses which do not vary proportionally to the load but follow a power greater than unity, as we shall see presently. In our case, as the coefficient must be equal to 10, the breaking load must be equal to 10 times the preceding values. We can then initiate the calculation of the various trusses which make up the structure of the wings. We shall proceed in the following order, computing: (a) bending moments, shear stresses and spar reactions at the supports. Determination of the neutral curve of the spars (6) front and rear vertical trusses (c) upper and lower horizontal trusses (d) unit stresses in the spars. (a) The spars can be considered as uniformly loaded continuous beams over several supports. In our case there are four supports for the upper spars as well as for the lower ones; the uniformly distributed loadings are the preceding. 280 AIRPLANE DESIGN AND CONSTRUCTION Let US note first, that in our bution of the spans of the rear spans of the front spars; thus the front and rear spars is in It suffices then to calculate the stresses and the reactions at spars; the same diagrams, by can be used for the rear spars, ing for the rear spars is equal spars. case as in others, the distri- spars is equal to that of the the only difference between the load per unit of length, bending moments, the shear the supports for the front a proper change of scales, In our case, the unit load- to 0.92 of that for the front .1 . ■4S67' . 8$ 76' 4^6- , e97e' , 4S67' J °i i 1 r r r r i' 1 i K 1 1 i i "1^ IN i ^^-•'•^! 'V k N"' -r iV ! \°i i A i 1 \ ! ! 1 B V ~A \ \ ! 1 i! 1 D 1 1 la, ib, Im, ! J Ic, id, in. ie, if. 25 SO In. Scale of Lengths Fig. 167. With this premise we shall give the graphic analysis based upon the theorem of the three moments, but we shall not explain the reason of the successive operations, referring the reader to treaties on the resistance of materials. First consider the upper front spar (Fig. 167); Jet XY be its length and A, B, C, D, its supports, made by the struts. Let each span be divided into three equal parts by means of trisecting lines aai, bbi, cci, etc. For each support with the exception of the first and last ones, the difference be- tween the third parts of its adjacent spans shall be deter- mined; and that difference is layed off starting from the support, toward the bigger span. In our case we subtract the third part of span BC from the third part of span AB, MAIN PLANES AND CONTROL SURFACES 281 and the difference is layed off starting from B toward A. Thus V is obtained. The line mini drawn through V per- pendicular to XY is called counter vertical of support. Analogously one-third of BC is subtracted from one-third of CD, and its difference is laid off from C toward D, fixing a second counter vertical of support nrii. Starting from A (Fig. 167) let us draw any straight line that will cut the trisecant hbi, and the first counter vertical of support mmi in the points E and F respectively. Draw the straight Une EB which prolonged will cut the first trisecant of the second span cci in the point G. Join G with F by a straight line which will cut XF at the point H. This point is called the right-hand point of support B. Starting from H we draw any straight line that will meet the second trisecant of the second span ddi and the second diagonal nui at the points M and A^ respectively. Find the point P by prolonging the straight line between M and C. Point 0, the right-hand point of the second support, is given by the intersection of line AT and hue XY. In order to find the left-hand points for the supports C and B, draw the straight hne PD which will interest the counter vertical nui at point Q. Point R where the lines MQ and XY intersect each other will be the left-hand point of support C. Starting from R draw the line RG which will cut the first counter diagonal at point S. Point T, the point of intersection of hues SE and XY will be the left- hand point of support B. The right-hand and left-hand points being known, we shall suppose that we load one span at a time, determining the bending moments which this load produces on all the supports. Summing up at every support the moments due to the separate loads, we shall obtain the moments origin- ated by the whole load. The moment on the external supports is equal to that given by the load on the cantilever ends, as it cannot be influenced by the loads on the other spans, owing to the fact that the cantilever beam can rotate around its support. The load on the cantilever spans however affects the other 282 AIRPLANE DESIGN AND CONSTRUCTION spans. To determine this effect we proceed in the follow- ing manner: Consider support A (Fig. 168); the moment wl- at this support is equal to ^> calling w the load in lb. per linear inch and I the length of the span in inches. Lay off, to any scale, the segment A A' = ^ * Let us then draw the straight line A'T; it will intersect the vertical line through support B at point 1 ; the segment IB measures, to the scale of moments, the moment that the load on the cantilevered span produces on support B. Scale of Uengtha. Scale of Moments. Fig. 168. Then draw the straight line 122; it will meet the vertical line through support C at T; the segment I'C measures, always to the scale of moments, the moment originated on support C by the load of the cantilevered span. The moment in D cannot be influenced by the cantilever load on X ^. Let us now determine the effect of the load on span AB, on the moment of the various supports. Draw FG perpen- dicular bisectrix oi AB and lay off, to the scale of moments, w "X. l^ a segment FG equal to — ^ — ; that is, equal to the moment which would be obtained at the center point of AB, by a, unit load w, if AB were a free-end span supported at the extremities. From T, the left-hand point of support B, MAIX PLANES AXD CONTROL SURFACES 283 raise a perpendicular which cuts line GB at W. Draw line AW to meet the perpendicular through support T at point 2. The segment B2 read to scale, will give the mo- ment on support B due to the load on AB. Point 2' is obtained by prolonging line 2R until it meets the perpendicular through C at 2'. Segment C-2' represents to the scale of moments, the moment on support C due to the load on AB. In order to find the effect of the load of span BC on the other spans, proceed analogously; that is lay off ML on the bisectrix of BC, equal to scale, to the moment ML = ^^' 8 Let us find points N and P as indicated in the figure and let us draw the line NP which prolonged will meet the per- pendiculars on supports B and C at points 3 and 3'. Seg- ments 5-3 and C-3' read to the scale of moments, will give the moments produced by the load of span BC on the sup- ports B and C respectively. Proceeding as for spans XA and AB we obtain the moments originated on BC by the loads on spans CB and BY. The construction is clearly indicated in Fig. 168. Resuming, we shall have the moment originated by canti- lever loads on the supports A and D, and the moment originated by the loads on all the different spans, on the supports B and C. For the point of support B the moment due to the canti- lever load is equal, read to the scale of moments, to dis- tance B-1, the moment due to the load on AB is equal to B-2, the moment due to the load on BC is equal to B-3, the moment due to the load on CD is equal to B-4' and that due to the cantilever load on Z)F is equal to B-5'. If we assume that the distances above the axis XY are positive and those below are negative, the total moment BB' on support B will be equal to the algebraic sum of the moments B-1, B-2, B-3, B-4:', and B-5'. Analogously the algebraic sum CC will represent the total moment on C. The total moment on the external supports will naturally remain the one due to cantilevers, 284 AIRPLANE DESIGN AND CONSTRUCTION and consequently equal to A A' and DD'. In order to find the variations of the bending moment on all the spans, the load being uniformly distributed, we must draw the para- bolse of the bending moments as though the spans were simply supported (Fig. 169). Scale of Moments Fig. 169. Fig. 170. Then the difTerence between the ordinates of the parabolas and those of the diagram AA' B' C D' D give us the diagram XA'a' B' y C c' D' YX which represents the diagram of the bending moment (Fig. 169). Knowing the diagram of the bending moments, it is easy MAIN PLANES AND CONTROL SURFACES 285 through a process of derivation applying the common methods of graphic statics, to find the diagram of the shearing stresses, and consequently the reactions on the supports (Fig. 170). The scale of forces is obtained by multiplying the basis H of the derivation, by the ratio between the scale of moments and that of the lengths. In Fig. 170 the scale of forces has been drawn, and on the supports the corresponding numerical values of the reactions have been marken. Furthermore, from the diagram of bending moments we can obtain the elastic curve, which will be needed later. 25 50Irv Scale ot Liinyths 8000 16000 in Scale ot Moments Fig. 171. ) 15.0 3a01nn(^ Scale at Deflections In fact, let us remember that the analytic expression of the bending moment is given by Ms = E X I X and consequently y rifd-f M„ dx' that is, by double integration of the diagram of M^ we obtain the deflections y, that is, we obtain the form which the neutral axis of the spar assumes, and which is called elastic curve (Fig. 171). We shall not pause in the process of graphic integration, as it can be found in treaties on graphic statics. 286 AIRPLANE DESIGN AND CONSTRUCTION We shall make use of the elastic curve for the determina- tion of the supplementary moments produced on the spars by the compression component of the vertical and hori- zontal trussings. 40 In. 5cale of Lengths Fig. 172. Figs. 167, 168, 169, 170 and 171 refer to the calculation of the upper front spar. In Figs. 172, 173, 174, 175 and 176 instead, the graphic analysis of the lower front spar is developed. 20 40 li Scale of Length; Fig. 173. On these figures, beside the unit loads which are already known, the scale of the moments, of the lengths and of the forces are also indicated. The preceding diagrams also give the bending moments. MAIN PLANES AND CONTROL SURFACES 287 12 I421nji(0) 6cal6 of Oeflec+iona FiQ. 176. 288 AIRPLANE DESIGN AND CONSTRUCTION the shearing stresses and the reactions on the supports for the rear spars; in fact it suffices to multipl}^ both the values of the forces and those of the moments by 0.92, as the spans are the same, and the loads per hnear inch of the rear spars are equal to 0.92 of the loads of the front spars. A special note should be made of the scales of ordinates for the elastic curve; these are inversely proportional to the product E X I, the elastic modulus by the moment of inertia, and consequently they vary from spar to spar. But we shall return to this in speaking of the unit stresses in spars. (5) Knowing the reactions upon the supports, it is possi- ble to calculate the vertical trussings. Since the front trussing has the same dimensions as the rear one, and since the reactions on the supports are in the ratio 0.92, it suffices to calculate only the first. Upper Spar Upper Spar Fig. 177. The vertical trussing is composed of two spars, one above, and the other below, connected by struts capable of resist- ing compression, by bracings called diagonals, which must resist tension, and by bracings called counter diagonals which serve to stiffen the structure (Fig. 177). In flight, the counter diagonals relax and consequently do not work; for the purpose of calculation we can consequently con- sider the vertical trussing as though it were made only of spars, struts, and diagonals; furthermore, because of the symmetry of the machine, for simplicity we shall consider only one-half of it, as evidently the stresses are also sym- metrical (Fig. 178); the plane of symmetry will naturally have to be considered as a plane of perfect fixedness. With that premise let us remember that for equilibrium it is first of all necessary that the resultant of the external MAIN PLANES AND CONTROL SURFACES 289 forces be equal to zero. The reactions upon the supports are all vertical and directed from bottom to top ; their sum is equal to 5695 lb.; now, this force is balanced by that part of the weight of the machine which is supported at point A and which is exactly equal to 5695 lb. Moreover it is necessary that in any case the appHed external force (reaction at support), be in equilibrium with the internal reaction; that is, as it is usually expressed in graphic statics, it is essential that the polygon of the external forces and of Fig. 178. the internal reactions close on itself. This consideration enables the determination of the various internal reactions through the construction of the stress diagram, illustrated, for our example, in Fig. 179. Referring to treaties on graphic statics for the demonstra- tion of the method, we shall here illustrate, for convenience, the various graphic operations. The values of the reactions on the supports individuated by zones ab, be, cd, and de are laid off to a given scale on AB, BC, CD, and DE (Fig. 179); from B and C we draw two parallels to the truss members determined bj^ the zones bh and ch respectively; in BH we shall have the 290 AIRPLANE DESIGN AND CONSTRUCTION 1000 2000 lbs. Scale •of Forces FiQ. 179. MAIN PLANES AND CONTROL SURFACES /44 29 In 291 500 100 lbs. Scale of Forces Fig. 180. 292 AIRPLANE DESIGN AND CONSTRUCTION stress corresponding to member bh, and in CH that corre- sponding to the member ch. From points H and D we draw the parallels to the members gh and gd; in HG and DG we shall have the stresses in hg and dg ; from points E and G we draw the parallels to the members determined by zones ef and gf; in EF and GF we shall obtain the stresses in these members; finally from points G and A we draw the parallels to the members individuated by zones gi and at, obtaining the corresponding stresses in GI and AI. The arrows of the stress diagram enable the easy determination of which parts of the truss are subjected to tension and which to compression. In Fig. 179, beside marking the scales of lengths and of forces, we have marked the lengths and the stresses corresponding to the various parts, adopting + signs for tension stresses, and — signs for compression stresses. By multiplying these stresses by 0.92 we shall obtain the values of the stresses of the rear trussing. The counter diagonals which do not work in normal flight, function only in case of flying with the airplane upside down. For this case, which is absolutely excep- tional, a resistance equal to half of that which is had in normal flight is generally admitted. The determination of stresses is analogous to that made for normal flight and is shown in Fig. 180. Based upon the values found in the preceding construc- tion. Table 34 can be compiled. That table permits the calculation of the bracings and struts. The calculation of the bracings presents no difficulties; it is sufficient to choose cables or wires having a breaking strength equal to or greater than that indicated in the table; naturally the turnbuckles and attachments must have a corresponding resistance. Table 35 gives the dimensions of the cables selected for our example. For the principal bracings we have adopted double cables, as is generally done in order to obtain a better penetration; in fact not only does the diameter of the cable exposed to the wind MAIN PLANES AND CONTROL SURFACES 293 Table 34 Front vertical truss Rear vertical truss Normal position Inverted position Normal 1 position Inverted position Member 1 Tension Com- pression Tension Com- pression ! Tension 1 1 Com- pression Tension Com- pression A-B 6120 130 4710 120 B-E 1400 .... 2800 1290 2580 F-C 5200 620 4780 570 A-C 6380 6870 D-C 200 2330 180 2140 A-D 1500 990 1380 910 B-B' 5780 3750 6320 3450 C-C 7600 2650 6990 2440 B-D 2800 2580 Table 35 Member Stress coef.lO Number of cables used Diameter of cable, in. Ultimate strength of cable, lb. Total ultimate strength. Coef. of safety £ A-C -f6380 2 He 4200 8400 13.1 I B-D -f2800 1 H 2000 2000 7.1 ■g B-E -h4700 1 %2 2800 2800 5.9 £ C-F +9200 1 %2 5600 5600 6.1 g A-C +5870 2 Ke 4200 8400 14.3 2 B-D +2580 1 H 2000 2000 7.7 » B-E +4320 1 H2 2800 2800 6.5 (S C-F +8500 1 %2 5600 5600 6.6 294 AIRPLANE DESIGN AND CONSTRUCTION result smaller, but it becomes possible to streamline the two cables by means of wooden faring. For the struts, which can be considered as solids under compression, it is necessary to apply Euler's formula which gives the maximum load W that a solid of length I with a section having a moment of inertia / can support In that formula a is a numerical coefficient and E is the elastic modulus of the material of which the solid is made. The theory gives the value 10 for coefficient a. We shall quickly see that practically it will be convenient to adopt a smaller coefficient in consideration of practical unforeseen factors. Let us remember that the struts, being exposed to the wind, present a head resistance which must be reduced to a minimum by giving them a shape of good penetration as well as by reducing their dimensions to the minimum. This last consideration shows, by what has been said in Chapter XVI, that for struts it is convenient to use mate- rials which even having high coefficients Ai and Ai have a high specific weight. Then the best material for struts is steel. In Chapter XVI a table has been given of oval tubes normally used for struts, with the most important characteristics, such as weight per unit of length, area of section, relative moment of inertia, etc. Let us apply Euler's formula to these tubes, remembering that for them I = td^, where t is the thickness and d is the smaller axis. We shall have W = ay Remembering then that the area of these struts is given with sufficient approximation by the expression A = Q.^ltd the preceding formula can be written as follows TT _ a XE 1 A ~ 6.37 ^ /IV (i) MAIN PLANES AND CONTROL SURFACES 295 where W —r = unit stress of the material A = ratio between that portion of the length which can be considered as free ended, and the minimum dimension of the strut. IIXIU 10 9 8 I \ «=47.,0%(!). A 7 \ \ \ \ '^;^ 4 ^^ i \ 3 ~ -^ . 4^^. /• \ •^^ '/ \ 2 ».^ \ ^V. ^ ^ /-» 10 20 30 40 50 60 TO 80 90 100 I _ d Fig. 181. Adopting pounds and inches as the unit, we have E = 3 X 10^ and consequently W 1 ^ = 47 X 10^ X a X jiy, (f 290 AIRPLANE DESIGN AND CONSTRUCTION Naturally this formula can be applied only for high values of the ratio ,; practically below the value , = 60 this formula can no longer be relied upon. In Fig. 181 the diagram corresponding to the preceding formula is given, drawing the diagram with a dotted instead of a full line for the values of -^ < 60. For those values the practical diagram is shown by a dot and dash line. In Tables 36 to 39 we have tabulated the results of some of the many tests on metal struts which have been made at our works. In these tables the practical value of coeffi- cient a of Euler's formula has been calculated; it is seen that while in some tests a has a value higher than 10, in general it gives lower values. That depends ujoon some struts being manufactured by hand and some being rolled, and also upon the thickness of the sheet and the dimensions of the sections being not always uniform. Based on aver- age values we can therefore assume that for properly manu- factured struts a coefficient a = 8 can be adopted for computation purposes. With this premise it is simple, when the ultimate stress which a strut must withstand, and its length, are known, to determine its dimensions. Moreover infinite solutions exist, since formula (1) when W and I are given, can be satisfied by infinite couples of values A and d. Evidently by increasing d, the value of A becomes smaller and consequently the weight of the strut diminishes; from that point of view it would be convenient to use struts having large dimensions and small thicknesses. However, the increase of d increases the head resistance of the airplane, and increases the power necessary to fly. Therefore it becomes necessary to adopt that solution which requires the minimum expension of power. If /3 is the weight per horsepower lifted by the airplane, y is the weight of one foot of strut of width d, k its coeffi- cient of head resistance as was definitely stated in Chapter MAIN PLANES AND CONTROL SURFACES 297 ^Ai >. Welded. Welded. Seamless drawn— poplar fill- mg. Seamless drawn. Welded. Seamless drawn. Welded. Seamless drawn. Welded. Welded. Seamless drawn. Welded. Seamless drawn. Welded. Seamless drawn. S i)\ *-pM « O 00 t> COCOiOCO-*00(NOiC0050I> CO o o oot^05t^i>0'-icooocoa3i> 1 11,200 5,720 8,770 oooooooooooo -*O(MiCi0^C000^O'*I- O --< 05 CO (M 00__ 0_ O^ O^ cO__ 05_ lO^ co" >o CO lo" lo" Qo" cT uri co' lo" co lo^ <:.g ss§ 00CO00»O00COiO00G0COC»G0 t^O0O>(N'-icOCOI:^l>00iCt^ o o o OOOr-H^OOOOOOO ^ H ^ fe| S- S' w OOOiOiOOOOOOOO l^^-*C500O5t^C3COiOCDQ0 ooiooiciocosooco— 'COCO ^' ^ (M" t^" rH' (N ^f th" ,-r rn" 5§ H -^d 00 -^ 00 lo -<*< CO CO^^ScOCOCO(M(M(M(N^ o o o oooooooooooo — i-3 lO lO lO O O (N IC ic to g ^ g 8 J2 8 § g S S 12 g C0_ O0__ CO CO__ iO_ t>-_^ 05_ C0__ O 0_ 05^ co_^ ccT co~ cd" co"co»o~io CO io^io^io~co — ITS (M (N O >Ot^iC-*t^»OTfiC-*iTt t- t> ?2 S ?^ S i i2 j: g S S f: g^ ~» C! IC lO lO §888^888^^88 i-H ,-H t^ (£. CO lO CC505 t^t^t^i^cooot^t^t^t^t^t^ o o o oooooooooooo Ci a O CD W •* CO CO ^ ^ ^ c? ^ ^ ^ ^ ^ ^ ^ ^ (N (N (M (N(M 298 AIRPLANE DESIGN AND CONSTRUCTION s Cl Seamless drawn. Welded. Seamless drawn. Welded. Welded. Welded. i 11 Welded. Seamless drawn. Seamless drawn. Welded. Welded. SS5^§2 s o o ^ §^§§§ t^ r^ i> t^ CO r^ CO t^ t^ CO l^ CO CO t^ o 1 c o o o o o o g i: S l^ g IS co' t-T co" i> CO co" CD' o" t-" o CO co' §8§|| ^„ ■*„ oi o, R, l-J" co" CD QC" lO <] c ||§|S| § ■* is 1 o ■ • O ■ 2 : :g : o ^ o o ^ o -^ o ^ ^ rt . • o • . g§§2§g ". «i -^^ «=- f^ -1 CO CC CO CC CO CO co"" 2g CO eo" 1 co" o o o o o O iC O) -* !M co' C* CO o o o o o o o o o o O O O O O ~ i-g O O OO O C^ lO 03 »0 ^ lO lO o TjT T^" iO~ rf TlT tJh" ■<*'' CO o TjT CO 00 -* (N (N ■^ -^ "* ■* CD -,-« O 00 O 00 CO "5 t^ 00 "3 t^ 00 C<> O ■* C.= 05 05 05 00 Oi OS g §§ g i§gg§ (N C^ C^ C<1 C<1 Cl C<1 C^ 04 M C<) (M 01 C^ C^l MAIN PLANES AND CONTROL SURFACES 299 . I 2 2 2 S 'oj a^ a> cs (— ( ty Qj ,_^ ,_H Q^ u.) OJ tiS bbS2 S S222 o o o o o o 00 ^ IN 00 ^ O O 0> t>- 00 00 CO «0 CO 00 00 00 00 o o o o o o o O ^ t^ lO IC 05 o o o o o o o CD O (M 00 O — < (M O iC 00 TfH C: OO Th ■* lO CO 00 lO IC t^ : :g o CO i S§ : 00 CO • • O t^ l> ■ • o • o o o o o o • • o o o O lO o o o o o CO (M t^ iM CO O 00 O CO —< CO 00 CO ^ (M (M C^ (M (M CO ^ o o o o o o o 00 CD 00 0> CD CO CO 00 CO CD O CO CO t^ o o o o o o o o o o o o o O O O O '^ o >0 CO lO lO (M 00 (M t^ lO lO O (M CD r- CO CO 00 oi lO lO lO IC -"ti Tf •* CO CO C5 •* ■* CO 05 rH —I CO CD (M Oi (M >0 lO -t^ 00 00 N- 01 00 GO CO C^ 'M t^ 02 CD CD CO >0 iC i-H CD lO lO ■* CO CO CD O O iM (M CO (N t^ r- Oi Oi Oi O^ 0> Oi lO lO lO IC lO lO o o o o o o d d d o d d Oi Gl Oi O O^ t^ Oi Oi Oi ^ Oi Oi d d d o o d (M C<1 (N Ca (N (N CO CD (N (N C<1 — < ^ t^ Tt< CD Tt< I> 1> t^ CO 1> l> t^ O^ 05 OS Oi Oi o o o o o o o o o o o o o o C^ C■ Welded. Seamless drawn. Seamless drawn. Seamless drawn— poplar fill- mg. Seamless drawn. Seamless drawn. Seamless drawn — fir filling. Welded. Seamless drawn. e o o o o 1-1 o ■* t- b-' (N d ^' t^ CO rtj t> 00 fel^ tJh" t~-" CO 00 (M CO (N ■* (N~ 00 ■<* (N" CO -<" c Oi Oi Oi d d> (6 <6 oi oi a> oi Oi <6 d d d d •« M 00 (N 00 00 00 00 05 00 Q ec CO CO CO CO CO CO CO CO MAIN PLANES AND CONTROL SURFACES 301 VII, V the speed of the airplane in m.p.h., and p the propeller efficiency, the total power p absorbed by a foot of strut will be equal to p = ^ + - X 267 X 10-9 k^V' Now the weight y is equal to 7 = 12 X A X 0.280 lb. = 3.36A lb. where A is expressed in square inches. In Chapter III we have seen that k = 3.5 for struts of the type which we are studying. Then, taking an average value p = 0.75 we shall have p = ?^y^ + 103.6 X 10-9 ^ys Formula (1) permits expressing A as function of d W ^ 47 X 10^ X a ^ consequently we shall have Supposing W, I, a, jS and V to be known, the preceding equation gives the expression of total power (that is, the resultant of the weight and head resistance), absorbed by one foot of strut as function of the minor axis d of its section. Evidently the designer's interest is to find the value of d that makes p minimum; but that value is the one which makes the derivative of the second term of the preceding equation equal to zero, that is, the one which satisfies the equation -2 x'-^'^^V 103.6X10- 7-0 from which wxr- ©' 13.8 X Let us remember that the symbols have the following significance : W = maximum braking load which a strut must support, I = length of strut, 302 AIRPLANE DESIGN AND CONSTRUCTION a = coefficient of Euler's formula, jS = ratio between the total weight and power of the airplane, V = speed of the airplane, For our example the weight of the airplane is 2130 lb. and its power is 300 H.P.; then /3 = 7.1 ; the foreseen speed is about 158 m.p.h. Furthermore for a we can adopt the value 8. Then the preceding formula becomes: d' = 61.5 X 10-9 WP (2) Euler's formula, for a =8, gives 5 = 3.76 X 10-' X -y]^, (3) (^) Equations (2) and (3) enable obtaining d and A, when W and I are known; then since A = 6.37^ the thickness t of the tube is easily obtained. The computations of the struts for the airplane in our example, Table 40, have been made with these criterions. Before passing to the calculation of the horizontal truss- ings it is necessary to mention the vertical transversal truss- ings which serve to unite the front and rear struts (Fig. 182). The scope of these bracings is that of stiffening the wing truss and at the same time of establishing a connection between the diagonals of the principal vertical trussings. Their calculation is usually made by admitting that they can absorb from ^ to % of the load on the struts. (c) The horizontal trussings have the scope of balancing the horizontal components of the air reaction. As we have seen, it is sufficient for the calculation, to assume for these horizontal components 25 per cent, of the value of the vertical reactions. As an effect of the stresses in the vertical trussings, a certain compression in the spars of the upper wings and a certain tension in the spars of the lower wings are developed. MAIN PLANES AND CONTROL SURFACES 303 As an effect of the stresses in the horizontal trussings we have a certain tension in the front spars and a certain com- pression in the rear spars. Table 40 Member Stress coef. 10 r/, lb. Length in. Diameter d (theo- retical), in. Thickness t (theo- retical), in. Diameter d (actual), in. Thickness t (actual), in. II I 11 III -1500 -2800 -5200 65 20 20 1 0.731 1 0.068 0.883 0.068 1.085 0.068 0.788 0.983 1.180 0.065 0.065 0.065 si 1 11 111 -1380 -2580 -4780 65 20 20 0.710 0.068 0.860 0.068 1.057 0.066 0.788 0.983 1.180 0.065 0.065 0.065 Fig. 182. Consequently in the various spars there is a distribution of stresses as shown in Table 41. 304 AIRPLANE DESIGN AND CONSTRUCTION Table 41 Upper front Upper rear Lower front Lower rear Effect of vertical trussing Compression Compression Tension Tension Effect of horizontal tr Tension Compression Tension Compression s 'SHr \ ■^^ISSir 5 \ / S ^ \/ /\ r / \ i / !S \ / •^ \ / § \ / >t '^ / \ / \ ■sq / -^Ql 0I£ ^? / ■\ ■■ S / \ c^ / \ ■sc /S',Z \ / ■^^noie ^ \ / ^ / \ \ . , / \ ^ ^v / "S-^ \ / - i \ / V \ \ ■Si <,9f ^QlSQt' ^ ^ 1 i^iesez "> MAIN PLANES AND CONTROL SURFACES 305 We see then that while there is partial compensation of stresses in the upper-front and lower-rear spars, in the other two spars instead the stresses add to each other. The spar which is in the worst condition is the upper-rear one, &cale of Forces STRESS DIAGRAM Fig. 185. which is doubly compressed. In order to take the stress from it, at least partially, it is practical to adopt drag cables which anchor the wings horizontally. Usually these drag cables anchor the upper wings only. Sometimes also the lower ones. 306 AIRPLANE DESIGN AND CONSTRUCTION 111 Fig. 183 the schemes of the horizontal trussings for the lower and upper wing are given. They are made of spars, a certain number of horizontal transversal struts, and of steel wire cross bracing. As we have already seen, in Fig. 183 the acting forces have been indicated equal to 25 per cent, of the vertical components. In Figs. 184 and 185 the graphic analysis of the horizontal trussings of the lower and upper wings have been given; as they are en- tirely analogous to those described for the vertical truss- ing, we need not discuss them. {d) Analysis of the Unit Stresses in the Spars. — This analysis is usually made following an indirect method, that is, under form of verification. Wc fix certain sections for the spars and determine the vmit load corresponding to the ultimate load of the airplane. After various attempts, the most convenient section is determined. Let us suppose that in our case the sections be those indi- cated in Fig. 186. The areas and the moment of inertia are determined first. The areas are determined either by the planimeter or by drawing the section on cross-section paper. The moment of inertia is determined either by mathematical calculation or graphically by the methods illustrated in graphic statics. Fig. 187 gives this graphic construction for the upper rear spar. Practically two principal methods of verification are used : A. The elastic curve method. B. The Johnson's formula method. A. This method consists of determining the total unit stress /r by adding the three following stresses : p 1. Stresses of tension or of pure compression fc =-r where P^ is the sum of the stresses Pl and P^ originated in the considered part of the spar by vertical and horizontal load, and A is the area of the section. M 2. Stress due to bending moments /j,/= -^ where M is the MAIN PLANES AND CONTROL SURFACES 307 308 AIRPLANE DESIGN AND CONSTRUCTION '■JI90 '•lOOZ MAIN PLANES AND CONTROL SURFACES 309 bending moment and Z is the section modulus. We shall remember that this modulus is obtained by dividing the moment of inertia I by the distance of the farthest fiber from the neutral axis. P X A 3. Bending stress due to the compression stress f^ = — A where P^ is the compression stress and A is the maximum deflexion of the span which is obtained from the elastic curve. In order to know A it is necessary to know the elastic modulus E of the material because this modulus enters into the equation which gives the scale of the elastic curve (see Figs. 171 and 176). By adding the values /c, j^j and/^ we obtain /r, which is the total unit stress, in our case corresponding to a load equal to ten times the normal flying load. If we wish to determine the factor of safety of the section it is necessary to know the modulus of rupture of the material ; this modu- lus of rupture divided by Y^^^ Jt gives the factor of safety. We have given in Chapter XVI the moduli of rupture to bending for various kinds of wood. For combined stresses of bending and compression stresses, it is necessary to adopt an intermediate modulus of rupture. Fig. 188 shows dia- grams giving the modulus of rupture as function of ratio ^ for the four following kinds of wood; Douglas fir, port- orford, spruce and poplar. In Table 42 all the preceding data for the sections of the spars most stressed has been collected. In this table Fi^ = stress due to vertical trussings. Pj) = stress due to horizontal trussings. Py = Pi, + P^ = total stress due to both trussings. For these stresses the — sign has been adopted when they are compression stresses and the -|- sign when they are ten- sion stresses. A = area of the section. Jc ^ E = elastic modulus of the material. 310 AIRPLANE DESIGN AND CONSTRUCTION o o o O O o o o o 9 o o MAIN PLANES AND CONTROL SURFACES 311 I = moment of inertia of the section. Z = section modulus. M = bending moment due to air pressure. M . Jm^'Y unit stress due to this bending moment. A = maximum deflexion of the span. Pr = moment due to compression stress P^. TO V A /^ = —^--. = unit stress due to the moment originated by the compression stress. S = total shearing stress. s = -r = unit stress to shearing. A. fc/fr = ratio between the compression stress and total stress. By using the diagrams of Fig. 188, this ratio enables us to determine the modulus of rupture, thence the factor of safety. B. The Johnson's formula method is based upon John- son's formula: "^ zx(i- KEI where I is the length of the span, i^ is a numerical coefficient and the other symbols are those of the preceding method. The value of coefficient K is dependent on end conditions and is = 10 for hinged ends = 24 for one hinged, one fixed = 32 for both ends fixed In Table 43 all the values of the quantities necessary for calculating the factor of safety by the Johnson's formula method have been collected. We see that the factors of safety are about equal to those found by the preceding method, with the exception of that corresponding to point B of the upper-rear-spar. This 312 AIRPLAXE DESIGN AND CONSTRUCTION Table 42 PT -*■ f. = I'T/A, K' s: lb. per sq. in. 5.70 - 5120 2.34 2190 - 3685} 5. 56 645 - 3020 2.34 1290 E, lb. per sq. in. Upper front spar ] ' C D -5120| -5120} +1435 -5780+2850 1.78X10« 4.02 2.02 20060 1.78X10' 2.831.84 1.78X10' |3.06}2.00 1.78X10' 2.83 1.84 65001 7200 [ 1210 Upper rear spar . 014.88 1.78X10' 2.5l}l.96 19000 B -4700 -1435 - 61352.58 2380 1.78X10' 2.10 1.64 5980 C -4700 -2865 - 756514.88 1550 1.78X10' }2.5l}l.96 6630 D -5320 -5445 -11765 2.58 4370 1.78X10' 2.101.64 1 1 ■ 1110 Lower front ^ spar - 200 - 200 +1110 + + 7600 +2220' + 05.70 200 1.98 910}5.56} 3820 1.98 I 100 165 4960 1.30X10' i3. 06^2. 00 12220 1.30X10' 12.52 1.84 7720 1.30X10' |3.06 2.00! 7800 1.30X10' '2.52 1.84' 5380 Lower rear spar A \ B - 1851-1110 C }- 185-2220 D +7000-3600 129c 240.^ 1.30X10' 1.92 1.49 615 I 1.30X10' 1.841.40 495 1.30X10' |l.92 1.4!) 1610 1.30X10' I.84J1. 40 11300 7140 7270 4970 • No bolt holes. discrepancy occurs because the coefficient K for this point should have been 32 instead of 24, as was assumed. In fact, from an examination of the elastic curve of the upper spars (Fig. 171), it is seen that point A is to be considered as an actual fixed point, and consequently for this point the coefficient 32 should have been taken. With this single exception, the two methods are practi- cally equivalent. Before leaving the calculation of the wing truss, the cal- culation of the shearing stresses and of the bending mo- ments which are developed in the ribs should be mentioned. MAIN PLANES AND CONTROL SURFACES 313 This calculation, which is usually made graphically is illustrated in Figs. 189 and 190. The rib can be considered as a small beam with two sup- ports and 3 spans; the supports being made by the spars. Diagram (a) of Fig. 189 gives the values of the pressures Table 42— (Continued) M/Z. lb. per sq. in. A, in. Pr.A. in. lb. j^=Pt.a/Z lb. per sq. in. s, lb. S lb. per sq. in. fc=fc+ \ in. 2 Modulus, lb. per sq. in. Factor safety Sec. 7880 1040 185 7880 0,000 9700 12.3 n. 3530 0.578 2960 1600 900 385 7320 0.299 8550 11.6 I 3600 0.578 2130 1065 720 125 5310 0.121 9250 17.4 ■ 660 0.106 320 175 350 150 2125 0.606 7450 35.0 I 9700 950 195 9700 0.000 9700 10.0 ■ * 3640 0.716 4390 2670 825 320 8600 0.270 8680 10.0 X 3380 0.716 5420 2840 660 135 7770 0.197 8900 11.5 f 680 0.131 1540 940 320 125 5990 0.723 7000 11.7 6110 4200 1030 180 6110 0.000 7900 12.9 1.43 290 160 900 455 4460 0.025 7850 17.5 900 160 4065 0.000 7900 19.4 300 150 7890 0.000 7900 10.0 7600 955 195 7600 0.000 7900 10.4 ■ 5100 1.55 2010 1435 830 395 7150 0.086 7600 10.6 I 4840 1.55 3730 2500 830 170 7835 0.062 7800 10.0 ? 4350 280 130 5960 0.000 7900 13.2 along the entire rib; the integration of this diagram gives diagram (6) of Fig. 189 whose ordinates correspond to the shearing stresses. In Fig. 190, diagram (o) represents diagram (6) of Fig. 189. In order to render this diagram more clearly it has been redrawn in Fig. 190 (b) referring it to a rectilinear axis and adopting a doubled scale for the shearing stresses. The integration of this diagram gives the diagram of the bending moments. Fig. 190 (c). The distributions of the shearing stresses and bending 314 AIRPLANE DESIGN AND CONSTRUCTION Table 43 Member Sec. PL, Ib.Pc, lb. Pl^Pd lb. A, sq. in. l.in. ^'^ if:.^ E, lb. per sq. in. J, in.« A B C D 5.70 2190 645 1290 1 2 R5 1. 78X10*4.02 1.78X10«2.83 1.78X10' 3.06 1.78X10' 2.83 Upper front spar -5120 -5120 -5780 + 1435 + 2850 - 5120 - 3685 - 3020 2.34 5.56 .3. 89.76 41.2X10«!l.84 24 89.76 29.7X10' 2.00 32 48.18 7. 0X10» 1.84 32 Upper rear spar A oi B -4700|-1435 C -4700j-2865 D -5320-5445 4.88 ■ 6135^2.58 ■ 756514.88 ■117652.58 2380 89.76 49.4X10' 1.64 1550 89.76 61.0X10' 1.96 4370 48.18 27. 3X10'|1. 64 1.78X10' 2.51 1.78X10'2.10 32 1.78X10'2.51 32 1.78X10' 2.10 Lower front spar 05.70 - 200 - 200 1.98 - 200 +1110 + 910 5.56 + 7600+2220+ 9820:1. . lOOi 165{ ,4960 c I I 2.00 .. 11. 62X10«ll. 84124 j 2.00 32 [1.84132 1.30X10' 3.061 1.30X10' 2.52 1.30X10'|3.06 1.30X10«i2.52 Lower rear spar OJ oj 04.88' 1.49.. 185|-1110|- 1295|2.11' 615j89.96JlO.5X10' I.40I24 1851-2220 - 240514.88 495,89. 96119. 5X lO^ll .49|32 + 7000 -3600 + 3400 2. 1.30X10^ 1.92' 1.30X106jl.84 1.30X10' 1.92 1.30X10' 1.84 moments being known the dimensions of the web and of the rib flanges can easily be determined. In Fig. 191 a general view of a very light type of rib is given. We shall now pass to the calculation of the tail system and the control surfaces. Figs. 192 and 193 give respec- tively the assembly of the fin-rudder group and the sta- bilizer-elevator group. The calculation of their frame is very easy when the distribution of the loads on the surface MAIN PLANES AND CONTROL SURFACES 315 is known. Consequently only the procedure for the cal- culation of these loads will be indicated. Let us first of all consider the fin-rudder group (Fig. 194). In normal flight as well as during any maneuver whatever, the distribution of the pressures on these surfaces is very Table 43 — (Contiriued) -l/.in. lb. KEI 1- pn KEI M s, lb. -1 lb. per sq.in. per sq.in. /^//< Mod. rupt., lb. per sq. in. Factor safety KEI '{^-KEl),\h. Sec. 20660 6500 7200 1210 7890 5350 4330 690 1040 900 720 350 180 385 125 150 7890 7540 4975 1980 0.291 0.130 0.651 9700 8600 9200 7300 12.3 11.4 37.0 ■ * 121.0X106 174.2X106 161.0X106 0.340 0.170 0.044 0.660 0.830 0.956 ■ I 19000 9700 8130 5900 880 950 825 660 320 195 320 135 125 9700 10510 0.226 7450 0.203 5250 0.883 9700 8850 8900 6600 10.5 8.5 11.9 12.6 I 189.5X106 143.0X106 119.3X106 5980 6630 1110 0.552 0.448 0.427 0.573 0.228 0.772 12220 6110 4300 3900 2930 1030 900 900 300 180 455 160 150 0110 4400 4065 7890 0.021 7900 7850 7900 7900 12.9 17.8 19.4 10.0 f ■ T 78.6X106 7720 0.021 7800 0.979 5380 57.4X106' 7140 80.0X1061 7210 4970 0.183J0.817 0.24410.756 7600 6240 6410 3550 955 195 7600 7900 10.4 830 395 6865 0.090 7000 11.0 830 170 6905 0.070 7G50 11.1 280 130 5160 7900 15.3 complex and varies according to their profile and their form. Practically, though, such high factors of safety are as- sumed for them, that it suffices to follow any loading hypo- thesis even if only approximate. For instance, as it is usually done in practice, the hypo- thesis illustrated in the diagram of Fig. 194 (c) can be adopted. We suppose that the unit load decreases linearly on the fin as well as on the rudder; in the fin it decreases from a maximum value u in the front part to a minimum 316 AIRPLANE DESIGN AND CONSTRUCTION 10 20 In Scale of Lengths LOADING DIAGRAM A2> 8.6 lb&./lln.In. Scale of Loads TABLE OF AREA WEI6HT5 IN POUNDS Area 1 8.2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 IT 2 Load 13.0 33.5 33.3 3T3 MO 3Q0 25.8 21.6 18.0 143 12.3 10 1 83 6.9 5.3 3e 05 31B Scale of Shears 168 336 lbs J I SHEAR DIAGRA^M Fig. 189. MAIN PLANES AND CONTROL SURFACES 317 10 20 in Scale of Lengths SHEAR DIAGRAM (1) J I I I 168 33& lbs Scale of Shears SHEAR DIAGRAM (2) 84 163 lbs. Scale of Shears. MOMENT DIAGRAM 550 1100 In lbs Scale of Moments. W Fio. 190 318 AIRPLANE DESIGN AND CONSTRUCTION value equal to 0.5 u in the rear part. In the rudder instead, the unit load decreases from u to zero. In order to determine the numerical value of u the aver- age value u„ of the unit load of the surfaces is usually given. This average value is assumed so much greater, as the airplane is faster; practically for speeds between 100 and 200 m.p.h. we can assume u^ = 0.167 expressing u^ in pounds per square foot. In our case we shall have about u„, = 25 lbs. per sq. ft. Then the surfaces of the fin and rudder are divided into sections (Fig. 194 (a)), and their areas are determined. In our case they are as given in the table of Fig. 194 (6); let us call a one of these areas and ku the corresponding unit load; the load upon it will be evi- dently aku. If A is the total area, we have that is :a X k X u = Au„ A XUr, :a X k The value u having been determined, we have all the MAIN PLANES AND CONTROL SURFACES 319 Fig. 194. ilHrLAXE DESIGN AND CONSTRUCTION AREAS IN SQ. FT. SECTIONS 1 2 3 4 5 e 7 d 9 10 II 12 13 14 S ELEVATOR 05Z 073 OdI 1.0 1.06 08 0.23 0.16 a/0 0.03 5.82 STABILIZa 0.80 0.80 1080 0.19 061 055 0.41 0.24 5.06 TOT/KU 1088 LOADING DIAGRAM (c) 5 6 7 B S> lO ft 12 13 14- WEIGHTS IN POUNDS \'>ECriOHS 1 2 3 U S 6 7 a 9 lO II 12 /3 14- s tUEVATORl 1.8 95 211 3A.5 46 6 596 13.7 95 5.9 i.d 204 {sTASIUZEIi 77.? 23.5 25.8 rj.d 252 23.0 18.2 11.3 m TOT^L. 381 FvG. 106. MAIN PLANES AND CONTROL SURFACES 323 elementary values aku, which in our case are as given in the table of Fig. 194 (d). These loads being obtained, we easily determine: (a) the center of loads of the fin, that is, what is usually termed the center of pressure of the fin, (6) the center of loads or center of pressure of the rudder, and (c) the center of loads of the entire system. It is then possible to determine the reactions on the various structures and consequently to make the calculation of their dimensions, following the usual methods. In Fig. 195 all the operations previously described are repeated for the stabilizer-elevator group, noting, however, that for this group we usually assume u„ = 0.22 X V that is, in our case Um = 35 lbs. per sq. ft. CHAPTER XIX STATIC ANALYSIS OF FUSELAGE, LANDING GEAR AND PROPELLER A. Analysis of Fuselage. — ^Let us consider the following particular cases: (a) Stresses in normal flight. (b) Stresses while maneuvering the elevator. (c) Stresses w^hile maneuvering the rudder. (d) Maximum stresses in flight. (e) Stresses while landing. (a) In normal flight the fuselage should be considered as a beam supported at the points where the wdngs are attached to it and loaded at the various joints of the trussing which make the frame of the fuselage. In these conditions it is easy to determine the shearing stresses and the bending moments when the weight of the various parts composing the fuselage or contained in it are known. Let us consider the case of a fuselage made of veneer. As w^e have seen in the first part of this book, such a fuselage has a frame of horizontal longerons connected by wooden bracings; this frame is covered with veneer, glued and nailed to the longerons and bracings. Let us suppose the frame to be the one shown in Fig. 196a. First the reactions of the various w^eights on the joints of the structure, and the reactions on the supports are calculated (Fig. 1966). It is then easy to draw the dia- gram of the shearing stresses (Fig. 196c), and of the bending moments (Fig. 196d), corresponding to the case of normal flight. (6) When the pilot maneuvers the elevator, the fuselage is subjected to an angular acceleration, which is easily calculated if the moment of inertia of the fuselage is known. 324 FUSELAGE, LANDING GEAR AND PROPELLER 325 215.7 la 15.15" I5J5" 23.10" 24.50" 23.20" 21.80" 2030" 18.30" n.50" 16.30" IQ.60'\ i is -1 6 I 7 :§ ^^ 5^^ ,oi ni isi 5 !i ^ 5t S S3 315 In. ^23IOIn\ ISi.lO In. 1 (") 1 > II § SPACE DIAGRAM . -| 30 60 In. Scale of Lencjths Ib^lbs SHEAR DIAGRAM 250 500 lbs Scale of Shears 5000 10000 Ib&.In Scale of Momenfa. 5 6 7 a 9 10 MOMENT DIAGRAM Fig. 19G. 326 AIRPLANE DESIGN AND CONSTRUCTION In Fig. 197 the graphic determination of this moment of inertia has been made; its result is / = 97,000 lb. X inch^. We shall suppose that a force equal to 1000 lb. acts suddenly upon the elevator. Then remembering the equation of mechanics do) ^ = ^'Xrf* where C = acting couple Ip = polar moment of inertia -J. = angular acceleration and as in our case C = 1000 X 177 = 177,000 lb. X inch / = 97,000 lb. mass X inch- v/e shall have do: 177,000 . oo 1 / 2 Tt = ^wm ^ '•'' '/''" This angular acceleration originates a linear acceleration in each mass proportional to its distance from the center of gravity and in a direction tending to oppose the rotation originated by the couple C. Thus, each mass will be sub- jected to a force, as illustrated for our example, in Fig. 198a. It is then easy to obtain the diagrams of the shear- ing stresses (Fig. 1986), and of the bending moments (Fig. 198c), originated by the forces of inertia which appear in the various masses of the fuselage, when a force of 1000 lb. is suddenly appUed upon the elevator. Let us note that the stresses thus calculated are greater than those had in practice; in fact for the calculation of the angular acceleration, the total moment of inertia of the airplane and not only that of the fuselage should have been introduced : therefore the angular acceleration found is greater than the effective one. However this approxi- mation is admissible, since its results give a greater degree of safety. FUSELAGE, LANDING GEAR AND PROPELLER 327 /= H.H'.Y = 100x50x19.4 = 97.000 lb. mass. K in^ Fig. 197. 328 AllWLANE DESIGN AND CONSTRUCTION 30000 eOOOO m/lbs Scale erf Mo me n+^. MOMENT DIA6RW4 r«3. 198. FUSELAGE, LANDING GEAR AND PROPELLER 329 (c) For maneuvering the rudder the same applies as for the elevator. The same diagrams of Fig. 198 may also be used for this case. SHEAR DIAGRAM FORTEN TIMES THE FUSELAGE WEIGHTS fTTTTTTMMIlL 6 ^ 8 9 10 II 12 .3 2 3 4^4'=' 5 Ullllllllllllllllll (b) SHEAR DIAGRAM FOR 762 LBS ON ELEVATOR rTTTTnTnTTTTTTTIIIII'llhTTi ^ 2 3 4*^ 4" 5 & 9 10 II 12 13 "' '"" '""T in] (c) SHEAR DIAGRAM FOR 300 LBS. ON RUDDER 'SHEAR DIAGRAM FOR COMBINATION OF RUDDER AND ELEVATOR LOADS AND TEN TIMES THE FUSELAGE WEIGHTS. {d) In order to calculate the maximum breaking stresses in flight, let us suppose that the breaking load is applied at the same time upon the wings, the elevator, and the 330 AiRPLAXE desk;:: and coxstruction rudder. This is equivalonl to make the following hypothesis : 1. to multiply the loads of the fuselage by 10, 2. to apply 762 lb. upon the elevator, 3. to apply 309 lb. upon the rudder. 30 60 In Scale of Lengths 30000 (60000 /lbs. Scale of Moments 2 3 4« 4" 5 6 7 ^ 9 10 II 12 MOMENT DIAGRAM FOR TEN TIMES THE FUSELA6E WEIGHTS ONLY. 2 3 4a4'> 5 6 7 8 9 10 11 12 13 MOMENT DIAGRAM FOR 762 POUNDS ELEVATOR LOAD ONLY. (^) ■^ ..rnTTlT!^ 2 3 4a 4b 5 e 7 8 9 10 II 12 MOMENT DIAGRAM FOR 306 POUNDS RUDDER LOAD ONLY. FiQ. 200. FUSELAGE, LANDING GEAR AND PROPELLER 331 MOMENT DIAGRAM FOR ELEVATOR LOADS AND TEM TIMES THE FUSELAOE WEIGHTS 40000 80OOO /bs. Scale of Moments 2 3 MOMENT DIAGRAM FOR COMBINATION OF ELEVATOR AND RUDDER LOADS AND TEN TIMES THE FUSALASE WEIGHTS Fig. 201. 332 AIRPLANE DESIGN AND CONSTRUCTION It is then easy to draw the diagrams of the shearing stresses in this case (Fig. 199, a, b, c), and consequently, through their sum, the diagram of the total shearing stresses in flight (Fig. 199rf). In order to calculate the maximum bending moments, it is necessary to consider separately those produced by vertical forces (loads on the fuselage and on the elevator), and those produced by horizontal forces (loads on the rudder). In Fig. 200 a, b, c, the bending moments are shown due respectively to 10 times the loads on the fuse- lage, to the load of 762 lb. on the elevator, and to the load of 306 lb. on the rudder. Fig. 201a shows a diagram obtained by the algebraic sum of the first two diagrams, Fig. 2016 shows the total dia- gram whose ordinates m" are equal to the hypotenuses of the right triangles having the sides corresponding to the ordinates m and m' of diagrams 200c and 20 la. Having obtained in this manner, the diagrams of the maximum shearing stresses and maximum bending mo- ments corresponding to the various sections, it is possible to proceed in the checking of the resistance of those sections. In Fig, 202 the checking for section 4-5 has been effectu- ated. For simplicity it is customary to assume that the longerons resist to the bending and the veneer sides to the shearing stresses. The stress due to shearing is given immediately, dividing the maximum shearing stress by the sections of the veneer. As for the stresses in the longerons, it is necessary to determine their ellipse of inertia. Let 1, 2, 3 and 4 be the four longerons constituting section 4-5. The maximum moment is equal to 216,600 in. lb., and its plane of stress makes an angle a with the vertical plain such that , _ Horizontal moment _ 16,400 _ ^ ^^^ ^^""^ ~ "Vertrcalmoment ~ 215:300 ~ """^^ Then a certain section is fixed for the longerons and with the usual methods of static graphics the moments of inertia of the four assembled longerons with respect to horizontal axis and to a vertical axis passing through the center of FUSELAGE, LANDING GEAR AND PROPELLER 30.70 Jn- 333 (a) TRANSVERSE SECTION AT 4-5 J v.y= 725 J^^ . "" A ■ X ^^ i / / k i / ►.^ i 1 o' U-i- 6 12 In Scale of Lengths. I I I 400 "600 In. Scale of Ellipse of Inertia X\ // / / \ \ ^. /// Tan ex. Mh 16400 Mv "" 215300 Ir = 825 in"^ {b) ELLIPSE OF INERTIA AT SECTION 4^5 Maximum Momenf ai- SccHon 216600 in lbs. Maximum Exireme Fiber Stress = ^.^^ - 4470 lbs/in^ Modulus of Rupture for Spruce =3700 lbs/in^ Facf or of Safety -^-x 10=217 Fio. 202. 334 AIRPLANE DESIGN AND CONSTRUCTION gravity of the system are determined (Fig. 202a). Then the eUipse of inertia may be drawn (Fig. 2026). The vector radius O'A' of such an eUipse which makes the angle a with the vertical gives the moments of inertia to be used in the calculations. In order to have the section modulus, it is necessary to draw B'O' the conjugate diameter to O'A'. From the center of gravity of the four longerons draw OB parallel to diameter O'B'; from the four points Mi, Mn, Ms, and Mi draw the parallels to OA, to meet the straight line OB in A^i, A^2, N3 and N^. By dividing the moments of inertia measured by O'A' by the largest of the 4 segments MiNi, M2N2, M3N3, MiNi the section modulus Zr is obtained. We can then compute the unit stresses and therefore the coefficient of safety. (e) In landing, the fuselage is supported by the landing gear and by the tail skid. The system of acting forces, with coefficient 1, is then that shown in Fig. 203. Fig. 204 shows the diagrams of the shearing stresses and bending moments corresponding to that case. Since, as it will be seen, the coefficient of resistance of the landing gear is usually taken between 5 and 6, it will suffice to multiply the preceding stresses by 6 and verify that the sections of the fuselage are sufficient. In our case these stresses result lower than the maximum considered in flight. B. Analysis of Landing Gear. — Let us consider the following particular cases: 1. Normal landing with airplane in line of flight. 2. Landing with tail skid on the ground. 3. Landing on only one wheel; that is, with the machine laterally inclined by the maximum angle which can be allowed by the wings. 4. Landing with lateral wind. Figs. 205, 206, 207 and 208 illustrate respectively the construction for those four cases, giving for each the ten- sion on compression stresses, the diagrams of the bending moments, and the member subjected to bending (axle and spindle). In the fourth case it has been assumed that the maximum horizontal stress is not greater than 400 lb. FUSELAGE, LANDING GEAR AND PROPELLER 335 15,—' FiQ. 203. 336 AIRPLANE DESIGN AND CONSTEVCTION 3r A r 1 ^ .5 s Ttl>.-i ^s ^ (N ^ S8§:2§SSS§8§S 8§§:2 °5|.2 f^^^ 2' 8"^' 2^8""^" 2^8'"'"" S'S^"^' ^ ^ rt ^ ,-4 j3 -^ .1-2 85 : :g5 : :8 8 : :8S§ : :i 1 ^£^ 19,2 11,0 19,2 11,0 27,7 14,8 ,h 2,0 5 h 2,01 bJ3 hO |S::|8::88::|°°::| oT cT .■ i oT o ! ; r>r rtT : : "^ : : "> ,_(i-H..,-HrH..(N,-l..hC ..hC c C . ^ -^ ^-'" ""£§• oo • -oo • -oo • -goo • -g ^ • ■ ^ • • •■ : m S § : : (£ CDOCOt^OO?Dt^COOcDt^^cDOCOI> ocooooiocooooio Sooooic'S ■* (M^OiMC<)^0(MC to -t^ m O) S.>< 2 «&>< 2 «a^ 2 ^•►cia^ 2 % -^ (N eo Tfi \ I 1 1 1 C c 1 342 AIRPLANE DESIGN AND CONSTRUCTION As for the criterions to be followed in the selection and computation of the shock absorbers, reference is to be made to what has been said in Chapter XVI. SECTION! A- A, SECTION S-B. Fig. 209. SECTION C-C AND D-D. C. Analysis of the Propeller. — In the following chapter it will be seen that for the airplane of our example the adoption of a propeller having a diameter of 7.65 ft. and a pitch of 9 ft. is convenient. We shall then see the aero- dynamic criterions which have suggested that choice. In this chapter we shall limit ourselves to static analysis of the propeller. This static analysis is usually undertaken as a checking; that is, by first drawing the propeller based upon data furnished by experience and afterward verifying the sections by a method which will be explained now. Supposing a propeller is chosen having the profile shown in Figs. 210, 211, 212 and 213. Fig. 210 gives the assembly of only one half the propeller blade the other half being perfectly symmetrical. Further- more it gives six sections of the propeller which are repro- duced on a larger scale in Figs. 211, 212 and 213. It FUSELAGE, LANDING GEAR AND PROPELLER 343 CM «^ ^ W^ VO 344 AIRPLANE DESIGN AND CONSTRUCTION Q J 2 3 Scale of Inches Fig. 211. /V V N^ \ h'0.72 ,n A= 7.37aq.in \ N S^ <^ ^ Ip- 19.7, ni Zp-S.ldm^ \ 1 \> \ V :> y Scale of Inche Fig. 212. <: ! h- 072 in A- 4.03sq.m. \ ^ '^^ -k H ^ Ip- 9.79 m* Zp-2.57m.^ 4"; ^ ^^ :^' i ^'-- > 18 3 I h - 055 in. A- 1.44 sqm /^ ^^ Ip- 2.18 m'' 7p-0.80m^ ^ ^ ^ ^ ^ > r:: ^ ' s ^ 5 3 7 Scale of Inches Fig. 213. FUSELAGE, LANDING GEAR AND PROPELLER 345 should be noted that in that type of propeller the pitch is not constant for the various sections, but increases from the center toward the periphery until the maximum value of 9 feet is reached which is the one assumed to characterize the propeller. The forces which stress the propeller in its rotation can be grouped into two categories: 1. Centrifugal forces which stress the various elements constituting the propeller mass. 2. Air reactions which stress the various elements consti- tuting the blade surface. If any section A of the propeller is considered, the forces which stress that section are then the resultants of the centrifugal forces and the resultants of the air reactions pertaining to that portion of the propeller included between section A and the periphery. In general, these resultants do not pass through the center of gravity of section A, so their action on that section produces in the most general case: 1. Tension stresses. 2. Bending stresses. 3. Torsion stresses. It is immediately seen that by giving a special curvature to the neutral axis or elastic axis of the propeller blade it is possible to equilibrate the bending moment in each section produced by the centrifugal force, with that produced by the air reaction. The stresses will then be those of tension and torsion, resulting thereby in a greater lightness for the propeller. We shall then proceed to find the total unit stresses and the curvature to be given to the neutral axis of the propeller blade. In order to proceed in the computations, it is necessary to fix the following elements : N = number of revolutions of the propeller, CO = corresponding angular velocity, Pp = power absorbed by the propeller when turning at A^ revolutions, 346 AIRPLANE DESIGN AND CONSTRUCTION A = density of the material out of which the propeller is to be made. In our case, N = 1800, and therefore CO = 2j^ = 188 1/sec. oO Furthermore Pp = 300 H.P. As for the material, the propellers can be made of walnut, mahogany, cherry, etc. Suppose that we choose walnut, for which A = 0.0252 lb. per cu. in. Let us now find the expression for the centrifugal force d^ which stresses an element of mass dM, and for the reaction of the air dR which stresses an element IdSoi the blade surface. The elementary centrifugal force d^ has, as is known, the expression d^ = dM X CO- X r since w^e can place dM = ^ X A X dr g where g is the acceleration due to gravity = 386 in. /sec. 2, A is any section whatever of the propeller, and dr is an infinitesimal increment of the radius. We shall then have d^ = "^ X i^- X A X r X dr 9 = 2.^ X A X r X dr from which ^ = 2.3 X A X r (1) dr Then by determining the areas of the various sections A, we shall be able to draw the diagram A = f (r) of Fig. 214, which by means of formula (1) permits drawing the other one d^ ., . whose integration gives the total centrifugal forces $ which stress the various sections (Fig. 215). FUSELAGE, LANDING GEAR AND PROPELLER 34< The elementary air reaction dR has the following expres- sion dR == KXdS X t/2 where K is a coefficient which depends upon the profile of the blade element and upon the angle of incidence, dS is a 16 20 24 2& 32 36 Radii -in Inches surface element of the blade, and U is the relative velocity of such a blade element with respect to the air. Calling I the variable width of the propeller blade, we may make dS ^ I X dr Fig. 215. on the other hand, velocity U is the resultant of the velocity of rotation r and of velocity of translation V, of the air- plane. The direction of these velocities being at right angles to each other we shall have [/2 = 0,2 X r2 + F2 348 AIRPLAXE DESIGN AND CONSTRUCTION therefore (IR = K X (co^ Xr- + Y'-) XlXdr from which dR dr = KX {o:- X r- + V) X I It is iminediately seen that it would be very difficult to take into consideration the variation of coefficient K from one section to the other, and therefore with sufficient Fia. 216. practical approximation K may be kept constant for the various sections and equal to an average value which will be determined. We note that dR being inchned backward by about 4° with respect to the normal to the blade cord, changes direction from section to section; it will consequently be convenient to consider the two components of dR, com- ponent dRt perpendicular to the plane of propeller rotation and component dRr contained in that plane of rotation (Fig. 216). The expression -j- can also be put in the following form : dr X (r^ + -') X ^ FUSELAGE, LANDING GEAR AND PROPELLER 349 350 AIRPLANE DESIGN AND CONSTRUCTION In our case to = 188 and V = 156 m.p.h. = 2800 in. per sec. (see page 374 for \'alue of V). On an axis ^A^ lay off the various radii (Fig. 217), make AB ^ - = loo = 14.9 perpendicular to AX, and from B draw segment BC. We shall evidently have BC' = AB' + AC' that is, BC' = ^ + r' Analogously by drawing BC, BC", etc. the squares of V- these segments will give the terms — r + r^. In this manner ~jj may be calculated, except for the constant K. Make ,. equal to CD, so that CD makes an angle of 4° with the prolongation of BC. Projecting D in E and F, we shall have dRr , dRt DE = -, and DF = -y- dr dr We may then draw the two diagrams -^-=/(r) and-^^-=/(r) whose integration gives the value of components Rr and Rt corresponding to the various sections; that is, gives the shearing stresses. For clarity, these diagrams have been plotted in two separate figures for components Rr and Ri, the former having been plotted in Fig. 217 and the latter in Fig. 218. The shearing stresses Rr and Rt being known, by means of a new integration, the diagrams of the bending moments Mr and M« can easily be obtained. It should be noted that the maximum value of Mr equals one-half of the motive couple. The power being 300 H.P. and the angular velocity w = 188, the motive couple will equal QfU) V r^rn 188 " ^^^^ "'• ^ ^^" ^ ^^'^^^ ^^'- ^ "'''^ FUSELAGE, LANDING GEAR AND PROPELLER 351 352 AIRPL.WE DESiaX AND CONSTRUCTION therefore Mr = I X 1 0,500 11). X inch = 5280 lb. X mch. The scale of moments is fixed in this manner and conse- quently that of the shearing stresses; and thus the value of the coefficient A' is also determined. Then, for each section, the resultant stress due to the centrifugal force, the shearing stresses Rr and Rt, and the moments Mr and M< due to the air reaction, are known. If the moment produced in any section whatever by the centrifugal force is somehow made to be in equilib- rium with the moments Mr and Mt, the deflection stresses will be avoided. Trac€ of ftJt Ptant of f*ofafion V - y/zlocttij of Aeroolana Let us first of all consider the moments Mt which are the greatest and consequently the most important, especially because they stress the blade in a direction in which the moment of inertia is smaller than that corresponding to the direction in which the blade is stressed by the bending moments Mr. dy Let us call -p the inclination of any point whatever of the neutral axis curve of the propeller. We shall then consider any section A whatever of the propeller blade, and the elementary forces d^ and dR applied to it. The elementary force d^ follows a radial direction, while the elementary force dRt follows a direction perpendicular to the plane of rotation of the propeller (Fig. 219); while FUSELAGE, LANDING GEAR AND PROPELLER 353 d$ is applied to the center of gravity of the element A X dr, the air reaction dRt is not applied to the center of gravity, but falls at about 33 per cent, of the chord. However, from known principles of mechanics, this force can be replaced by an elementary force dRt applied to the center of gravity, and by an elementary torsion couple dTt. The effect of this couple will again be referred to, and for the moment we shall suppose dRt applied to the center of gravity. Let us assume then the condition d$ _ dy dRt ~ dr Fig. 220. that is, that the resultant of c?$ and dRt be tangent to the neutral curve of the propeller blade. Under these condi- tions, supposing that this be true for every element A X dr of the propeller, all the various sections will be stressed only to tension. Since we may write d^ _ d^/dr dRt ~ dRt/dr it is easy to draw the diagram f = /« and, by graphically integrating this diagram, obtain y =f{r) which gives the shape that the center of gravity axis of the propeller blade must have in elevation (Fig. 220). 354 AIRPLANE DESIGN AND CONSTRUCTION With an analogous process, the shape in plan is found by considering the forces d^ and dRr] in Fig. 221 the rela- dv tive diagrams have been drawn for , = /(r) and y = /(r). Thus the propeller may be designed. In Fig. 210 the neu- tral axis has been drawn following this criterion. 1 j \ |jr^ ^ .^: -4- --U T i i M 24 2G Radii in Inches -fk Let us now determine the unit stresses corresponding to the case of normal flight. These stresses are of two types: 1. tension stresses, 2. torsion stresses. 1 600 i -^ 400 200 1 i n : 3 ! 4 ; 6 j — / "^ ^ ^¥ R?'R? 1 i ! ^ ^v / '' 1 i ^ 1 I 1 / 1 i 1 ^ \^ 1 i i 1 K, i i s Ni 1 i i s, 20 24 28 Radii in Inches Tension stresses are easily calculated, in fact, for every section A they are equal to f, = _ _ In Fig. 222 the diagram of /i obtained by the preceding equation has been drawn. FUSELAGE, LANDING GEAR AND PROPELLER 355 As to the torsion stresses, they depend only upon the air reaction. Let us consider a section A and the air reac- tion dR which acts upon the blade element I ■ dr correspond- ing to this section. Evidently dR = {dR:- + dRr')^- The point of application of dR falls, as we have seen, at 0.33 of the width of the blade l; therefore dR will in general produce a torsion about the center of gravity; let us call Fig. 223. h the lever arm of the axis of dR with respect to the center of gravity; the elementary torsional moment will be dT = hX dR = hX idR^ + dRr^y' and consequently f-x[(fr-(fi The values of h are marked on the sections (Figs. 211, 212 and 213) ; the values ,- and -~ are given by the diagrams of Figs. 217, 218; thus in Fig. 223 the diagram may be drawn of dT «. V and by integrating, that of T = fir) 356 AIRPLANE DESIGN AND CONSTRUCTION 12 3 4 5 6 48 40 32 ~' 16 1400 400 ?00- 0: 16 20 24 28 32 36 40 44 Fig. 224. 2 3 4 5 6 . 1 j 1 1 ! ! 1 i ^ < 1 k i i / 1 i \| ^ j / 1 1 1 j ) i i ! 1 i 1 i 40 32 16 5 16 20 24 Fig. 225. 32 36 40 44 16 20 24 28 32 36 40 44 Fig. 226. FUSELAGE, LANDING GEAR AND PROPELLER 357 It is now necessary to determine the polar moments Ip of the various sections ; to this effect it suffices to determine the ellipse of inertia of the various sections by the usual methods of graphic analysis; then calling I^ and ly the moments of inertia with respect to the principal axis of inertia, we will have /, = {IJ + Iy^y For each section (Figs. 211, 212 and 213), we have shown the values of the area, of the polar moment Ip and of Zp = —■ In Fig. 224 the diagram Ip for the various sections and the diagram -- = Zp have been drawn. Dividing, for each section, the corresponding values of the total moment of torsion T by the values of the section modulus for torsion Z, we shall have the values /2 of the unit stresses to torsion (Fig. 225). It is immediately evi- dent that this method is exact only when the neutral axis of the propeller is rectilinear and in the direction of the radius, which, however, does not correspond to practice. In effect though, as the torsion stresses represent a small fraction of the total stresses, the approximation which can be reached is practically sufficient. When the unit stresses /i and /a to tension and torsion are known, the total stress ft is determined by the formula ft = 0.35 X /i + 0.65 X ifi' + 4 X a^- X /2^)^' where ^ modulus o f rupture in tension _ ,^ 7 1 .3 modulus of rupture in shearing Then the diagram which gives ft for the various sections may be drawn (Fig. 226). It is seen that the value of the maximum stress is equal to 1280 pounds per square inch; that is, to about I9 the value of the modulus of rupture. As a safety factor between 4 and 5 is practically suffi- cient for propellers, it may be concluded that the aforesaid sections are sufficient. CHAPTER XX DETERMINATION OF THE FLYING CHARACTERISTICS Once the airplane is calculated and designed, it becomes possible to determine its flying characteristics. The best method for this determination would undoubtedly be that of building a scale model of the designed airplane and of testing it in an aerodynamic laboratory. This, however, is often impossible, and it is therefore necessary to resort to numerical computation. Let us remember that the aerodynamical equations bind- ing the variable parameters of an airplane are W = 10-'' \AV' and 550Pi = L47 X 10-^5^ + cL- \ s. s s "^V~~ -- o^ \ mmb = ^ : ^^ = = ■:zzi fefr 1 izz: : z^ § ^ P 1 = 'ZZZ. :z^ EEESE--L^ / \ ^ : Z — ::. .^^ X E = = = •is-*-^ = = V- V -■ ^ \ ^± — A — — r - ^ — \ _ -N ' \^ <-.\o o:^ ;>^ S^Q^ s. ^ \ ^ _^Sv:. \ -^ s^\W ^ s- iX \, s v"^ s v> \ rv-'^^ N^ s s' n^ ~ °« V °c.^ -<> '; -^ r- \ ' '\::. (^ \ K •{ iS s\ s \ . \ \^ ^\^ o \ f;f-X=^r. \, s,^/^ ^^^^'v. \ V s N A A% \ ^ \ N.^ A !o CV i S \ s ■■■% "■y \ ^ s \ ^ ' Q s\ S - ^ ll^' \ A % A A... 360 AIRPLANE DESIGN AND CONSTRUCTION Then, X, 5 and o- being known, it is possible to obtain a pair of values of A and A corresponding to each value of i, and the logarithmic diagram of A as function of A can then be drawn. Let us suppose that X and 8 are given by the diagram of Fig. 155 (Chapter XVII). The value of a is calculated by remembering that cr = 2X X .4 that is, it is equal to the sum of the head resistances of the various parts of which the airplane is composed. This, however, does not always hold true, because of the fact that the head resistance offered by two or more bodies close to each other and moving in the air is not always equal to the sum of the head resistances the bodies encounter when mov- ing each one separately, but it can be either greater or smaller. Thus, an exact value of the coefficient Z / 2 y Z y 04 X 0.4 ^ ^ 7 if 01 . . n __ & 9 n D Fio. 229. 2.0 6 y 'o .4 0.2 propellers of the best known type to-day, with the indica the value of maximum efficiency, adopting as units, how tion of the values ~, a = \jy^ and yz corresponding to 364 AIRPLANE DESIGN AND CONSTRUCTION ever, m.p.h. for V, r.p.m. for n, feet for p and D, and H.P. for P. Since we want p = 0.815, and consequently we have seen that Fmax. = 1^3 m.p.h., the diagrams of Fig. 229 allow us to obtain the number of revolutions and the diam- eter of the propeller. In fact for p = 0.815 we find ^ = 11.4 X 10-3 1.18 D V Knowing that V = 153 and P = 300 H.P. we have as unknowns n, D and p, whose values are defined by the preceding equations. Solving these equations we obtain: n = 1690 revolutions per minute, D = 7.92 feet, and p = 9.35 feet. Since the number of revolutions found is very near to the average R.p.m. of the engine, it will be convenient in our case to connect the propeller directly with the crank-shaft. Having obtained the propeller, it is necessary to know the characteristic curve of the propeller family to which it be- longs. It should be remembered that all propellers having the same blade profile and the same ratio between pitch and diameter, have the same characteristics (see Chapter IX). Let the characteristics of a family to which our propeller belongs be those given in the logarithmic diagram of Fig. 230. Then with the same criterions which have been explained in Chapters, IX, XIII, and XIV, it is possible to draw the diagram of pP2 as a function of V for any altitude ; for instance, the altitudes 0, 16,000, 24,000, and 28,000 ft. For this purpose the diagrams have been drawn in Fig. 230, DETERMINATION OF THE FLYING CHARACTERISTICS 365 which give the values -^ffb corresponding to these altitudes and in Fig. 231 the diagrams of P2 of the same heights. 6x10''^ 8)(I0'3 10x10'^ 12x10'^ 14x10'' V nD 60 70 80 90 100 150 200 t I I I I I I I lihiillrtri l I I I I I I J_U V. m.p.h. Fig. 230. By using these diagrams those of Fig. 232 have been drawn from which it is seen that the maximum velocity at sea level is only 150 m.p.h. with a corresponding useful power of 225 H.P. This depends upon the fact that a pro- 366 AIRPLANE DESIGN AND CONSTRUCTION peller has been directly connected which should have been used with a reduction gear having a ratio of , o^„ We will Fio. 231. immediately see that if we wish to adopt a direct connection it is more convenient to choose a propeller which, although DETERMINATION OF THE FLYING CHARACTERISTICS 367 ^ V- — \ s. \ \ \l V- == = = - ^ i 1 = E = = = = = = S - ; — ^ \ E — \ \ 1 I § = = = = = E^^ iz: - r- -N ^ = = EE :.S = = = = "V — — -N Sr — L'* * V — — ' — \ s\ ■i- ^ rs S V s y \V V- \ cC \ s^ \- r"^ s. ^ o o\ G- A — . \ . :^ \ S \ \ S \ ^ s o \ \ =- ° s \ — — "- ...fc — V, \J 1 \ \, V l^v?^ \1 N o^ .--^^.^^ X ^ &^>xz^-^^-:^ \ w K'^ k^ £Si 56 In ^.^ ^ \ X o _ ^^^K^ ^ \ ^^Vl V \ W^ \ \ r a \: 1^ - 9.^ \ >J ^s ^ ^ \ 1 "^ } ^ <'.^r 1 ^ ".■\ \" >Li.^ \ ^ \r- \ -^\ \ ^- \ '^.-H H 368 AIRPLANE DESIGN AND CONSTRUCTION belonging to the same family, is of smaller dimensions so as to permit the engine to reach the most advantageous num- ber of revolutions and therefore to develop all the power of which it is capable. It is interesting, however, to first study the behavior of the propeller having a diameter of 7.92 ft. in order to compare it to that of a smaller diameter. The diagrams of Fig. 232 show that the maxinmm hori- zontal velocities at the various altitudes with the propeller of 7.92 ft. of diameter are at ft., 150 m.p.h. at 16,000 ft., 148 m.p.h. at 24,000 ft., 144 m.p.h. at 28,000 ft., 138 m.p.h. These diagrams allow us to obtain the differences pP2 — Pi and therefore to compute the values of the maximum chmbing velocities v at the various heights. These veloc- ities are plotted in Fig. 233; on the ground the ascending velocity is equal to 29.5 ft. per second. At 28,000 ft. it is equal to 1.7 per second; that is, equal to a Uttle more than 100 ft. per minute; the height of 28,000 ft. must then be considered as the ceiling of our airplane if equipped wdth the above propeller. From the diagram of y = f{H) it is easy to obtain that of - = f{H) (Fig. 234a), and therefore by its integration, we obtain that of t = f{H), which gives the time of climbing (Fig. 2346). It can be seen that with this particular pro- peller, the airplane can reach a height of 28,000 ft. in 3000 seconds; that is, in 50 minutes. Let us now suppose that a propeller is adopted of such diameter as to permit the engine to reach its maximum number of revolutions. By using the diagram of Figs. 227 and 230 we find with easy trials and by successive approxi- mation that the most suitable propeller will have a diam- eter of 7.65 ft. and therefore as w = 1.18, a pitch of about DETERMINATION OF THE FLYING CHARACTERISTICS 369 w'U ~ ""^ ~ ■ ' ~ k > 1 V ■ \ > ^ 25 \ k \ V V > V \ 20 L \ 1 \ \ \ \ V 15 ^ V \ V \ s r > V 10 \ s \ s \ \ \ 5 \ \ V V , ^ 10000 20000 30000 H=F4-. 370 AIRPLANE DESIGN AND CONSTRUCriON ~ ~ ~ ~ ~ - n t; ' 1 To'SS'F ; 1 i T lO T\ £ 03 w / P / r -lf» / A / J / / y ^ ' 1 <■ I 0^ - — - " o - _ 10000 20000 30000 H=F+. ^""" -r- -r- -T ± J 4- / t X 7 J r-^gV cL ^ _. ^0- ^3 ^ -/^ ^ z « ^ -^ S J n- / / z 7 &00 - ^ y > ^ : .^^ ^•^ ^ ^ ** ..j** ^ -"'^ 10000 20000 30000 H=F+ FiQ. 234. DETERMINATION OF THE FLYING CHARACTERISTICS 371 T t.., ■ ^ li! 1 1 1 1 1 : ~ Tt- - < \ f^ ^ T ' \i V- g \ OJ ^ 1 Li...i,t E:::::i.7 5::::; =1= :: :: ■< . s 'I ?,:::::1||.::: *::::^::.^: --4=^- S_ K:i^ff:tf|:: 5::iii: - 'i" vTb'^ 11""" 4: \ ' . S l^p ^i ::;;;:;;::;:i::::E^?s mi : 3: :S s:==i = =^ ^ X — °OJ ~ .J _L :;:: v"-^ :^i:i-iog7 '"■"'-<»-. \ ^ •-* ^ \ 4.^ -A'"" \ ^ \ 0^ ?^ :::::::t,> .1^^ s 5^ - d . ^^ - - A n> % \ s '^ \ . <1 3 372 AIRPLANE DESIGX AND CONSTRUCTION \ 30 25 20 I ?^ lOOOO 20000 30000 M=F+. FiQ. 236. DETERMINATION OF THE FLYING CHARACTERISTICS 373 .-H pe A U.to r-'7gV ^ ~o 59 F f i ^ 0.2 j- T' / X i' . "^ y ^»*^ 10000 20000 30000 H= F+. 7 f 7^- 55 ^ y ;^ ^ 1 z 1 y^ y"^ ^'^ ^^ y^ y ^y'^ - ^^^ ^•^^ 10000 20000 3000C H=F+ Fig. 237. 374 AIRPLANE DESIGN AND CONSTIUCTION 9 ft. This propeller is the one for which the static analysis was given in the preceding chapter. For such a propeller the logarithmic diagi-ams of pP^, the diagram v = f{H) and those of - = /(//) and t == f(H) have been plotted in figures 235, 236 and 237a6 respectively. The diagrams of Fig. 235 show that the new maximum velocities are at ft., 156 m.p.h. at 16,000 ft., 155 m.p.h. at 24,000 ft., 150 m.p.h. at 28,000 ft., 144 m.p.h. The diagram of Fig. 236 shows that at an altitude of 28,000 ft., V = 3.7 ft. per second = 222 ft. per minute; that is, the ceiling has become greater than 28,000 ft. The diagram of Fig. 237 finally shows how the height of 28,000 ft. is reached in 2400 seconds; that is, in only 40 minutes. The second propeller, therefore, is decidedly better than the first one. The question now arises: What is the maximum load that can be hfted with our airplane? It is therefore neces- sary to suppose the efficiency of the propeller to be known. Supposing p = 0.815, then the maximum useful available power will be 244 H.P. Let us again examine the diagram A = /(1. 47 A) (Fig. 238). For our airplane at the point corresponding to 244 H.P. on the scale of powers, draw a perpendicular to meet tangent t in B drawn from the diagram parallel to the scale of velocities. From B draw the parallel BC to the scale of powers. Point C gives the maximum theoretical load which the airplane could hft, and which in our case would be about 7300 lb. The corresponding velocity is measured by segment BD which, read on the scales of velocity, gives 7 = 132 m.p.h. Practically, however, the airplane cannot lift itself in this condition as it is necessary to have a certain excess of power in order to leave the ground. DETERMINATION OF THE FLYING CHARACTERISTICS 375 o ''^ - \ ■ CO - :::::^;;;::::: : : - \ o — ^ - in T \ - d . < I— 1 \ ^ 1 ---- -^-— o - : ;:;::d:^ ::_::_ 2 . o o '< = - = 5 ;:::-:::;:EE:-sPg=-^-- cL . DC — Tb- ^ ^ Vri ■ V __ _|V- A; O*- \ ■ - A ^X \ °^ QrT^ 'M ^ g, \ r^ ^ \r^ \N^ NS Jv^*^ \ A ^sn \ >K ... _ v^ o , o \ ^ \ 7^S ^ _ ^ ;^-S\ - V \ ?^ - .-} a \ ^ \ "\ L^H Q — ^ _1_ \ Cl -A CO Or?^ H- -^ ^ r-> ... C7^N^ o " ^ H;_ Q) > -4-^J^ ■ D c Y X_. O -L^/^U --^l---^---------_£-^ ~~^ """'^ -f^'^ol >^ ^v" _y^o::: f h N. %n 1 1 <:o° =o2 376 AIRPLANE DESIGN AND CONSTRUCTION Supposing then we fix the condition that the airplane should be able to sustain itself at a height of 10,000 ft. As H = 60,720 log ^ for H = 10,000 we will have m = 0.085, therefore in this case the useful power becomes 0.815X0.685X300 = 167.5 H.P. Let us then draw a perpendicular from A' corre- sponding to 167.5 H.P. to meet tangent t in B'. From B' draw the parallel to the scale of power. From origin of the diagram draw a segment 00' parallel to the scale of M and which measures ^ = 0.685; from 0' raise the per- pendicular until it meets the horizontal line in C drawn from BB'; from C draw the parallel to 00' up to C"; this point defines the value of the maximum load which our airplane could lift up to 10,000 ft. and which in our case is about 4100 lb. The corresponding velocity is measured by B'D and is equal to 116 m.p.h. Let us now study what the effect would be of a diminu- tion of the lifting surface. Until now we had supposed that A =265 sq. ft.; that is, we had a load of 8 lb. per sq. ft. Now supposing this load is increased up to 10, 12, 14, and 16 lb. per sq. ft. respectively; that is, the lifting surface is reduced from 265 sq. ft. to 214, 178, 153 and 134 sq. ft. successively. For each of such hypotheses it will be necessary to calculate the new values of A and A ; the results of these calculations are grouped in Table 47. By means of this table the diagrams of Fig. 239 have been drawn; let us then suppose that in each case a pro- peller having the maximum efficiency of 0.815 has been adopted. The useful power will be 244 H.P. ; drawing from A, the point which corresponds to this power, the parallel p to the scale of velocity, on the intersection with this line and the diagram we shall have the point which defines the maximum velocities; drawing the tangent t parallel to the scale of V from each of the various curves the points of tangency which determine the minimum velocities will be obtained. DETERMINATION OF THE FLYING CHARACTERISTICS 377 o ~ o " o o in ci • c - - - ^ \ > q ^ - <2/' \ ^ \ 5: ^ = --: \ --\ — U : — \-7: 1 i = j --^ 1 = ^ > ^ ^|:-:4: rmi 1 E in — §- :r. o = = ^ p ^ = =^^ -^- ^ — ^ -\ — — ' — - - d \^fc r^ --\ vV*' MA^a.^ \ ^ . -^ K*"^ K% 'k N. ^ 5s, ^ Lf) s \ % k s» \ \ - O v \ s o \ s^ <> «" """ " ^ s ]^ sf^ ^ -5k -.-.-.-.Ih -.- ^ F^ s;::.|;::- s s^ |;...,.|:: ?? c ^ i:|::-S -, ci A ^ \ \ ^ \ -o \ CrA / \ \ \ -^ A I -^lllilll . o 5.9 378 AIRPLANE DESIGX AND CONSTRUCTION Table 47 = I and W + W" + W" we shall ha\'e W Xi =- IX W Let us proceed analogously for the case of Fig. 240. In this manner two lines v' and v" are obtained whose inter- section defines the center of gravity. To eliminate eventual errors and to obtain a check on the work it is convenient to determine the third line v'", by / \ Fig. 248 balancing the machine on the wheels; v'" will then be the vertical which passes through the axis of the wheels (Fig. 247). The three Unes v', v" and v'" must meet in a point (Fig. 248). Ill The flight tests include two categories of tests, that is; A. Stability and maneuverability tests. B. Efficiency test. A. The purpose of the stability tests is to verify the balance of aeroplane when (a) flying with engine going, and when volplaning, (h) in normal flight and during maneuvers. Chapter XI has stated the necessary requisites for a well- balanced airplane, therefore a repetition need not be given. SAND rESTS^WEIGHINa^FLIGHT TESTS 391 W & H £ ^ 1 ^3 < ^ £ 1 1-1 - - ll II a o 1 a ft 1a - i < i w - - Q E i Q a c ^ If CO ^ ca E Average Motor speed r.p.m. - 1 - 111 a •^1 P JO uottBJnQ 1 ^' •draa^ JaiT3Ai 1 gntp'B8J -uiopaadg •uida 3™TX •duia'} J3:>B^ ~|" § ' iuip^aj -uiopaadg 01 ■md-a auiix 1 •duI^:^ la-i^M. "f SuipuaJ -mopaadg •ui dH 1 auiTX 8 ■duiai Jai'BAi. 3uipT53J -mopaadg ii ■uidH — - auiTX 1 CD •dina'j Ja^BA^. ^" ^g autpTJaJ -uiopaaclg "^ •uidH •■3 auiix 1 •dina^ Ja-JB^ Suip^ai -uiopaadg 1" mdy^ a.uTX § •duia^ MiB_/v^ anipBaJ -mopaadg •uidH a-HtX s •draa^ M'ivj^ SuipTsaj -uiopaadg ui-dy^ aui'X 1 •duta^ -laiBAi Sutpcaj -mopaadg ■uid-a s^TX ; 1 392 AIRPLANE DESIGN AND CONSTRUCTION The same may be said of maneuverability tests, whose scope is to verify the good and rapid maneuverabihty of the airplane without an excessive effort by the pilot. B. The scope of the efhciency tests is to determine the flying characteristics of the airplane, that is, the ascensional and horizontal velocities corresponding to various loads and types of propellers which might eventually be wanted for experiments. Table 51 gives examples of tables that show which factors of the efficiency tests are the most important to determine. APPENDIX The following tables are given for the convenience of the designer: Tables 52, 53, 54, 55 and 56 giving the squares and cubes of velocities. Table 57 giving the cubes of revolu- tions per minute and per second. Table 58 giving the 5th powers of the diameters in feet. Table 52. — T.\ble of Squares and Cubes of Velocities V y2 V3 Miles per hr. Ft. per sec. Miles per hr. Ft. per sec. Miles per hr. Ft. per sec. 50 73.33 2,500 5,377.7 125,000 394,430 51 74.80 2,601 5,595.0 132,651 418,510 52 76.27 2,704 5,817.1 140,608 443,670 53 77.73 2,809 6,042.0 148,877 469,640 54 79.20 2,916 6,272.6 157,464 496,790 55 80.67 3,025 6,507.6 166,375 524,970 56 82.13 3,136 6,745.3 175,616 553,990 57 83.60 3,249 6,988.9 185,193 584,280 58 85.07 3,364 7,237.0 195,112 614,270 59 86.53 3,481 7,487.5 205,379 647,890 60 88.00 3,600 7,744.0 216,000 681,470 61 89.47 3,721 8,004.9 226,981 716,200 62 90.93 3,844 8,268.2 238,328 751,830 63 92.40 3,969 8,537.8 250,047 788,890 64 93.87 4,096 8,811.8 262,144 827,140 65 95.33 4,225 9,087.8 274,625 866,340 66 96.80 4,356 9,370.2 287,496 907,040 67 98.27 4,489 9,657.0 300,763 948,990 68 99.73 4,624 9,946.0 314,432 991,920 69 101.02 4,761 10,205.0 328,509 1,030,920 70 102.67 4,900 10,541.0 343,000 1,082,260 71 104 . 13 5,041 10,843.0 357,911 1,129,090 72 105.60 5,184 11,152.0 373,248 1,177,580 73 107.07 5,329 11,464.0 389,017 1,227,450 74 108.53 5,476 11,779.0 405,224 1,278,350 75 110.00 5,625 12,100.0 421,875 1,331,000 76 111.47 5,776 12,426.0 438,976 1,385,080 77 112.93 5,929 12,753.0 456,533 1,440,220 78 114.40 6,084 13,088.0 474,552 1,497,200 79 115.87 6,241 13,426.0 493,039 1,555,654 80 117.33 6,400 13,766.0 512,000 1,615,203 393 394 AIRPLANE DESIGN AND CONSTRUCTION Table 53. — T^vble of Squauk;:; and Cubes of Velocities V ^ ■J I -3 Miles per hr. Ft. per see. Miles per hr. Ft. per sec. Miles per hr. Ft. per sec. 81 118.80 6,561 14,113 531,441 1,076,680 82 120.27 6,724 14,465 551,368 1,739,690 83 121.73 6,889 14,818 571,787 1,803,820 84 123.20 7,056 15,178 592,704 1,869,960 85 124.67 7,225 15,543 614,125 1,937,700 86 126.13 7,396 15,909 636,050 2,006,570 87 127.60 7,569 16,282 658,503 2,077,550 88 129.07 7,744 16,659 681,472 2,150,190 •89 130.53 7,921 17,038 704,960 2,224,000 90 132.00 8,100 17,424 729,000 2,299,970 91 133.47 8,2S1 17,814 753,571 2,377,670 92 134.93 8,464 18,206 778,688 2,456,550 93 136.40 8,649 18,605 804,357 2,537,720 94 137.87 8,836 19,008 830,584 2,620,650 95 139.33 9,025 19,413 857,375 2,704,800 96 140.80 9,216 19,825 884,736 2,791,310 97 142.27 9,409 20,241 912,673 2,879,650 98 143.73 9,604 20,658 941,192 2,969,220 99 145.20 9,801 21,083 970,299 3,061,260 100 146.67 10,000 21,512 1.000,000 3,155,180 101 148.13 10,201 21,943 1,030,301 3,250,340 102 149.60 10,404 22,380 1,061,208 3,348,070 103 151.07 10,609 22,822 1,092,727 3,447,750 104 152 . 53 10,816 23,265 1,124,864 3,548,670 105 154 . 00 11,025 23,716 1,157,625 3,652,260 106 155.47 11,236 24,171 1,191,016 3,757,850 107 156.93 11,449 24,627 1,225,043 3,864,720 108 158.40 11,664 25,091 1,259,712 3,974,340 109 159.87 11,881 25,558 1,295,029 4,086,030 110 161.33 12,100 26,027 1,331,000 4,199,000 APPENDIX 395 Table 54. — Table of Squares and Cubes of Velocities r 1-2 T'3 Miles per hr. Ft. per sec. Miles per hr. Ft. per sec. Miles per hr. Ft. per sec. Ill 162.80 12,321 26,504 1,367,631 4,314,820 112 164.27 12,544 26,C85 1,404,928 4,432,770 113 165.73 12,769 27,466 1,442,897 4,552,010 11-1 167.20 12,996 27,C56 1,481,544 4,674,220 115 168.07 13,225 28,450 1,520,875 4,708,580 116 170.13 13,456 28,944 1,560,8C6 4,924,790 117 171.60 13,689 20,447 1,601,613 5,053,080 118 173.07 13,924 29,953 1,643,032 5,184,000 119 174.53 14,161 30,461 1,685,159 5,316,310 120 176.00 14,400 30,976 1,728,CC0 5,451,780 121 177.47 14,641 31,496 1,771,561 5,589,520 122 178.93 14,884 32,016 1,815,848 5,728,620 123 180.40 15,129 32,544 1,860,867 5,870,^60 124 181.87 15,376 33,077 1,906,624 6,015,060 125 183.33 15,625 33,610 1,953,125 6,161,700 126 184.80 15,876 34,151 2,000,376 6,311,120 127 186.27 16,129 34,697 2,048,383 6,462,C20 128 187.73 16,384 35,243 2,097,152 6,616,080 129 189.20 16,641 35,797 2,140,689 6,772,720 130 190.67 16,900 36,355 2,197,000 6,931,820 131 192.13 17,161 36,914 2,248,0!; 1 7,0:2,280 132 193 . 60 17,424 37,481 2,299,968 7,256,320 133 195.07 17,689 38,052 2,352,637 7,422.860 134 196.53 17,956 38,624 2,406,104 7,500,7tO 135 198.00 18,225 39,204 2,460,375 7,762,3£0 136 199.47 18,496 39,788 2,515,456 7,936,570 137 200.93 18,769 40,373 . 2,571,353 8,112,120 138 202.40 19,044 40,966 2,628,072 8,291,470 139 203.87 19,321 41,563 2,685,619 8,473,440 140 205.33 19,600 42,160 2,744,000 8,656,800 ^') 396 AIRPLAXK DESIGN AND CONSTRUCTION Tablk 55. — Tahi.k of Squares axd Cubes or Velocities Miles per hr. Ft. per sec. Miles per hr. Ft. per sec. Miles per hr. Ft. per sec. 141 206.80 19,881 42,760 2,803,221 8,844,050 142 208.27 20,164 43,376 2,863,288 9,034,000 143 209.73 20,449 43,987 2,924,207 9,225,330 144 211.20 20,736 44,605 2,985,984 9,420,670 145 212.67 21,025 45,229 3,048,625 9,618,750 146 214.13 21,316 45,852 3,112,136 9,818,220 147 215.60 21,609 46,483 3,176,523 10,021,800 148 217.07 21,904 47,119 3,241,792 10,228,200 149 218.53 22,201 47,755 3,307,949 10,435,900 150 220.00 22,500 48,400 3,375,000 10,648,000 151 221.47 22,801 49,049 3,142,951 10,862,800 152 222.93 23,104 49,698 3,511,808 11,079,100 153 224.40 23,409 50,355 3,581,577 11,279,500 154 225.87 23,716 51,017 3,652,264 11,523,300 155 227.33 24,025 51,679 3,723,875 11,748,200 156 228.80 24,336 52,349 3,796,416 11,977,600 157 230.27 24,649 53,024 3,869,893 12,209,900 158 231.73 24,964 53,699 3,944,312 12,443,600 159 233.20 25,281 54,382 4,019,679 12,682,000 160 234.67 25,600 55,070 4,096,000 12,923,300 161 236.13 25,921 55,757 4,173,281 13,166,000 162 237.60 26,244 56,454 4,251,528 13,725,800 163 239.07 26,569 57,154 4,330,747 13,663,900 164 240.53 26,896 57,855 4,410,944 13,915,800 165 242.00 27,225 58,564 4,492,125 14,172,500 166 243.47 27,556 59,278 4,574,296 14,432,300 167 244.93 27,889 59,991 4,657,463 14,693,400 168 246.40 28,224 60,713 4,741,632 14,959,600 169 247 . 87 28,561 61,440 4,826,809 15,229,000 170 249.33 28,900 62,166 4,913,000 15,499,700 APPENDIX Table 56.— Table of Square.s and Cubes of Velocities 397 Miles ; hr. 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 Ft. per sec. Miles per hr. Ft. per sec. ' Miles per hr. 1 1 Ft. per sec. 250.80 29,241 62,901 5,000,211 15,775,800 252.27 29,584 63,640 5,088,448 16,054,500 253.73 29,929 64,379 5,177,717 16,334,850 255.20 30,276 65,127 5,268,024 16,620,500 256 . 67 30,625 65,880 5,359,375 16,908,500 258.13 30,976 66,631 5,451,776 17,199,500 259.60 31,329 67,392 5,545,233 17,495,000 261.07 31,684 68,158 5,639,752 17,794,000 262 . 53 32,041 68,922 5,735,339 18,094,300 264.00 32,400 69,696 5,832,000 18,399,800 265 . 47 32,761 70,474 5,929,741 18,709,800 266 . 93 33,124 71,252 6,028,568 19,029,750 268.40 33,489 72,038 6,128,487 19,335,400 269.87 33,856 72,830 6,229,504 19,655,500 271.33 34,225 73,620 6,331,625 19,975,500 272.80 34,596 74,420 6,434,856 20,301,900 274.27 275.73 34,969 35,344 75,224 76,027 6,539,203 6,644,672 20,631,500 20,962,900 277.20 35,721 76,840 6,751,269 21,300,000 278.67 36,100 77,657 6,859,000 21,640,750 280.13 36,481 78,473 6,967,871 21,982,500 281 . 60 36,864 79,299 7,077,888 22,330,500 283.07 37,249 80,129 7,189,057 22,682,000 284 . 53 37,636 80,957 7,301,384 23,034,750 286.00 38,025 81,796 7,414,875 23,393,500 287.47 38,416 82,639 7,529,536 23,756,000 288 . 93 38,809 83,481 7,645,373 24,120,000 290 . 40 39,204 84,332 7,762,392 24,490,850 291.86 39,601 85,182 7,880,599 24,861,500 293.33 40,000 86,043 8,000,000 25,239,000 398 AIRPLAXE DESIGN AND CONSTRUCTION Table 57. — Table of Cubes op R.p.m. and R.p.s Per min Per sec. Per min. Per sec. 500 8.33 125.0X10" 578.7 550 9.17 166.4X10" 768.5 600 10.00 216.0X10" 1,000.0 650 10.83 274.6X10" 1,271.4 700 11.67 343 . X 10" 1,588.0 760 12.50 421.9X10" 1,953.3 800 13.33 512.0X10" 2,370.4 850 14.17 614.1X10" 2,843.2 900 15.00 729 . X 10" 3,375.0 950 15.83 857.4X10" 3,969.4 1,000 16.67 1,000.0X10" 4,629.6 1,050 17.50 1,157.6X10" 5,359.1 1,100 18.33 1,331.0X10" 6,162.0 1,150 19.17 1,520.0X10" 7,025.0 1,200 20.00 1,728.0X10" 8,000.0 1,250 20.83 1,953.1X10" 9,042.1 1,300 21.67 2,197.0X10" 10,171.0 1,350 22.50 2,460.4X10" 11,364.0 1,400 23.33 2,744.0X10" 12,704.0 1,450 24.17 3,048.6X10" 14,114.0 1,500 25.00 3,375.0X10" 15,625.0 1,550 25.83 3,723.9X10" 17,241.0 1,600 26.67 4,096.0X10" 18,963.0 1,650 27.50 4,492.1X10" 20,797.0 1,700 28.33 4,913.0X10" 22,746.0 1,750 29.17 5,359.4X10" 24,812.0 1,800 30.00 5,832.0X10" 27,000.0 1,850 30.83 6,331.6X10" 29,313.0 1,900 31.67 6,859.0X10" 31,7r)5.0 1,950 32.50 7,414.9X10" 34,329.0 2,000 33.33 8,000.0X10" 37,037.0 2,050 34.17 8,615.1X10" 39,885.0 2,100 35.00 9,261.0X10" 42,874.0 2,150 35.83 9,938.4X10" 46,011.0 2,200 36.67 10,648.0X10" 49,296.0 2,250 37.50 11,390.6X10" 52,736.0 2,300 38.33 12,167.0X10" 56,329.0 2,350 39.17 12,977.9X10" 60,083.0 2,400 40.00 13,82 1. OX 10" 64,000.0 2,450 40.83 14,706.1X10" 68,084.0 2,500 41 67 15.625. OXIO'-' 72,338.0 APPENDIX 399 05 O r-l r-l C5 CO "I rH 05 00 o -* 'g* t^ Tt^ CTi CO CO CO 00 CT> (N ^ O i> (» o^ oo_ co^ (N 00__ CO (N t-Tko" O" lO lO" Co" 00 t-^ oo"" '^'^ co~ o fO iO o 2 ?5 ^ :S S o. -< rH 0> (N CO (N CO CO t^ O (M 00 CD CO t>- i^ ijj CO t^ a> OJ 00 U3 00 »o iC 00 t^ CO o t~- ic ^_ 0_ CD_^ o CO ^" 00 (M" O CD oo co' o~ o •^"~ c^ »o cr. Tt< (M TtH 1-1 (M CO ot^o. i t^ --H 00 (M CO lO Tj* Tt< 00 lO o .-H O CD -rr t^ lo '^ 00 ^ ^ 02 t^ O lO O 00 00 (M (>J CO CD -* 00 o CD^CC 1^05 10 02 (N CD CO c<\ -^ cc ^ ^ CO 00 00 lo .-H (N CO Tt< CD 05 t^ ec lo CO t^ CO ^ o o CO CO o ci lO ^ CO (M CO 00 lo 00 05 Eh «5 lO lO CO o lo 00 o lo c^ CO 00 g d lO (N lO t^ — 1 CO O h- lO CO CO (M -* 00 CO — 1 -H CO CO (M P^ ^ (N CO -* CO C2 g p; cc CO o ^ 00 00 lo CO CO CO rH §§ CO t^ t- (M CO t- o t^ CO H ■o t- CO_^ C0__ 0__ rH_ rH '^^ «i «=, lO ^ fO^ T^ t-J" t>r ,— 1 lo" oo" o' '*~ ^ o (N ■<# t^ (M O O TJH Tt< 03 T-H (N CO Tt< CO X P h (N t- 03 (M 0> CD '^ CD O Tt< rH oi CO TlH t^ t^ O TC lO t^ 1-1 00 CO o lo ^ O -- .-H IB =>' rjT O" (N 1-h'" co" ^" c^' co~ (N of CD" K C- C^ 03 OT CO ^ CD H Tf CD X 1 Ph (M ■* ^ O c; t^ CO -H X ^ ^ 00 !M CO 05 CO (M O CO lO t^ ^ a ^ CI t-~ CO lO O CO lO ^ 05 Tj< 5 -* o~ o" c£ oT lo '^ — ' co" b-- x" Q (M CO CD ^ 00 00 1 ^ ^ (N 5< S X 30 U5 C5 Tl< 00 00 -rh — 1 CO CO X ^ O CO "* t- o o CO t- 1-} N CO CD cr. ^ C<\ C<> 1^ CO CO !< CO 05 Oi t^ >C O CD O o" t^ — H T-H CO CD rH t^ r^ O t- -H ^ i-H (N ■<*< iC X O CO (N 00 CO ^ ^ rfH lO X t^ lO 'f ^ CO O O (M t- ■* ^ o CO Tt* -H CO CO eo" oo~ oo" ^^ c 00 C5 O -H (M CO th lO c^ t^ O CD rr O lO CO 02 (M t^ >— 1 t-- 00 t^ O O O X (N X CO o CO 1> Sg|§85^" -h'^ t-T oT r^ CO ic 1-H --H C<1 CO »o l^ lO CO t^ 00 03 O ^ (M CO -^ lO INDEX Aerodynamical Laboratory, 90 Aerodynamics, elements of, 87,-101 Ailerons, 33 construction of, 35 Air pump pressure feed, 58 Aluminum, 234 ses of, 234 Angle of drift, 89 of incidence, 89 Axis, direction, 19 pitching, 19 principal, 19 rolling, 19 B Banking, 31 angle of, 32 Biplane, effects of, system, 12 structure, 15 Cables, 225 splicing, 226 Canard type, 27 Ceiling, 195-203 Center of gravity, 273 position of, 273 Climbing, 188-203 influence of air density on, 189 speed, 130-133 time of, 191-194 Compressors, 70 Control surfaces, 19 sand test of, 286 Copper, uses of, 234 Cruising radius, 204-220 factors modifying, 214 Cubes, tables of, 393-397 Dihedral angle, 35 Dimensions of airplane, increasing the, 209 Dispersion, angle of, 73 Distribution of masses, 21 Drag, definition of, 1 Drift. 1 Efficiency, factors influencing lift- drag efficiency, 2 of sustaining group, 102 problems of, 161-166 Elastic cord, 256-258 curve method of spar analy- sis, 306-311 work absorbed by, 257-258 Elevator, 20 computation, 322 function, 22 size of, 20 Engine, 51 center of gravity of, 56 characteristics of, for airplane, 51 function of, at high altitudes, 68-71 types of, 51 Fabrics, 247-256 Fifth powers table, 399 Fin computation, 314 Flat turning, 29 Flying characteristic determination, 358-378 Flying in the wind, 151-159 Flying tests, efficiency, 392 401 402 INDEX Flyins? maneuverability, 391 stability, 390 Flying with power on, 115-133 Forces acting on airplane in flight, 45 effect of, 45-46 Fuselage, 37-43 reverse curve in, 39 sand test of, 384 spar analysis of, 332 static analysis of, 324-334 types of, 39-40 value of A' for, 39 Materials for Aviation, 221-200 Metacentric curve, 137 Monotoque fuselage, 39 Motive quality, 165 Mufflers, ()7 Multiplane surfaces, 211 o Oil tank, position of, 58 Gasoline, multiple, tank, 58 piping for, feed, 60 types of, feed, 58-60 Glide, 102-114 angle of, 104 spiral, 111-114 Glues, 260 Great loads, 204-220 Incidence, angle of, 88-89 Iron and steel in aviation, 222-234 Landing gear, 44-50 analysis of, 334-342 position of, 46 sand test of, 386 stresses on, 46-47 type of, 44 Leading edge, 6 function of, 6 Lift, 1 Lift-drag ratio, 2 efficiency of, 2 law of variation of, 6 value of, 2 M Maneuvrability, 134-150 Marginal losses, 10 Pitot tube, 91 Planning the project, 261-275 Pressure zone, 1 Principal axis, 19 Propeller, 72-85 efficiency of, 79-85 pitch, 74 profile of, blades, 75 static analj'sis of, 342-357 types of, 73 width of, blades, 74 R Radiators, 61-67 types of, 62 Resistance coefficients, 96-98 Rib construction, 16 Rubber cord, 47-48 binding of, 49 energy absorbed Ijy, 47 Rudder, 36 balanced, 3() static analysis of, 315 Sand test, control surface, 386 fuselage, 384-385 landing gear, 386 wing truss, 379-384 Shock a])sorbers, 47-48 uses of, 47 Spar analysis, 276-288 Speed, 167-187 means to increase, 168 INDEX 403 Spiral gliding, 111-114 Squares, tables of, 393-397 Stability, 134-150 directional, 141 intrinsic, 147 lateral, 140 transversal, 141 zones of, 139 Stabilizer, computation of, 322 dimension of, 20 effects of, action, 137 function of, 20 mechanical, 147-150 shape of, 20 Static analysis, of control surfaces, 315-323 of fuselage, 324-334 of main planes, 276-314 Streamline wire, 225 Struts, fittings, 18 computations, 294-29(3 tables, 297-300 Sustentation phenomena, 1 Synchronizers, 73 Tail skid, 49, 50 uses of, 50 Tail system computations, 314-323 Tandem surfaces, 211 Tangmt flying, 121 Tie rods, 226 Trailing edge, function of, 9 Transmission gear, 56 Transversal stability, 30 Triplane, effect of, sj^stem, 12 Truss analysis, 2SS-292 Tubing, tables for round, 229-231 table of moment of inertia for round, 231 of weights for round, 230 tables of streamline, 232-233 U Unit loading, 12, 278, 279 Useful load increase, 212 \'arnishes, 259 finishing, 259 stretching, 259 Veneers, 241-254 tables for Haskelite, 246-254 ^^' Weighing the airplane, 389 Wind, effect of, on stability, 156 Wing, analysis of, truss, 276 element of, efficiencj^ 9 elements of, 3 sand test of, 379 unit stress on, 306-314 Wires, steel, tables, 224 streamline, 225 Wood, 234-254 characteristics of various, 236- 239 Library N. C. State College m itr hii it f'H m