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MISCELLANEOUS CONTRIVANCES 284 H APPENDIX . . 341 INDEX . . . 355 INDEX TO APPENDIX ........ 360 713782 ;ineerii ibrary Engineering Lit) ELEMENTS OF MECHANISM. CHAPTER I. INTRODUCTORV. A MACHINE is an assemblage of moving parts, constructed for the purpose of transmitting motion or force, and of modifying, in various ways, the motion or force so transmitted. In order to form a definite idea of the meaning which attaches to the word ' machine,' it may be useful to refer to an example commonly met with such as an ordinary sewing machine. The apparatus is rightly called a machine, as being capable of doing work of one definite kind, under the simple condition that some natural source of energy shall bear upon it and set the working parts in motion. Upon looking into its construction we should find a fixed framework supporting combinations of movable parts, whereof some are employed in actuating a needle and shuttle, while others carry forward the material which is to be stitched. The movable parts are constrained to take certain definite motions, which are arranged beforehand, while some natural force, such as the power of the hand or the foot, is applied to the proper recipient, and then the machine does work as a necessary consequence of the action of the motive power. In commencing the systematic study of machinery, it will be readily understood that certain simple relations of motion are traceable between the prime mover which starts the machinery B 2 Elements of Mechanism. and the pieces which execute the work ; and it is also clear that, in practice, relations governing the transmission of force must exist as certainly as those which govern the transmission of motion The considerations relating to force may often occupy the mind of the mechanic in a greater degree than those which refer to mo- tion; but in reducing the subject to analysis it will be found con- venient to separate the two points of view, and to confine our attention in the first instance mainly to Theoretical Mechanism that is, to an examination of the various contrivances and arrange- ments of parts in machinery whereby motion is set up or modified and to disregard or postpone any enquiry into the mechanical laws which control the forces concerned in these movements. But as the present work is intended for use and study by practical men, the author will to a small extent break through this general rule, and will take occasion, where the enquiry would be useful, of pointing out also the manner in which certain pieces of mechanism have served a compound object in transmitting exact and definite amounts of motion, while dealing at the same time with refined and subtle distinctions as to the method of transmitting force. We have now to consider and arrange the method according to which our enquiries are to be carried on, and if we were to pause for a moment and look back upon that rapid creation of machinery which followed so closely upon the splendid invention of the steam engine by Watt, we should naturally expect that some uniform arrangement for applying steam power would be adopted by common consent, and that this arrangement would powerfully influence the art of constructive mechanism. Accordingly we find that in applying the power derived from steam for the purpose of driving machinery in our mills and factories, it is the practice to connect the engine with a heavy fly wheel, the rotation of which is made as uniform as possible, and then to carry on, by lengths of shafting, the uniform motion of the fly wheel to each individual machine in a factory. Suppose, for example, that we were visiting a cotton mill, and were examining and endeavouring to comprehend the action of a complete piece of machinery, such as a power loom for weaving calico. We should at once see that every moving part, acting to produce the required result, derived its motion from the uniform Introductory. 3 and constant rotation of a disc or pulley, outside the machine itself, and communicating by means of a band with the shafting driven by the engine, and thus it would become obvious that the problem of making a machine resolved itself mainly into a question of the resolution or transfer of circular motion in every variety of manner, and subject to every possible modification. We shall therefore commence, in the present chapter, with some general observations on the conversion and transfer of motion in the simple primary forms under which it is to be regarded at the outset of the study of mechanism, reserving a more complete discussion of the different divisions of the subject for the remaining chapters, each of which will treat of movements of a particular class. It will soon become apparent that, by combining, transferring or modifying simple modes of motion, an almost endless variety of mechanisms may spring into existence, and our object will be to classify and arrange these mechanisms in such a manner that the reader may acquire a fair knowledge of what has been already accomplished, and may trace the principles which have been de- veloped in the construction of many well-known machines. To the geometrician a straight line or a plane surface are crea- tions of the mind. Euclid, more than 2,000 years ago, had as complete a conception of a straight line or a plane as we have at the present day, but he could not realise his conceptions even approximately, by reason that accuracy of surface was at that time a thing unknown. And even now how few among young mechanics are aware of the exact conditions under which truth of surface has been originated, or that a difference of length of 40000 of an inch is a quantity which can be palpably and unmistakably measured by a workshop instrument, without the aid of a microscope or magni- fying lens of any kind. It is not enough to describe machines on paper, and to say that they will effect such and such results. The science of mechanism is a practical science; it must be more than a speculative creation ; the principle of each movement must be embodied in shaped pieces of suitable material, and there must be some method of testing the exact form or dimensions of the several parts. It follows that a knowledge of the steps which have 4 Elements of Mechanism. given to the mechanician the two aids upon which he mainly relies, viz. truth of surface and the power of measurement, will form an essential portion of the subject-matter upon which we have to treat. ART. i. To commence with a few enquiries relating to the motion of a point in space a point being that ideal thing called a material particle, which is defined in a Cambridge text-book as being 'a portion of matter indefinitely small in all its dimensions, so that its length, breadth, and thickness are less than any assign- able linear magnitude ' we shall treat the motion of such a point as a simple matter of geometry, all its movements being exact of their kind. There are three primary cases : I. The point may move in a straight line. In such a case the direction of its motion remains constant, being that of the line in which it moves. II. The point may move in one plane, but may continually change the direction of its motion. III. The point may change its direction so as to move in a curved line of any kind. In the second case the point is said to move in a plane curve, for, according to geometers, a curve is a line traced out by a moving point, which is continually changing the direction ot its motion, and a plane curve is one which lies in a given plane. Conceive that a point is describing a plane curve AB ; then the straight line in which the point would move at P, if it there ceased to change the direction of its motion, is called the tangent to the curve AB at the point P that is to say, the direction of the motion of a moving point is at each instant the tangent to its path when the path is a curve. It should be borne in mind that we have spoken of a material particle as the moving thing, because a material particle has the property of mass, and cannot change its velocity or direction abruptly unless it be subjected to the action of an infinite force, though of course it may be moved abruptly in the sense of being set in motion from a position of rest. Velocity. 5 In the third case the point may move in a curve traced upon the surface of a cylinder, and not lying in one plane, as, for ex- ample, if it followed the outline of the banister of a cylindrical staircase. The definition of a tangent previously given applies generally to all curves, and if a point be moving in a straight line at any instant it must also be moving in some definite plane; hence we may describe the motion now under consideration by saying that the direction of the curve at any instant is continually changing by the twisting of the plane in which the point is then moving about its tangent line, ART. 2. We have next to speak of the velocity of the moving point. So long as a point is moving continuously, we can form an idea of the rate at which it is changing its position relatively to other points which are assumed to be at rest. This rate of change of position is the velocity of the moving point. Velocity may be either uniform or variable. The word 'uniform' indicates that the lengths of path de- scribed by the moving point in equal times are always the same, and the word ' variable ' is applied when the rate of change of position of the moving point is continuously altering. The latter term would not apply to a step-by-step movement. When the velocity of a point is uniform, it is measured by the length of path passed over in a unit of time that is, in technical language, by the space described in the unit of time. The unit of space is usually one foot, and the unit of time is one second. Hence if a point be moving uniformly with a velocity v, it will describe v feet in any second, and will describe tv feet in / seconds. Let s be the space described in / seconds, then s = tv, or z>=~. Def. A foot- second of velocity is the velocity which would cause a point to move uniformly through a foot in every second. If the velocity of a point be n foot-seconds the point will move uniformly through n feet in each second of time. 6 Elements of Mechanism. ART. 3. If the velocity of the point be variable we must look to the rate at which s changes as / flows on, or to the ratio be- tween the so-called fluxions of s and /. The word ' fluxion ' was introduced at the time of Newton. Suppose that in time A/ the point describes a space As, and that its velocity in the same time increases or decreases continu- ously and becomes v + &v. The space it actually describes lies between the spaces it would describe if its initial and final velo- cities were continued uniform during time A/, or vt, AJ, (v are in order of magnitude, and so are Now let A/, and consequently As, A?; be diminished indefinitely, in which case let -- become -,-, then the first and third terms A/ dt become equal to that lying between them, or v = -/. dt Hence v = - T for variable motion. dt In other words, velocity, when uniform, is measured by the space described in a unit of time ; when variable it is measured by the space which would be described in a unit of time, if the point retained throughout that unit the velocity which it has at the in- stant considered. The above statements apply equally whether the point be moving in a straight line or in a curved line of any kind. It is further apparent that the velocity of a point at any instant may be represented by a straight line, for the direction of motion of the point will be the direction of the line, and the numerical measure of the velocity will determine the number of units of length in the line. Inasmuch as a straight line can be drawn in any direction from a point, and since it is usual to describe the straight line as positive when it is drawn in one direction from a point, and negative when it is drawn from the same point in the opposite direction, so velo- cities can be similarly described as positive or negative, according to their directions in the same straight line. Resolution of Velocities. ART. 4. We pass on to the resolution and composition of velocities. Conceive that a point is moving uniformly in the straight line PQR with a velocity ?', and let P, Q, R be three positions of the moving point Take any two straight lines Ox, Qy inclined at a given angle, and lying in a plane passing through PQR ; draw P/, Q?, Rr respectively parallel to Oy ; then pq : pr \ \ PQ : PR, and the motion of p along Qx will be in a constant ratio to the motion of P along PR. The point / is called the projection of P on Ox, and the velo- city of/ is said to be the velocity of P resolved along Ox. In like manner, if a straight line AR be taken to represent the velocity existing in a point at any instant, and Ax, Ay be any two straight lines intersecting in A, and if the parallelo- gram APRQ be completed, the sides AP, AQ will represent the resolved velocities of the point in directions Ay, Ax. It is usual to speak of AP, AQ as the compo- nents of the velocity AR ; and, again, AR is called the resultant of the component velocities. Hence the following proposition : Prop. If there be impressed simultaneously on a particle at A two velocities which would separately be represented by the adjacent sides AP, AQ (fig. 3) of the parallelogram APRQ, the actual velocity of the point will be represented by that diagonal AR which passes through the point A. Cor. i. Let the angle PAQ be a right angle, and let RAQ=, FIG. 3. FIG. 4 . FIG. 5. AR = v. Then will AQ = v cos a, AP = QR = v sin a. 8 Elements of Mechanism. Cor. 2. Let velocities represented by the three sides of a tri- angle taken in order viz. AQ, QR, RA be impressed at the same instant on a point ; then no motion will ensue, or the point will remain at rest ; for the velocities AQ, QR are equivalent to a single velocity AR, and the velocities AR, RA are equal and opposite, and therefore destroy each other. ART. 5. It has been stated that the most simple case of motion is that of a point describing a straight line with a uniform velocity, also that the motion of a point in a straight line may be the aggregate of two movements in lines at right angles to each other, and that this is true whether the motion of the point in a straight line be uniform or variable. But one imperative condition must be observed, viz. that the amounts of the corresponding movements in the two perpendicular lines must be in a fixed ratio to each other. When this condition fails, the point will describe a curve and not a straight line. The whole learning of analytical geometry proceeds on the doctrine of the composition of motion. If we wish to represent a curve by means of a relation between symbols, called an equation to the curve, we begin by drawing two straight lines xQx', j'Oy at right angles to each other, and employing them as lines of reference. For example, let the curve pass FlG- 6> through point 6 ; take P any point in the curve, and draw PN perpendicular to CXr ; let ON=^, NP==>>; then, if y be to x in a fixed ratio, the point P must of necessity lie in a straight line * passing through O. Whereas if the ratio between y and x varies for every position of the moving point P, that point will describe a curve. Let P describe a circle whose centre is C, and let CO=a ; join CP: then NC=OC-ON=a-x But CP 2 =PN 2 + NC 2 . Circular Motion of a Point. FIG. 7. The above relation is satisfied only by points lying in the circle, and gives an analytical representation of the particular curve to which reference has been made. The lines x, y are called the co-ordinates of the point P, and the axes xOx', yOy' are the axes of co-ordinates ; also the signs + and are employed to indicate the position of P in any par- ticular quadrant ; thus if P were situated anywhere in the quad- rant x'Oy, the corresponding values of x and y would both be negative. ART. 6. We are now in a position to discuss the nature of cir- cular motion, and may premise that the belief held by the ancients with regard to it was fanciful in the extreme, and is obviously untenable It was said that the motion of a point in a circle was simple, in the sense that it was not made up by putting together other separate movements, a doctrine in direct opposition to that just laid down. The modern belief is that the point P, while describing the arc OP of the circle OBDE, may have been the subject of two independent movements, one from O to N in the direction of the diameter OD, and the other from N to P in a perpei dicular direction. Thus let OCD, BCE be two diameters of a given circle at right angles to each other, P any point in the circumference, PN, PM perpen- diculars on OC and BC respectively. Let P describe the circumference with a uniform velocity ; then the point N will travel to and fro along OCD with a variable velocity, while at the same time the point M moves at a varying, rate up and down through BCE. The motion of the point N is distinguished by a technical name, according to the following definition : Def. When a point P moves itniformly in a circle, the ex- tremity N of the perpendicular PN let fall from P upon a fixed diameter OD has a simple harmonic motion. It appears that this is nearly the case with such bodies as the satellites of Jupiter when seen from the earth. The term ' harmonic motion ' has been chosen for designating one component of circular motion because it represents approxi- mately the motion of a particle in the various media in which waves of sound, light, and heat are propagated. Thus a point at the end of the leg of a vibrating tuning-fork has a simple harmonic motion very approximately. We conclude that circular motion is of a compound character, and is capable of resolution into its elements. If it be thus re- solved, and if one equivalent be suppressed, so that the motion of N is substituted for that of P, we obtain the fundamental case of the conversion of circular into straight-line motion. And, further, we regard circular motion as compounded of two simple harmonic motions in lines at right angles to each other, but so related that one component comes to rest when the other is in the middle of its swing. It will be found that harmonic motions enter into the analysis of many forms of mechanism, and that no progress can be made without some knowledge of the laws here sketched out. The following technical terms are introduced in order to state the nature of the movement correctly. Def. The amplitude of a simple harmonic motion is half the distance between two extreme positions. In other words, it is the radius of the auxiliary or bounding" circle. Def. The period is the interval of time between two successive passages through the same position in the same direction that is, it is the time of describing the complete circle. Def. The phase is that fraction of the period which has elapsed since the moving point was at its extreme position in the positive direction. In applying these definitions we should say that M and N have the same amplitude, that the have the same period, and that they differ in phase by of the period ; whence we finally conclude that uniform circular motion is compounded of two simple har- monic motions of equal period and amplitude, taking place in lines at right angles to each other, and differing in phase by one- quarter of the whole period. Jj tju,^^***^*- K*~*~O^b-tA^ x -t>l>l^/f ^ lislsisi^ **X/1^L^- ^7^; .Simple Harmonic Mation. 7 1 1^ t/(j4yt "** -&**t *W /3 , ( = AP, w =^=*L. AC 7* Angular Velocity. 19 which is the equation connecting the uniform linear velocity of the point P with the angular velocity of the disc or rotating body. It follows that when a body is rotating uniformly, the linear velo- city of any point of it increases directly as the distance from the axis of rotation. Ex. i. A wheel 6 feet in diameter turns uniformly on its centre 20 times in a minute : what is the linear velocity of a point in its circumference ? We shall now apply the notation of foot-seconds, which are written f.s. for brevity. Here the angular velocity = "" = ^-, 60 3 and the linear velocity of a point at a distance of 3 feet from the centre of rotation = x 3 foot-seconds, = 27T f.S. Ex. 2. How far from the centre will a point lie which is moving at the rate of one mile per hour ? The linear velocity of this point = LL^. 3__ f. s . 60x60 The angular velocity of this point o /. required distance = I76ox g x -Lfeet, 60 X 60 2JT = == J- feet nearly, ion- 10 ART. 1 6. Again, if 6 be the angle described by CP in / seconds, we have tat, or w = -. If the angular velocity be variable, it may be proved by reason- ing precisely similar to that adopted in Art. 3, that the angular velocity w is given by the equation d\ " = dt ] that is to say, - J. represents the angular velocity of a rotating at body when the motion is variable. c2 20 Elements of Mechanism. ART. 17. Of two moving pieces that which transmits motion is termed the driver, and that which receives it is the folloiver. Conceive that the driver and follower have each a simple motion, either of translation or rotation ; then the ratio of their comparative velocities is called the velocity ratio between them. ART. 1 8. It will now become necessary to consider the- manner in which a motion of rotation of a solid body may be transferred from one axis to another, and it is apparent that the most simple case occurs when a circular disc or plate moves another in its own plane by rolling contact. In such a case the uniform motion of the axis A conveys a perfectly even and uniform motion to the other axis B. If A and B were circular plates with flat edges, and very accurately adjusted, it would be quite possible for A to move B by friction alone, the two plates rolling smoothly and evenly upon each other with- out any slipping of the surfaces in contact ; but we could not expect A to overcome any great resistance to motion in B ; or, in other words, we could not in practice convey any considerable amount of force by the action of one disc upon the other. The transmission of energy being an essential condition in machinery, the discs A and B are provided with teeth, as in the annexed figure, and the machariv/endeavours so A J l fVJ ^ " 1*7^ to form and shape the teeth that the motion shall be exactly the same as if one circle rolled upon another. Herein consists the perfection of wheelwork : a perfectly uniform motion of the axis A is to be con- veyed by means of teeth to the axis B ; and the motion of B, when Spur Wheels. 21 tested with microscopic accuracy, is to be no less even and uniform than that of A. Since, then, it appears that the motions of A and B are exactly the same as those of two circles rolling upon each other, such imagi- nary circles may always be conceived to exist, and are called the pitch circles of the wheels in question. They are represented by the dotted lines in fig. 21. The pitch circle of a toothed wheel is an important element, and determines its value in transmitting motion. Suppose that two axes at a distance of 10 inches are to be connected by wheelwork, and are required to revolve with veloci- ties in the proportion of 3 to 2. Two circles, centred upon the respective axes, and having radii 4 and 6 inches, would, by rolling contact, move with the desired relative velocity, and would, in fact, be the pitch circles of the wheels when made. So that what- ever may be the forms of the teeth upon the wheels to be con- structed, the pitch circles are determined beforehand, and must have the proportion already stated. It appears also that when the number of teeth upon a wheel is indefinitely increased, the wheel itself degenerates into the pitch circle. So much of the tooth as lies within the pitch circle is called its root QI flank, and the portion beyond the pitch circle is called the point or addendum. The pitch of a tooth is the space ac upon the pitch circle cut off by the corresponding edges of two consecutive teeth. Spur wheels are represented in fig. 20, and are those in which the teeth project radially along the circumference. In a. face wheel, cogs or pins, acting as teeth, are fastened per- pendicularly to the plane of the wheel ; in a crown wheel the teeth are cut upon the edge of a circular band ; and annular wheels have the teeth formed upon the inside of an annulus or ring, instead of upon the outer circumference, 22 Elements of Mechanism. A straight bar provided with teeth is called a rack, and a wheel with a small number of teeth is termed opinion. Gearing and gear are the words used to indicate the combina- tion of any number of parts in a machine which are employed for a common object. Toothed wheels are said to be in gear when they are capable of moving each other, and out of gear when they are shifted into a position where the teeth cease to act. ART. 19. The spur wheels, before described, are suited to con- vey motion only between parallel axes ;_it often happens, however, that the axes concerned in any movement are not parallel, and as FIG. 22 a consequence they may, or may not, meet in a point. If the axes do not intersect we proceed by successive steps, and con- tinually introduce intermediate intersecting axes, and thus we are led to the use of inclined wheels whose axes meet each other, and which are known as bevel wheels. It is easily proved in geometry that two right cones which have a common vertex will roll upon each other, and the same would be true of the frusta of two cones such as LM and NR, which are represented as having a common vertex in the point O. The rolling of the cones will allow us to consider any pair of circles in contact and perpendicular to the respective axes as the pitch circles of the frusta, and teeth may accordingly be shaped upon them so as to produce the same even motion as that which exists in the case of spur wheels. This fact about the rolling of two cones becomes very clear Rolling Circles. 23 when enquired into, and it is evident that if one of the cones be flattened out into a plane table, by increasing its vertical angle up to 1 80, the property of rolling will not be interfered with. But in that case the common vertex will be a fixed point in the table, and, accordingly, if we roll a cone upon a table, the vertex ought not to move in the least degree as the cone runs round. It is quite easy to test the matter in this way, and if the table be smooth and level the apex will remain perfectly stationary, although the com- itself is free to run in any direction. The principle under discussion is sometimes applied in the construction of machinery ; there is a large circular saw in the arsenal at Woolwich which is driven by the rolling contact of the frusta of two cones, and upon examination it will be found that the directions of the axes of the two frusta meet exactly in the centre of the revolving circular saw. Equal bevel wheels whose axes are at right angles are termed mitre wheels. ART. 20. Prop. When two circles roll together, their uniform angular velocities are inversely as the radii of the circles. This proposition is exactly analogous to that which obtains when a point describing the circumference of one circle passes off into another circle of different diameter, and the proof is the same. Let the circles centred at A and B move by rolling through the corresponding angles PAD and QBD. Let AD=rt "I PAD=9 1 then PD=flfl, QD=A/>, but PD=QD a But the angular velocities of the circles, being uniform, are as the angles described by each of them in the same given time, . angular vel. of A__BD angular vel. of B a AD ' which proves the proposition. 2 4 Elements of Mechanism. ART. 2t. Two simple questions relating to the transfer of motion by wheelwork remain to be determined. i. Let two axes be parallel, and let m be the velocity ratio to be communicated between them. If a be the distance between the axes, and r, r 1 be the radii of the two pitch circles A and B, the condition of rolling gives us = ;? = ve ' r 1 m vel. of A Also r+r 1 n and r 1 = .. m + n whence r and i' are known in terms of m, n, and a. 2. Let the axes meet in a point, and let it be required to con- struct two cones which shall communicate the same velocity ratio by rolling contact. We now refer to fig. 24, and assume that DN, DM are the radii of the bases of the cones LCD, HCD, whose angular velocities are as the numbers m and n respectively. Let MCN=a, NCD=0, then DN^ CD sin 0, DM=CDsin (a-fl)-, PM_sin(a-Q) ' DN sin But - , since the inverse ratio of the radii of the bases DN n of the cones is the velocity ratio between the axes, ) =sin cot 0-cos , n sin tf .. whence tan 6 n sin a COS c If

, (m>n), n sin o sin a we have tan 6 Pulleys and Belts. 25 whence is expressed in a form adapted for logarithmic compu- tation. Cor. If (=QO, we have tan 0= . m ART. 22. Belts or straps, otherwise called bands, are much used in machinery, in order to communicate motion between two axes at a distance from each other. In this case an endless band is stretched over the circumference of a disc or pulley upon each axis, and the motion is the same as if the discs rolled directly upon each other. The usual form of the ^ pulley is shown in fig. 25. fes ^ It is a common practice to convey steam power by means of shafting a-nd wheelwork to the various floors of a mill, and then to distribute it to the separate machines by the aid of straps or belts. These straps adhere by friction to the surfaces of pulleys, and work with a smooth and noiseless action ; but they are subject to two principal objections, which may or may not be counterbalanced by their other advantages. The friction of the axes upon their bearings is increased by the double pull of the strap, arising from its tension, and there is a liability to some change in the exact- ness of the transmission of motion by the possible stretching or slipping of the band. The drawing shows the method of communicating motion from one axis to another at a distance. The diameters of the large and small pulleys A and B are respectively as 3 to i, and the result is that when A makes 40 revolutions B makes 120 revo- lutions. The velocity ratio is precisely the same as if A moved B by rolling contact. 26 Elements of Mechanism. The strap may be open or crossed. In the former case A and B rotate in the same direction, and in the latter case they rotate in opposite directions, as indicated by the arrows. ART. 23. The term band is applied either to a flat strap or a round cord indifferently. The best material for round bands, such as are used in light machinery, is no doubt catgut, and then the band is fitted with a hook and eye to make it continuous. It must work in a pulley with a grooved rim, or it would slip off, and this groove prevents our shifting it easily from one pulley to another. The power of readily shifting a driving band is often an indispensable condition, and can be obtained at once by the use of a flat belt, which will hold on to its pulley with perfect security if we only take care to make the rim slightly convex, as shown in fig. 25, instead of being concave. No groove is neces- sary, or indeed admissible ; and, upon entering a workshop where steam power is employed, we see each machine driven by a flat belt riding upon one of a set of two or more pulleys with perfectly smooth edges. The belt has no tendency to slip off, and it is shifted with the greatest ease from one pulley to another when pressed a little upon the advancing side by a fork suitably placed. The reason for making the rim of a pulley slightly convex will be apparent if we examine the case of a tight belt running upon a revolving conical pulley. The belt embraces the cone, and tends to lie flat upon the slant surface, thus becoming bent into the form AB, the portion B being somewhat nearer to the base of the cone than the portion A. The cone, during its revolution, exerts an effort to carry B onward in a circle parallel to its base, and the con- sequence is that the belt tends to remain upon the slant surface of the cone, and to rise higher rather than to slip off. In like manner, if a second cone of equal size were fastened to the one shown in the drawing, the bases of the two cones being joined together, the belt would, if its length were properly ad- justed, work its way up to the part where the bases met, and Pulleys. 27 would ride securely upon the angular portion formed by their junction ; but this is the same case as that of a slightly convex pulley, for it is evident that a little rounding off of the angle at the junction of the bases would convert this portion of the double cone into a convex pulley. Thus the action becomes perfectly intelligible. ART, 24. The fast and loose pulleys are an adjunct of the driving belt. They consist of two pulleys placed side by side, as in fig. 28, whereof one, A, is keyed to the shaft, CD, to which motion is to be con- veyed, and the other, B, rides loose upon it. When the strap is shifted from the loose E to the fastened pulley the shaft will begin to rotate, otherwise it remains at rest, the loose pulley alone turning round. The shaft EF is the driver, and carries one broad pulley keyed upon it. The band is shifted by a fork, which, as before stated, is made to press laterally upon c its advancing side. The advancing portion of the band must always lie in the plane of the pulley round which it is wrapped, but the retreating portion may be pulled on one side without causing the band to leave the pulley. This rule applies whether the band is round or flat. ART. 25. It is by observing this condition that a band may be used to communicate motion between two axes which are not parallel, and which do not meet in a point. Problems such as these are interesting, as presenting difficul- ties to be overcome by a knowledge of principles. Suppose that we are required to arrange that a band working over a pulley upon one given axis shall drive another pulley upon an axis at right angles to the first. Here we intend that the pulleys should be placed one above the other as in the sketch. As the band goes round we have to provide that its advancing portion shall always lie in the plane of the pulley upon which it works. The easiest way of proceeding is to draw a straight line, AB, upon paper, and to place circles AB 28 Elements of Mechanism. EF and DC representing the pulleys in contact with AB upon each side of it. Draw now the lines ED, CF, to represent bands passing round the circles, and however you may bend the two planes containing the pulleys by folding the paper about AB as an edge, it is clear that the advancing portion of the strap will continue to lie in the plane of its pulley so long as the motion occurs in the direction of the arrows. Reverse the motion, and the strap will leave the pulleys at once. ART. 26. It may be useful here to enquire how the necessary size or strength of the strap is ascertained when energy is trans- mitted, and we take the following example : Suppose that a force capable of doing work which is techni- cally estimated at 5 horse-power is to be carried on by a strap moving with a velocity of 600 feet per minute over a suitable pulley. The work done by 5 horses is 5 x 33000 foot pounds per minute, and the work done by the strap must be the same. Let P be the pull upon the strap in pounds, then P x 600 is the work done by the strap in one minute, Telodynamic Transmission of Power. 29 .'. P x 600 = 5 x 33000, If the velocity of a point in the strap had been reduced to 300 feet per minute, P would have been 550 Ibs. ; if it had been in- creased to 3,000 feet per minute, P would have been 55 Ibs. ; and thus we recognise the well-known mechanical principle, that the slower the movement by which any given amount of energy is transmitted, the greater must be the strength with which the moving parts are constructed. In carrying out this principle successful attempts have recently been made to transmit the driving power of turbines or water- wheels to considerable distances, by means of a slender wire rope moving at a high velocity, and the method is called the telodynamic transmission of power. The first experiment was made in 1850 by Mr. C. F. Him, at Loyelbach, near Colmar, Alsace. A band of steel 172 yards long, jj 1 ^ inch thick, and 2 inches broad was slung as an endless band over two pulleys, each 6^ feet in diameter, which were placed at a distance of 84 yards, and made 120 revolutions per minute, giving a speed of 28 miles per hour in the band. There were practical objections to the use of a flat band ; nevertheless the plan was successfully adopted for a year and a. half, and transmitted 12 horse-power to 100 looms. Since that time the flat rope has been replaced by a round rope made of steel wire. It may be interesting to refer to some operations carried on at Schaff hausen, on the Upper Rhine. Here the water-power is taken from three ordinary vertical-flow turbines, each 9^ feet in dia- meter, and driven by a fall of water varying from 12 to 16 feet. Each turbine makes about 48 revolutions per minute, and the whole can develope collectively about 750 horse-power. The wire rope is f inch in diameter, made of the best Swedish iron, and having 72 wires in the rope. It starts by running upon pulleys driven by the turbines, and these pulleys are each 15 feet in diameter, and make zoo revolutions per minute, giving a linear velocity to the rope of about 53 miles per hour. No less than 17 factories in different positions have been supplied with motive 30 Elements of Mechanism. power from one set of turbines, and it is stated that the total length of transmission is 3,300 feet. For a more homely illustration we may mention the locomotive workshops of the London and North-Western Railway at Crev/e, where a cotton rope |ths of an inch in diameter, and weighing about \\ oz. per foot, has been carried along the length of a workshop with a velocity of 5,000 feet per minute, and so em- ployed for actuating a traversing crane which is adapted for lifting a weight of 25 tons. The velocity of 5,000 feet per minute would be reduced, by suitable mechanism, to that of i foot 7^ inches per minute, and the requisite work would be done by subjecting the whole cord to no greater strain than that of 109 Ibs. ART. 27. Guide pulleys are sometimes used, and they are constructed as follows : Conceive that a band moving in the direction of AB is to be diverted into another direction, CD. There are two cases to be considered. i. Let AB and CD meet in E. At the angle E place a small guide pulley whose plane is coincident with the plane AED. This pulley obeys the required condition, and will answer its purpose. FIG 31. FIG 32. 2. If AB and CD do not meet, or do not meet within a reasonable distance : Draw any straight line, EF, cutting both AB and CD. In the plane AEF, and at the angle E, place a guide pulley, E, and do the same thing at F by fixing a guide pulley in the plane EFD, and thus the strap will be carried on. One advantage in the use of guide pulleys will be found in the fact that they enable us to overcome the inconvenience of not being able to reverse the motion when the planes of the pulleys are inclined to each other. Guide Pulleys. Thus, conceive that two pulleys work in the planes ZAx, ZAy inclined to each other at an angle xAy. In the line of the inter- section of the planes, viz., AZ, take any two convenient points, H and B, place one guide pulley at H in the plane CHD, and another at B in the plane EBF, then the band CHDFBE will run round the two main pulleys securely in either direction. This is evident, as we have done nothing to infringe the necessary condition, each advancing and re- treating portion of the band will, in both cases, be found in the plane #/ of the pulley upon which it rides. Instead of bands we may employ chains to communicate motion from one axis to another, and there is one instance where a chain is always so used, viz., in the transfer of the pull of the spring from the barrel to the fusee of a watch. Here the form of chain, is the type of most others of the heaviest construction, con- sisting of one flat plate or link riveted to two others, which are placed one above and the other below it, and thus the chain con- sists of one and two plate-links alternately. When a chain of this sort is used to transmit great force, it is called a gearing chain, and the open spaces formed by the two parallel links engage with pro- jections on the wheel or disc over which it runs, rendering it impossible for the chain to slip. One practical objection to the use of chains, where great accuracy is required, consists in the fact that the links are liable to stretch, and that the pitch or spacing may lose its exactness, the result being to cause jar and vibration in the working. ART. 28. Instead of confining the motion of a body to simple translation in a straight line, or to simple rotation about a fixed axis, we shall now suppose that the body moves by sliding along a plane, in such a manner that any straight line in it, as AB, passes into the position A'B', which lies in the same plane with AB, but is not parallel to it 32 Elements of Meclianism. In such a case AB has a motion both of translation and rota- tion. Its centre describes some line straight or curved, which indicates a motion of translation, and at the same time the line itself changes the direction in which it points, or is subject also to a motion of rotation. We shall now prove that the motion of AB may be repre- sented by supposing it to be rota- FlG - 34- ting at each instant about some point O, which point is continu- ally changing its position, and is therefore called the instantaneous centre of motion. The curve which the instanta- neous centre describes is called the centrode, from two Greek words signifying ' the path of the centre.' It will be understood that if the actual motion existing at the time considered were to be permanent, it would be a motion of rotation about an axis through O, which is therefore called the instantaneous axis. In order to find the position of O, let the motion from AB to A'B' be infinitesimal, and join AA , BB . Bisect AA' in E, and draw EO perpendicular to it ; bisect also BB' in F, and draw FO perpendicular to it. O will mark the instantaneous axis concerned in the motion from AB to A'B'. Join OA, OA', OB, and OB'. Then since OA=OA', OB=OB', and AB=A B' /. angle AOB = angle A'OB'. Take away the common angle A'OB and we have angle AOA' = angle BOB'. Therefore while A is rotating about O into the position A', the point B is also rotating about O into the position B', or O gives the position of the instantaneous axis. ART. 29. In some simple cases, O is a fixed point, but it may still have the property of an instantaneous axis. This happens when a body is fixed to one end of a rotating arm, the centre of motion of the arm lying away from the body. Instantaneous Axis. 33 For example, let an arm placed horizontally be made to rotate about a vertical axis through one end, and let a wheel B be locked to the other end of the arm, in such a manner that its plane is horizontal. As the arm goes round, a person inspecting the apparatus from a little distance will see the wheel B turning on its axis, and if he watches a mark upon the rim, he can entertain no doubt about this fact. The truth is that although the wheel B does not move rela- tively to the arm, it is, nevertheless, the subject of two distinct motions, whereof one consists in a rotation about an axis through its centre, and the other is a motion of translation, whereby the centre of the wheel describes a circle whose radius is the distance between the centre of B and the axis of rotation of the arm. This is an example of the resolution of a compound move- ment into its simple elements, and the instantaneous axis remains permanently in the axis about which the arm rotates. If the wheel B were looked at from the centre about which the arm revolves, no motion of rotation could be recognised. B would appear to have a motion of translation which carried it round in a circle. The very same thing happens in the case of the moon. As- tronomers tell us that only about one-half of the face of the moon has ever been seen by those upon the earth's surface, and they explain the fact by saying that the moon turns once upon its axis during the period of a single revolution in its orbit round the earth ; or, in other words, that it moves as if it were fixed to a rigid bar stretching from the earth to the moon. ART. 30. Case III. If the motion of rotation of a body about an axis be combined with a motion of translation of the axis itself in the line of its direction, any point in the body will describe a curve which cannot lie in one plane. Such a movement may be obtained from a single pair of elements, as in the example of a nut on a screwed bolt. After toothed wheels, the screw plays the most important part in mechanical appliances, and indeed it is difficult to over-estimate its value or utility.' The screw bolt and nut are used to unite the various parts of machinery in close and firm contact, and are D 34 Elements of Mechanism. FIG. 35. peculiarly fitted for that purpose ; then, again, the screw is em- ployed in the slide rest and in the planing machine to give a smooth longitudinal motion, the same purpose for which it aids the astronomer in measuring the last minute intervals which are recognisable in the telescope. In the screw press we rely upon it to transmit force, we use it in screw piles to obtain a firm founda- tion for piers or lighthouses, and as a propeller for ships it has given a new element of strength and power to our navy. The definitions relating to the screw are the following : If a horizontal line AP, which always passes through a fixed vertical line, be made to re- volve uniformly in one direc- tion, and at the same time to ascend or descend with a uni- form velocity, it will trace out a screw surface APRB, in the manner indicated in the sketch. The points of intersection of this generating line with any circular cylinder whose axis coincides with AB, will form a screw thread, PR, upon the surface of the cylinder. The//'/^ of a screw is the space along AB, through which the generating line moves in completing one entire revolution. Also AB is called the length of the screw surface APRB, and the angle PRQ represents the angle of the screw. In the diagram, AP is shown as describing a right-handed screw , if it revolved in the opposite direction during its descent, it would describe a left-handed screw. ART. 31. The screw thread used in machinery is a project- ing rim of a certain definite form, running round the cylinder, and obeying the same geometrical law as the ideal thread which we have just described. In practice the pitch of a screw bolt is usually estimated by observing the number of ridges which occur in an inch of its length ; thus we speak of a screw of one-eighth of an inch pitch as being a screw with eight threads to the inch. The Thread of a Screw. 35 FIG. 36. If a single thread were wound evenly round a cylinder, and the path of a thread marked out, we should have a single-threaded screw ; whereas, if two parallel threads were wound on side by side, we should obtain a double-threaded screw. The object of increasing the number of threads is to fill up the space which would be unoccupied if a fine thread of rapid pitch were traced upon a bolt, and thus to give the bolt greater strength in resisting any strain which tends to strip away the thread. In- creasing the number of threads makes no difference in the pitch of the screw, which is dependent on any one continuous thread of the combination. The ordinary screw-propeller is a double-bladed screw, and has sometimes three or even four blades, which correspond to the multiple threads here spoken of. ART. 32. The two principal forms of screw-thread used by engineers are the square and the V thread ; they are given in the sketch, and in applying them we should understand that there are three essential characters belonging to a screw-thread, viz., \\spitch, depth, and form; and three principal conditions required in a screw when completed, viz., power, strength, and durability. It is easy to see that no one can de- clare exactly what power, strength, or du- rability is given by a screw-thread of a certain pitch, depth, or form, when traced out upon a given cy- linder. The problem is indeterminate, and must remain so ; we cannot lay down any rule for determining the diameter of a screw bolt required for any given purpose, nor can we say what should be the precise form of thread. It is the province of practical men to determine any such questions when they arise, being guided in their judgment by experience and by certain general considerations which we propose now to examine. i. The power of a screwed bolt depends upon the pitch and form of the thread. If the screw-thread were an ideal line running round a cylin- 36 Elements of Mechanism. der, the power would depend solely on the pitch, according to the relation given in all books on mechanics, viz. : weight x pitch = power x I Circumference of the circle described I by the end of the lever-handle. If the thread were square we should substitute for the ideal line a small strip of surface, being a portion of the screw surface shown in fig. 35, which would present a reaction P to the weight or pressure everywhere identical in direction with that which occurs in the case of the ideal thread. Hence, if there were no friction, we should lose nothing by the use of a square thread in the place of a line. A square-threaded screw is, therefore, the most powerful of all, and is employed commonly in screw presses. But if the thread were angular, the reaction Q which supports the weight or pressure would suffer a second deflection from the direction of the axis of the cylinder over and above that due to the pitch, by reason of the dipping of the surface of the angular thread, and we should be throwing away part of the force at our disposal in a useless tendency to burst the nut in which the screw works. In this sense, the square thread is more powerful than an angular or V thread of the same pitch. 2. The strength depends on the form and 'depth. This statement is obvious. In a square thread half the material is cut away, and the resistance to any stripping of the thread must be less than in the case of the angular ridges. Again, if we deepen the thread we lessen the cylinder from which the screw would be torn if it gave way, and thus a deep thread weakens a bolt. 3. Finally, the durability of a screw-thread depends chiefly upon its depth, that is, upon the amount of bearing surface ; and in the case of a screw which is in constant use, as, for example, in the slide-rest of a lathe, it would be well for the young mechanic to satisfy himself upon this point by ascertaining the amount of bearing surface given by the fine deep thread which is found upon the screw working in the slide-rest of a well-made lathe. Probably the finest specimen of minute workmanship in screw-cutting will be found in the screws provided by Mr. Simms The Endless Screw. 37 for moving the cross wires or web across the field of view of a micrometer microscope. There are 150 threads to the inch, the diameter of the bolt being about ^th of an inch ; the head of the screw is a graduated circle read off to 100 parts, and the movement of the wires pro- duced by turning the screw-head through the space of one graduation is quite apparent. Upon examining the thread with a microscope, we should see a fine angular screw, consisting of a number of comparatively deep-cut ridges, having the sides a little inclined and the edges rounded off. In the year 1841, Sir J. Whitworth proposed a uniform system of screw threads for bolts and screws used in fitting up steam engines and other machinery. This system has been adopted, and has given rise to the so-called Whitworth thread, about which it is only necessary here to say that tables are published giving the pitches for screws with angular threads on bolts of given diameter, and further that the angle of inclination of the sides of the thread is constant, being 55, with one-sixth of the depth rounded off at the top and bottom. ART. 33. A worm wheel is a wheel furnished with teeth set obliquely upon its rim, and so shaped as to be capable of engaging with the thread of a screw ; the revolu- tion of the endless screw or worm AB will then impart rotation to the wheel C, and the wheel will advance through one, two, or three teeth, upon each revolution of AB, according as the thread thereon traced is a single, double, or triple thread. This reduction of velocity causes the combination to be particularly valuable as a simple means of obtain- ing mechanical advantage, and, as we have stated, the number of threads upon the screw determines the number of teeth by which the wheel will advance during each revolution of AB. In the transmission of force the screw is always employed to FIG. 37. 38 Elements of Mechanism. drive the wheel, and necessarily so, because the friction would prevent the possibility of driving the screw by means of the wheel, even if the loss of power were disregarded ; but in very light mechanism, where the friction is insensible, the wheel may drive the screw, and then the screw is frequently connected with a re- volving fly, and serves to regulate the rate at which a train of wheels terminating in the worm wheel may run round. ART. 34. The annexed lecture diagram, taken from Sir J. Anderson's collection, shows an endless screw and worm wheel as applied in lifting heavy weights. FIG. 38. The machine is called a lifting jack, and will exert considerable power through a space of a few inches. Sometimes an apparatus of this kind consists only of a screw enclosed in rigid casing, and rotated by a long handle, but the drawing shows a piece of me- chanism which is rendered more powerful by the introduction of a second screw and worm wheel between the lever handle and the weight raised. It will be seen that the casing encloses a vertical square- threaded lifting screw, having a head and claw marked SS. Upon the screw is fitted a strong massive nut in the form of a worm The Lifting Jack. 39 wheel, one half of which is shown in either view of the apparatus, the remaining half being cut away in order that the disposition of other working parts may be better understood. The worm consists of an endless screw on a spindle terminat- ing at C, and rotated by a lever handle HH. It is apparent that the lever handle and worm drive the worm wheel, and further that the rotation of the worm wheel or nut imparts a longitudinal motion to the lifting screw. In order that the apparatus may work, provision is made that the lifting screw shall not rotate, the nut in which it works is fixed in position in the casing, and can freely turn without shifting, the result being that S rises or falls slowly as the handle rotates. An example, set out upon the diagram, and solved upon the principle of work, will give a better insight into the mechanical construction. The friction of the working parts is neglected in order to obtain a simple numerical result. Let the pitch of the lifting screw be 1-25 inches, let the worm wheel have 16 teeth, and let the circumference described by H be 87-5 inches. Then motion of handle after 16 tums=87'5 x 16 inches, = 1400 inches, motion of screw at same time= i '25 inches, /.motion of H is to motion of S as 1400 to 1^25, as 1 1 20 to i. Let pressure on handle=2o Ibs. /.weight raised by screw=2ox 1120 Ibs. = 22400 Ibs. = 10 tons. ART. 35. In concluding this chapter we may mention that the doctrine of resolved motion enables us to deduce the relation between two balancing forces when acting on the straight lever. This relation follows as a direct consequence of the principle of work. Let ACB be a straight lever whose fulcxum is C, and let the forces P and W acting perpendicularly to the lever at the points A and B balance each other. Let the lever be now tilted into the position aCl>, and draw am, bn, perpendiculars on ACB. Then the resolved motion am 4 o Elements of Mechanism. represents the displacement of A in direction of P, while nb re- presents the displacement of B in direction of W. FIG. 39. But the forces P and W 2 balance, and no work is done, therefore tf* J?-^l- T^ P*am-Wx&n=0. ^_bn _C_CB 01 W~^~~Crt~CA' FIG. 40. which is the well-known condition of equilibrium. ART. 36. Bell-crank levers serve to change the line of direc- tion of some small motion, and are of universal application. They consist simply of two arms standing out from a fixed axis so as to form a bent lever. i. Suppose it to be required to construct a bell-crank lever so as to change the direction of some small motion from the line BD into the line DA, where BD and DA meet in a point D. Draw DC, dividing the angle at D into two parts whose sines are in the ratio of the velocities of the movements in the given directions. This may be done by setting up perpendiculars anywhere on BD and DA in the required ratio, and drawing straight lines through their extremities parallel to BD and DA re- spectively. These parallel lines will intersect somewhere in DC, and will determine that line. In DC take any convenient point C, and draw CA, CB, per- pendicular to DA and DB respectively, then ACB will be the bell- crank lever required. This is the construction, and it can be immediately verified, for the arcs described by the extremities A and B, when the lever ACB is shifted through a small angle o, will be represented by AC x a and CB x a respectively, and will measure the velocities of the points A and B. TT velocity of A_ACx_AC_sin CDA velocity of B CBx CB~sin CDB' Bell-crank Levers. FIG. 41. It is evident that the movements in DA and DB are very nearly rectilinear, and will become more so the further we remove C from the point D. Any play which may be necessary at the joints A and B, by reason that the ends of the levers really describe small circular arcs, may be easily provided for in the actual arrangement. 2. To change the direction from one line to another not inter- secting it. Draw PQ, a common perpendicular to the lines AD and BE ; through Q draw QH parallel to DA ; construct a bell-crank lever, a cb, for the movements as trans- ferred to the lines BQ, QH; draw ce parallel to PQ and equal to it, and further make e d parallel and equal to cb. The piece a ced will be the lever required; what has been done is this, a bell-crank lever ^ a cb has been formed by the rule given above in order to transfer the motion from BE to QH, and then the motion in QH has been shifted into another line DA parallel to itself. Elements of Mechanism. FIG. 42. CHAPTER II. ON THE CONVERSION OF CIRCULAR INTO RECIPROCATING MOTION. ART. 37. In discussing the nature of harmonic motion we have necessarily been led to consider the most elementary form of apparatus for converting the circular motion of a pin moving in one plane into the reciprocating motion of a guided bar. t The annexed sketch, taken from a model belonging to the School of Mines, shows an apparatus having a pin, P, fastened to a disc of wood, and capable of being rotated by a handle at the back. A horizontal slotted bar, EF, attached at right angles to a vertical guided bar, AD, completes the arrangement. Such an ap- paratus has already been referred to, and it is apparent that the bar rises and falls with a true harmonic motion as the pin, P, moves round uniformly in a circle. We pass on to analyse the conversion of circular into recipro- cating motion by means of the crank and connecting rod. A crank is merely a lever or bar movable about a centre at one end, and capable of being turned round by a force applied at the other end ; in this form it has been used from the earliest times as a handle to turn a wheel. When the crank is attached by a connecting rod to some reciprocating piece, it furnishes a combination which is extremely useful in machinery. In the next chapter we shall see that the crank and connecting The Crank and Connecting Rod. 43 rod is one of the principal contrivances for converting recipro- cating into circular motion ; the student will understand that any such distinction as to the effect of the contrivance is one of classi- fication only, regard being had to the direction in which the moving force travels. The arrangement is often used under both aspects in one and the same machine ; as in a marine engine, where the piston in the steam-cylinder actuates the paddle-shaft by means of a crank and a connecting rod, and the motion is then carried on, by a crank forged upon the same shaft, to the bucket or piston of the air-pump. It was in the year 1769 that James Watt published his in- vention of ' A Method of Lessening the Consumption of Steam and Fuel in Fire-Engines,' the main feature of which was the condensation of the steam in a vessel distinct from the steam- cylinder. The steam-engine was at that time called a fire-engine, and was used exclusively in pumping water out of mines. The steam piston and the pump rods were suspended by chains from either end of a heavy beam centred upon an axis, the action of the steam caused a pull in one direction only, and the pump rods being raised by the agency of the steam were afterwards allowed to descend by their own weight. In this shape the steam-engine was entirely unfitted for actu- ating machinery, and it was not until after the impulse given by Watt's invention was beginning to be felt that it became apparent that the expansive force of steam could be made available as a source of power in driving the machinery of mills. While Watt was occupied with the great problem of the con- struction of double-acting engines, which eventually he fully solved, it happened that one James Pickard, of Birmingham, in the year 1780, took out a patent for a 'new invented method of applying steam or fire engines to the turning of wheels,' in which he proposed to connect the great working beam of the engine with a crank upon the shaft of a wheel by means of a spear or connecting rod, jointed at its extremities to the beam and crank respectively. It is probable that Watt had foreseen this application of the crank as early as the year 1778, and had intended to apply the combination as a means of carrying on the power from the end of 44 Elements of Mechanism. the working beam to the fly-wheel. Being forestalled, however, by the patentee, he did not dispute the invention, and contented himself with patenting certain other methods of obtaining a like result, among which will be found the sun and planet wheels de- scribed in a subsequent chapter. This latter invention served his purpose until the patent for the crank had expired, and then it was that the arrangement which we are now about to discuss came into general use. The manner of employing the crank and connecting rod in the FIG. 43. locomotive engine is shown in fig. 43. The crank CP is made a part of the driving wheel of the engine, the connecting rod PQ is attached to the end of the piston rod QR, and the end Q is constrained to move in a horizontal line by means of the guides HK, LM. In this engine the crank is the follower and not the driver, but the combination is the same whether the circular motion of CP causes the reciprocating motion of Q, or whether the recipro- cation of Q imparts a circular motion to CP. ART. 38. We now pass on to discuss the primary fundamental piece of mechanism de- rived from a simple tri- angle. Since a triangle is an immovable figure, and will not 'rack,' as mechanics express it, some provision must be made for varying the dimensions of at least one side. Fis. 44- The Crank and Connecting Rod. 45 In our combination, the crank CP is of fixed length, as is also the connecting rod PQ, but the side CQ is of variable length. When CP performs complete revolutions, it is clear that Q will reciprocate in the line CQ. We shall presently see that the motion of Q is of an aggregate character, or that Q is the reci- pient of two distinct movements which are simultaneously im- pressed upon it. ART. 39. To determine the relative positions of the crank and connecting rod during the motion. FIG. 45. Let CP be the crank centred at C, PQ the connecting rod, and let the point Q be constrained to move in the straight line CED. Draw PN perpendicular to CD, and let CP=#, PQ=, also let the angle PCQ=C, and PQC=Q. Then CQ = CN + NQ = a cos C + cos Q. =4 m . sin Q sin c sin C b b 2 C .-. CQ = a cos C + N/ 2 a a sin 2 C, which gives the position of Q for any value of C, that is, for any given position of the crank CP. Cor. i. LetC = o, /. CD =a+; C= i8o,/.CE= - a + b; whence DE = CD CE = 2a. The space DE is called the throw of the crank. Cor. 2. If the position of Q be estimated by its distance from D, we have DQ = CD - CQ= a + b-(a cos C+b cos Q) = a (i-cos C) + (i-cos Q). 46 Elements of Mechanism. But we have just shown that an expression such as a (i cos C) represents the resolution of circular into reciprocating motion, and we infer that the motion of the point Q is compounded of the resolved parts of two circular motions, one being that due to the motion of P through an angle C in a circle round C, the other, that resulting from the motion of P in a circular arc through an angle Q, and produced by the swinging of the rod PQ about one end Q as it moves to and fro. Hence the connecting rod introduces an inequality, which pre- vents the motion of the point Q from retaining that evenness and regularity of change which was found in the motion of the point N (Art. 7). We now see by analysis that this inequality, whereby the motion of Q differs from that of N, is equal to b (i cos Q). Ex. Let CP = 10 inches, PQ 5 feet, as in the engine in fig. 43 ; find the position of the piston when the crank is vertical. -cos Q) = 10 + 60(1-^35) = 10 +'84 nearly, or Q is nearly six-sevenths of an inch in advance of the centre of its path when the crank has made a quarter of a revolution from the line CD. As the connecting rod is shortened - the inequality increases, and the motion becomes more unequal. Take a very extreme case, where PQ=CP=#, .-.DQ = tf(i cos C)-f a(\ cos Q). Let now C=6o, then Q=6oalso, because the triangle CPQ is now isosceles, /.cos C=-=cos Q /.DQ=20 a=a, or Q has moved through half its path, while CP has turned through an angle of only 60. Ratio of Velocities. 47 When 0=90, the point Q comes to C, and there the motion ends, for the crank CP can now go on rotating for ever without tending to move Q. Co r. 3. If the 'connecting rod could be prolonged until it be- came infinite, we should have PQ always parallel to itself, or Q = o, and in that c;ase the travel of Q would be represented by the equation DQ=(i cos C). A crank with a connecting rod of infinite length is an imagi- nary creation, but we shall presently see that an equivalent motion e obtained in various ways. 'ART. 40. To determine the velocity ratio of P to Q : V be the velocity of P, v the velocity of Q. From C a straight line at right angles to CQ, and let it meet QP produced in H (fig. 45). Then since PQ is a rigid rod, the resolved velocity of P along QP is equal to the resolved velocity of Q along QP. Let PCQ=C, PQC=Q, CPH=a. Then P may be taken to move in a tangent at P, and therefore its resolved velocity in PQ is V sin a, also the resolved velocity of Q in the same direction is v cos Q. Hence V sin o=z> cos Q, . #_sin n _sin a_CH ' 'V~cos Q~sin H~~CP~' The same result may be obtained by analysis, for making PCQ=0, PQC- & we have ,, a s0+ If COS sin x CQ PQ cos $ sin sino _ sin o _ CH cos-p sin sin H CP ' 48 Elements of Mechanism. ART. 41. The relative velocities of P and Q may be set out in a diagram. Take CP : PQ:: i I 4, which will give a figure of convenient dimensions, and let Q,^ be two positions of the end of the con- necting rod. Produce QP, qp to meet the vertical diameter FCK FIG. 46. in H and h respectively, the line BCD being horizontal. Join CP, C/, and measure off CS = CH, and Or = Ch. Then, as before, z>_CH_CS V~CP~CP" In like manner, -=--, and so for all other values, vel. of p C/ whereby the curve CS.y will indicate the comparative velocities of the crank pin and of the end of the connecting rod during one single stroke. Also, by a similar construction, the lower loop may be set out, preserving the proportion, Vel. of r:vz\. of R::C/: CR, and making CT = Ct. If PQ were kept parallel to itself throughout the motion we should have in effect a crank with an infinite link, and the looped curve would become a circle described on the radius FC as a diameter, as in the case of the simple harmonic motion. The in- equality introduced by the obliquity of the connecting rod is thus rendered apparent by comparison of the distorted loop with the true circle. Ex. To compare the positions of Q when CP makes a given angle of x with CA, CB respectively. The Eccentric Circle. 49 Let D and E be the extreme positions of Q, then DQ=0(i cos Q} + b(\ cos ) EQ=20 DQ=0(i+cos Q} b(i cos ). Also, let d be the circular measure of x, then $ can be found in terms of a, b, and 0. Take the case of a direct acting engine, where a = i foot, b = 6 feet, and letjc=i. Calling Q,Q' the required positions of the end of the connecting rod, we have cos 0= -9998477, cos < = -9999958, /.DQ = -oooi775 feet, EQ'=-oooi27i feet. ART.' 42. The eccentric circle supplies a ready method of ob~ taining the motion given by a crank and link, and we proceed to examine it with the intention of ascertaining by what expedients we are enabled to vary the lengths of the particular crank and link which exist in every form of the arrangement. And, first, we notice that the length of the crank is in every case equal to the distance between the centre of F|G 47 motion and the centre of the eccentric circle ; it is, in fact, the line CP in each of the annexed drawings. Let us consider the motion shown in fig. 47, where a circular plate, movable about a centre of motion at C, imparts an oscillatory movement to a bar, QD, which is capable of sliding between guides in a vertical line, DQ, pointing towards / C. Since PQ remains constant as the plate re- / volves, it is evident that Q moves up and down in the line CD, just as if it were actuated by the crank, CP, and the connecting rod, PQ. The length of the connecting rod is in this case, therefore, equal to the radius of the rotating circle. It is obvious that an arrangement of this kind would be little used, by reason of the oblique thrust on the bar QD. A second form which, however, is of the greatest possible value, is deducible at once from that last examined. 5O Elements of Mechanism. Instead of allowing the end of the bar, QD, to rest directly FJG g upon the circumference of the circle, suppose that bar to terminate in a half-hoop which fits the circle, as shown in the drawing ; let the rod point to the centre of the circle, and let one end, Q, be compelled to move in a line point- ing to C. As the circle revolves it is evident that we have a crank, CP, just as before ; but we have, in addition, a link which is now re- presented by PQ, and which may extend as far as we please beyond the limits of the circular plate. We thus obtain a combination which is sometimes described as a mechanical equiva- lent for the crank and connecting rod. The form usually adopted in practice is derived at once from the arrangement just described. A circular plate is completely encircled by a hoop, to which a bar is attached ; this bar always points to P, the centre of the plate, and its extremity drives a pin, Q, which is constrained to move in the line CQ. The plate is movable about a centre of motion at C, and we have already explained that PQ remains constant during each re- volution of the plate, or that the resulting motion impressed upon Q is that due to a crank, CP, and a link, PQ. As before, the throw of the eccentric is the same as that of the crank, viz., a space equal to the diameter of the circle whose radius is CP. We should remark that P, the centre of the plate, may be brought as near as we please to C, the centre of the shaft, and that the throw of the eccentric may be reduced accordingly ; but that we are limited in the other direction, for the shaft must be kept within the boundary of the plate, and the plate itself must not be inconveniently large, considerations which are sufficient to prevent our increasing CP in any great degree. The eccentric circle may also be regarded as a simple form of cam (see Art. 58), but we have examined it here on account of its being identical in principle with the crank and connecting rod. The object of the complete hoop is to drive Q in alternate The Eccentric. directions. In some cases Q is brought back by a spring, and then only half the hoop is required. An instance occurs in modern forging machines, where the motion is very small and rapid. ART. 43. Having thus explained the principle of construc- tion adopted in the eccentric, it remains to show the contrivance as made and applied in an engine. In the annexed diagram, which is taken from a small oscillating engine, the circle C repre- sents a section of the crank shatt, C being its centre. Upon the shaft are fitted two circular half-pulleys of cast iron, which are bolted together, and have a centre at P. Two half-hoops of brass, tinted in the sketch, and united together by bolts and double nuts at E and H, carry the eccentric bar, which actuates a pin at B connected with the valve lever. The engine being designed for a river boat, and therefore requiring to be reversed at pleasure, there is a strap, ab, to prevent the eccentric rod from falling away from the pin while the valve is being moved by hand. Also, in this case, the eccentric pulley rides loose upon the shaft within cer- tain limits defined by stops, and there is consequently a disc, D, forming a counterbalance to the weight of the pulley, which pr,e- vents it from falling out of position during the disengagement of the pin at B. ART. 44. It will be understood that the crank and connect ing rod labour under the disadvantage of entailing a division ot the shaft whenever it is required to place the crank anywhere ex- cept at one end, for the connecting rod is continually traversing over the centre of the shaft. If, therefore, a crank is wanted in some intermediate portion of an axle or shaft, the axle must be cranked in the manner shown in fig. 50, or be divided, and the two cranks or arms will be con- Elements of Mechanism. FIG. 50. nected by a pin. These cranks and the pin are frequently forged in one solid mass upon the shaft, and shaped afterwards by the machinery of the workshop. The annexed sketch is taken from a lecture diagram in Sir J. Anderson's collection, and repre- sents a small vertical engine suitable for driving light machinery. The steam cylinder is marked C, and H is the slide case, the piston rod being connected with the crank pin by the connecting rod PR. The slide valve is worked by an eccentric, shown at E, and the eccentric rod attached to the valve spindle is marked ED. The great value of the eccentric arises from the circumstance that it enables us to derive the motion, which would be given by a crank, from any part of a shaft without the necessity of subdividing it. This is particularly noticeable in the me- chanism here set forth, where the crank of small throw which is re- quired for moving the steam slide- valve is furnished by the aid of an eccentric keyed upon the main shaft. \\RT. 45. It is hardly necessary to explain that when a circle revolves about an axis perpendicular to its plane, and a little out of the centre, it will be enveloped in all positions by a somewhat larger circle, the increase of the radius being equal to the eccen- tricity. This fact has been applied to a useful purpose in the produc- tion of gun-stocks by machinery, as, for example, at the Small Arms Factory at Enfield. The practice has been to form a har- dened steel block with cavities, shaped so as exactly to correspond with the cavities which it is intended to carve out of the gun- The Eccentric Circle. 53 stock. The clearing out of the recesses is effected by revolving drills, making some 6,000 revolutions per minute, which are carried over the work, and are guided on the copying principle by a dumb tracer, which travels over every portion of the steel block. The tracer and the rotating drill exactly correspond in size. For more delicate portions of the work smaller bits with corresponding tracers are employed, and thus the wood is carved out with the details of form which are to be found in the pattern. But now comes a difficulty. The tracer and pattern are of hardened steel, and each tracer has its corresponding bit or drill, which is an exact revolving counterpart thereof, so long as its cutting edges are not worn away. But when the drills wear by sharpening, and become smaller, the copy will be defective. In order to compensate for this source of error, the conical hole in the end of the running spindle into which the drills are fitted is made slightly eccentric. The drill is also eccentric on its shank to the FlG - si- same amount. It is therefore possible to bring the axis of the drill itself exactly in the same line as the axis of the running spindle, in which case the drill will revolve in a cylinder of its own size. Or the axis of the drill may be set on one side of the axis of the running spindle, in which case the drill will carve out a cylinder larger than itself. The end of the spindle and drill are graduated round the circumference, and thus the amount of eccentricity, and the con- sequent enlargement of the drill spindle, can be adjusted with the utmost nicety. In the drawing A represents the centre of the running spindle, B is the centre of the conical hole made therein, C is the centre of the drill spindle. When the drill is at work C describes a circle round A. But C can be shifted into any position round B as a centre, and therefore C can be brought either to coincide with A, or can be placed at any distance from A, lying between the limits zero and AC. Hence the result which has been stated. ART. 46. There is yet another method of arranging the eccen- .94 Elements of Mechanism. trie circle, which gives us the combination of a crank with an infinite link. We here intend, as explained in a former article, that the re- ciprocating piece shall be driven by a crank and connecting rod in such a manner that the connecting rod shall always remain parallel to itself, a re- sult v hich could only happen in theory if the connecting rod were indefinitely lengthened. Suppose the roller at Q to be replaced by a cross bar QR, standing at right angles to DC. As the circle revolves, it will cause the bar to reciprocate, CP will remain constant, and PQ will always be at right angles to RQ, and will therefore remain parallel to CD in all positions. But this is the motion of a crank with an infinite link. If C were in the circumference of the circle, the motion would be just the same, except that now the crank would be the radius of the eccentric circle. We shall presently notice a useful illus- tration of this particular case. (Art. 48.) Take the following as an example of the movement under dis- cussion. In Sir J. Anderson's machine for compressing elongated rifle bullets, which was in use during the Crimean war, there are punches fixed at the two ends of a strong massive rod, to which a reciprocating motion in a horizontal line is imparted, and a piece of lead is compressed into the required form at each end alternately. The object of one part of the mechanism is to cause this rod to reciprocate, and the movement is obtained as follows. A small circle centred at C represents in section a shaft caused to rotate by the power of an engine. Upon this shaft a short cylindrical block is forged so as to form part of- it, and is after- wards accurately turned into the form of a circular cylinder, whose axis passes through P upon one side of the original axis. A rectangular brass block, RS, is bored out to fit the larger cylinder and slides in the rectangular frame FG, to which the Examples of Harmonic Motion. 55 cylindrical pieces AB, DE, which carry the punches, are attached. The whole is put together in the manner shown, and it requires very little effort to understand that the rotation of the eccentric cylinder round an axis through C will impart to RS simultaneous FIG. 53- movements in a horizontal and vertical direction, whereof one, viz., that in a direction perpendicular to AE, will be inoperative, and the other will be communicated directly to FG, and so to the rods carrying the punches. Thus AB and ED will be made to oscillate in the guides indicated in the sketch. In this case, then, the eccentric circle, whose centre is at P, is caused to rotate about the point C, and gives motion to the sides of the frame FG, just as it moved the bar QR in the last article, and hence the resulting motion is that of a crank, CP, with an infinite link. The contrivance of the ratchet wheel at the right hand will be explained in a subsequent chapter. ART. 47. The crank with an infinite link also appears under the guise of a swash-plate. Here a circular plate, EF (fig. 54), is set obliquely upon an axis, AC, and by its rotation causes a sliding bar, PQ, whose direction is parallel to AC, to oscillate continually with an up and down movement, the friction between the end of the bar and the plate being relieved by a small roller. We must now try and ascertain what is the law which governs this motion, and we observe that since PQ FIG. 54. 5G Elements of Mechanism. remains always parallel to AC, the actual path of Q, as projected upon an imaginary plane through the lowest position of Q, and perpendicular to AC, will be a circle. If this be so, it follows that the path of Q upon the plate itself will not be a circle, but an oval curve, and, as a matter of geometry, we can prove that the line CQ will vary in length as Q rises or falls during the rotation of the plate, in the precise degree necessary for the description of the curve known as an ellipse. In fig. 55 let EQF represent the actual path of Q on the plate, and let the circle ERD be the projection of this path upon a plane perpendicular to the axis AC. Draw QM perpendicular to EF, QR perpendicular to the plane ERD, and RN perpendicular to ED, which is the diameter of the circle ERD. Join MN, and suppose the plate to rotate through an angle EAR = A, and thus to carry the roller at Q through a vertical space equal to RQ. Then RQ = MN = AC x = AC(i -cos A); or the motion is that of a crank AC with an infinite link. This is a curious result. Hitherto, in the motion of a crank with an infinite link, the reciprocation has always taken place in a plane perpendicular to the axis of rotation, but here we get the very same movement in a plane which contains the axis instead of being perpendicular to it. ART. 48 It is sometimes required that the reciprocating mo tion shall be intermittent, or have intervals of rest. This motion may be provided for by placing a loop at the point where the eccentric bar engages the pin. It is evident that the pin will only move when one end of the loop takes it up but in Intermittent Motion. 57 FIG. 56. doing this a blow is struck, which it may be well to avoid, and hence an intermittent motion has been obtained in a much better manner by a movement adapted for working the slide-valves of a steam-engine. We can readily see that if any portion of the plate in Art. 42 be shaped in the form of a circle round C, such portion will have no power of moving the sliding bar. Let the pin P assume the form of a circular equilateral triangle, CAB, formed by three circular arcs, whose centres are in A, B, and C respectively, and let it be embraced by a rectangular frame attached to a sliding rod. As CAB revolves round the point C, the portion CB will raise the plate ; the point B will next come into action, and will raise the plate still higher ; the upper edge of the groove will then continue for a time upon the curved surface AB, which is a circular arc described about C as a centre, and here the motion will cease ; the plate will next begin to fall, will descend as it rose, an interval of rest will succeed, and thus we shall produce an intermittent movement, which may be analysed as follows : Suppose the circle described by B to be divided into six equal parts, at the points numbered i, 2, 3, 4, 5, 6. FIG. 57. As B moves from i to 2, the frame remains at rest ; from 2 to 58 Elements of Mechanism. 3 the arc CB drives the frame, the centre of motion of the eccen- tric circle being now a point in its circumference, and the hori- zontal bar is driven as it would be by a crank CB with an infinite FIG. 58. link (see Art. 46) ; from 3 to 4 the point B drives, and the motion is again that of a crank, CB, with an infinite link (see Art. 46) ; i.e., the motion from 3 to 4 is the same as that from 2 to 3, except that it is decreasing in velocity instead of increasing. From 4 to 5 there is rest, then an increase of motion from 5 to 6, and finally a decrease to zero as B passes through the arc 6 to i and completes an entire revolution. ART. 49. Circular may be converted into reciprocating motion by the aid of escapements. An escapement consists of a wheel fitted with teeth which are made to act upon two distinct pieces or pallets attached to a re- ciprocating frame, and it is arranged that when one tooth escapes, or ceases to drive a pallet, the other shall commence its action. One of the most simple forms is the following : A sliding frame, AB, is furnished with two projecting pieces at C and D, and within it is centred a wheel possessing three teeth, P, Q, and R, which tends always to turn in the direction indicated by the arrow. FIG. 59. The upper tooth, P, is represented as pressing upon the pro- jection D, and driving the frame to the right hand : when the tooth Escapements. 59 P escapes, the action of Q commences upon the other side of the frame, and the projection C is driven to the left hand. Thus the rotation of the wheel causes a reciprocating movement in the sliding piece, AB. It is clear that the wheel must have i, 3, 5, or some odd num- ber of teeth upon its circumference. ART. 50. The Crown Wheel escapement was invented for the earliest clock of which we possess any record. The form of the wheel is that of a circular band, with large saw-shaped teeth cut upon one edge ; the vibrating axis, AB, carries two flat pieces of steel, a, b, called pallets, which project from the axis in directions at right angles to each other, and engage alternately with teeth upon the opposite sides of the wheel. In order to ensure that each pallet shall be struck in succession, either the wheel musbhave an odd number of teeth, or the axis of the pallets must be set a little out of the central line. Suppose the wheel to turn in the direction towards which the teeth incline, and let one of its teeth encoun- ter the pallet b and push it out of the way ; as soon as b escapes, a tooth on the opposite side meets the pallet a and tends to bring the axis AB back again : thus a reciprocating action is set up, which will be very rapid unless AB is provided with a heavy arm, CD, at right angles to itself. Such an arm possesses inertia, so that its motion cannot be suddenly checked and reversed, and a recoil action is set up in the wheel, which materially subtracts from the utility of the contrivance. For it will be seen that the vibration of CD cannot be made to cease suddenly, and that the wheel must of necessity give way and re- coil at the first instant of each engagement between a tooth and its corresponding pallet. The more heavily CD is loaded at a distance from the axis the more slowly will the escapement work, and the greater will be the amount of the recoil. 6o Elements of Mechanism. Here we have an invention which has done good service to mankind. It was used in the first clock which was ever made, and dealt out time through the step-by-step movement of the wheel with pointed teeth. This wheel, urged on by a weight, and hampered always by the vibrating bar, whose pallets were perpetually getting into the way of its teeth, moved round with a slow, intermittent, and step- by-step movement, checked and advancing alternately, but solving for mankind, in a clumsy though tolerably accurate manner, the great problem of the mechanical measurement of time, and giving birth, by the idea suggested, to those marvellous pieces of mechanism which have finally resulted in the modern astronomical clock and the chronometer. Being, however, in some particulars, a defective or imperfect contrivance, it has gradually sunk from one level to another ; it has disappeared from clocks and from watches also, and is now seldom to be met with except in the homely contrivance of the kitchen-jack for roasting meat. ART. 51. We purpose, before going further, to examine a little more particularly the mechanism of an escapement, so as to gain some idea of the refinements of its construction when ap- plied to the best made clocks or watches, and we will review very rapidly the elementary facts which are to be found in the books on mechanics. An imaginary or simple pendulum is a conception of mathema- ticians, and is defined as being a single particle of matter, P, sus- pended by a string, DP, without weight. FIG. 61. This particle may swing to and fro in a vertical plane under the action of the pull of the earth, and the oscillation of the pendulum is the whole movement which it makes in one direction before it begins to return, viz., ACB The time of a small oscillation is the period of this movement, and is given by the for- mula : where / = time of a swing in seconds, length of the string in feet, ^=32-2 feet. The Pendulum. 6 1 Make /= i, and we have the length of a pendulum oscillating seconds, which is a little less than a metre, being equal to 39 -14 inches. The discovery of the so-called pendulum law was made by Galileo, who noticed that a lamp swinging by a chain in the me- tropolitan church at Pisa made each movement in the same time, although gradually oscillating through smaller arcs while coming to rest. For ordinary purposes the time of a swing may be supposed to depend only on the length of the pendulum, and not at all upon the arc through which it swings. It is practically very nearly the same for arcs up to 2 or 3 degrees on each side of the vertical, and it may be shown by calculation that the error introduced by assuming the law to be strictly true, in the case of a pendulum moving through an arc of 5 degrees from the vertical position, would only amount to 2rirVo tn P art f tne ^ me f a swing. A seconds' pendulum in a well-made clock swings through about 2 degrees on each side of the vertical. The question now arises, how is the law of the swing of this imaginary pendulum to be applied in the regulation of clocks, and wherein does a solid heavy pendulum, which we must of necessity employ, resemble or differ from an ideal pendulum? Conceive a straight uniform bar of iron to be suspended close to one end upon small triangular wedges of hardened steel, technically called knife edges, which rest upon perfectly horizontal agate or steel planes, and then to be set swinging. It is evident that each particle of the bar will endeavour to observe the pen- dulum law stated above, and will tend to swing in different times. But all the particles must swing together, and the result is that a sort of compromise takes place between the different tendencies, and the \vhole bar swings as if its material were all collected into one dense point at a certain distance from the knife edge, which is known as the centre of oscillation. Thus the solid pendulum swings as an ideal pendulum would do whose length was equal to the distance between the centres of suspension and oscillation. This is the theory of the rigid pendulum, which has been in- vestigated by Huyghens, and by others after him, and has led to many interesting experiments in applied mechanics. 62 Elements of Mechanism. ART. 52. A method of showing roughly to a class the position of the centre of oscillation is the following : The drawing represents a wooden sword, having a pin at C passed through its handle and resting in two small upright sup- ports, in such a manner that the sword can freely turn in a vertical plane about the point C. The sword is divided into two parts at B in the proportion show r n, and there is a tightening screw at B, which may be set to make it somewhat difficult to bend the sword out of shape. The sword is then allowed to hang freely as a pendulum from C, and a ball suspended by a silk thread at C is swung by the side of it. When the sword and the bullet swing together the position of the centre of the bullet is marked at O upon the sword. The experiment can now be completed. An anvil, P, is placed under the sword as in the figure, the sword is raised, and is allowed to fall and strike upon P. It will bend as shown. If CP be greater than CO it will bend with its end upwards, whereas if CP be less than CO it will bend downwards, just as if CA were a thin wire carrying a weight, O, upon it. The whole weight of the sword is for the purpose of the blow collected in the point O, and if CP be equal to CO the sword will not bend at all. ART. 53. The application of the pendulum in the manner in which it is now employed in clockwork forms an important branch of mechanism, and a great advance was made by Dr. Hooke, the contemporary of Newton, who devised the so-called Anchor Escapement. Looking at the contrivance without regard to its connection with a pendulum, we find in it a wheel with pointed teeth, which is centred at E, and tends always to turn in the direction indi- cated by the arrow. Escapements. 63 A portion of this wheel is embraced by an anchor, A C B, centred at C, the extreme ends of which are formed into pallets, A m and B n : these pallets FIG. 63. may be flat or slightly convex, but they are subject to the condition that the perpendicu- lar to A m shall pass above C, and the perpendicular to B n shall pass between C and E. The point of a tooth is repre- sented as having escaped from the pallet B n after driving the anchor to the right hand; and the point g, by pressing against A ;//, is supposed to have already pushed the an- chor a little to the left hand, and thus the wheel can only proceed by causing a vibratory motion in the anchor, A C B. If the escape wheel engaged only a light metal anchor, such as that shown in the drawing, the rapidity of the vibration set up, as above described, would be very great ; but in a clock the object is to provide that the wheel shall advance by slow and regular steps, and the anchor is therefore controlled by the inertia of a com- paratively heavy swinging pendulum. There is one uniform method of connecting the anchor ancj the pendulum which can be seen in any clock. The pendulum, consisting often of a compound metal rod with a heavy bob, is swung by a piece of flat steel spring, and vibrates in a vertical plane a little behind that in which the anchor oscillates. To the centre of the anchor is attached a light vertical rod, having the end bent into a horizontal position, and terminating in a fork which embraces tl e pendulum rod. It follows that the anchor and the pendulum swing together as one piece, although each has a sepa- rate point of suspension. The same recoil is experienced upon each swing of the pen- dulum as that which we noticed in the last article, and the con- trivance is commonly known as the Recoil Escapement. 64 Elements of Mechanism. ART. 54. The exact character of the action which takes place between the pendulum of a clock and the scape wheel has been the subject of a long and interesting mathematical investigation, and before mentioning the results which have been arrived at, we may state in general terms the nature of the problem. The going part of a clock consists of a train of wheels tending to move under the action of a weight or spring : if the last wheel of the train were left to itself, it would spin round with great ve- locity, and we should fail in obtaining any measure of time. The escapement is one part of a contrivance for regulating the velocity of the train of wheels, but the escapement alone is not sufficient : we require further a vibrating body possessing inertia, the motion of which cannot be suddenly stopped or reversed. Such a body is found in the pendulum, and a very intricate mutual action exists between the pendulum and the scape wheel. The function of the pendulum is to regulate and determine the periods and amount of onward motion in the scape wheel, whereas the office of the wheel is to impart such an impulse to the pendu- lum at each period of this onward movement as may serve to maintain its swing unimpaired, and may cause it to move with the same mathematical precision which would characterise the vibra- tions of a body swinging in vacuo, and uninfluenced by any dis- turbing causes. It is remarkable that in the case of the ideal pendulum, where there is no artificial resistance and no friction, the movement is in theory perpetual. In the case of the rigid pendulum the friction at the point of suspension and the resistance of the air would gra- dually diminish the arc of swing, and the movement would slowly subside and die away, although it might be many hours before it became quite imperceptible. The mechanism of a clock must therefore so act upon the pendulum as to maintain its swing un- impaired by these resistances, and it should be borne in mind that the swing of the pendulum is identical with that of the anchor in the last-mentioned escapement, whence it follows that any impulse or check given to the anchor is felt at once as an impulse or check upon the pendulum. Refer now to the recoil escapement described in Art. 53, and conceive that the escape wheel is urged onwards in the direction The Pendulum. 65 of the arrow by the force of the clock train, so as to press its teeth slightly against the pallets of the anchor, the pendulum being hung from its point of suspension by a thin strip of steel, and vibrating with the anchor in the manner already stated. Let the arc A E C D B be taken to repre- sent the arc of swing of the centre of the bob of the pendulum. As the pendulum moves from B to E the point q of the escape wheel rests upon the oblique surface A m of the pallet, and presses the pendulum onward until the point of the tooth escapes at the end of the pallet. For an instant the escape wheel is free, and tries to fly round, but a tooth is caught at once upon the opposite side by the oblique edge B n, and the escape wheel then presses against the pendulum and tends to stop it, until finally the pendulum comes to rest at the point A, and com- mences the return swing. What now has been the action ? From B to E the force of the train as existing in the escape wheel has been acting with the pendulum and has performed its proper office in assisting to main- tain the swing ; whereas from E to A this force has acted against the pendulum. So also on the return swing, the escape wheel will ac f with the pendulum from A to D, and against it from D to B. The action then is alternately with and against the pendulum, and it might be supposed that the injurious effect of a pressure against the pendulum would be entirely corrected by the main- taining force in the other part of the swing ; but this is not the case. The pendulum no longer moves with what we may call its natural swing, as a free pendulum would oscillate, and any varia- tion in the maintaining force will disturb the rate of the clock. The matter has been carefully analysed by mathematicians, and they have shown that the principle of this escapement is radi- cally bad, because it is impossible to remedy entirely the harm which is done by continually interfering with the swing of the pendulum. There occurs also the useless expenditure of energy. It is almost superfluous to remark that no mechanical arrangement will ever bear a close scrutiny when it is so constructed as to throw away work. F 66 Elements of Mechanism. ART. 55. The dead-beat escapement W3& invented by Graham, and at once removes this primary objection. It is, however, most worthy of note that the change in construction which abolishes the defects due to the recoil, and gives the astronomer an almost perfect clock, separates the combination entirely from its original conception, viz., that of an apparatus for converting circular into reciprocating motion. No such conversion can be effected by Graham's escapement. The improvement is made clear by the sketch, and the student will observe that the pallet A has its lower edge in the form of a FIG. 65. circular arc, Ag, whose centre is C, and again that the upper por- tion of the pallet B is also a circular arc struck about the same centre. The oblique surfaces gm, np complete the pallets. Take the case shown in the diagram, which is enlarged so as to make the action more apparent. As long as the tooth is resting on the circular portion nr of the pallet, the pendulum is free to move, and the escape wheel is locked. Hence in the portion EA, and back again through AE, there is no action against the pendulum except the very minute friction which takes place between the tooth of the escape wheel and the surface of the pallet. Through Dead-beat Escapement. 67 a space ECD the point of the escape wheel is pressing against the oblique edge np and is urging the pendulum forward. Then at D the tooth upon the opposite side falls upon A^, and the escape wheel is locked ; from D to B, and back again to D, there is the same friction which acted through EA or AE ; whereas from D to E the point of a tooth presses upon qm and urges the pendulum onward ; at E another tooth is locked upon the pallet B, and thus the action is reproduced in the order in which it has been described. It follows that any action against the pendulum is eliminated, or, more correctly, is rendered as nearly as possible harmless, and the difference between the ' recoil ' and the ' dead beat ' will be understood upon contrasting the three enlarged diagrams, which sufficiently explain themselves, the lower sketch referring to the recoil escapement. The term ' dead beat ' has been applied because the seconds' hand which is fitted to the escape wheel stops so completely when the tooth falls upon the circular portion nr. There is none of that recoil or subsequent trembling which occurs when a tooth falls upon B;z and is driven back. The actual construction of the dead-beat escapement having been explained, it only remains for us to state two of the principal conclusions which follow from a theoretical inquiry into the mo- tion of the pendulum. 1. All action against the pendulum should be avoided, and if some such action be inevitable, it should at any rate be reduced to the smallest amount that is practicable. 2. The maintaining force should act as directly as possible, and the impulse should be given through an arc which is bisected by the middle point of the swing. That is, the arc of impulse DCE should be bisected at the lowest point C. This latter condition cannot be exactly fulfilled, because the point of the escape wheel must fall a little beyond the inclined slope of the pallet in order that it may be locked with certainty. ART. 56. In a printing telegraph instrument the recoil escape- ment has been employed to control the rapidity of motion in a train ot wheels, and the number of vibrations of the anchor are 68 Elements of Mechanism. appreciated by listening to the musical note which it imparts to a vibrating spring. The anchor ACB (fig. 66) is centred at C, and vibrates rapidly as the scape wheel E revolves ; a strip of metal, F, carries on the oscillation to a steel spring which gives the note, and the velocity of the train can be regulated by an adjustable weight attached to the spring. Again, the same escapement forms part of the mechanism of an Alarum dock, where a hammer is attached by a bar to the an- chor, and blows are struck upon the bell of the clock in rapid suc- cession as the scape wheel runs round. FIG. 66 FIG. 67. ART. 57. The teeth of the wheel in an anchor escapement are sometimes replaced by pins, in which case the form of the anchor may be so altered that the action shall take place upon one side of the wheel, as shown in fig. 67. ART. 58. Circular may be converted into reciprocating mo- tion by the aid of cams. The term ' cam ' is applied to a curved plate or groove which communicates motion to another piece by the action of its curved edge. Such a plate is shown in fig. 68, and, as an illustration, we shall suppose that the portions al>, ca are any given curves, and that be is a portion of a circle described about the centre of motion. It is easy to understand that as the cam rotates in the direc- Cams. 69 tion of the arrow, the roller P at the end of the lever AP will be raised gradually by the curved portion ab, will be held at rest while be passes underneath it, and, finally, will be allowed to fall by the action of ca. In this way a cam may be made to impart any required mo- tion, and may reproduce in ma- chinery those delicate and rapid movements which would otherwise demand the highest effort of skill from a practised workman. ART. 59. The circular motion being uniform, the recipro- cating piece may also move uniformly, or its velocity may be varied at pleasure. i. Suppose that the reciprocating piece is a sliding bar, whose direction passes through the centre of motion of the cam-plate. Take C as this centre, let BP represent the sliding bar, and let A be the commencement of the curve of the cam -plate. The curve AP may be set out in the fol- lowing manner. With centre C and radius CA describe a circle, and let BP produced meet its circum- ference in the point R. Divide AR into a number of equal arcs Aa, ab, be, &c. Join Ca, C, O, &c., and produce them to/, q, r, &c., making ap, bq, cr, &c., respec- tively equal to the desired movements of BP pi in the corresponding positions of the cam- plate. The curve hpqr . . . P will represent the curve required. This curve will often present in practice a very irregular shape, but in the particular case where the motion of PB is required to be uniform, it assumes a regular and well-known form. Let CA = a, CP = r, PCA = 0, and let BP move in such a Elements of Mechanism. FIG. 70. manner that the linear velocity of P shall be constantly m times that of the point A, in other words, let RP = m . RA. Now RP = r a, and RA = ad. .'.r a = m . ad, which is the equation to the spiral of Archimedes. 2. We will next examine the case where the centre of motion of the cam-plate lies upon one side of the direction of the sliding bar, and we shall find that the method of setting out the curve changes accordingly. Suppose that the direction of BP passes upon one side of the centre of motion C, draw CR perpendicular to BP produced, de- scribe a circle of radius CR, and conceive the motion to begin when A coincides with R. As a matter of theory such an extreme case is possible, and we will imagine it to exist in order to obtain the equation which represents the complete curve. Practically, the cam would be more effective in straining the bar than in moving it when the point P was near to the point R. Divide AR into the equal intervals A a, afr, be, &c., but now draw ap, bq, cr, &c., tangents to the circle, and equal in length respectively to the desired movements of BP during the corresponding periods of motion of the cam-plate. The curve Kpqr . . . P will be that required, and the analytical representation of it is the following : Let CP=r, CA=a, ACP=0, PCR=), RP = a tan 0, .*. a (6 + /)) = ma . tan 0. FIG. 7 Now cos = -, and tan = tan V:;- COS 2 Cams. whence + cos ~ l - Cor. Let m=i, or RA=RP, which would happen if AP were a stretched string unwound from the circle. The curve traced out by the end P of the string becomes in this case a well-known curve called the involute of the circle, and our equation takes the form which is the equation to the involute of a circle. ART. 60. The heart wheel has been much used in machinery, and is formed by the union of two similar and equal cams of the character discussed in the first part of Art. 59. A curved plate, C, shaped like a heart, actuates a roller, P, which is placed at the end of a sliding bar, or which may be at- tached to a lever PAB, centred at some point A, and connected by a rod BD to the recipro- cating piece. The pe- culiar form of the cam allows it to perform complete revolutions, and to cause an alter- nate ascent or descent of the roller P with a velocity which may be made quite uniform. Since a cam of this kind will only drive in one direction, the follower must be pressed against the curve by the reverse action of a weight or spring. ART. 6 1. In order to illustrate in a lecture the power of cams to produce any required movement, the late Professor Cowper ar- ranged a model which would write the letters R I, selected pro- bably as a compliment to the Royal Institution. The principle of this combination of cams will be readily under- stood if we remember that the successive movements of a point in directions parallel to two intersecting lines will suffice to enable the point to take up any position in the plane of the lines. .72 Elements of Mechanism. The bars shown in the drawing have fixed centres at A and B, and it is apparent that if we were to remove the cam and fasten the joint R to the plane, we should be able to give P a vertical movement by the swing of the arms AE and RD. In the same way, if we fastened E to the plane and liberated R, the arm EP could swing about E, and P would then describe a small circular arc which would closely approach to a horizontal line. FIG. 73 Connect now the bars with the cam as in the sketch, and press the pointers at Q and R against the curves of the respective cams. Let these cams revolve slowly about a centre, C (marked as a round spot in the smaller cam-plate), in the direction shown by the arrow, and the required letters will be traced out by the pencil at P. In the figure the letter R has just been completed, and the pencil is about to trace out the lower tail of the letter I. The two darkened lines in the cams are arcs of circles about the centre of motion where nothing is being done, the pencil re- maining at rest while the cam rotates through a small angle. This example shows us that a combination of two cam-plates actuating a simple framework of levers will give the command of any movement in a plane perpendicular to the axis of rotation. We shall presently see how to obtain a motion parallel to the same axis, and thus we can secure any required movement in space. Compound Cams. 73 ART. 62. In like manner two simultaneous rectilinear move- ments in lines at right angles to each other are obtained in a well- known form of sewing machine by the operation of grooved cams upon the face of a plate. The sketch is taken from a lecture diagram. 1. The needle bar a, carrying the needle N, is constrained to move up and down in a vertical line between guides: it is driven by the lever BDP, which has a centre of motion at D, and is con- nected by a roller P with the groove marked nn in the cam-plate. The centre of motion of this plate is at C, and is marked by a strong dot. If the groove were formed in a circle round C the needle would remain at rest, and it receives its required motion under the constraint of the cam. The dotted lines show the extreme posi- tions of the lever during the motion. 2. The shuttle S moves to and fro in the path or ' race ' be. FIG. 74. It is actuated by rod SV passing between guides df and ter- minating in a pin V which runs in the groove ;//. As the cam- plate rotates the pin V moves to and fro in a horizontal line, the extreme positions both of V and S being indicated by dotted lines in the diagram, and thus the needle and shuttle operate together in the manner required. The direction of rotation of the cam- plate is marked by an arrow. ART. 63. The movement of the needle in a sewing machine 74 Elements of Mechanism. is sometimes obtained from a peculiar form of cam, of the type of the heart wheel described in Art. 60. The peculiarity consists in driving the cam by a pin P, placed on the face of the circular plate. In the ordinary heart wheel the pin or roller, P, moves up and down while the heart-shaped piece rotates upon its centre. Here the heart-shaped piece moves up and down while the pin rotates. The result is that we find only FIG 75 * ne a P ex f the heart instead of the complete outline. The contrivance is well worth study- ing as an example of the combination of move- ments. The drawing shows the needle bar AB, carrying a needle at N, and guided by openings in the fixed frames aa and// The bar is attached to the darkly-tinted heart-shaped piece connected with it at the boss marked C, and the result is that the heart and the needle bar move as one thing. The pin P is attached to the circular plate, which rotates about its centre of figure, and it is clear that the needle bar is now nearly at rest in its lowest position, and will remain so until P gets round to the upper edge of the heart. The bar will then rise, and continue to rise while P is appar- ently descending the sloping side into the posi- tion shown by the dotted lines. The needle bar will then have reached its highest position, and will afterwards descend. ART. 64. In the striking part of a large clock the hammer may be raised by a cam, and may then be suffered to fall abruptly. The figure represents the cam devised for the Westminster clock ; the hammer rises and falls with the lever, AC, and the cam is so formed that its action com- mences at the extremity of the lever, and never departs sensibly from the same point ; the cam, al>, is a circle whose centre is at the point of intersection of the tangents to the rim of the wheel at a and c. FIG. 76. Punching Machine. 75 ART. 65. The lever punching machine is worked by a cam resembling that which we give as an example by Mr. Fletcher, of Manchester. The cam is here shown attached to the axis of the driving wheel, and the lever, which carries the punch in a slide connected with its shorter arm, is centred on the pin at E. FIG. 77. The curve of the cam is adapted to raising the longer arm of the lever bar in of a revolution of the driving shaft, it allows the lever to fall in the next 5 of a revolution, and finally leaves the punch raised, as shown in the sketch, during the remaining 2 of a revolution, thereby giving the workman an interval of time for adjusting the plate of iron before the next hole is punched. That this action occurs will be quite evident upon inspecting the form of the cam, and it will also be seen that the cam is pro- vided- with a circular roller B, which determines the form of the driving surface while the work is being done, and which is merely an arrangement for lessening the friction just at the time when the greatest pressure is being exerted. ART. 66. Cams are employed when it is required to effect a movement with extreme precision. Thus in a now obsolete ma- chine of Mr. Applegath for printing newspapers, the sheet of paper Elements of Mechanism. used to start upon its journey to meet the type at a particular in- stant of time ; an error of one-twelfth of a second would cause the impression to deviate half a foot from its correct position, and would throw two columns of letter-press off the sheet of paper. The accuracy with which the sheet was delivered was therefore very remarkable, and was insured by the assist- ance of the cam represented in the diagram (fig. 78) As C revolves, the roller at B drops into the hollow of the plate, thereby determining the fall of the lever AB, and by it the fall also of another roller which starts the paper upon its course to the printing cylinder. ART. 67. Hitherto we have considered the cam to be a plane curve or groove, but there is no such restriction as to its form in practice. Let us examine the following very simple case : FIG. 79. F A > CD is a rectangle with a slit RS cut through it obliquely: a pin P fixed to the sliding bar AB works in the slit. If the rectangle CD be moved in the direction RS, it will impart no motion to the bar AB ; but if it be moved in any other direction, the pin P will be pushed to the right or left, and a longitudinal movement will be communicated to the bar AB (fig. 79). Cams. 77 The contrivance here sketched is of frequent use in some form or other, and we may point out its application in the rifling bars used at Woolwich in the manufacture of rifled guns. In work of this kind, where the greatest accuracy is demanded, the bore of the gun acts as a guide to the head of the rifling bar, and the cutter does its work while the bar is being twisted and pulled out of the gun. It is essential, therefore, to keep the cutter within the head while the bar is being inserted preparatory to the removal of a strip of the metal, and to bring it out again at the end of the stroke. In order to arrive at this result the bar is made hollow, and the tool-holder in the rifling head, shown in fig. 80, is made to move in and out laterally by means of a pin P working in an inclined slot, RS, in the internal feed rod. As the feed rod is pushed through a small definite space in either direction along the axis of the bar, the cutter will also move in or out in the direction of the dotted line AB. In discussing this motion there are two FIG. 81. cases to consider. R. 1. Suppose that CD is moved at right angles to AB. Draw RN perpendicular to AB. A_ Th travel of CD = RN n travel of AB PN =tan RPN. 2. Let CD move in a direction inclined at any given angle to the direction of the groove RS. Draw RN in this direction, and we , FIG. 82. have P travel of CD = RN travel of AB NP = sin RPN sin NRP' In other words, the velocity ratio of CD to AB is expressed by the fraction sin RPN sin NRP' a ta ^ es tne f rm tan RPN when the angles at R and P make up a right angle. Elements of Mechanism. ART. 68. Next let CD be wrapped round a cylinder ; it will form a screw-thread, and the revolution of the cylinder upon its FIG g axis will be equiva- A T , lent to a motion of the rectangle at right angles to the bar, in the manner shown in the preceding article. We shall have, therefore, by the arrangement in the figure, a continuous uniform rectilinear motion of the bar AB during the revolution of the cylinder upon which the screw thread is traced. If the pitch of the screw be constant, the motion of PB will be uniform, and any change of velocity may be introduced by a proper variation in the direction of the screw-thread. If the screw be changed into a circular ring, AB will not move at all. It is, then, a matter of indifference whether the cam be a groove traced upon a flat plate or a spiral helix running round a cylinder. In the first case motion ensues when the groove departs from the circular form, and the distance from the centre varies ; in the second case motion ensues the moment the groove deviates from the form of a ring, whose plane is perpendkular to the axis. As an illustration of a cam of the latter character, we may refer to the diagram, which shows a form very much used where a small motion of a lever is re- quired; the lever ACB is centred upon the point C, and will com- mence to move as soon as the pin at the end A reaches that portion of the ring which departs from the circular form. 7V0&. This kind of cam has the property of giving a motion parallel to the axis upon which it is shaped. ART. 69. As, an example of two cam grooves in juxtaposition, one formed upon a flat plate and the other upon a cylinder, we may refer to a machine which was formerly used for shaping coni- cal box-wood plugs of the kind required for expanding an elon- gated bullet into the grooves of a rifle. FIG. 84. Cams. 79 Bullets of this class have a hollow recess at the base, into which a plug is fitted, and when the powder is fired the thin sides of the recess are forced into the grooves of the rifle by the action of the plug. At the present time these plugs are made of compressed clay, but formerly they were cut from a rod of box-wood by ma- chinery. The wood is first cut up into small square rods by means of circular saws, and one of the wooden rods is fixed in a saddle, as shown at H in fig. 85. Our drawing is taken from a lecture diagram by Sir J. Anderson. FIG. 85. The next operation is to bring a revolving cutter, which carves out the shape of the plug, upon the end of the rod H. This is effected by the lever PCE, centred at C, and having the end E jointed to a spindle which carries the cutter at its opposite end. A driving pulley M is rotated at a high velocity by means of a strap and pulley not shown in the drawing, and the longitu- dinal movement to and fro of the spindle, with its cutter and driving pulley, is effected by a grooved cam on the cylinder A. It will be apparent that this cam ought to be, and, in fact is, shaped precisely after the fashion of the cam in fig. 84. 8o Elements of Mechanism. As soon as the cutter had been forced down so as to form the end of the plug, it was withdrawn, and the final operation was to cut off the shaped piece. In the earlier machines a circular saw made a transverse movement, to cut off the plug, whereby the plugs and the chips were mixed together in the same heap, and Sir J. Anderson has stated that the labour for their separation cost more than for their manufacture. But soon it was arranged that the whole saddle HR carrying the plug rod should be shifted bodily back through a small space against a circular saw S, whereby the plugs might be dropped into a separate box. The circular saw S is driven by a pulley N, and the trans- verse motion of HR is effected by the cam-plate B, which acts upon a roller at Q and connecting rod QT. The saddle HR is placed upon the top of a weighted rocking frame, and it is apparent that the cam B would give the transverse movement required. D is a driving spur wheel which gives motion to the cam shaft. There is also a contrivance which we do not explain for pushing the rod H forward by the length of a plug at the end of each stroke. ART. 70. We subjoin a further example, devised many years ago, in which a reciprocating movement is imparted to a frisket frame in printing machinery, and it will be presently seen that the required result can be obtained in a much more simple manner. The use of a cam-plate allows of an interval of rest at each end of the motion, and enables the printer to obtain an impression, and to place a fresh sheet of paper upon the form. Here AH is the reciprocating frame attached to the combina- tion of levers GFEDCB by the link AB (fig. 86). At the end of the lever, FG, is a sliding pin which travels along the grooves in the flat plate centred at O, and determines, by its position, the angular motion of the levers about the fixed centres at F and C. Where the groove is circular, which occurs in those portions which are to the left hand of the vertical dotted line, the levers remain at rest, and they change into the position shown by the dotted lines when the sliding pin passes from the outer to the inner channel. The pin is elongated in form, as shown at G', Cams. 8 1 and is thus capable of passing across the intersections of the groove. Precisely the same character of movement may be obtained by the aid of a helical groove traced upon a revolving drum. The intervals of rest occur when the groove assumes the form of a flat ring, whose plane is perpendicular to the axis of the drum. A right and left-handed screw-thread is traced upon the worm barrel, AB. which revolves Pm RT in one uniform direction a pin attached to the table of a printing machine fol- lows the path of the groove upon the barrel, and its form is elongated so as to enable it to pass in the right direction at the points where the grooves intersect. The interval of rest commences with the entry of the pin into the flat ring at either end of the barrel, and may be made to occupy the whole or any part of a revolution of AB, according as the grooves enter and leave the ring at the same or different points. This construction dispenses with the complicated system of levers, which constitutes such a serious defect in the other ar- rangement. Mr Napier has patented an invention which causes the in- G 32 Elements of Mechanism. Fig. i. terval of ' rest ' to endure beyond the period of one revolution of the barrel. At the entrance to the circular portion of the groove a mov- able switch is placed, and it is provided that the switch shall be capable of twisting a little in either direction upon its point of support, and also that the pin upon which the switch rests shall admit of a small longi- tudinal movement pa- rallel to the axis of the barrel, the pin itself be- ing urged constantly to the right hand by the action of a spring. In fig. i the shuttle is seen entering the cir- cular portion of the groove, and twisting the Fig- 3- switch into a position which will allow the shuttle to meet it again, as in fig. 2, and to make a second journey round the circular ring. The spring which presses the point of support of the switch to the right hand will now cause it to twist by means of the reaction which the passing pin affords, and the consequence will be, that the switch will be left in the position shown in fig. 3, and will guide the shuttle into the helical portion of the groove. Thus the p'eriod of rest will be that due to about one and two-thirds of a re- volution of the barrel. ART. 71. We remark, in conclusion, that when the mechanic causes the moving body to be influenced by a pin which exactly fits the groove along which it travels, it is obvious that the moving body will take the exact position determined by the pin ; on the other hand, where the cam is merely a curved plate pushing a body before it, there is no certainty that this body will return unless it be brought back by a weight or spring. Hence it arises that double cams have sometimes been employed in machinery, and we take the next example from an early form of power- loom. AB is the treadle, E and F are the cam-wheels or tappets, which revolve in the directions shown by the arrows, and in such Cams. relative positions that the projections and hollows are always exactly opposite to each other. As the cams rotate, the treadle, AB, is alternately elevated and depressed, and the threads of the FIG. 89. FIG. 90. warp are opened so as to permit the throw of the shuttle during the operation of weaving the fabric. ART. 72. Cams are frequently employed for the purpose of opening or closing with rapidity the valves of a steam cylinder, or other valves concerned in what is termed the expansive working of steam, that is, the cutting off the supply of steam before the end of the stroke of the piston. In a movement of this kind the cam is re- quired to lift the valve rapidly from its position of rest, then to hold it up for a time while the steam is passing through, and next to allow it to drop into its seat and remain at rest. The cam usually operates upon one end of a lever, the other end of which is connected with the valve, and it is apparent that it will suffice to surround the crank shaft of the engine by a plate or QZ Elements of Mechanism. cylinder having a circular portion ef, on which the end of the valve lever rests when the valve is closed, and a raised portion, AB, also circular, upon which the end of the valve lever runs when the valve is to be opened. Thus there are two circular portions which determine the opening and closing of the valve, and an arbitrary sloping portion connects AB with ef, and determines the rapidity with which the changes take place. For some purposes, as where steam is to be expanded in vary- ing degrees, the raised portions are of different lengths, as AB, AC, AD, arranged in successive steps, one behind the other, whereby the valve may be held open for different periods. Also, it is manifest that the cam may lie on the face of the plate instead of being part of its edge, and that in effect two portions of flat plates rotating about a common axis perpendicular to each, and raised one above the other, with a sloping surface connecting them, would be a mechanical equivalent for the cam described. Such a cam-plate was used by Sir W. Fairbairn. ART. 73. Where the cam- plate is required to effect more than one double oscillation of the sliding bar during each revolution, its edge must be formed into a cor- responding number of waves. There is an example in telegraph commutators, the interruptions of the current being caused by the vibrations of a lever, PCQ, centred at C, and whose angular position is determined by a pin travelling in the groove. As the wheel revolves, it can im- press any given number of double oscillations upon the lever. ART. 74. We have hitherto confined our attention to simple examples in the geometry of motion : we shall now extend our view of the subject, and shall consider the communication of motion when the driver is a toothed wheel or pulley rotating con- tinuously upon a fixed centre or axis ; and in order to generalise still further, we shall suppose the reciprocating motion to be either FIG. 91. Reversing Motion. 8 5 rectilinear or circular. In this manner we shall be enabled to bring under one point of view a great variety of useful mechanical contrivances. The student will be aware that in the transfer of force by machinery, the moving power is carried from one piece of shafting to another, throughout the whole length and breadth of the factory ; it passes from point to point, enters each separate machine, and gives movement to all the several parts which may be prepared for its reception. Now it must be remembered that the engine is never reversed, and that the power continues to flow onward in one uniform direction. Take the case of a machine for planing iron : here the princi- pal movement is that of a heavy table sliding forwards and back- wards, and carrying the piece of metal which is the subject of the operation. There are two methods of obtaining the desired result : the power may be poured, as it were, into the machine by a stream running always in one direction, and the reciprocation may be provided for by the construction of the internal parts, or the flow of the stream may be reversed by some intermediate arrangement external to the machine itself. ART. 75. The former method is that usually adopted, and we shall now examine those machines where the reciprocation de- pends upon the internal construction of the moving parts. And, first, we shall discuss a very simple and useful reversing motion which is obtained by a combination of two or three spur wheels, and which depends upon an obvious fact. FIG. 92. Let A,B,C represent three spur wheels in gear ; it will be seen that A and B turn in opposite directions, while A and C turn in the same direction. If then we connect two parallel axes by a 86 Elements of Mechanism. combination of two and three spur wheels alternately, and pro- perly arrange our driving pulleys, so that the power shall travel first through one combination and then through the other, we shall have a movement which has been adopted by Collier in his planing machines, and which has been subsequently much used by other makers. The power is now derived from the shafting by means of a band passing over a drum on the main shaft and over one of the three pulleys, E, I, F, at the entrance into the machine. Of these pulleys E is keyed to the shaft, I rides loose upon it, while F is attached to a pipe or hollow shaft, through which the shaft connecting E with A' passes, and which terminates in the driving wheel A. FIG. 93. There is also a second shaft B'C, which carries the toothed wheels B' and C. B is an intermediate wheel riding upon a separate stud. When the band drives the pulley E, it is clear that A' and B' turn in opposite directions ; whereas the motion is reversed when the band is shifted to F, for in that case A and C turn in the same direction. When the driving band is placed upon I, the machine remains at rest. Reversing Motion. 87 The rotation of B'C may be made much more rapid in one direction than in the other, and the construction is therefore par- ticularly useful in machinery for cutting metals. The slow movement occurs while the cutting tool is removing a slip of metal, and the return brings the table rapidly back into the position suitable for a new cut. ART. 76. This contrivance is in common use, and the draw- ing is from a machine arranged for cutting a screw-thread in the interior of the breech of an Armstrong gun. In this case the driving pulleys are placed between the wheels A and A', and are formed in such a manner that the pulley F and the wheel A make one piece, and ride loose upon the shaft HK, as do also, in their turn, the pulley E and the wheel A' : the wheel M is keyed to HK, so as to rotate with it, and is further attached by a coupling to the muzzle of the gun which is to be operated upon. FIG 94 When the strap is upon E, the motion travels from A' to B', and so on to L and M, causing the gun and the shaft HK to rotate together slowly in one direction ; whereas, upon shifting the strap to F, the motion passes from A to C through a small inter- mediate wheel, and thence to L and M, whereby the rotation of the gun is reversed, and a higher speed is introduced. 88 Elements of Mechanism. The object of the machine is to copy upon the interior of the breech of the gun a screw-thread which is formed upon the end R of the shaft HK. For this purpose the shaft HK is screwed, as shown, and a slide-rest carrying a cutter advances longitudinally along the gun, with a motion derived directly from a nut which travels along the screw-thread formed upon R. Since the cutter can only re- move the metal while passing in one direction, there is a loss of time during the return motion, which it is the object of this com- bination to reduce as much as possible. ART. 77. The same combination, slightly modified, is adopted generally in planing machines, and is valuable by reason of the uniformity of the movement, the rate of advance of the table being perfectly constant. It also possesses the important advantage of causing the table to traverse with a quick return movement when the cutter is not in action. We give so much of the machine as will explain the method of reversing the motion of the table. When the strap is upon the pulley F, the wheel A turns in one direction. When the strap is FIG. 93. upon the pulley E, the motion passes to B, which turns with E, and thus the axis, CD, is made to revolve in the opposite direction with a reduced velocity. Quick Return Movement. 89 The wheels A and D both engage with another wheel which actuates the table, and the reversal takes place when the moving power is transferred from the wheel A to D; but inasmuch as it would be difficult to give a clear representation of the movement without a sketch in perspective, an additional lecture diagram has been prepared wherein the toothed wheels are represented by circular discs with smooth edges. FIG. 96. Taking the two diagrams together, it will be apparent that the pulleys, E, I, and F, are arranged by the side of the first toothed wheel, A, whereby either F drives A, or E drives B, C, and D. It has been stated that the wheels A and D both engage with another wheel which actuates the table, and the second drawing shows this additional wheel, H, engaging both with A and D on one side, and connected directly with a compound or stepped wheel, K, which is placed under the rack R. The object being merely to indicate the position and arrange- ment of the working parts, the long rack which lies underneath the whole bed of the table is here marked as a short piece R rolling upon the disc K. ART. 78. We shall now examine another class of reversing motion's, and shall commence in the most elementary manner. Conceive a disc E, having a flat edge, to run between two parallel bars, AB and CD, arranged in a rectangular frame ; and 9 o Elements of MecJiamsm. conceive, further, that the frame can be raised or depressed so as to bring AB and CD alternately into contact with E (fig. 97). If the disc rotates always in the direction of the arrows, it will move the frame to the left when brought into contact with CD, and to the right when brought into contact with AB. We have therefore a reversing motion within the limits of the frame. FIG. 97 FIG. 98. In order to make the motion continuous, it will only be ne- cessary to alter the bars into circular strips or discs, as shown in fig. 98, and we shall reverse the motion of the vertical axis by bringing the upper or lower discs AB and CD alternately into close contact with the driver E. In this way we obtain the first idea of a reversing motion, and it only remains for us to improve the general construction and arrangement of the working parts so as to make it practically useful. And we should observe that inasmuch as the rolling action of cones is more perfect than that of circular discs, for the reasons already explained^ in the introductory chapter, it will be better to substitute cones for the discs, in the manner shown in fig. 99, and the reversal will occur, just as before, when AB and CD are alternately brought into frictional contact with the driving cone E. The geometrical condition of rolling will demand that the vertex of the driving cone E shall coincide with that of AB in one position of contact, and with that of CD in the other ; hence the vertices of the two cones AB and CD must be separated through a small space equal to that through which the common axis is shifted. If we desire to transmit force beyond the limit at which the cones would begin to slip upon each other, we must put teeth Reversing Motion. upon the rolling surfaces, as in fig. 100, and we thus obtain a re- versing motion which has been used in spinning machinery. FIG. 99. FIG. ioo. Here a bevel wheel, E, is placed between two wheels, A and C, which are keyed to the shaft whose motion is to be reversed, the interval between A and C being enlarged so that E can only be in gear with one of these wheels at the same time ; the reversal is then effected by shifting the piece AC longitudinally, so as to allow E to engage with A and C alternately. ART. 79. A reversing motion which depends upon the shift- ing of wheels in and out of gear is not perfect as a piece of mechanism ; we must try, therefore, to convert it into another, so arranged as to give the reversal by passing a driving clutch from one wheel to the other, the wheels concerned in the movement remaining continually in gear and fixed in position. For this purpose we employ one working pulley F, keyed upon the shaft ED ; by its side we place a second pulley I, which rides loose upon the shaft, and which carries the driving band when no work is being done. The wheels A and C ride loose upon the shaft ED, and the intention is to impart the motion of the shaft ED, which f is driven by steam power, to the wheels A and C alter- nately. We now fit upon the shaft E a sliding clutch N, having projections which serve to lock it to J" 1 Q2 Elements of Mechanism, A or C as required, and we place also a projection, or feather, upon the inner part of the clutch which slides in a corresponding groove formed in the shaft, so that N must always turn with ED. It is clear, therefore, that if we allow the clutch N to engage with A we shall communicate to B a rotation in one direction, and that, further, we shall reverse the rotation of B if we connect C with N, for the student will see that in this combination A and C must always rotate in opposite directions, and that the rotation of B as derived from A must be different from that which B would derive from C. This reversing motion may be commonly seen in steam cranes. The shaft ED is then driven directly by a steam-engine attached to the crane, and the sliding clutch may be locked to either bevel wheel by a friction cone, and is pushed to the right or left by means of a lever which grasps it without preventing its rotation. There is another application in screwing machines where a rapid reversal is required. In this case the ,;haft ED is reversed by the action of the bevel wheels, instead of imparting its rotation to each of them in turn. The driving pulley F being attached by a pipe to the wheel A, the reversal is effected by shifting the clutch, and thereby locking the shaft ED to the wheels A and C alternately. ART. 80. We have next to examine the application of this reversing motion in planing machines, and shall describe the com- bination of three pulleys with three bevel wheels which has been adopted by Sir J. Whitworth. In pursuing an inquiry into machinery of this character we may remark that the principle of machine copying, whereby a form contained in the apparatus itself is directly transferred to the material to be operated upon, is the distinguishing feature of all planing machines. The application of this principle is perfectly general, and, as a rule, wherever a process of shaping or moulding is well and cheaply performed by the aid of machinery, we find that some skilful and carefully arranged contrivance for transfer- ring a definite form is contained within the machine. In the earliest form of planing machine, a method of carrying the cutter along parallel bars was adopted, and the present practice is to employ perfectly level and plane surfaces called Vs, which are Reversing Motion. 93 placed on either side of the machines, and are shaped exactly as their name indicates ; their form gives a support to the table, pre- vents any lateral motion, and allows the oil required for lubrica- tion to remain in a groove at the bottom, from whence it may be worked up by the action of the machine. The table has projecting and similar Vs which rest upon the former, and the object of the mechanism is primarily to cause the table to move in either direc- tion along the grooves, and thus to copy upon a piece of iron supported thereon, and carefully bolted down, an exact plane sur- face which possesses the truth of the guiding planes. Whether it may be better to move the table by a rack and pinion or by a screw is a subject upon which different opinions are held, and at all events the quick return movement which is given by a combination of spur wheels, as already described in Art 77, is extremely valuable. To recur to Sir J. Whitworth's arrangement, we find that he effects the required movement by rotating a screw which runs along the central line of the bed, and which imparts to the table a perfectly smooth traversing motion, equal of course in exactness, if not superior, to that which could be obtained by the best-con- structed wheelwork. There are now three pulleys, E, I, and F, whereof I is an idle pulley, and rides loose upon the shaft ; E is keyed to a shaft ter- minating in the bevel wheel C, and F fits upon a pipe through which the shaft connecting E and C passes, and which terminates in the bevel wheel A. B is a bevel wheel at the end of the shaft whose direction of rotation is to be reversed. 94 Elements of Mechanism. It is clear that the motion of the wheel B is reversed when the driving strap is shifted from E to F. One objection to this movement consists in the fact that it does not permit the motion of B to be more rapid in one direction than in the other, and in order to economise the steam-power to the fullest extent, a method of rotating the tool-box was adopted by which means the cut was made while the table traversed in either direction. This reversal answers very well in planing ordi- nary flat surfaces. It may, however, be so arranged as to obtain a quick return by making A and C of equal size, and by causing them to gear re- spectively with two unequal wheels upon the axis of B, or a com- bination of spur and bevel wheels may be employed. ART. 8 1. The contrivance just described is shown in fig. 103, as applied in a machine for rifling guns, and the method adopted is precisely that so generally employed in planing machines. The three pulleys and the three bevel wheels are connected to- Reversing Motion. 95 Aether in the manner already indicated, and the bevel wheel B, by its rotation, causes a saddle S carrying the rifling bar to move along the screw in the direction of its length. A bell crank lever, MLN, controls the bar PQ, which carries a fork used to shift the strap, the arms of the lever lying in different horizontal planes, while a movable piece, R, fixed at any required point of the bar, NR, is caught by a projection on the saddle as it passes to the right hand, and thus the bell crank lever is actuated, and the strap is carried along from E to F. A weight falls over when this is taking place, and gives the motion with sharpness and decision, so as to prevent the strap from resting upon I during its passage. On the return of the saddle to the other end of its path, a similar projection again catches a second piece upon the sliding bar NR, and the strap is thrown back from F to E. This bell crank lever, as employed for shifting the strap, is worthy of notice ; it consists of two arms, LN and LM, lying in different planes and standing out perpendicularly to an axis. It is a contrivance which affords a ready means of transferring a motion from one line, RN, to another, PMQ, which lies in a perpendicular direction at some little distance above it (Art. 36). ART, 82. Where the reciprocation is effected by a contrivance external to the machine, two driving bands may be employed : of these one is crossed, and the other is open, and it has been already pointed out that the followers will turn in opposite direc- tions, although they derive their motion from a single drum, which, being driven directly by the engine, must rotate always in one direction. The form which the arrangement assumes in practice is shown in the sketch, fig. 104, where PQ is the driving shaft carrying the drums R and S. Confining our attention at first to the left-hand diagram, we observe that one of the driving bands is represented as crossed, and that the rotation of the lower shaft HL is to be derived from each band alternately. There are three pulleys, whereof A and B are each loose upon the shaft, and are about twice as broad as C, which is a working pulley. 96. Elements of Mechanism. The bands are shifted by two forks, and remain always at the same distance from each other. In the diagram the crossed strap is upon the idle pulley, and the open strap is on the working pulley, the result being that the shafts PQ and HL rotate in the same direction. When the bands are shifted a little to the left both straps will lie on the respective idle pulleys, A and B, and the shaft HL will cease to rotate. Whereas, upon shifting the straps still more to the left the crossed strap comes upon C, and the shaft HL begins again to rotate, but in the opposite direction to that in which it moved previously. Here the rate of rotation is the same in either direction, but it may be varied by connecting the drum S with two pairs of pulleys, viz., D, E and F, K of unequal size, upon the lower shaft NM. The extreme pulleys D and K are the working pulleys, and the reversal is effected just as before, but NM rotates more rapidly in one direction than in the other. ART. 83. There is yet another most ancient contrivance for changing circular into reciprocating motion, which will repay the trouble of analysing it. It is deduced from the same triangle as that concerned in the motion of the crank and connecting rod, but the varying dimensions of the sides are arrived at in a dif- ferent manner. One simple form is obtained when the points C and Q in the triangle CPQ become fixed centres of motion, the Quick Rettirn Motion. 97 FIG. 105. crank CP being less than CQ, and the extremity P of the crank CP moving in a slot or groove running along the line QR. The drawing shows an arm CP centred at C, and conveying motion to the grooved arm QR by means of a pin, P, which fits into the groove. As CP revolves with a uniform velocity, and in a direction opposite to the hands of a watch, it will cause QR to swing up and down to equal distances upon either side of the line QC, but with this peculiarity, that the upward swing will occupy less time than the down- ward swing. The motion of QR will be variable, its ve- locity changing at every instant, and we must endeavour, in the first instance, to discover an expression for its rate of motion as compared with that of CP. According to well- established rules, we estimate the relative rates of motion of two revolving pieces by comparing the sizes of the small angles de- scribed by either piece in a very minute interval of time reckoned from any given instant. ART. 84. Let now P/ represent the small arc described by P in a very minute interval of time, such as the -nnn^h P art f a second. Join Q/>, C/, and draw Pn perpendicular to Q/>. FIG. 106. Then angular ve! of QR . *& in the , imi angular vel. of CP L PC/ = limit of 98 Elements of Mechanism. But ~Pn = P/ cos/P = P/ cos RPC = P/ cos (C + Q). angular vel. of QR_CP cos (C + Q) ' ' angular vel. of CP ~~ PQ We may test this formula in the usual way ; for instance, let C + Q = 90, in which case QR touches the circle, then cos (C + Q) = cos 90 = o ; .*. angular vel. of QR=o, or QR stops, as we know it must do. Next, let C = o, Q = o, or let P be crossing the line CQ, then cos (C + Q)= cos. 0=1. . angular vel. of QR = CP ' 'angular vel. of CP QP' or the vel. of QR is as much less than that of CP as QP is greater than CP, a result which is evidently true. If it be required to find the position of QP when P is at any given point of its path, we have the equation CP sin C whence the angle PQC is known in terms of C. If we draw QD, QH, tangents to the circle described by P, it will be evident that the times of oscillation of the arm will be FIG. 107. unequal, and will be in the same proportion as the lengths of the arcs DLH, DKH. The ratio of the angular velocities of QR and CP may also be obtained by analysis. Let CQP = fc PCQ = 0, CP = a, CQ = c, Quick Return Motion. 99 .-.o=sm (0+0) cos ^~sin cos (9 + 0) j i +^J J /.sin cos (0 + 0) = {sin (0+0) cos 0-cos (0 + 0) sin 0}- ART. 85. Another way of looking at the motion, which is technically known as a slit-bar motion, is to consider that PQ is a line of indefinite length jointed at P to the crank CP, and con- strained to pass through an opening at the fixed point Q. As before, the condition that CP is less than CQ must maintain, or the line PQ will no longer oscillate. This again is the movement in an oscillating engine, where the steam cylinder is swung upon trunnions, and the crank CP is con- nected by a piston rod to a piston moving up and down in the cylinder. Whichever of the above forms may be selected as an illustration, the movement is precisely the same. FIG. 108. Taking CPQ as the triangle, draw CR perpendicular to QP produced, and draw PH parallel to CR. CP cos (0 + 0)_CP RP_RP_CH PQ PQ CP PQ HQ angular vel. of QR _ CH r ' angular vel. of CP HQ' Referring to the oscillating cylinder, let V be the linear velocity of the crank pin P, and v the velocity of the piston, which moves in a cylinder swinging on trunnions at Q, and which, for sim- plicity, we will indicate by the point S in the drawing. ioo Elements of Mechanism. Then the velocity of P resolved along PQ is equal to the velocity of S. ..VsinRPC = z>, v CR r ' V = CP' which gives the velocity ratio of the piston to the crank pin in an oscillating engine. ART. 86. Hitherto we have supposed CP to be less than CQ, and the result has been that QR swings about the point Q in un- equal times ; but we will now arrange that CP shall be greater than CQ, in which case QR will sweep completely round with a circular but variable motion. We shall, in fact, have solved the problem of making a crank revolve in such a manner that one half of its revolution shall occupy less time than the other half. Now this is a very important result, and is of great value in machinery, because if the crank be made to perform its two half revolutions in unequal times, it follows that any piece connected with it by a link may be caused to advance slowly and re- turn more rapidly ; a movement which, as we have already pointed out, is peculiarly useful in machines for cutting metals. Constructing as before, let C be the centre of the circle described by P. Then the equation tan gives the position of QP when that of the crank is assigned. A] angular vel. of QR _ P_^ _._ P/ P/ cos CPQ CP S angular veL of CP ~~ PQ ' CP ~ .~PQ~~ P/ Cor. i. If CQ be small, the angle CPQ will be small also, and we shall have cos CPQ = i nearly ; in which case the angular vel. of QR varies as ~, while that of CP remains constant. Cor. 2. When CQP is a right angle, we have Quick Return Motion. 101 FIG. no. cosCPQ=, that is, the angular vel. of QR = the angular vel. of CP. This happens twice during a revo- lution, and gives the line of division of the inequalities of the motion of QR. Hence, if we draw DQH perpendicular to CQ, and cutting the circle described by P in the points D and H, the times of each half-revolution of QR will be in the proportion of the arcs DKH and DLH. Cor. 3. The angular ve locity ratio between QP and CP may be set out in a sim- ple geometrical form just as in the previous case. Draw PH perpendicular to QP, and QS parallel to PH. FIG. in. Q Then angular vel. of QP _ CP angular vel. of CP ~ PQ -CP x PQ PQ PS = CP PS QH ART. 87. If BR be made to carry a link, RQ, as in the case of the crank and connecting rod, the linear motion of Q will be the same in amount as if BR revolved uniformly, but the periods of each reciprocation will in general be different. (Fig. 112.) The difference in the times of oscillation will depend upon the direction of the line in which Q moves. The best position for that line is in a direction perpendicular to CB. We have shown that the times of oscillation are always as the arcs DKH and DLH, and it is also evident that the inequality 102 Elements of Mechanism. between these arcs is greatest when DH is perpendicular to CB, and diminishes to zero when DH passes through CB. We have now an arrangement very suitable for effecting a quick return of the cutter in a shaping machine. Let one end of a connecting rod be made to oscillate in a line perpendicular to CB, or nearly so, and let the crank BR be driven by an arm, CP, which revolves uniformly in the direction of the arrow, we at once perceive that Q will advance slowly and return quickly, the periods of advance and return being as the arcs DLH and DKH. FIG. 112. FIG. 113. ART. 88. Such a direct construction is not very convenient for the transmission of force, and it has been so modified by Sir J. Whitworth in his Shaping Machine, that the principle remains unchanged, while the details of the moving parts have undergone some transformation. This machine is analogous tp a planing machine, but there is no movable table ; the piece of metal to be shaped is fixed, and the cutter travels over it. The object is to economise time, and to bring the cutter rapidly back again after it has done its work. The arm CP is here ob- tained indirectly^ by fixing a pin, P, upon the face of a plate, F, which rides loose upon a shaft, C, and is driven by a pinion, E. Shaping Machine. 103 As the wheel F revolves upon the shaft represented by the shaded circle, the pin moves round with it, and remains at a con- stant distance from its centre. A hole, B, is bored in the shaft, C, and serves as a centre of motion for a crank piece, DR, shown in fig. 114. The connecting rod, RQ, is attached to one side of this crank piece, and the pin, P, works in a groove upon the other side. Thus the rotation of the crank causes the end Q to oscillate backwards and forwards, and to return more rapidly than it advances. The length of the stroke made by Q must be regulated by the character of the work done, and is made greater or less by shifting R farther from or nearer to B. This adjustment does not affect the FIG. 114. inequality in the relation between the periods of advance and re- turn which the machine is intended to produce. (x-<^ ' ART. 89. As a further illustration of this slit-bar motion, we give a sketch of a curvilinear shaping machine used at the Crewe Locomotive Works. There have been instances of unequal wear of the tyres in the leading wheels of locomotive engines, which have been traced to the circumstance of the wheel itself being a little out of balance ; that is to say, the centre of gravity of the wheel did not exactly coincide with its centre of figure. In one case a wheel was found upon trial to be 9 Ibs. out of balance. Now we learn in mechanics that a weight of W Ibs. describing a circle of radius (r) with a velocity of (v) feet per second, will, I0 4 Elements of Mechanism. during its whole motion, exert continually a pull upon the centre in the direction of a line joining the body and the centre, which will be measured in pounds by the expression 32-2 x r Suppose a wheel 3 ft. 6 in. diameter to run at a velocity of 50 miles an hour. In this case v will be equal to if?, and r will be -, and . _ = 32-2 x r 9 x 32-2 x I 8 x 12100 7 x 9 x i6'i whence the pull of only one pound weight at a distance of 3$ ft. from the centre will amount to rather more than 95 Ibs., and a weight of 9 Ibs. would produce a pressure upon the bearing of rather more than 75 cwts. ; and then, in the time of a half- revo- lution, viz., about y^th part of a second, the same pressure in the opposite direction. FIG. 115. It is, of course, only at high speeds that the defects due to want of balance become serious, and this numerical result shows very plainly the necessity of great care in the construction of wheels which are required to run at a high velocity. Mangle Wheels. 105 The machine intended to shape the curved inner face of the rim of locomotive wheels has the quick return movement which we have just discussed. The point B is the centre of motion of the lever bar, and coincides with the centre of the circuJar portion forming the inner surface of the rim of the wheel W. The tail end of the lever has a long slot in which the crank pin P works : this pin is attached to the driving disc centred at C, and the length of the stroke can be adjusted by shifting P in the direction of the radius CP. ART. go. Mangle wheels form a separate class of contriv- ances for the conversion of circular into reciprocating motion. A mangle wheel is usually a flat plate or disc furnished with pins projecting from its face ; these pins do not fill up an entire circle upon the wheel, but FIG. n6. an interval is left, as shown at F and E. A pinion, P, engages with the pins, and is sup- ported in such a manner as to allow of its shifting from the inside to the outside, or conversely, by running round the pins at the open- ings F and E. The pinion, P, always turns in the same direction, and the direction of rota- tion of the mangle wheel is the same as that of P when the pinion is inside the circular arc, and in the opposite direction when the pinion passes to the outside. The mangle wheel may be converted into a mangle rack by placing the pins or teeth in a straight line. Here the pinion must be so suspended as to allow of its shifting from the upper to the under side of the rack. As to the velocity ratio between the wheel and pinion, it will FIG io6 Elements of Mechanism. be shown hereafter that the inner and outer pitch circles coincide in the case of a pin wheel, and therefore that the relative rotation of the mangle wheel to the pinion is the same in both directions. If the pins be replaced by a curved ring furnished with teeth, the mangle wheel will move more rapidly when the pinion is upon the inside circum- ference, and by giving certain arbitrary forms to this annulus, the velocities of advance and return may be modi- fied at pleasure. Contrivances such as this are seldom met with at the present time. ART. 91. Sometimes the pinion is fixed, and the rack shifts laterally. An excellent form of this arrangement was introduced by Mr. Cowper, and serves to give a reciprocating movement to the table in his printing machine. FIG. 119. The rack HF is attached to the system of bars in the manner exhibited in the diagram. A and C are centres of motion, and are the points where the bars are attached to the table. AG and CE are bisected in B and D, and are joined by the rod BD ; the rack HF is attached to the bars AG and CE by the connecting Mangle Motion. 107 links GH and FE, and it must be remembered that it is the in- tention to obtain this so-called mangle motion by the reverse process of fixing the pinion and causing it to drive a continuous rack which runs upon each side of it alternately. The precise value of the contrivance consists in the arrange- ment of the bars, which will be understood upon referring to the section upoh Parallel Motion, and it will be seen that when the pinion has pushed the rack to either end of its path, the bars will so act 'as to move together, and will shift this rack HF to the opposite side of the pinion, without allowing it to deviate from a direction coincident with that in which the table is moving. This is the object of the contrivance, and, as we have said, the method by which the result is arrived at will be apparent when the subject of parallel motion has been examined. Thus the table carrying the parallel bars and the rack oscillates backwards and forwards, while the pinion, which transmits the force, remains fixed in space. When this machine was applied to the printing of newspapers, the table moved at the rate of 70 inches in a second, and its weight, including the form of type, would be about a ton and a half. When urged to its highest speed the machine would give 5,500 impressions in an hour, which is about the greatest number attainable under a construction of this kind ; the true principle in rapid printing being that announced in the year 1790 by Mr. Nicholson, who proposed to place the type upon a cylinder having a continuous circular motion, and upon which another cylinder holding the paper should roll to obtain the impression. But although Mr. Nicholson enunciated the principle nearly a hun- dred years ago, and took out a patent for a mode of carrying it out, there is a wide difference between saying that a thing ought to be done, and showing the world how to do it in a practicable manner; hence it was not until late years that Mr. Applegath, and finally Mr. Hoe, were enabled so to arrange their cylinder printing machines upon the principle of continuous circular motion as to satisfy the wants of the daily papers, and to print some twelve or fourteen thousand sheets ir| an hour. To recur to our shifting rack, it must be remarked that by reason of the great weight of the table, and the rapidity with which io8 Elements of Mechanism. it moves, it would be quite unsafe to leave the rack and pinion in the present unassisted condition ; a guide roller therefore deter- mines the position of the pinion relatively to the rack, while the rack itself shifts laterally between guides. But since, theoretically, the rods would cause HF to move always in a direction parallel to itself, they practically enforce the desired movement in the path of the guides, with as little loss of power as possible. ART. 92. If it be required that the reciprocation shall be in- termittent, *>., that there shall be intervals of rest between each oscillation, we may employ a segmental wheel and a double rack, as shown in fig. 120. The teeth upon the pinion engage alternately with those upon either side of a sliding frame, and the motion is of the character required. The intervals of rest are equal, and are separated by equal periods of time. A pin upon the wheel and a guide upon the rack will ensure the due engagement of the teeth. FIG. 120. A mechanical equivalent to the above is found in the use of two segmental wheels and a single rack (fig. 121). These segments must be equal, but they may be placed in different relative positions upon the discs to which they are attached ; and, as a consequence, the intervals of rest may be separated by unequal periods of time. These segmental wheels have been employed in the earlier days of mechanism, and there was a well-known instance in Mr. Cowper's printing machine, where a segment of a wheel engaged with a small sector at each revolution, and so fed on the sheets of paper by the push given while the segments were in action. Segmental WJieets. 109 Sir J. Whitworth has proposed the for the reversal in a machine for cutting further example of the use of these wheels, which, however, should always be avoided if possible. There is only one driving pulley, and two segmental wheels are keyed upon the driving shaft. They are close together in the machine, and for the sake of the ex- planation we have placed one above the other. The object is to effect the reversal of a shaft C : the segmental wheels A and A' have teeth formed round one half of each circumference, and the toothed segments are in situa- tions opposite to each other, as in fig. 122. When the action of A ceases, that of the wheels A and C, or the wheels A', action, i.e., we have a reciprocation of C. of the case given in Art. 75. subjoined arrangement screws : we take it as a A' begins, and we have B, and C alternately in This is a direct example HO Elements of Mechanism. CHAPTER III. ON LINKWORK. ART. 93. In the present treatise the term 'linkwork' is ap- plied to combinations of jointed bars movable in one plane, the joints being pins, whose axes are respectively perpendicular to the plane in which the bars move. The crank and connecting rod is an example of linkwork, but the present chapter deals principally with combinations of three or more links. As a fundamental case, and one which will repay the trouble of examining it, we take two cranks or levers centred at a distance from each other, mov- able in one plane, and connected at their extremities by a jointed bar. Such a combination is repre- sented by CPQB, where C and B are fixed centres of rotation f B formed by two parallel cylindric pairs, the arms CP, BQ being cranks or levers movable about axes through C and B, while PQ is technically known as a link or coupling rod, and is attached to the respective cranks by pins. There are, in fact, four parallel cylindric pairs, and three movable bars. Prop. When two unequal arms or cranks are connected by a link, as in fig. 123, the angular velocities of the arms are to each other inversely as the segments into which the link divides the line of centres. i. This may be proved by reference to an instantaneous axis. Upon causing the combination to rack a little, it will be found Two Cranks and a Link. in that P begins to describe a circle round C, while Q begins to describe a circle round B, whence it becomes evident that the instantaneous centre, F IG . , 24 . about which PQ is ro- tating at any instant, lies in the point O, where CP and BQ meet when produced. Produce PQ to meet BC in E, and draw CS, R J2/ OF, BR respectively per- pendicular to PQ. Let G PCE=0, QBC=, and when the figure racks a little, let CP, BQ describe the small angles dd and d^. At the same time P and Q will each describe the same small angle n about the centre O. . CP ^ = BQ . ^9_BQ OP_BR OF_BR_BE * V<2~OQ X CP~OF X CS~ CS~CE' 2. The same may be proved by the resolution of velocities. Let P and Q shift to / and q during the smallest conceivable interval at the beginning of the motion. Then the resolved part of the motion of Q in the direction QP is ultimately equal to Q? cos ?QP, FlG " I25 ' =Q? sin BQR = BRx angle QB^. So also the resolved part of the motion of P in the direction QP=CS x angle PC/. But in the first instant of the motion these resolved parts are equal to each other, because PQ remains for a brief space parallel to its first position. 112 Elements of Mechanism. :. BR x angle QB?=CS x angle PC/, . angle QB^_ CS ' * angle PC/~BR' But although the angular velocities of the arms BQ and CP change continuously, yet they will be at any instant in the same proportion as the limiting ratio of the angles described by these arms in a very minute interval of time, the relative motions of the arms not being supposed to change during that interval. TT angular vel. of CP_angle PC/ angular vel. of BQ angle QB^ _BR = B:E ~cs CE' 3. By analysis, the same may be proved. Let CP=al CS=/n PQ 8 and as before, and the angle CEP being equal to //. d=. a cos (0-t d) + c cos ^-b cos (< -f \L) '. by differentiation we have o a sin (0 + ip) (dQ + d^) c sin ^ +b sin .'. csin But k /i = ^ /.o = Four-bar Motion. 113 . angular vel. of CP_BR_BE ' 'angular vel. of BQ~CS ~CE' which proves the proposition in its general form. ART. 94. Let it be required that one crank (viz. CP) shall sweep round in a circle, while the other (viz. BQ) oscillates to and fro through a given angle. In the diagram, the point P describes a circle, while the point FIG. 127. Q oscillates to and fro in a circular arc, the points H and E mark- ing its extreme positions. Then CH = PQ-CP, CE = PQ+CP. Also since CP makes complete revolutions, it is essential that CP and PQ should come into a straight line before BQ and PQ have the power to do so. Hence we must have CB + BQ greater than CE, and CB-BQ less than CH. It will be readily seen, upon testing this statement, that if CB be taken equal to PQ, the crank CP will revolve, and BQ will oscillate, so long as CP is sensibly less than BQ. The angle of swing of BQ increases also as CP becomes more nearly equal to BQ, and tends to reach two right angles as a limit. Ex. Let CP 2, CB = PQ= 15, BQ= 8, to prove that BQ will oscillate through 30 from a position perpendicular to CB. When BQ is at the end of its swing on the right hand we have, since CE=iy, 114 Elements of Mechanism. 2 x 15 x8 .'. CBQ = 90, or BQ is vertical. When BQ is at the end of its swing on the left hand we have, since CH=i3, cos CBQ= 22 5 6 A-i6 9= i2o = i 2 X 15 X 8 240 2 .'. CBQ = 60, which proves the statement. ^ ART. 95. It has been stated that an extreme case occurs when CP = BQ, and when the connecting link PQ is also equal to the distance between the centres, viz., CB. Under these circumstances one crank, as CP, will make com- plete revolutions while the other, viz , BQ, oscillates through 180. But at the same time CPBQ may form a parallelogram, whose opposite sides and angles are equal, and if any provision be made for retaining PQ parallel to CB, the crank BQ will no longer os- cillate but will perform complete revolutions. In the diagram, CP, BQ represent two cranks connected by FIG I2g a link PQ, which is equal to CB, then it is apparent that CPQB forms a paral- C B lelogram so long as PQ remains parallel N. \ to itself, or that the parallelism of PQ ^ ^ is the condition which ensures the joint rotation of the cranks. It is/ also apparent that when the driving crank comes upon the line of centres the joint CPQ will bend if there be any re- sistance to the motion in the follower, and BQ will then return, while CP alone continues its rotation. This state of things is shown in fig. 1 29, where CP sweeps round in a circle, while BQ oscillates through 180. The difference is FIG. 129. that in the first case PQ remains parallel to itself, and that in the 15 second case it oscillates through an angle 20, such that But the oscillation is put an end to by superposing a second Four-bar Motion. FIG. 130. V4 combination exactly similar to the first, as in fig. 130, where the bell crank lever PCP' is connected with a second identical bell crank QBQ', by means of the equal links PQ, P'Q'. In this way, when PQ is passing through the dead points, P Q' will hold it in a parallel position, and each connecting rod will prevent the other from taking that oblique position which is destructive of the required motion. This is the principle of the coupling link between the two driving wheels in a locomotive engine. There are always two links, one on each side of the engine, and the cranks are of course at right angles. The necessity of the second pair of cranks, with their link, is obvious upon a little consideration, and may be made very clear by constructing a small model of the arrangement ; it is only necessary to make the links move in different planes so that they may be able to pass each other. ART. 96. We next observe that any point in PQ will describe a circle equal to either of the circles described by P or Q, so long as PQ remains parallel to itself, and hence that a third crank equal to either CP or BQ, and placed between them, would be driven by PQ, and would further prevent PQ from getting into an oblique position at the dead points, or would produce the same result as the second pair of cranks with the link in the locomotive engine. Again, the same would be true of any point R in a bar SR connected rigidly in any way with PQ, the point R would de- scribe a circle equal to either of the primary circles so long as PQ remained parallel to CB. Also, if a crank were supplied at ER, the three cranks would go round together, and PQ would remain parallel to itself. I 2 FiG. Elements of Mechanism. FIG. 132. Conceive now that three equal cranks, P/, Q^, Rr, are cen- tred at equal distances along a circle PQR, as shown in fig. 132, and let a second circle pqr, equal to PQR, be jointed to the cranks at the points, /, q, r. If the circle pqr be shifted so fhatj the cranks are allowed to rotate, each of them will describe a circle, the respective cranks will always re- main parallel to each other, and the circle pqr will move in such a man- ner that any line drawn upon it re- mains always parallel to itself. Hence the circle pqr may be employed as a driver to rotate all three cranks at the same time, and while doing so, it will itself sweep round without the slightest movement of rotation upon its own centre. It has what is sometimes called a motion of circumduction. ART. 97. A very small alteration in the construction will give a combination which has been useful in rope-making ma- chinery, and which was the first movement suggested for feathering the floats of paddle-wheel steamers. Let the centres of the two circles PQR and pqr be made fixed centres of motion, and let P/, Q^, Rr, remain as before. A power of rotation will now be given to both the circles, and P/>, Q^, Rr will be the connecting links which always remain parallel to CB, the line of centres. That is to say, the rotation of the circle PQR, about the centre C, will cause an equal rotation in the circle pqr, about its centre B, and P/, Qff, Rr, must remain parallel to CB. It is the same combination that we started with, under a different aspect, by reason that the proportionate si^e of the pieces has been FIG, 133. Feathering Paddle Wheel. 117 changed. The two circles have been enlarged and brought to- gether, so that their circumferences overlap, and CP, B/> are the parallel cranks. Regarding the contrivance as a method of feathering the floats in paddle wheels, we find that, in the year 1813, Mr. Buchanan patented a form of paddle wheel in which one circle, as PQR, carried the floats, and another circle, pqr, rotating with the former, held these floats always in a vertical position, and so made them enter the water edgeways, instead of striking it obliquely with the flat surface, as is the case in an ordinary paddle wheel. ART. 98. This wheel of Buchanan has not been used for very sufficient reasons. It is not a good arrangement for the floats to enter the water in an exactly vertical line, because the motion of the vessel must compound with that of the floats, and the sup- FIG. 134. posed vertical path will not be one in reality, any more than It would be in the case of a stone dropped from the same vessel into the water. The stone appears to fall in a vertical line, but is really projected forwards. According to this view the float should enter the water at an angle such that its line of direction will pass through the highest Elements of Mechanism. point of the wheel, this being the direction of the resultant of the two equal velocities impressed upon a point in the wheel, the one being that due to the vessel, the other being due to the rotation of the wheel. This result may follow very closely from the construction in the drawing, which represents Morgan's wheel, where the floats are connected by rods with a ring that rotates round a fixed centre in the paddle-box. The floats are attached to small cranks and pivoted upon centres, one of them (the lowest in the drawing) being driven by a rigid bar which springs from a solid ring. Each float passes the lowest point in a vertical position, and is some- what inclined when entering or leaving the water. ART. 99. In the conversion of circular into reciprocating motion by two cranks and a connecting link, it is a condition of the movement that the connecting link shall swing through a given angle. This fact has been usefully applied in wool-combing machinery, and in 1852 Messrs. Lister and Ambler patented (No. 13,950) an improved arrangement for transferring wool from one carrying comb to another in the act of working the same. FIG. 135. The diagram represents the combination CPBQ, the connect- ing link QP being prolonged to E, and carrying a comb at its extremity. H and F are two fixed combs, and the object in view is to detach a lock of wool from H by the comb E, and to transfer it to F. The proportions of the parts are so chosen that CP per- L em ie lie's Ventilator. 119 forms complete revolutions, and the positions of the passing comb are shown. In the left-hand figure E is in the act of rising through the wool at H, and detaching it, while the other sketch shows E as about to deposit the tuft on the distant comb F. ART. 100. The changes in the position of PQ may also be applied to a useful purpose when both the arms CP and BQ make complete revolutions. A remarkable instance occurs in a mechanical ventilating machine proposed some years ago by M. Lemielle as a substitute for furnace ventilation in mines. FIG. 136. The apparatus consists of a circular chamber of masonry com- municating by an air passage A with the shaft of the mine on one side, and having a discharge opening D on the other side. Within the chamber is placed a revolving drum centred on an axis at C, and having shutters or vanes, as PQ, connected by rods, as BQ, with a second fixed axis at B. It is apparent that CPBQ forms our well-known combination, I2O Elements of Mechanism. and, with the proportions in the drawing, it is also clear that CP, BQ will both make complete revolutions. In doing so, PQ opens out and closes up again, sweeping out before it the air in the open portion of the chamber, and driving the mass before it from A to D. The several positions of PQ are shown at pq y p'q' , and it will be seen that PQ begins to open out on the left-hand side, and gets to its extreme position at or near to pq. In practice there are at least three vanes employed, as shown. A ' Lemielle ventilator ' has been constructed and set to work where the chamber is 14 ft. in diameter and 7 ft. deep, with a fan making some 37 revolutions per minute, and discharging in that ,e 25,000 cubic feet of air. ART. 101. In sewing machines the 'shuttle race' or path of the shuttle is commonly a straight line, but the shuttle is some- times caused to oscillate to and fro in a circular arc. The drawing shows an arrangement for this purpose, founded upon the above combination of two cranks and a connecting link. Here the shuttle S is carried in a frame which oscillates on a fixed centre at B. The driver is a pin P placed on the face of a cam-plate whose centre is at C, and connected by a link PQ to the stud Q attached to the frame carrying the shuttle. The plate carrying P is not circular, and its edge A is used as a cam driver for another part of the operation of the machine. FIG. 137. It is apparent that we have assigned such dimensions to the working parts, viz., CP, BQ, and PQ, as will produce the result that Q will oscillate while P performs complete revolutions. Hand Shearing Machine. 121 Accordingly, the left-hand diagram shows two extreme posi- tions of S and of the frame. It will be seen that P describes a circle of radius CP while Q moves through a portion of the dotted circular arc formed round the centre, B. The carrying backwards and forwards of the shuttle is thus arrived at very simply. ^. ART. 102. The combination CP, BQ, and PQ appears in th form of a hand machine for cutting metals. The diagram will make the construction quite apparent. The lever BH, having a handle at H and a fulcrum at B, is connected by the link PQ with a second piece CP carrying a knife-blade. This latter piece has a centre at C, and the cutting is performed by depressing the handle H. FIG. 138. As a question of mechanism it will be found that the angular motion of CP is much less in amount than that of BH, and the principle of work comes in and affords an easy explanation of the power of the machine. A skeleton diagram will show the mechanical advantage, Let S be the force applied at H, R the resistance in cutting the metal as felt at D. Then, if T be the thrust in QP, we have TxBQ = S xBH, RxCD = TxCP. /. R x BQ x CD = S x BH x CP 122 Elements of Mechanism. Ex. Let CD = BQ = i, CP = 4, BH = 10, ThenR = Sx -^4 i x i = 408, which is a much better result than would be obtained by the use of a single lever CH carrying a knife. ART. 103. The combination of two unequal arms with a con- necting link forms the celebrated Stanhope levers which have been so generally employed in printing presses worked by hand. From the invention of the Art of Printing in the year 1450 till the year 1798 no material improvement was made in the printing press. The earliest representation of a press occurs as a device in \ books printed by Ascensius. There is scarcely any difference be^ V tween it and a modern press, and it is truly a matter of astonish- v ment that so long a period as 350 years should have rolled on\\ without some improvement being made in so important a machine, x The wooden press consists of two upright pieces of timber > joined by transverse pieces at the top and near the bottom ; a screw furnished with a lever works into the top piece, and by its descent forces down a block of mahogany, called the 'platten,'and thus presses the sheet of paper upon the type, which is laid upon a smooth slab of stone embedded in a box underneath. In the year 1798 Lord Stanhope constructed the press of iron instead of wood, and at once transferred the machine from the hands of the carpenter to those of the engineer ; he further added a beautiful combination of levers for giving motion to the screw, causing thereby the platten to descend with decreasing rapidity and conse- quently increasing force, until it reached the type, when a very great power was obtained. These levers consist of a combination of two arms or cranks, CP, BQ, connected by a link, PQ, in such a manner that the con- necting link shall come into a position perpendicular to one of the arms at the instant that it is passing over the centre of motion of the other arm. In order that this may happen, it is evident that the various pieces must satisfy the relation. PQ_CP= N /CB 2 -BQ 2 . The Stan hope Levers. 123 For the convenience of the workman who is employed upon the press, a handle, CA, is attached to the crank, CP, and moves as part of it, but the intro- FlG I3g duction of this handle does not affect the principle of the movement, which, re- garded as a question in mechanics, depends simply on the combination of CP, BQ, and PQ. If, now, a force, F, be applied at the end of the handle, AC, so as to turn the crank, CP, uniformly in the direction indicated, the arm, BQ, will, under the conditions already stated, move with a continually decreasing velocity until it comes to rest, and then any further motion of CP will cause BQ to return. The lower diagram shows the levers in this extreme position, and the graduated scales at P and Q indicate the relative angular movements of CP and BQ. Now the motion, interpreted with relation to the transmission of force, implies that the resistance at Q necessary to balance the moving power which turns the crank is increasing rapidly as the rotation of BQ decreases, and that there is no limit theoretically to the pressure which will be felt as a pull at Q by reason of the force F. In practice this extreme pressure is exerted through so very small a space that the theoretical advantages are scarcely realised, but the arrangement is exceedingly useful as applied in the printing press. The lever BQ is there employed to turn the screw which acts upon the platten ; tne workman gives a pull to the handle AC, and by doing so causes the platten to descend with a motion which is at first considerable, and afterwards rapidly dies away. Thus the limited amount of power which is being exerted comes out with greatly magnified effect in impressing the paper upon the type. a, ^ 124 Elements of Mechanism. * ** ART. 104. In order that this contrivance may be better under- FIG. 140. stood, take the annexed sketch to c B represent it, and draw CN perpen- S[\ \ dicular to PQ. J^ \ \ \ A force F acting at A in a direc- s N \ s o tion perpendicular to CA would be balanced by a force S acting in PQ, such that S x CN = F x CA, o FxCA vS : ~~CN~' ' This force S, necessary to balance F, would be supplied by the resistance to motion in the arm BQ, and would, in fact, be the pull felt at Q. Now, as the arms turn, the link PQ gets nearer and nearer to C, and CN becomes less and less until it has no appre- ciable magnitude, and the consequence is that S increases enor- mously in the last instant of the motion. Ex. Let F = 20 Ibs., CA 20 inches, CN = T Vth of an inch, we have ART. 105. In the Stanhope levers the work is completed just as PQ overlaps CP, and there is no advantage to be gained in carrying the motion further ; but regarding the combination in its general form, it has been shown that the joint CPQ straightens into a line twice while CP is making a complete revolution. When this happens we obtain a subordinate combination of levers, which is known as a knuckle joint, or toggle joint, and is commonly used in hand printing presses, as well as in machinery for punching and shearing iron. The first form of the joint is shown in Fig. 141, where two arms, CP, PQ, generally equal in length, but which may be unequal, are jointed together, the point C is fixed, and the end Q exerts a pressure which may be carried on by a piece moving in the direction CQ. Toggle Joint. 125 The force F, which produces the result, is supposed to act upon the joint at P, and the reaction S, which is felt at Q, will be trans- mitted also to P in the direction QP. Let the thrust, so set up in QP, be called R, and draw CN perpendicular to PR. Then, by the principle of the lever, we have R x CN = F x (perpendicular from C upon F). But as CPQ straightens, CN diminishes, and ultimately becomes equal to zero while the product Fx (perpendicular from C upon F) remains approximately constant. It follows, therefore, that the product of R x CN remains nearly constant while CN diminishes to nothing, but this can\pnly happen if R increases in the same proportion that CN diminishes, that is, without limit. Hence the power of the combination. It is upon this principle that the heavy chain of a suspension bridge cannot be stretched into a straight line ; it would break long before it straightened. In the same way, if the joint be doubled back, the point Q being fixed, and the force F being supposed to act in a line per- pendicular to CQ, we shall have the pull upon C, which we may call R, felt as a pres- sure in the line CP. If now QN be drawn per- pendicular to the direction of R, as felt at P, the principle of the lever gives us the equation RxQN = FxQM, and R becomes infinite when QN vanishes, or when CP is passing over Q. ART. 1 06. If the toggle joint be regarded merely as a me'ans of communicating a slow motion to the end of one of its pair of levers, it is not without practical utility. But inasmuch as any force which may be transmitted through such a joint must be intensified at the period of slow motion, we shall generally find that some work is being done which demands an increase of pressure. The joint is often attached to a revolving crank and connecting tod as shown in the diagram, where we have the combination 126 Elements of Mechanism. CPQB as in Art. 105, with the addition of a rod EQ jointed to Q, and guided so that the end E moves in the line BR. FlG . I43 . In this case C should be in the ver- tical line passing through Q when the joint is straightened, and PQ should be just long enough to reach the lowest point of the circle at the same instant. It will follow that the joint can, under these circumstances, straighten itself only once during a complete revolution of CP, and the contrivance may then be applied so as to obtain a decreasing motion of the point E, and thereby to transmit a pressure which greatly and rapidly increases. An example occurs in printing machinery, where a knuckle joint, actuated by a crank exactly as described above, is employed to depress the platten upon the impression table, and so to effect, in a large machine worked by steam power, the same thing which is done on a smaller scale by the pull of the lever of a hand press. In the case last treated, the point Q never passed below the line BR, and thus the joint only straightened itself once during a revo- lution of CP ; it is possible, however, to cause this straightening to occur twice in each revolution of the crank, and to effect this, it is only necessary to shift the point C a little nearer to BR, so that the joint may straighten when P is upon either side of the lowest or highest point of its circular path. The sketch shows the knuckle joint as applied to a movement of this character in a power loom (fig. 144). In this case the joint will straighten when P arrives at certain points on either side of the vertical line CA, as shown by the posi- tions of CP, CP', and thus we shall find that the point Q falls below BE, as well as rises above it, and that there will be two positions of P upon either side of A in which BQE becomes a straight line. In weaving, the thread of the weft requires to be beaten up into its place after each throw of the shuttle ; and in some cases, as in carpet weaving, two beats are wanted instead of one. The arrangement which we are now discussing has been used Multiplied Oscillation. 127 to actuate the movable swinging frame, or batten, which beats up the weft, and the result is that two blows are given in rapid suc- cession. FIG. 144. In the figure referred to, EF is the batten, movable about F as a centre, and it is clear that when the crank takes the positions CP, CP', the joint BQE will straighten, and, as a consequence, the batten will be pushed as far as it can go to the left hand, or a beat- up of the weft will take place. We thus solve the problem of causing a reciprocating piece to make two oscillations for each complete revolution of an arm with which it is connected. ART. 107. This principle of obtaining two vibrations of a bar for each revolution of the driving-crank may be extended still further, and we will alter the construction so as to obtain four vibrations instead of two. The arrangement of the knuckle joint BQE and the crank CP remains as before, but the arm QE is now connected with a second knuckle joint FDL, and the piece AK, which is to receive four 128 Elements of Mechanism. vibrations, is centred at A, and is attached at the point L to the linkDL. FIG. 145. It is clear that the joint FDL will straighten itself four times in each revolution, viz., when the crank CP is in the positions marked i, 2, 3, 4, upon the circle, and thus AK will make four complete vibrations for each revolution of the crank. In other words there are two positions of the joint BQE in which FDL straightens, and each of these positions of BQE is obtained by two distinct positions of CP. By recurring to the earlier part of the chapter, we shall under- FlG J46 stand that a cam -plate movable about C, and shaped as in Fig. 146, may be employed to drive the bat- ten, and may replace the above com- bination, being, in point of fact, a mechanical equivalent for it. The roller P is then connected with levers attached to the batten, and the beat-up occurs when P passes through the hollows upon each side of the projection at C. Vatt's Parallel Motion. 129 ART. \c$r The \Parallel Motion used in steam-engines was the invention of Janies Watt, and was thus described by himself in the specification of a patent granted in the year 1784 : ' My second new improvement on the steam-engine consists in methods of directing the piston rods, the pump rods, and other parts of these engines, so as to move in perpendicular or other straight or right lines, without using the great chains and arches commonly fixed to the working beams of the engine for that pur- pose, and so as to enable the engine to act on the working beams or great levers both by pushing and by drawing, or both, in the ascent and descent of their pistons. I execute this on three prin- ciples. . . . The third principle, on which I derive a per- pendicular or right-lined motion from a circular or angular motion, consists in forming certain combinations of levers moving upon centres, wherein the deviations from straight lines of the moving end of some of these levers are compensated by similar deviations, but in opposite directions, of one end of other levers.' The annexed sketch is copied from the original drawing depo- sited in the Patent Office. FIG. 147. LI J 1 T iJ AB is the working beam of the engine^ PQ the piston rod or pump rod attached at P to the rod BD, which connects AB and another bar, CD, movable about a centre at C. ' When the working beam is put in motion the point B de- scribes an arc on the centre A, and the point D describes an arc on the centre C, and the convexities of these arcs, lying in opposite directions, compensate for each other's variation from a straight K 1 30 Elements of Mechanism. line, so that the point P, at the top of the piston rod, or pump rod, which lies between these convexities, ascends and descends in a perpendicular or straight line.' ART. ^o^-This invention being an example of our combi- nation of two cranks and a connecting link, we proceed to discuss it in a careful manner, and to examine its peculiar features. The lines AB and CD in the diagram represent two rods movable about centres at A and C, and connected by a link, BD. If BD be moved into every position which it can assume, the path of any point P in BD will be a sort of figure of eight, of which the portions which cross each other are nearly straight lines. At the beginning of the motion let the rods be so placed that the angles at B and D shall be right angles. FIG. 148. FIG. 149 We shall now endeavour to discover that point in BD which most nearly describes a straight line, and in doing so, we first re- mark that BD begins to shift in the direction of its length, and therefore that the straight line in question must coincide with BD. The exact position of the so-called parallel point, that is, the point P in Watt's diagram, is determined very simply by analysis, and we shall give the investigation immediately. But we can readily predict where it must be found. As stated by Watt, the points B and D describe circular arcs about the centres A and C, the convexities of these arcs lying in opposite directions, and if AB and CD be equal, the parallel point P must be so placed that its tendency to describe a curve with a convexity approaching to that of the path of B is exactly neu- tralised by its tendency to describe another curve with a like con- vexity in the opposite direction due to its connection with CD. Watt's Parallel Motion. \ 3 1 Hence P must lie in the middle of BD, and being solicited by two equal and opposite tendencies, it will follow the intermediate course, which is a straight line. If, however, AB and CD are un- equal, the path of the point P will be affected by the increased convexity due to its connection with the shorter arm CD, and in order to escape from this effect it will be necessary to move P away from D, and to bring it nearer to the arm AB, whose ex- tremity traces out a curve of less convexity. It may be expected, since we are dealing with circular arcs, that the point P should now approach B in a proportion identical with that given by comparing AB with CD, or that we should have BP CD It is very easy to construct a small model, and to verify in this way the principle of Watt's parallel motion. If the arms AB, CD, be equal, but the describing point P does not bisect BD, but is brought near to the end D, as in fig. 149, the regular looped curve will become distorted and will incline towards C, in the manner shown in the diagram. ART. no. Refer now to fig. 150, and suppose the rods to be moved from the position ABDC into another position A.l>dC. FIG. 150. \ FIG. 151. Draw bm, dn perpendicular to AB and CD respectively, and let P be the point whose position is to be determined. Let AB = r, b? = x, BA = 6, 132 Elernen ts of Median ism. We shall suppose in what follows that the motion of AB and CD is restricted within narrow limits, and shall deal approximately with our equations, by putting , rf> sin - = -, and sin - =-, 22 22 ., x f>P Em the V == ^ = D _r (i cos0) s (i cosp) 2 sin 2 ? s 2 sin 2 ?? 2 = s nearly. JY/ But the link only turns through a very small angle, which may be considered to be nothing as a first approximation, in which case the vertical motion of B is equal to that of D, .-. bm dn, or r sin = 5 sin 0, whence r 6 = s nearly. * * y s r 2 /' P _ CD r Pd AB ' />., the point P divides BD into two parts which are inversely as the lengths of the nearest radius rods. In the case considered, which is that which occurs in practice, the parallel point lies in the connecting link, but if the rods be arranged on the same side of the link, as shown in fig. 151, the required point will lie in BD produced, and on the side of the longer rod. Suppose now the rods to be moved into the position AbdC, and draw bp, dl perpendicular to BD and BD produced re- spectively. ' _ bp _ r (i-cos 0) _ r 2 *~ dl~ s (i cos y) ~ 7y* Watt's Parallel Motion. Also r =stp , by parity of reasoning, '^P = ^ X ^ = ? 133 and the point P obeys the same general law whether it be found in the link itself or in the prolongation of the line of its direction. ART. 1 1 1. We have supposed that sin Q = Q and sin

in terms 1 3 }. Elements of Mechanism. of 0, and we shall see that the deviation sought for depends upon the difference of the cosines of and 6. FIG. 152. As before, observing that s = r, we have J$m = r (i cos 0), m b = r sin 0, d n = r (i cos ), Dn = r sin (j>. Let BD = /, then / + mb = I cos a + ~Dn, or / + r sin 6 = I cos o + r sin = ?- sin d + / (i cos a), (i). Now a being the angle through which BD is twisted, and being moreover very small, we shall have /a = B/ t dn^ very nearly, = r (i cos 0) + r (i cos 0) = 2 r ( i cos 0), since is nearly equal to 9. /. a = 2 r (i cos 0) very approximately. By substituting in equation (i) we can calculate with con- siderable accuracy, and then the deviation of P from the vertical and can therefore be ascertained. Ex. Let 6 = -, and assume r s = 50 in., / = 30 in. 9 .'. n = --' (-0603074) = : - (-0603074) = 2010247, or represents the angle 11 31'. Substituting in equation (i) we have sin = sin 2o + 2(r cos 11 31') = -3420201 + 3 ('0201333) = -3541001. .*. represents an angle of 20 44' nearly. Kence the deviation of P from the vertical = 5 (COS 20 -COS 20 44) = 25 (-0044544) = T ^th of an inch approximately. Similar Curves. 135 It may be shown that this amount of deviation is again capable of reduction if we cause the centres of motion, A and C, to approach each other by shifting them horizontally through small spaces. ART. 112. The point B, whose motion has been examined, is usually found at the end of the air-pump rod. We have now to obtain a- second point, also describing a straight line, and suitable for attachment to the end of the piston rod. We require, in the first instance, to know when two curves are similar, and in a Cambridge treatise on Newton's ' Principia ' the test of similarity is stated in the following terms : Two curves are said to be similar when there can be drawn in them two distances from two points similarly situated, such that, if any two other distances be drawn equally inclined to the former, the four are proportional. Ex. Thus all parabolas are similar curves, and all ellipses with the same A - eccentricity are similar curves. Let A, A', be the vertices, S, S', the foci of two parabolas. Then SA, S'A', are two lines drawn from two points similarly situated, viz., the foci of the curves. Let SP, S'P' be radii inclined at the same L V to SA, S'A' respectively. ThenSP= 2SA -, i+cos 0' . i-f cos SP = SA ' ' S'P' S'A" whence the curves are similar, and there is no exception to this rule. Those who are conversant with the properties of a parabola know very well that it represents, with great exactness, the path of a stone thrown obliquely into the air, and gives the theoretical form of the path of a projectile when unaffected by the resistance of the air. 136 Elements of Mechanism. The similarity of all such curves to one another is by no means evident upon cursory observation, but it is at once established by this simple reasoning. In the case of ellipses, we proceed in a similar manner, and now S and S' represent the foci of two ellipses of eccentricity e and e 1 respectively. i + e cos S'A' . s /p/ i+e' cos 6 ' Let now e d, the eccentricities being identical, ^ or the curves are similar only under the condition stated. *f ART. ^y. Without any further enquiry into the nature of A the curves which satisfy the condition of similarity, we will pass on to examine an extremely useful instrument called a Pantograph^ which is formed as a jointed parallelogram with two adjacent sides prolonged to convenient lengths, and is used to enlarge or reduce drawings according to scale. This parallelogram was incorporated by Watt into the inven- Kr tion of the parallel motion, and gave it that completeness which ity has at the present time. We have now to show that the pant^ graph is an apparatus for tracing out similar curves. y In the diagram, let BQRC represent a parallelogram whose sides are jointed at all the angles, and having the two adjacent sides BC, BQ, lengthened as shown. Take a point S, somewhere in BC produced, as a centre of motion, place a pencil at any point P' in the side RC, produce SP' to meet BQj or its prolongation in P, place another pencil at P, when it will be found that by moving about the frame over a sheet of paper, and at the same time allowing the joints free play, it will be possible to describe any The Pan tog rap/i. 1 3 7 two curves that we please, and these curves will be similar to each other. Our definition tells us that the two pencils will trace out similar curves if we can show that SP always bears to SP' the same ratio that two other fixed lines radiating from S, and to which SP and SP' are equally inclined, also bear to each other. Conceive, now, that SCB originally occupied the position SA'A, and draw the line SA'A as a fixed line upon the paper, then we shall always have gp>= s ^=g^7, which is a constant ratio, and the angle ASP is equal to the angle A'SP', therefore the points P and P'will trace out similar curves so long as SPT remains a straight line. This is, therefore, the only condition which we have to observe in using the instrument. - ART. 114. We are now in a position to complete the Parallel Motion of a Seam Engine, for if one of the points describe a straight line, the other must do the same. To the system, ABCD, we superadd the parallelogram BFED; ABF being the working beam of the engine. FIG. 155. The usual construction is to make the arms AB and CD equal to each other, in which case P, which is the point to which the air-pump rod is fixed, will be in the centre of BD ; whereas the second point E, which we are about to find, lies in AP produced, and is the point of attachment of the end of the piston rod. In order to find the side BF in this parallelogram BFED, assume that AB = r, CD = j, BF = x. Then -= by similar triangles ABP, DPE. T PB 1 38 Elements of Mechanism. p\T) A T) ^ Also -=---= -, by property of the parallel motion, PD CD s x r r 1 .'. -=-, or x= , r s s which equation determines the proportion between BF, AB, and CD, in order that the second point sought for may lie at the vertex, E, of the parallelogram BFED. Thus we see that the complete arrangement consists of two distinct portions incorporated together. 1. The combination of AB, CD, and BD, which compels some point P to describe a straight line> the position of this point de- pending upon the relation between AB and CD. 2. The Pantograph ABFED, which has some point in FE, here for simplicity selected at E, that must necessarily describe a straight line parallel to the path of P. And, as we have stated, the points of attachment of the ends of the air-pump and piston rods to the main beam of the engine are thus provided for. ART. 115. Where a beam engine is used in a steam vessel the beam must be kept as low down as possible, and the motion is altered as in the figure, but it is precisely the same in principle. Beginning with ABDC, a system of two arms and a connecting link, we obtain the parallel point P ; we then construct the panto- graph CDBFE so as to arrive at the point P', whose path is similar to that of P. Here CE represents the beam of the engine, and P' is the point to which the end of the piston rod is attached. Draw GH parallel FlG . I56 . to FE, and let Z ^ BD = I FF= *' HD = GB = BP BP CD s .,HD =1 D= , 2 s r r FFFFEH A Parallel Motion. 139 r+s whence thej^osition of the point P' is ascertained. AR^J^/ A Parallel Motion may also be useful in machinery. In the old process of multiplying engraved steel plates at the Bank of England, which was practised before the art of electro- typing was understood, it was necessary to roll a hardened steel roller upon a flat plate of soft steel with a very heavy pressure, and so to engrave the plate. The difficulty of maintaining this pressure during the motion of the roller upon the surface was overcome by the aid of the parallel motion shown in the drawing. FIG. 157. The system of jointed bars allowed the heavy frame C to traverse laterally, while the necessary pressure was obtained by a pull upon the end B of the lever AB, which lever was movable round A as a centre of motion, and was further connected at B with somersource of power. ART. (lira A straight line motion which is founded on the propositiorr in Euclid that the angle in a semicircle is a right 140 Elements of Mechanism. FIG. 158. angle has been suggested by Mr. Scott Russell. It is derived from the ordinary crank and connecting rod. Let the rod RQ be bisected in P, and jointed at that point to another rod CP, which is equal in length to PQ. Suppose the point C to be fixed as a centre of motion, and the end Q to be con- strained to move to and fro in the line CQ, then R will move up and down in a straight line pointing also toC. Since CP = PQ = RP, the point P will be the centre of a circle passing through C, Q, and R. Also RPQ is a straight line, and must therefore be the diameter of the circle, whence the angle ^ RCQ is a right angle, or the point R must always be situated in a /\r straight line through C perpendicular to CQ. That is, the path of R is a fixed straight line pointing to C. The motion fails after CP has rotated through two right angles \ from the upper vertical position. The point Q then gets back to C and remains there. The motion is rather one for copying a straight line than for generating it, for the truth of the straight line described by R will be neither greater nor less than that of the guide CQ which directs its motion. ART. 1 1 8. The movement maybe analysed on the doctrine of harmonics. Taking the rods in the position shown, it is appar- FIG. 159. ent that CP has moved from C/ and that P has completed a harmonic mo- tion ;//P in a horizontal direction towards the right hand. At the same time PR has turned round P towards the left hand through a circular arc rR. which is exactly the same as the corresponding path of P. The harmonic motion of R is therefore #R in a horizontal direction towards the left hand. But R receives the motion of P in addition to its own proper Straight Line Motion. 141 movement round P as a centre, and the horizontal components of these movements are equal and opposite. Hence R describes a vertical straight line. Hereafter it will be shown that the backward rotation of RP at the same rate as the forward rotation of CP may be provided for by wheelwork, and that a straight line motion may be obtained without any guide along CQ. ART. fj^-Since the path of P is a circle round C it follows that if AB7T>F, represent grooves on a plane surface, and the rod RQ has pins at R and Q FlG l6o- working in the grooves, the D circular motion of CP will cause R and Q to oscillate to and fro in AB and DF through spaces equal to 4CP. And the converse is also true. B This suggests an in- structive model. For if CP and PQ repre- sent an ordinary crank or connecting rod, CP being equal \.o PQ, it will be found that the rotation of CP brings Q up to C and then the motion fails. The rods CP and PQ merely continue to maintain a circular motion round C. Whereas if QP be produced to R, such that RP = PQ, and a pin at R works in grooves lying in DF but not continued quite down to the centre C, the rotation of CP will cause Q to move from a distance 2CP on the right, up to C, and then to pass to a distance 2CP on the left of C, thus having a throw equal to four times the length of the crank. ART^5MP- Another form of parallel motion was devised for marine enfiries before the principle of direct action was so gene- rally adopted. It was fitted to the engines of the ' Gorgon ' by Mr. Seaward, and has since been applied in a modified form to small stationary engines, which are convenient in the workshop, and are known as Grasshopper engines ; but except so far as the I 4 2 Elements of Mechanism. latter application is concerned, it has not been regarded with par- ticular favour. It is, however, remarkable as illustrating a mechanical principle for reducing the friction upon an axis, by causing the driving pres- sure and the resistance to be overcome to act upon the same side of the centre of motion ; for here the connecting and piston rods are both attached to the rocking beam upon the same side of its axis. In this respect it has an advantage, for in ordinary beam engines the pressure upon the fulcrum of the beam is the sum of the power and the resistance, whereas here it is the difference of these forces, and the friction is proportionally diminished. It is derived from Scott Russell's motion by replacing the guide by a portion of a circular arc of comparatively large radius. For the purpose of explanation we refer to the diagram, where the line HSP corresponds to the line RPQ in the last article, the point P moving very approximately in a horizontal line by reason of its connection with PQ, which has a centre of motion at Q, and the point H being that which most nearly describes a straight line. The system of rods is then TS, HSP, PQ, the points T and Q being centres of motion. Draw SR and HK perpendicular to TP, and SV perpendicular toHK, -.a\ cp _ STR=0 :b } * - r 'SPR=$> FIG. 161. and let Then TR = acos0 = a ("1 2 sin 2 -^ (O 2 \ i J nearly, Peaucellur's Invention. cos

P, of two teeth in contact at P ; then the tooth aP will press against bP so that the perpendicular to the surfaces in contact at P shall pass through D, and the relative angular velocities of two pieces centred at A and B, and fur- nished with these teeth, will be the same as those of the two pitch circles. As the wheels rotate, we find that the point of contact P tra- vels along the upper small dotted circle starting from D. Tn the same way the points of contact of teeth to the left of ADB travel along the lower dotted circle up to D, and it is, therefore, essential to form the teeth in the manner which we are about to describe by somewhat extending our con- struction. We have now to make complete teeth upon both wheels, and to provide that either A or B may be the driver. As far as we have gone we have described the point of a tooth upon A and the flank of one upon B, and have supposed A to drive B. If the conditions were reversed, and B were to drive A, we should have to obtain from one describing circle the curves suitable for the point of a tooth upon B and the flank of one upon A. This describing circle is not necessarily of the same size as the former one, but it is very advantageous to make it so, and we shall therefore assume that the teeth upon A and B are formed by the same describing circle. Let the describing circle G trace PQ, SR upon A, and /^, sr upon B, then the complete teeth can be made up as shown in the diagram on the next page (see fig. 193). 176 Elements of Mechanism. By preserving a constant describing circle, any wheels of a set of more than two will work together, as, for example, in the case of change wheels in a lathe. FIG. 193. FIG. 194. It remains to discuss the character of the teeth as dependent upon changes in the form or configuration of the hypocycloidal portion of the curves. ART. 149. If we trace the changes in form of the hypocycloid, as the describing circle increases in size until its diameter be- comes equal to the radius of the circle in which it rolls, we shall find that the curve gradually opens out into a straight line. It is indeed a well-known geometrical fact that when the diameter of the circle which describes a hypocycloid is made equal to the radius of the circle within which it rolls, the curve becomes a straight line. Let C be the centre of the describing circle at any time, and let P be the corre- sponding position of the describing point (fig. 194). Suppose that P begins to move from E, so that the arc PQ shall be equal to the arc EQ. Join CP, AE; let EAQ = 0, PCQ = . ThenPQ = arcEQ, or Teeth of Wheels. 177 But AE = 2 CQ, /. CQ x = zCQ x 8 or = 28. Now cannot be equal to 26 unless P coincides with R in the line AE, in which case the diameter EAD is the path of P. This property of a hypocycloid is taken advantage of in Wheatstone's photometer, where an annular wheel is constructed, and a second wheel of half its diameter is made to run very rapidly upon the internal circumference : a small bead of glass, silvered inside, is attached to a piece of cork fitted on this internal wheel The bead would give the images of two lights held upon either side of it. When the wheel revolves these small images or spots of light become luminous lines of light, whose brilliancy can be compared, and made equal, by shifting the apparatus towards the weaker light This contrivance is a philosophical toy, it is not used. ART. 150. The first particular case of the general solution is the subject of the present article. It will be remembered that the hypocycloid determines the flank of the tooth upon either wheel : if, therefore, the radius of the circle describing the hypocycloid be taken in each case to be half that of the corresponding pitch circle, the teeth will have straight, or radial flanks, as they are commonly called. The method of setting out the teeth is the following : Let A and B be the centres of two pitch circles which touch in the point D. Let a circle, F, whose diameter is equal to BD, roll upon the circle A, and generate the epicycloid QP : this curve determines the form of the driving surface of the teeth to be placed upon A. Let another circle, G, whose dia- meter is equal to AD, roll upon the circle B, and generate the epicycloid qp : this curve determines the driv- ing surface of the teeth to be placed upon B. FIG. 178 Elements of Mechanism. Here of necessity the describing circle is not of the same size FlG 196 when tracing out the points of the teeth upon A and B ; but, by reason that the same circle gives the point upon A and the flank upon B, or con- versely, and that the flanks in each case are straight lines, the condition in Art. 146 is still fulfilled. The annexed figure shows us these teeth with radial flanks, the straight edges of the teeth pointing towards the centres of the respective pitch circles. ART. 151. As the circle describing the hypocycloid goes on increasing until it becomes equal to the circle in which it rolls, the curve passes from a straight line into a curve, and finally degenerates from a small half-loop shown in the sketch down to an actual point. FIG. 197. It appears also that the same hypocycloid is generated by each of the circles A and B, which are so related that the sum of their diameters is equal to the diameter of the circle in which ^hey roll. ART. 152. The second particular case of the general solution occurs when the hypocycloid degenerates to a point : we then Teeth of Wheels. 179 FIG. 198. obtain a wheel with pins in the place of teeth, and derive a form which is extensively used in clockwork. There is a very old form of pin wheel, called a lantern pinion, where the pins are made of round and hard steel wire, and are supported be- tween two plates, in the manner shown in the sketch. This form has been much used by clockmakers, because it runs smoothly, and has the merit of combining great strength with durability. The pin must have some sensible diameter, but we will first suppose it to be a mathematical point. We have just seen that when the hypocycloid becomes a point, the describing circle must be taken equal to that within which it is supposed to roll. As before, let A and B be the centres of two given pitch circles which touch each other in the point D. Let a circle, F, equal to B, roll upon the circle A, and generate the epicycloid PQ. This curve will determine the acting surface of the teeth to be placed upon A, which will work against pins to be placed at equal intervals on the circumference of the circle B. Thus we shall have epicycloidal teeth upon the driver, working with hypocycloidal teeth on the follower, but these latter teeth are pins, or mere points theoretically, instead of being curved pieces of definite form. Here it is perfectly ap- parent that the condition upon which we rely is again fulfilled. The pin must have some size, and we shall take into account the size of the pin by supposing a small circle, equal to it in N 2 i8o Elements of Mechanism. sectional area, to travel along the theoretical path of the point, and to remove a corresponding portion of the curved area occu- pied by the epicycloids. Assume that QP represents the acting surface of a tooth which drives before it a point, P (fig. 200). Make P the centre of a circle equal to the size of the pin: suppose this circle to travel along PQ, having its centre always in the curve : remove as much of the tooth as the circle intercepts, and the remainder will give the form of the working portion. We shall presently find that in practice the pins are always placed upon the driven w/iect, and as this rule is never broken, for reasons to be stated hereafter, we shall assume it to exist when we come to apply our solution to the case of a rack and pinion. ART. 153. If either of the wheels becomes a rack, that is, straightens into the form of a bar, the radius of the pitch circle must be infinitely large ; and we shall now take up the inquiry as to the changes introduced into the shape of the teeth by this transition from the circle into a straight line. The curve which we have called an epicycloid changes into a cycloid when the rolling circle runs along a straight line instead of upon the outer circumference of another circle, Teeth of Wheels. 181 It is, in fact, the curve described by a point in the rim of a wheel as it runs along a level road or rail. It is shown in fig. 201, and possesses some very interesting properties with reference to the swing of a pendulum ; it is, there- fore, a curve very familiar to those who study mechanics. So far as the general solution in Art. 148 is concerned, the changes will be the following. Conceive that the circle A is enlarged till it becomes a straight line ; then the circle G, which rolls upon the inner and outer circumferences of the circle A, tracing thereby the points and flanks of the teeth upon A, will in each case generate the same curve, viz., a cycloid. Thus the teeth upon B will remain as before, and each face of a tooth upon the rack A will be made up of two arcs of cycloids meeting in the pitch line. ART. 154. In Art. 150, where the teeth have radial flanks, the matter is not quite so simple, for the describing circles which give the radial flanks are in each case to be of one-half the dia- meter of the pitch circle in which they respectively roll ; and here one of the pitch circles is infinite, whence it follows that a circle half its diameter is infinite also, or may be re- garded as a straight line. The curve traced out by one ex- tremity of a straight line rocking upon the circumference of a given circle, is, of course, the same as that described by one end of a string PQ, which is kept stretched while it is unwound from the circumference of the circle. The end of the line, or the end of the string, is at first at the point A in the curve AP, and the curve is traced out while the line rocks in one direction, or during the un- winding of the string. This curve AP is a very well-known curve, and is called the involute of a circle. We have met with it before, and we proceed to show that in the case of a rack and pinion having teeth with radial flanks, the driving surfaces of the teeth upon the pinion will be the involutes of the pitch circle of the pinion in question. 182 Elements of Mechanism. ART. 155. To make this matter clear, we refer to fig. 203, and observe that the circle F, rolling upon a straight line, generates a cycloid and gives the form of the driving sur- faces of the teeth upon the rack: the circle G becomes infinite, and E/ changes to a straight line. The change which the rest of the con- struction undergoes is simply the substitution of the invo- lute qp for the corresponding epicycloid, the circle G having passed into a straight line. The change is scarcely visible to the eye, but the form of the teeth is shown in the diagram, where the curved portions in the rack are cycloids, the radius of the describing circle being half that of the pitch circle of the pinion, and the curves upon the pinion are the involutes of its own pitch circle. ART. 156. Where pins are substituted for teeth in either the rack or the pinion, we construct in accordance with the rule that the //>/.$ are always placed upon the follower. FIG. 205. FIG. 206. FIG. 204. i. Let the rack drive the pinion. Here the circle A becomes infinite, and the curve PQ passes Teeth of Wheels. 183 into a cycloid, so that the teeth upon the rack are cycloidal, as shown in fig. 205. 2. Let the pinion drive the rack. Here the circle B becomes infinitely large, and CP changes into a straight line, the curve PQ passing into the involute of a circle, with the result exhibited in fig. 206, where the teeth of the driver are the involutes of a circle and are known as involute teeth. ART. 157. The last case to be brought before the reader is derived from a property of this involute of a circle, and the teeth are very easily obtained, but are not used in practice, on account of their being unsuited for the transmission of any considerable forces. We proceed to show that the geometrical requirements of our construction are fulfilled completely by involute curves. Let A and B represent the centres of two pitch circles touch- ing at the point D, as shown by the dotted lines, and with B as a centre, and any line BQ, less than BD, as radius, describe another circle. Through D draw DQ touching this smaller circle, draw AR perpendicular to QD produced, and with centre A and radius AR describe a circle touching QR in the point R. If now we take any point P in QR, and describe the involutes EP and FP by winding two portions of strings, such as PQ and PR, back again upon their respective circles, we shall have two forms of imaginary teeth in contact, viz., EP and FP, such that (1) These teeth have a common perpendicular to their surfaces at P, viz., RPQ. (2) This perpendicular cuts the line of centres in a fixed point D. But these are the conditions which we are seeking to fulfil. No more direct illustration of our leading proposition could be conceived than this one. 184 Elements of Mechanism. The lines PR and PQ are the respective radii of curvature of the involute curves in contact at P, while RQ, which is equal to RP + PQ, is the link of constant length connecting the arms AR and BQ. The angular velocities of AR and BQ are therefore as BQ to AR, or as BD to AD, and this ratio remains constant so long as the curves EP, FP remain in contact. ART. 158. In order to construct the teeth we must draw our FlG 2og pitch circles touching in D, and then select some angle BDQ at which to draw the line RDQ. When this angle is de- termined, we obtain the circles of radii BQ, AR, by dropping perpendicu- lars upon RDQ from the centres A and B, and we then describe teeth of the required pitch by constructing the invo- lutes of these two cir- cles respectively. We observe, of course, that a great latitude is introduced from the circumstance that AD and BD remain con- stant while AR and BQ may have different values. In teeth of this kind there is no difference between the point and the flank : the whole of each edge of a tooth is one and the same curve, viz., the involute of one of the two arbitrary circles. And further, the points of contact of two teeth must lie either in the line RDQ, or in a second line passing through D, and touching both the circles upon the opposite sides. ART. [59. To adapt this solution to the case of a rack and pinion, we note that one of the circles becomes infinite, and, fur- ther, that the involute of the infinite circle of radius AR is a Straight line perpendicular to its circumference, or perpendicular Teeth of Wheels. 185 to QD. Hence the teeth of the rack are straight lines perpendi- cular to the direction of QD. The direction of DQ is arbitrary; but when it has once been assumed, the radius BQ will be determined, and involute teeth can be formed upon B, the teeth of the rack being straight lines in- clined to the pitch line at an angle equal to BDQ. ART. 160. There are now sundry general points for consi- deration. We may inquire, where does FIG the action of two teeth begin, and where does it leave off? Referring to the solution in Art. 148, we observe that if the motion takes place in the direction of the arrows, and the describing circle be placed so as to touch either pitch circle in D, the contact of two teeth com- mences somewhere in #D, travels along the arc aDl>, and ceases somewhere in D& Since aD lies entirely without the pitch circle B, it is clear that the action in aD is due solely to the fact that the teeth upon B project beyond the pitch circle B, and similarly that the action in D& depends upon the projections or points of the teeth upon A. It is further evident that the greater the number of teeth upon the wheels, the closer is their resemblance to the original pitch circles, and the more nearly is their action confined to the neigh- bourhood of the point D. By properly adjusting the amount to which the teeth are allowed to proiect beyond the pitch circles, and also their num- 1 86 Elements of Mechanism. bers, we can assign any given proportion between the arcs of con- tact of the teeth upon either side of the line ADB. Where the teeth upon B are pins, there is comparatively very little action before the line of centres, and there would be none at all if the pins could be reduced to mere points, as in that case there would be nothing projecting beyond the pitch circle B. Again, since the line DP in Art. 147 is a perpendicular to the surfaces in contact at P, it follows that the more nearly DP re- mains perpendicular to ADB, the less will be the loss of the force transmitted between the wheels. Here we have an additional reason for keeping the arc of con- tact as close as possible to the point D. There is a sensible loss of power as soon as the line DP differs appreciably from the direc- tion perpendicular to AB. It is on this account that involute teeth are not used in ma- chinery calculated to transmit great force. The line RPDQ in Art. 157 is always inclined to the line ADB at a sensible angle, and a direct and useless strain upon the bearings of the wheels is the result. ART. 1 6 1. In combinations of wheel work, the accurate po- sition of the centres must be strictly preserved. All the solutions given above, with one exception, entirely fail if there be any error in centring the wheels ; they are totally vitiated if anything arises to deprive them of their geometrical accuracy. The exception occurs with involute teeth : the position of the centres determines the sum of the radii of the pitch circles, and the wheels will work accurately as long as the teeth are in contact at all. We see too that teeth with radial flanks are not suitable for a set of change wheels ; the describing circles of one pair of wheels are derived directly from their pitch circles, and cannot be adapted to any other pair in the series. Where, however, the solution in Art. 148 is employed, the describing circles may be made the same for all the pitch circles,, instead of varying with each one of the series, and in that case any pair of wheels will work truly together. As regards the strength of the teeth, we remark that this quality is influenced by the size of the describing circle. If the diameter of the describing circle be less than, equal to, Teeth of Wheels. 187 or greater than the radius of the pitch circle, we shall have the flanks as shown in the sections a, b, c of the sketch. It is evident that a small describing circle makes the teeth strong, and that it would be unwise to have them weaker than they are with radial flanks. The form of involute teeth being somewhat similar to that of a wedge, the teeth of this character are usually abundantly strong. It will be proved, when we treat of rolling curves, that the surface of one tooth must always slide upon that of another in contact with it, except at the moment when the point of contact is passing the line of centres. This matter should be well understood, the teeth are per- petually rubbing and grinding against each other ; we cannot pre- vent their doing so : our rules only enable us so to shape the acting surfaces that the pitch circles shall roll upon each other. Nothing has been said about the teeth rolling upon each other; it is the pitch circles that roll ; the teeth themselves slide and rub during every part of the action which takes place out of the line of centres. Since, then, the friction of the teeth is unavoidable, it only re- mains to reduce it as much as possible, which will be effected by keeping the arc of action of two teeth within reasonable limits. Generally, the friction before a tooth passes the line of centres is more injurious than that which occurs after the tooth has passed the same line : the difference between pushing a walking-stick 188 Elements of Mechanism. along the ground before you and drawing it after you has been given as an illustration of the difference between the friction before and after the line of centres ; but this difference is less appreciable when the arc of contact is not excessive. Where a wheel drives another furnished with pins instead of teeth, the friction nearly all occurs after the line of centres ; hence such pin wheels are very suitable for the pinions in clock- work. ART. 162. When the axes are not parallel we must employ bevel wheels, the teeth upon which are formed by a method due to Tredgold. FIG. 212. Let FEDH, KEDL, represent the frusta of two right cones, whose axes meet in C, and which are therefore capable of rolling upon each other. Let it be required to construct teeth upon two bevel wheels which shall move each other just as these cones move by rolling contact. Teeth of Wheels. 189 Draw ADB perpendicular to DE, meeting the axes of the cones in the points A and B. Suppose the conical surfaces, HAD, BDL, to have a real exist- ence, and to be flattened out into the circular segments DR, DS : these segments will roll upon each other just as the circular base HD rolls upon the circular base DL. Hence these segments will serve as pitch circles, upon which teeth may be constructed by the previous rules : such teeth may be formed upon a thin strip of metal, and their outline can then be traced upon the surface of the cone terminating in A. Similarly, if ^Ea be drawn perpendicular to ED, the circle of radius Ad = Ea will be the pitch circle for the teeth upon the conical surface EaF. The teeth will taper from D to E, and the intermediate form will be determined by a straight line moving parallel to itself, and originally passing through the points D and E. It is stated in Buchanan's account of this method that ' the length of the teeth, the friction of them, and the peculiar advan- tages of the different modes of forming them, may be considered on the developed pitch lines in the same manner as if they were the pitch lines of spur wheels ; consequently every remark that applies to the one, applies to the other. Indeed, the only difficulty in this construction of the teeth of bevelled wheels consists in applying the patterns correctly to the conic surface whereon the ends of the teeth are to be described.' 90 Elements of Mechanism. CHAPTER VI. ON THE USE OF WHEELS IN TRAINS. ART. 163. When a train of wheels is employed in mechanism, the usual arrangement is to fasten two wheels of unequal size upon every axis except the first and last, and to make the larger wheel of any pair gear with the next smaller one in the series. FIG. 213. Let A be the driver, L the extreme follower, and conceive that L makes (e) revolutions while A makes one revolution ; number of revolutions of L in a given time "number of revolutions of A in the same time ' It will be convenient to distinguish (e) as the value of the train, and the ratio which it represents may be at once found when the numbers of teeth upon the respective wheels are ascertained. Suppose that A, B, C, D, &c., represent the numbers of teeth upon the respective wheels, thus we infer from the condition of rolling that number of revolutions of B in a given time _ A number of revolutions of A in the same time~B ' and similarly for each pair of wheels : ACE K Wheels in Trains. 191 It may frequently simplify our results if we regard e as positive or negative according as A and L revolve in the same or in oppo- site directions : thus, in a train of two axes, e would be negative, and in a train of three axes it would be positive. The comparative rotation of wheels is estimated in various ways, thus : Let N, n be the numbers of teeth upon two wheels, such as A and B. R, r their radii. P, / their periods of revolution. X, x the number of revolutions made by each wheel in the same given time. It is easy to see that Note. A belt and a pair of pulleys supply a mechanical equivalent for spur wheels : the belt may be open or crossed, and in either case the number of revolutions of B in a given time the number of revolutions of A in the same time _ diameter of A "diameter of B ' The crossing of the belt merely reverses the direction of one of the pulleys. Whence it follows that two pulleys with a crossed strap are equivalent to two spur wheels in gear, but that if the strap be open the combination is equivalent to three spur wheels. Ex. Suppose that we have a train of five axes, and that 1. A wheel of 96 drives a pinion of 8. 2. The second axis makes a revolution in 1 2 seconds, and the third axis in 5 seconds. 3. The third axis drives the fourth by a belt and a pair of pulleys of radii 20 and 6 inches. 4. The fourth axis goes round twice while the fifth goes round three times. 96 12 20 3 Here < = * 8 -x--x y x*= 144, or the last axis makes 144 revolutions while the first axis goes round once. 192 Elements of Mechanism. ART. 164. An example of the communication of motion direct from the fly-wheel of an engine to a rotating fan by means of pulleys and bands is given in one of Sir J. Anderson's diagrams. Here the beam of the engine vibrates through the arc a&, and the crank pin at the end of the connecting rod describes a circle, the diameter of which is to that of the fly-wheel A as 4 to 12, or as i to 3. Let the mean pressure on the crank pin = 6000 Ibs., then the tangential pressure at circumference of fly-wheel is equal to 2000 Ibs. It will be seen that the motion is carried from the fly-wheel A to the pulley B by an open strap, then from the pulley C to D by means of a crossed strap, then from E by an open strap to F, the fan. In each case there is an increase of the speed of rotation of the driven pulleys, and a corresponding decrease in the driving pressure. According to the numbers set out on the diagram the fly-wheel makes 20 revolutions per minute, and the fan makes 1600 revolu- tions in the same time, the rate of increase being arrived at by a comparison of the respective diameters of the drivers and followers. In like manner the tension of each band may be deduced Wheels in Trains. 193 from the principle of work, by observing that the product of the tension of a band into its linear velocity is constant. Whence, linear velocity of band on C I linear velocity of band on B ; ; 8 I 3. Therefore, tension of band on C=^ * 2000 Ibs. = 750 Ibs. In like manner, tension 8f band on E=- x 750 Ibs. =300 Ibs. The results are tabulated on the diagram as follows : Diameter Revolutions Pressure A 12 2O 2OOO B 3 80 2COO C 8 80 750 D 2 320 750 E 5 3 20 300 F i I6OO 300 ART. 165. It is very obvious that a wheel and pinion upon the same axis is a combination equivalent to a lever with unequal arms, and modifies the force which may be transmitted through it, and, further, that a single wheel is equivalent to a lever with equal arms, and produces no modification in the force which may pass througrrtt. So, therefore, when any number of separate wheels are in gear, no two wheels being upon the same axis, they are equivalent to a single pair of wheels, viz., the first A, and the last L : the inter- mediate wheels act as carriers only, and transfer the motion through the intervening space. FIG. 215. This also appears from the formula, where we find that ABC K A : B X C X D X ' L~L' 194 Elements of Mechanism. which is the same result as if A and L were alone concerned in the movement. >^&RT. 1 66. If, however, a single wheel, such as B, be inter- posed between two other wheels A and L, although B will not modify the force transmitted, nor alter the velocity, it may be useful in changing the direction in which the wheel L would otherwise revolve. An intermediate wheel so introduced is tech- nically called an idle wheel, and we give instances where this intermediate wheel serves a very useful purpose in causing twc other wheels to rotate in the same direction with precisely the same velocity. 1. The student will remember the peculiar heart-shaped cam for driving the needle bar in a sewing machine, as well as the combination of two cranks and a link for giving motion to the shuttle, and he will find on looking back that in each case the driver was a pin rotating in a circle. The machine from which these movements were taken illus- trates a combination of three spur wheels in gear, the central wheel being the driver, and the other two wheels being equal and re- volving in the same direction with equal ve- locities. There is a small fly-wheel driven by hand, f / \ on the axis of which is the spur wheel B, and 11 I the object being to cause two parallel axes at ' A and C to rotate in the same direction, and at the same rate, it is arranged that equal f ( ^ \ spur wheels, A and C, shall both gear with B, vv ) in the manner shown by the pitch circles marked in the sketch. 2. The Blanchard turning-lathe, of which a portion is shown in the sketch, is used for shaping the spokes of wheels, gun-stocks, shoe-lasts, and other pieces of an arbitrary form, which no one could imagine, until the method was explained, as being the sort of objects that would probably be turned in a lathe. But the solution is, that this copying principle admits of end- less application, and it will be seen that if we place two lathes side by side, and cause the actual cutter in the one to copy exactly the WJieels in Trains, 195 form which an imaginary cutter is tracing out upon a model in the other, we shall reproduce upon a piece of wood placed in the second lathe the precise pattern which exists as the copy. In the drawing the mandrels of the two lathes are shown at F and G, the dark oval at F representing a section of the spoke of a wheel, and being, in fact, an exact copy in iron of the thing to be manufactured. The spoke F is attached to the wheel A, while B is an intermediate wheel 'or driver, and C is another wheel of the same size as A. FIG. 217. The unfinished spoke is placed parallel to the copy, along the axis of the wheel C, and the function of the intermediate wheel, or driver B, is to cause the material to revolve in the same direc- tion and at the same rate as the pattern. A sliding frame, K, carries a tracing wheel, I, with a blunt edge, which is kept pressed against the pattern by a weight or spring, and also contains the cutters, H, which are driven at a speed of about 2,000 revolutions per minute by an independent strap. The circle described by the extremities of the cutters is pre- cisely the same size as the circle of the tracer, and it follows that the exact form which the tracer feels, as it were, upon the pat- tern, will be reproduced by the whirling of the cutters against the material, G, and that the spoke may be completed by giving a slow motion to the combination in a direction parallel to the axis of the pattern. Sometimes the tracer and cutters are mounted upon a rocking o 2 196 Elements of Mechanism. FIG. 219. frame, instead of upon a slide rest, but the principle of the ma- chine is not changed thereby. An ; ntermediate wheel may also be useful when two parallel axes are so close together that there is not space for the ordinary spur wheels. In such a case the axes A and C may be connected by a third wheel, B, and will of course revolve in the same direction. The wheel, B, is elongated so as to gear with both A and C, and is called a Marlborough wheel. The axes might also be connected without wheelwork, as we shall see hereafter. ART. 167. Speed pulleys are so called because they allow of the, transfer of different velocities of rotation from one shaft to another : they are much used in engineers' factories. They are made in a series of steps, as shown in the diagram, one pulley being the counterpart of the other, but pointing in the opposite direction. If the steps be equal, as is com- monly the case, the sum of the radii of each pair of opposite pulleys will be a constant quantity. It is a geometrical fact that when two circles are placed with their centres at a given distance, and are so related that the sum of their radii remains constant, an end- less crossed band connecting both the circles will not vary in length in the smallest degree during the change in the actual dia- meter of each circle. Hence a crossed strap will fit any pair of the pulleys in our series with perfect exactness. The proof is the following : Let A and B represent two pulleys whose radii a.re AP and Speed Pulleys. 197 BQ, and assume that AP + BQ remains unchanged while AP and BQ respectively increase and diminish, and let AP = *, BQ=^, PAC = DBQ = 0. Then CPQD = xQ t yd + PQ, But it is clear that if AP be increased by a given quantity, and BQ be diminished by the same quantity, we shall not change the length of PQ, by reason that the alteration will only cause PQ to move through a small space parallel to itself between the lines AP, BQ, which are also parallel. Also x +y is constant by hypothesis, /. CPQD remains unaltered in length so long as our condition holds good. It may be interesting to examine this matter a little further, and to find an expression for the length of an open band connect- ing t\vo given pulleys. We will assume that we are dealing with step pulleys, the sum of the radii being za in every case, and x being the depth of the step or steps, or the quantity by which either radius differs from the assumed value of the semi-sum of the radii FlG- 220- Let AP=a+x, PAa = = QC/>, and let /= length of the band PQRS. Then the curved portions of the band resting upon the pulleys are (T + 23) (a + x) and (TT 20) (a x) respectively. + (rr-28)(-.x Now --^4 = cos 0, AC .'. PQ = AC cos = f cos 0, ,*. 2X = AC sin = c sin 0, . /= 2ira + 2cd sin + 2^ cos d. 198 Elements of Mechanism. It is evident that / is no longer constant, and that it must neces- sarily change when x or changes: still the variation of length may be so little as to be disregarded under the ordinary propor- tions occurring in a workshop. Since would seldom represent an angle so large as 10, and we have pointed out in Art 1 1 1 how small a difference exists be- tween sin 8 and within even larger limits, we will assume that sin = 0, then cos 0=1-2 sin 2 ^= i- ^ = i- 6 *. 2 4 2 / / fl2 \ Therefore /= 2*0, + 2OT-f 2C (\ -- J, Call /' the value of / when x=o, or = o But 2X = which expresses the difference of the lengths in a convenient form. It is apparent at once that / is greater than /'. It has been stated that this difference is very trifling in many cases, and the following example is an illustration. Let the diameters of the steps of the pulleys be 4, 6, 8, 10, 12 inches respectively, and let / be the length of strap for the pair of 12 and 4, while /' is the length for the equal pair of 8 and 8, the distance between the centres of the pulleys being 6 feet. Then /-/'= 4 (^4) 2 =i^=2 inch whkh is rather Jes3 than 72 72 9 of an inch. In practice, open bands are usually preferred to those which are crossed. The latter embrace a larger portion of the circum- ference, and are therefore less liable to slip, but they rub and wear away at the point where they cross. Speed Pulleys. 199 ART. 1 68. As an example of the use of speed pulleys, we refer to the contrivance sketched in fig. 221, which is to be found in every large lathe, and is useful in other machinery, where it is required to obtain increased power or a diminished speed. It enables the mechanic to change the velocity of the mandrel of the lathe, and gives another simple example of the use of wheels in trains. There is a driving shaft overhead, provided with a cone pulley, and with fast and loose pulleys, which receive the power from the engine : a second cone pulley, F, is fitted on the spindle of the lathe, and rides loose upon it : to this cone is attached a pinion G, which drives a wheel H, and so the motion is communicated by the pinion K to the wheel L, which is fastened to the mandrel of the lathe, and turns with it. The result is that the wheel L revolves much more slowly than the cone pulley F, and that the speed of the mandrel is reduced by the multi- plier ** where G, xl X Li K, H, L represent the numbers of teeth upon these wheels respec- tively. Where the lathe is worked at ordinary speeds, the wheels H and K are pushed out of gear by sliding the piece HK in the 2OO Elements of Mechanism. direction of its axis, as shown in the lower diagram, and the cone pulley, F, is fastened to L by a pin. This pin must of course be removed as soon as the slow mo- tion comes into work. As this movement is very similar to the gearing in a crane, we shall presently examine the application of these trains in raising heavy weights, and shall see how they may be applied so as to reduce velocity, and thereby to increase the amount of force which is called into play. After what has been stated it is scarcely necessary to point out the express use of conical pulleys : they form an obvious modification of step pulleys where the change is continuous in- stead of being abrupt. There are two forms, one where the oblique edges of a section are parallel straight lines, and the other where the convexity of one section exactly fits into the concavity of the other. If the band be crossed we have seen that it will retain the same tension in every position upon the cones. If it be open, it will be less stretched at the middle than at either end, ac- cording to Art. 167. When the obliquity is small, the difference becomes absorbed in the elas- ticity or ' sag ' of the band ; otherwise it must be provided for by giving convexity to one or both of the cones. The rotation of the upper cone being uniform, it is evident that the rotation of the lower cone will decrease as the strap is shifted towards the right hand. One of the cones is sometimes replaced by a cylindrical drum, in which case the strap must be kept stretched by a tightening pulley. As an illustration, we refer to the use of these conical pulleys in the manufacture of stoneware jars and other large earthenware vessels, where a mass of clay is fashioned into the required form upon a rotating table, and the workman varies the speed of the Clock- Train. 201 table according to the requirements of the work by shifting the driving strap along a pair of cones. ART. 169. A common eight-day clock affords a familiar illus- atioh of the employment of a train of wheels. We have marked the disposition of the wheelwork in a clock of this character, and the various wheels are named in the sketch. The great wheel turns round once in 12 hours, and may have 96 teeth. Suppose it to engage with a pinion of 8 teeth on the axis or arbor of the centre wheel, this pinion will turn twelve times while the great wheel turns once, and is capable of carrying the minute hand. Let the pendulum swing 60 times in a minute, or be a seconds' pendulum, the scape wheel will then have 30 teeth, and will be required to turn once in a minute. Hence the value of the train from the centre to the scape wheel should be 60 ; and in constructing the train we observe that if the pinions on the axes of the second and scape wheels have each of them 8 teeth, the centre and second wheels may have 64 and 60 teeth. In such a case we should have ^ 6 8x8 0==6 - In order that the clock may go for 8 days, the great wheel must be cap- able of turning 16 times before the maintaining power is exhausted. It is easy to see that if the speed of the scape wheel at one end of the train be increased, and if we are at the same time limited in respect of the number of rotations of the great wheel, it will be con- venient to introduce a new axis into the train FIG. 223. and, accordingly, 202 Elements of Mechanism. an additional wheel and pinion is found in the train of a watch, where the balance wheel, which performs the function of a pen- dulum, makes at least 120 vibrations in a minute. Another illustration of a train of wheels is found in the method of driving the hour hand of a clock or watch, and in order to understand it we have only to observe that in a clock or watch the minute hand is fastened to the arbor or axis of the centre wheel, and that the hour hand is attached to a pipe which fits upon this axis, and derives its motion from the minute hand. This appears from the diagram, and all we have to do is to connect the pipe and axis by a train of wheels which shall reduce the velocity in the ratio of i to 1 2. FIG. 224, HOUR HAND H- The drawing is taken from a small clock, and represents the train of wheels employed. The pinion K, attached to the axis of the minute hand, drives H, whence the motion passes through G to L, and thus to the hour hand, which is fastened to the pipe on which L is fitted, and which corresponds to the mandrel of the lathe. The value of e in the train is given by the equation _KxG_28x 8 , ~H x L~ 4 2 x 6 4 ~~ Tir ART. 170. The mechanism of a lifting crab for raising weights affords an elementary example of the use of a train of wheels. The diagram is from Sir J. Anderson's series. On the right hand there is an elevation of the crab, showing the upper shaft, AB, to which the driving lever handles are attached. The radius of the circle described by the extremity of the lever handle is to that of the spur wheel C as 5 : i. Lifting' Crab. 203 Again, the radius of the spur wheel D is to that of the drum as 5 : i. Hence the power is to the weight raised as 25 : i. Let P=6o Ibs., then resistance overcome by a rope wound round the drum=6ox 25 Ibs. = i5oo Ibs. FIG. 225. Mo Mo. The diagram is analysed for the use of a teacher. The mark- ings 'Dis. i,' 'Dis. 5,' &c., indicate the relative distances at which the forces act, and the markings ' Mo. 5,' ' Mo. 25,' give the key to the mechanical advantage gained. Thus, when the rope at- tached to the weight is pulled in by one foot, the corresponding motion of a point in the circumference of the large pitch circle on the axis of the drum is 5 feet, while the motion of the end of the driving handle is 25 feet. Hence, by the principle of work, 60 x 2$=x x i, where x is the resistance overcome, therefore x=i$oo Ibs. The arrangement of wheelwork in a crane for raising the heaviest weights would be something of the character shown in fig. 226, with this difference, that the wheels would be broader and more massive as we approached the axis on which the weight directly acts. 2O4 Elements of Mechanism. We take a case in which four men, each exerting a force of 15 ibs., could raise a weight of somewhat more than 4 tons. As we are only examining the theoretical power of the combi- nation, we will neglect the loss of power by friction. The men act upon the winch-handles, and the lengths of the arms of these handles are shown as being equal to the diameter of the drum on which the rope or chain is coiled. This gives a leverage of 2 to i. Fic - 22fi - We next observe that a large an d small wheel are placed upon ^ each axis, and calling e the value Jj of the train, we have the relation, 20 40 20 e= x 2 x TOO I2O IOO Hence the wheelwork multiplies the power 75 times, while the proportion between the length of the winch -handles and the radius of the drum multiplies the power by 2, and thus we re- duce the velocity of the weight which is being lifted 150 times as compared with the rate at which the ends of the handles will move ; that is to say, the power exerted upon the weight is 150x60 Ibs., or 9000 Ibs., which is larger than 4 x 2240 Ibs., or larger than 4 tons expressed in pounds. ART. 171. It is a maxim among mechanics that all screws which are required to be perfectly accurate must be cut in a lathe, and there is a geometrical reason for this statement, de- pending upon the varying inclination of the screw surface at dif- ferent distances from its axis. In cutting a screw-thread upon a bolt without using a lathe, we employ pieces of a nut which would exactly fit the screw Screw Cutting, 205 when finished, in order to carve out the thread. These pieces, which are called dies, are made of soft steel in the first instance, but are afterwards hardened and tempered, and have cutting edges. They are pushed forward by wedges towards the axis of the bolt, during the operation of cutting the thread. It follows that the angle of a ridge upon the die or cutter begins to trace out the screw-thread upon the bolt. But this angle corresponds to the inside line, in the hollow between two ridges, when the screw is completed. We begin, therefore, by tracing out a line, which is slightly different in inclination from the line of the thread that we require. The inclination of the thread, when the cutter begins its work, is not theoretically the same as when it leaves off. The difference is scarcely appre ciable, or even recognisable, in small screws ; but it exists not- withstanding, and we encounter in screw-cutting a practical diffi- culty which has never been absolutely overcome. We can only avoid this difficulty by having recourse to the lathe. In order to make this statement more intelligible, we refer to the sketch where RPM, RQN represent two right-angled triangles concerned in the formation of a screw-thread of a given pitch. Let PM=QN, and conceive that the triangle RPM is wrapped round a right cylinder, the circumference of whose base is RM, then RP will form a screw-thread ,, r IG. 227. whose inclination is the angle PRM. In like manner RQN may be wrapped round a larger cylinder, the circumference of whose base is RN, in which case RQ will be the screw-thread lying at an inclination QRN. Thus, for screw-threads of the same pitch the inclination is less M w as the cylinder on which the thread is traced becomes greater. Bearing this in mind, let D ABCE represent a section of a d' which is to be employed to carve out a screw-thread on a cylinrs whose axis is HH. The cutting edges at C and A first c 206 Elements of Mechanism. upon the cylinder, and they correspond to the angles of the thread marked c, a, respectively. They therefore begin by tracing out a thread whose inclination is greater than it should be, and it is manifest that this difficulty is incurable so long as we are operating with ordinary dies. ART. 172. The principle of construction of the screw-cutting lathe will be apparent from the sketch. Here the copying principle receives one of its most valuable applications. The maker of a lathe furnishes a screw, shaped with the greatest care and exactness, and places this screw in a line parallel to the bed of the lathe. The lathe now carries within itself a copy, which can be re- produced or varied at pleasure, for by means of it we can advance the cutter so as to carve out any screw that we may require. The screw-thread which forms the copy is traced upon the F IG . 228. axis CD, and has a definite A B pitch assigned to it by the I maker. This screw carries a nut, N, and, disregarding the actual construction, we " i will suppose that the nut, N, is furnished with a pointer, P, capable of tracing a screw-thread upon another axis, AB. Conceive, now, that AB and CD are connected by a train of wheels in such a manner that they can revolve with any required relative velocities. Upon each revolution of CD the nut advances through a space equal to the pitch of the screw. If AB also revolve at the same rate as CD, and in the same direction, the point P will describe upon AB a screw-thread exactly similar to that upon CD. If AB revolve more or less rapidly than CD, the pitch of the screw upon AB will be less or greater than that upon CD. A train which would conveniently connect the axes is shown in fig. 229. Here C is the axis of the leading screw, and A carries the bar which is to be the subject of the operation ; it is, in fact, e mandrel of the lathe. we Let E, F, K, H represent the numbers of the teeth upon the Screw-Cutting Lathe. 20? wheels so distinguished, and let e be the value of the train, and suppose AB to make (tri) revolutions for (n) revolutions of CD, we have therefore ExK But e-. FxH pitch of screw on AB n pitch of screw on CD m . pitch of screw upon AB_E x K ' 'pitch of screw upon CD FxH FIG. 229. The guiding screw being right-handed, the above arrangement is suitable for cutting right-handed screws. To cut a left-handed screw it is essential that AB and CD shall revolve in opposite directions. Now AB revolves with the mandrel of the lathe, and therefore the direction of the rotation of CD must be reversed. This is effected by interposing an idle wheel between H and K, which re- verses the motion of the guide screw, CD, and makes the nut travel in the reverse direction. There is a double slot or groove upon the arm which carries K, in order to allow the adjustment of this idle wheel. 208 Elements of Mechanism. A set of change wheels is furnished with these lathes, and a table indicates the wheels required for cutting a screw of any given number of threads to the inch. The screw upon CD having two threads in an inch, the numbers of teeth to be assigned to E, F, H, K, are given in the table of which a specimen is subjoined. No. of threads per inch * F K H 12 60 90 2O 120 iH 60 85 2O 90 13 90 90 2O 130 34 60 90 2O 90 is! 80 IOO 2O no 14 90 90 2O 140 Ex. Let the pitch of the screw upon CD be \ an inch, and let it be required to cut a screw of -j^-inch pitch upon AB, or a screw with 13 threads to the inch. Here e = , which is satisfied in the following manner : E x K go x 20 F x H 130x90' In the case of the micrometer screw with 150 threads to the inch, mentioned in the introductory chapter, a lathe is employed for cutting the thread. The guiding screw has 50 threads to the inch, and the mandrel of the lathe rotates faster than the guiding screw in the proportion of 3 to i. This change of velocity is effected by two wheels having these proportions, and connected by an intermediate wheel, the position of the centre of which can be altered so as to suit principal wheels of different sizes. The cutter which shapes the thread has a fine pointed edge, and the screw is nearly finished in the lathe, but is finally rendered perfect in form by screwing it through a pair of dies. This latter Wheels in Trains. 20$ operation has a tendency to alter the pitch of the screw by per- manently stretching the metal of which it is made, and should therefore be resorted to as little as possible. A screw with 150 threads to the inch, and furnished with a graduated head reading off to hundredths of a revolution, would measure a linear space of or T-^Tnrth of an inch. 150 x 100 ART. 173. We pass on now to an enquiry into the construe* tion of a train of wheels for any given purpose ; and here it is ne- cessary to point out that mechanicians are tied down by practical considerations, whereby it often happens that an arrangement, which is quite simple and feasible in theory, would nevertheless prove utterly absurd if any attempt were made to carry it out in practice. One very simple example will explain what we mean. Suppose it to be required to communicate motion from one axis, A, to another, C, and that C is to make 60 revolutions while A makes i revolution, as in the clock train. If A be made to drive C directly, it is clear that the number of teeth upon A must be 60 times as great as the number upon C, so that if C have 8 teeth, A must have 480 teeth. This would involve the use of two wheels side by side, one of which was 60 times as large as the other, to say nothing of the practical difficulty of dividing the larger wheel so as to form the teeth, and accordingly no such combination is to be found in any clock train. But the insertion of an intermediate axis relieves us at once from the difficulty. Place such an axis, which we may call B, between A and C, and fasten upon it two wheels of 8 and 60 teeth respectively, give 64 teeth to A and 8 teeth to C, and the value of the train becomes ~ ~, or 60, the necessary result being obtained with perfect O X O ease and complete simplicity of construction. ART. 174. We see, then, that this problem of connecting two axes by a suitable intermediate train of wheels is an arithme, tical problem which may, of course, in some cases prove extremely troublesome, and may demand a considerable amount of arithme- tical ingenuity. P 2io Elements of Mechanism. The value of e being assigned as a fraction, the only thing to be done is to resolve the numerator and denominator into their prime factors, and then to compose the best train which may sug- gest itself. Thus, let it be required to connect two axes so that one shall revolve n times while the other revolves once. Assume some value for n, say 720. Then e= n= 10x9x8, _ Sox 72 x64 .-8x8x8 ' which gives a probable solution for the train. If any of the factors appear unmanageably large, we may ap- proximate to the value of e by continued fractions, and seek other factors which present less difficulty. If the value of e be an in- teger, we have seen that it must still be split up into factors, and must be further multiplied and divided by the numbers of teeth in each pinion. Thus, suppose the two axes are to be connected whereof one revolves in 24 hours, and the other in 365 days 5 hours 48 minutes 48 seconds, as in Mr. Pearson's orrery. Since 24 hours = 86400 seconds, and 365 days 5 hrs. 48 min. 48 sec. = 31556928 seconds. IQX 9 X5 Here 269 is an inconveniently large number, and 5 is certainly too small. The wheel of 269 teeth cannot be got rid of without altering the entire ratio, but the pinions of 9 and 5 teeth may be changed into others of 18 and 10 teeth. 10 x 10 x 18 We might have approximated to e by an algebraical process and have derived the fraction 94063 3i<;i:6Q28 : J > as representing * ?p - - very closely, Wheels in Trains. 2 1 1 But 260 4x5x13 ' _44X 89x97 8x 10 x 13' which avoids the higher number 269, and corresponds to a period of 365 days 5 hours 48 min. 55-4 sec. Thus we have a train in which the numbers are suitable. Every possible arithmetical artifice Is resorted to in cases of this kind, and the ratio 2 ^ has been dealt with after the following manner, in virtue of the discovery that 269001 9x9x9x9x41, and it is not very difficult to get upon the necessary track, for we see at once that 269001 is divisible by 9 because the sum of its digits is so divisible, and again 9 001 = 29889, which is again divisible by 9 for a like reason, and thus we soon arrive at the last quotient after all the successive divisions by 9, viz., 41. Since 2 - 6 9 = _J62_, i 10 x 10 x 10 269001 = very nearly. 10 x 10 x 10 The numerator can now be split up into 3 factors, which will express the numbers of teeth in the 3 wheels of a train, and we may consider that ,= 1 6 _9 = 269001 earl I 10X10X10 8__x 81 X4 1 10 x 10 x 10' an approximation which would introduce an error of only one revolution in 269000. ART. 175. It is also a matter of enquiry to ascertain the smallest number of axes which may be concerned in the trans- mission of any required motion, since we do not want to employ more wheels than are necessary. The smallest number of teeth which are to be allowed upon a pinion must be given, as well as the largest number to be allowed upon any wheel 212 Elements of Mechanism. Suppose that no pinion is to have less than 6 teeth, and no wheel more than 60, and let us trace the values of e. With two axes e = = 10. 6 If the numerator be diminished, or the denominator be in- creased, the resulting value of e is lessened, or, in other words, 10 is the greatest possible value of e when two axes are employed. With three axes the greatest value of e is - x -, or 100, and with four axes it is 1000, and so on. Let e have some value between 10 and 100 ; we observe that three axes will suffice, and that each wheel must have less than 60 teeth in order to reduce e from 100 to 60. Thus e =^ x 4= 60. 6 6 Again, let e = 3-A-ss 12 if, and suppose 180 and 12 to be the O limiting numbers of teeth upon a wheel and pinion respectively. In the train which is about to be composed, we shall now find that this extension of the limits of the numbers of teeth upon the respective wheels and pinions will give us the power of arranging the train without increasing the number of axes. Here =1, and x =15x15 = 225. 12 12 Now 1 2 if is less than 225, and therefore three axes will suffice, as in the train represented by c __ i8c? x 146 18 12 ' We may work out this arithmetical reasoning by the use of symbols, and then our solution will apply to every case which can occur. Assume now that p represents the least number of teeth upon a pinion, w the greatest number upon a wheel, and let x represent the number of fractions in e. If all the fractions making up the value of e were equal to each Wheels in Trains. 213 other and had the greatest admissible value, then e would reach its limiting value, and we should have e= x H x . . . . to * factors = (^ P P P \p whence log e x (log w log /), log e ""log w - log/ Now x will probably be a fraction, in which case the next in- teger greater than x + i will represent the required number of axes. Ex. Let e = $-$-, w = 180, p = 12, . ^_ . x = log 365 - log 3 "/ log 15 = i + a fraction. Now the integer next greater than x -f i is 3, therefore 3 axes will be required. We observe that it is not necessary to find the actual value of x, but simply to ascertain the integer next greater than it. ART. 176. It is sometimes a matter of enquiry how often any two given teeth will come into contact as the wheels run upon each other. We will take the case of a wheel of A teeth driving one of B teeth where A is greater than B, and let ~ when re- B b duced to its Lowest terms. It is evident that the same points of the two pitch circles would be in contact after a revolutions of B or b revolutions of A. Hence the smaller the numbers which express the velocity ratio of the two axes, the more frequently will the contact of the same pair of teeth recur. i. Let it be required to bring the same teeth into contact as often as possible. Since this contact occurs after b revolutions of A or a revolu- tions of B, we shall effect our object by making a and b as small as possible, that is, by providing that A and B shall have a large common measure. Ex. Assume that the comparative velocity of the two axes is 2 1 4 Elements of Mechanism. intended to be nearly as 5 to 2. And first make A = 80, B = 32, in which case we shall have 4=^=5 exactly, B 32 2 or the same pair of teeth will be in contact after five revolutions of B, or two revolutions of A. 2. Let it be required to bring the same teeth into contact as seldom as possible. Now change A to 81, and we shall still have -= very nearly, B 2 or the angular velocity of A relatively to B will be scarcely dis- tinguishable from what it was originally. But the alteration will effect what we require, for now = , which is a fraction in its B 32 lowest terms. There will therefore be a contact of the same pair of teeth only after 81 revolutions of B or 32 revolutions of A. The insertion of a tooth in this manner was an old contrivance of millwrights to prevent the same pair of teeth from meeting too often, and was supposed to ensure greater regularity in the wear of the wheels. The tooth inserted was called a hunting cog, because a pair of teeth, after being once in contact, would gradually separate and then approach by one tooth in each revolution, and thus ap- pear to hunt each other as they went round The clockmakers, on the contrary, appear to have adopted the opposite principle. Finally, we would remind the reader that everything which we have said here about wheels in trains is true, whatever be the direc- tions of their axes. We only care to know the relative sizes of the pitch circles and the directions in which they turn : any part of the train may be composed of bevel wheels without affecting our results. Aggregate Motion. 2 r $ CHAPTER VII. I / IZZT" AGGREGATE MOTION. ' / / ART. 177. We have seen that every case of the curvilinear motion of a point is of a compound character, resulting from the superposition of two or more rectilinear motions. It often happens in machinery that some revolving wheel or moving piece becomes the recipient of more than one independent motion, and that such different movements are concentrated upon it at the same instant of time. The motion is then of a compound or aggregate character, and we propose to classify under the head of ' Aggregate Motion ' a large variety of useful contrivances. We commence with two or three simple examples. The well-known frame called Lazy Tongs is a contrivance depending upon aggregate motion. The rapid advance of the ends A and B is due to the fact that these points are the recipients of the sum of the resolved parts of the circular motion which takes place at each angle. Consider the angular joints at the ends of the first pair of bars which carry the handles : these ends of the bars describe circles, just as the points of a pair of scissors would do. Either of these motions in a circular arc may be resolved as in Art. 7 and one of the components so obtained will be carried to the end of the combination. The same thing happens at every joint of the series, and thus A and B receive the aggregate of all these separate movements. A wheel rolling upon a plane is a case of aggregate motion ; 2 1 6 Elements of Mechanism. the centre of the wheel moves parallel to the plane, the wheel itself revolves about its centre, and these two simple motions give the aggregate result of rolling. Thus, in the case of the driving-wheel of a locomotive, each point on the tyre becomes a fulcrum upon which the rest of the wheel turns, and is for an instant absolutely at rest. The centre of the wheel has the velocity of the train, while a point in the upper edge moves onward with twice that linear velocity. Simple as this matter is, it puzzles some persons when they first think about it. In the same way, if a beam of timber be moved longitudinally upon friction rollers, the travel of the beam will be twice as great as that of the rollers. So, again, in moving heavy guns, the men employ what is called a wheel purchase ; that is, they fasten one end of a rope to the spoke of a wheel of the gun-carriage, and make the rope run round the rim. This gives them the leverage of the spokes of the wheel, and the power exerted is exactly one-half of what it would be if the rope were attached directly to the axis of the wheel, in virtue of this principle that the linear velocity of the upper part of the rim is twice that of the centre of the wheel. In some printing machines the table is driven by a crank and connecting rod, and the length of its path may be doubled by applying the principle under discussion. Here a wheel, Q, is attached to the end of the connecting rod PQj so that it can turn freely on its centre, Q. FJG 23I . Let the wheel revolve between the two racks A and B, whereof A is fixed to the framework of the machine, while B carries the reciprocating table. The rack B receives the motion of Q in its twofold character, and moves through exactly twice the space that it would describe if connected simply with the point Q. The size of the wheel makes no difference in the result, for in Differential Pulley. 217 all cases the velocity of a point in the upper edge will be twice that of the centre. ART. 178. We may confirm our views of the nature of rolling motion by seeing what would happen if the fulcrum, round which the wheel turns, were raised above the level of the road. We have now a contrivance by which a carriage may be made to move faster than the horse which draws it, a startling method of stating the fact which has been sometimes adopted. The in- ventor was a Mr. Saxton, who patented a Differential Pulley in the year 1832 (No. 6,351), with a view of obtaining great speed in railway caniages propelled by a rope. By the use of this inven- tion, the consumption of the rope, proposed to be wound up at a stationary engine house, would be much less than if the carriage were attached in the ordinary way. Let two wheels of different diameters (say as 6 to 7) be centred on a common axis at C, and be fastened together, and let an FIG. 232. endless rope be wound round the wheels and pass over pulleys at E and F in the manner shown in the diagram, the rope taking a turn round each of the pulleys. Conceive now a pull to be exerted on the rope at A, in the direction AF, then the tension of the string will cause an equal and opposite pull to be felt at B in the direction BE, and thus the compound pulley has a tendency to turn about D, the middle point of AB. This tendency in the pulley to turn about the point D causes the linear motion of C to be very much greater than that of any point in the rope : for example, when B moves through a small 2 1 8 Elements of Mechanism. space B^, the centre C will advance through G", which upon our supposition is thirteen times as great, so that when one yard of rope is wound up, the carriage will have travelled through 13 yards. The carriage may be at once stopped by disconnecting the pulleys. ART. 179. The differential screw is another instance of aggre- gate motion, and is a favourite with writers on mechanics, inasmuch as it gives theoretically a mode of obtaining an enormous pressure by the action of a comparatively small force. It is constructed on the following principle : two screw threads of different degrees of inclination are formed upon the same spindle AB, the spindle itself passing through two nuts, whereof one, E, is part of a solid frame, and the other, D, can slide in a groove along the frame. Let P, Q represent the pitches of the screws at E and D ; then upon turning AB once the nut D is carried forward through a space P, and is brought back again through a space Q : it therefore advances through the difference of these intervals. (Fig. 233.) There is a form of the differential screw described in the fifteenth volume of the 'Philosophical Transactions,' which is known as Hunter's Screw. Here one screw is a hollow tube acting as a nut for the second screw in the manner shown in fig. 234. The smaller screw is attached to a piece D sliding in the frame, and is not allowed to rotate : upon turning the screwed pipe AB, the piece D will move through a space equal to the difference of the pitches of the two screw threads. If one screw thread were right-handed and the other left- handed, the nut would travel through a space, P + Q, upon each revolution. ART. 1 80. A right and left-handed screw are often seen in combination, for the purpose of bringing two pieces together. Aggregate Motion. 219 FIG. 235. There is a very common instance in the coupling which is used to connect two railway carriages. Upon swinging round the arm AB, the screws which are moved by it bring the nuts E and F at the ends of the coupling links closer together, or cause them to separate. This is obviously a most convenient arrangement. The lever arm and weight at B serve a two- fold purpose: they enable the railway servant to screw up the combina- tion easily, so as to put a pressure upon the buffer-springs, and the weight B prevents the screws from shaking loose during the running and vibration of the train. There is another instance of the use of a right and left-handed screw in combination which is found in the valve-motion of Nasmyth's steam-hammer. Here a right and left-handed screw are placed side by side, and are connected by spur-wheels so that they rotate in opposite directions. Two nuts fastened together engage with the separate screws, and both rise and fall at the same time, being both ad- vanced in the same direction by screws which rotate in opposite directions. ART. 1 8 1. Any system of pulleys will form an example of aggregate motion. Taking the single movable pulley in fig. 236, it is apparent that when W is raised one inch, the centre of the block rises an inch, and therefore the end P of the line DP is shifted one inch. But at the same time the circular sheave of the pulley runs upon the line AB, just as a wheel runs upon a plane, and by turning on its centre until an additional inch of string has come in contact with it, will transfer the end P through another space of one inch, whereby, on the whole, P moves through two inches. FIG. 236. 220 Elements of Mechanism. ART. 182. Another contrivance for lifting heavy weights by a small expenditure of power is the Chinese Windlass. Here a rope is coiled in opposite directions round two axles A and B, of unequal size : the rope is consequently unwound from one axle while it is being wound up Flt " 2 ^ v 7 ' by the other, and the weight may rise as slowly as we please. Let R, r be the radii of the axles, then W moves through TT (R r) upon each revolution of the axles. The practical objection to this windlass consists in the great length of rope required during the opera- tion. In the ordinary windlass the amount of rope coiled upon the barrel represents the height through which the weight is raised, whereas here we begin by winding as many coils on the smaller barrel as the number of turns which we intend to make with the winch -handle, and then at the close of every turn a length of rope equal to 2?rR is coiled upon the larger barrel, by which expenditure the weight has only been lifted through v (R r). Ex. Let R= u, r= 10, then the amount of rope wound up in any number of turns bears the same proportion to the space through which the weight is raised that 22 bears to i. This is a sufficient commentary on the invention regarded as a practical contrivance. ART. 183. The object of Westorfs Differential Pulley -block is to avoid this difficulty about the expenditure of rope. In the Chinese Windlass, one end of the rope is supposed to be fastened to the axle A, and the other end to the axle B. If, however, these two ends were brought together, the supply of rope necessary for B might be drawn from that coiled upon A, and the expenditure would be really 2* (R r). There would be many inconveniencies attending this arrangement in practice, but it has been put into a working shape in the manner shown in the drawing. Differential Pulley. 221 In Weston's pulley-block there are two pulleys A and B, nearly equal in size, turning together as one pulley, and forming the upper block : an endless chain supplies the place of the rope, and must of course be prevented from slipping by projections which catch the links of the chain. The power is exerted upon that portion of the chain which leaves the larger pulley, the slack hangs in the manner shown in the sketch, and the chain continues to run round till the weight is raised. The combination is therefore highly effective. Of the detached sketches, one shows a section of the differen- tial block, and the other is intended to explain the mechanical principle involved. Let C be the centre of the compound block, draw the hori- zontal diameter ABCDE, and let it be noted that the string or chain is unable to slip upon the surface of either pulley. Let P represent the tension of AP, W the weight raised, T the tension of the string or chain at E, 222 Elements of Mechanism. Also let R, r, be the radii of the respective blocks, Then TxCE=Px AC + T x BC, or TxR=PxR + Tx?" . . . (i) Also 2 T=W ..... .... (2) /. W(R-r)= 2 PxR, Ex. 2x15 or W= 3 oP. Regarding the question as an application of the principle of work, the diagram sets out the calculation as follows : Since the diameter of the pulley A is 15, we shall assume that a point in the chain passing over A moves through a space 30, or that P has a motion 30. Also the diameter of B is 14, whence it follows that the chain passing over B will have a motion 28 in the opposite direction. Note. In fig. 238, taken from the 'Anderson ' series, the symbol ' Mo.' stands for the word ' motion.' Hence the motion of the chain round the pulley which sup ports W is (30-28), and the motion of W itself is 3 2 = i. Hence motion of W : motion of P : : i : 30, or W : P::3o : i. ART. 184. The subject of Epicydic trains will now occupy our attention, and we shall discuss some of the most useful applications of that peculiar arrangement of wheelwcrk which is technically so designated. An epicyclic train differs from an ordinary train in this par- ticular : the axes of the wheels are not fixed in space, but are attached to a rotating frame or bar, in such a manner that the wheels can derive motion from the rotation of the bar. There are certain fundamental forms which consist of trains of two or three wheels ; the first wheel of the train is usually con- centric with the revolving arm, and the last wheel may be so likewise. It should, however, be understood that any number of inter- Epicyclic Trains. 223 mediate wheels may exist between the first and last wheels of the train, and that the wheels in the train may derive the whole of their motion from the arm ; or they may receive one portion from the arm and the remainder from an independent source. The elementary form of a train is exhibited in the annexed diagram, and the peculiarities which result from compounding any independent motion with that which arises from the rotation of the arm will demand some careful and attentive study. Here it will be seen that the wheel B, or the wheels B and C, are attached to a bar which is capable of revolving about the centre FIG. 239- of the wheel A, the axis of this latter wheel being firmly held in one position. ART. 185. In order to understand movements of this kind let us take a simple case to begin with. Suppose that there were only two wheels in the train, viz., A and B, and let A be locked so that it cannot rotate ; suppose, further, that A has 45 teeth, and that B has 30 teeth, and let us inquire how many rotations B will make while the arm is carried round once. We might at first imagine that the wheel B would rotate %% or f times by running round upon A ; but this is only a part of its movement. The wheel B has also been carried round in a circle about A by reason of its connection with the arm, and having turned upon its axis once more on that account, it has really made % turns, instead of |, during one revolution of the arm. In confirmation of this view, let us consider the case of three wheels, A, B, and C, whereof A and C are equal. As the arm goes round, we conclude that C will turn once in the opposite direction to the arm by the rolling of the wheels, and it will turn once in the same direction as the arm by reason of its connection 224 Elements of Mechanism. therewith ; the aggregate result being that C will be carried round in a circle without rotating at all upon its own axis. FIG. 240. The motions of the wheels B and C in an epicyclic train are shown in the sketch. The arm is supposed to have revolved through an angle of 45, and it will be seen that B has turned round through a right angle, while C has not rotated at all. We propose now to examine the motion by the aid of analysis. Remembering that there may be any number of wheels in the train, of which A is the first, and L the last wheel. Conceive that the arm makes a revolutions! dufi the ^ the first wheel A makes ,// revolutions > iod Qf d the last wheel L makes n revolutions f and let e be the value of the train. Then the first wheel makes (ma) revolutions relatively to the arm, and the last wheel makes (n d) revolutions relatively to the same arm, or, in other words, L makes (na) revolutions for (m a) revolutions of A. Recurring to our definition of the value of a train (see Art. 163), we at once deduce the equality n a e = - ma There are three principal cases to consider ; i. Let A be fixed, or ;;/ = o, or n = a (1 ~e) and a = -- Epicyclic Trains. 225 Let L be fixed, or =o, otm a(i--\ and = V e) ei 3. Let neither A nor L be fixed, we have now the formula m a whence emea=.n a, or n=me+(ie)a. In applying these formulae we must remember that e is positive when the train consists of 3, 5, or an odd number of wheels, and negative when there are 2, 4, or an even number of wheels. Ex. i. Let there be two equal wheels, A and B, in the train, and conceive A to be locked, or let A be a dead wheel^ as it is termed. Here m = o, and e = i, or the wheel B makes two rotations for each revolution of the arm. Ex. 2. Let there be three wheels, A, B, and C, whereof A and C are equal, and let A be a dead wheel as before. Here ;// = o, e = i, whence n = a (i x x or the rate of diminution of n is twice that of a. It now becomes easy to obtain any required reduction in the Differential Motion. 235 velocity of C. A reduction in the velocity of H must first be effected by shifting a driving strap along a conical pulley, and the velocity of C will be reduced twice as much as that of H. Mr. Houldsworth's invention consists, therefore, in imparting to the wheel C two independent motions which travel by different routes, and which, after combination in the manner just investi- gated, are capable of producing the desired differential motion. ART. 193. In order to fix our ideas, let us calculate the motion in the following ex- FIG. 250. ample : Suppose A, B, C to repre- sent three equal wheels, and let A be fixed to a shaft AD, which carries a conical pulley provided with grooves at a, b, c, d, e, where the diameters are 4, 5, 6, 7, 8. EF is another shaft carry- ing a second conical pulley which is the counterpart of the first, and terminating in a wheel F, whose diameter is half that of H. A crossed band connects the two cones, and the axis AD is made to revolve with a uniform velocity. It is required to ascertain the motion of C when the strap is shifted along the conical pulley. i. Let the strap be placed at a, the angular velocity of H will be that of AD, and we have a = > 4 2 or C moves in the opposite direction to A, and with half its velocity. e r er 2. Let the strap be at b, the velocity of H will be ~ that of , e AD (here e = - x - = ~- according to Art. 163), T r tyH i C/// 2/7/ therefore a i , and n=^ m = > r 4 7 7 236 Elements of Mechanism. hence C still moves in the opposite direction to A, but less rapidly, in the ratio of 2 to 7. 3. Place the strap at c, when e increases to I, and a becomes equal to - ; , /. n = 2 x m m = o, or C stops altogether, its motion being entirely destroyed. 4. Place the strap at d, and we have a = x - = ~~, im 2m whence n-=2a m = m = > that is, C and A move in the same direction with velocities in the ratio of 2 to 5. 5. Finally adjust the strap at e, and the velocity of H will be the same as that of AD. Here a = m, and n = 2m m = w, or the motion of C is precisely the same as that of A. The principle of this invention may now be understood, although it is difficult to appreciate such a movement thoroughly without the assistance of a model. It only remains to present to the student a representation of so much of an actual machine as will embody the cones and the differential train of wheels. The diagram exhibits the manner in which simple elementary movements may be combined together so as to form a train of mechanism, the arrangement of which, before it is properly understood, might appear to be very complex and intricate (see fig. 251.) The operation of spinning, so far as it is carried on by the mechanism before us, is effected by passing a partially twisted fibre or roving through a tube, called a flier, attached to the end of a spindle, and then causing both the flier and the bobbin to rotate with a high velocity. Before the fibre reaches the fliers it is elon- gated or drawn out by a combination of rollers, moving at different speeds and called drawing rollers ; it is therefore of necessity fed on at a fixed uniform rate. The flier and the bobbin both rotate together, and thus twist the roving, but they also rotate at somewhat different speeds, by which arrangement it is provided that the joint operations of twist- Differential Motion. 237 mg the thread and of winding it up upon the bobbin shall go on together. A bobbin with its spindle and flier is shown in the sketch. It will be seen that the roving passes down through the hollow ver- tical arm and is carried to the bobbin by a finger ; the finger is pressed against the bobbin by the centrifugal action of a small elongated piece which runs down the side of the arm, and which, by its tendency to get as far as possible from the axis of the spindle during its rotation, keeps the finger pressed against the surface of that portion of roving which is already wound upon the bobbin. This part of the apparatus has formed the subject-matter of a most lucrative invention. As the winding goes on the bobbin rises and falls, and the flier winds the fibre in uniform layers upon the bobbin. Thus the spindle and flier rotate together, and they are driven by skew-bevels, whereof one is shown at the bottom of the drawing. The bobbin rotates independently of the spindle, and is also driven by skew-bevels, whereof one is shown just underneath the bobbin. Note. As to skew-bevels, see Art. 239. These bevel wheels are in direct communication with the spur wheels marked ' to spindles ' and ' to bobbins ' in the drawing. It will be understood that the winding on will take place when the spindles and the bobbins move at different velocities, and that either may go faster than the other. We shall take the case in which the bobbins precede the fliers. Since the spindles with their fliers move at' a fixed velocity, while the bobbins are continually filling with the rovings and becoming larger, we infer that the bobbins will require a smaller amount of rotation relatively to the fliers, in order that the wind- ing up of the fibre, which is being fed on at a fixed rate by the drawing rollers, may take place uniformly. Hence, if the bobbin runs in advance of the flier, the speed of revolution has to be diminished as its diameter becomes larger. Refer now to the sketch, and it will be seen that the power may pass through the combination of bevel wheels to the three spur wheels placed in a line at the extremity of the 'driving axis' and connected with the cone marked as the 'driver.' The driving power then crosses over to the follower, and enters the 238 Elements of Mechanism. combination of bevel wheels by the small pinion upon the axis of the lower cone which gears with the large spur wheel marked H, which latter wheel rides loose upon the driving axis. The combination of four bevel wheels is exactly analogous to that discussed in Art. 191, the two wheels B and B are equivalent to a single wheel, and prevent the one-sided, unbalanced action which would otherwise occur. The wheel A is fixed to the driving shaft, the wheel C rides loose upon it, but is fastened immovably to the spur wheel marked ' to bobbins,' the function of which has been already explained. We have to prove that the combination of the two cones with the spur and bevel wheels is capable of gradually reducing the velocity of the bobbins as they fill up with the roving. Assume that the cones are equal in section where the strap is placed, then the speed of the first cone will be reduced to ^ by the combination of three spur wheels starting from the driving axis, and thus the pinion which drives H will move at | the speed of the driving axis. But H is five times as large as that pinion, hence the velocity of H is ^th that of the driving axis. The wheel H also rotates in the opposite direction to the driving axis. Take now the formula n = m e + (i e) a. Here e - i, since A, B, and C are equal, and a = as we have just shown ; 10 5 6m Hence the speed of the bobbin pinion is to that of the flier pinion as 6 to 5, or 18 to 15, the negative sign merely showing that the loose wheel C revolves in the opposite direction to the driving axis. The student may be surprised to find that all this apparently reducing arrangement has ended in making the last spur wheel in Differential Motion. 239 240 Elements of Mechanism. the train turn faster than the driving axis ; but an explanation is found in the fact that the rotation of H takes place in the oppo- site direction to the driver, that is, in the same direction as the loose wheel C, and accordingly we shall find that if the velocity of H be reduced we shall also reduce the inequality between the velocity of the bobbins and spindles. Conceive now that the strap is shifted towards the right hand until the sections of the cones are in the proportion of 2 to 3; that is, nearly as far as the spur wheels. The velocity of H will be reduced two-thirds, and will become equal to y^th that of the driving axis. 17 m IS 2 -^ 15 or trie relative speed of the bobbin pinion to that of the flier pinion is reduced from 18 to 15, and now stands at 17 to 15. It is hoped that the complete action of the apparatus is now sufficiently explained, and there is only one refinement in con- struction which remains to be pointed out. It will be seen that the upper cone is slightly concave and the lower one convex: this configuration is adopted because the absolute increase in the diameter of a bobbin bears a ratio to the actual diameter which is not constant, but is continually diminishing in a small degree. The mechanic must not forget or overlook any material point in working out his design. ART. l(^> Epicyclic trains may be employed to produce a very slow motion upon the following principle : Let A, B, C, D represent the numbers of teeth in a train of wheels in gear arranged as in the diagram. If A = D, and B = C, then A and D will rotate with the same velocity in the 'same direction; but if the equality between (A, D) and (B, C) be slightly disturbed, we shall produce a small change in the value of the train. Suppose, for example, that A is less than D, Slow Motion. 24! or that A=3i, D=32 ; and, again, that B is less than C, or that B = 125, C = 129 : then e, the value of the train, will be = AC _ 3_x_i9 ^ 3999 BD 125 x 32 4000' Also, the more nearly the equality is maintained between (A, D) and (B, C) respectively, the more nearly will the angular velo- cities of A and D be the same, or the more nearly will e be equal to unity. Thus if B = D =; 100, A = 101, C = 99, Let us now arrange A, B, C, D in an epicyclic train, and carry back the wheel D so that it shall turn upon the same axis as A. The turning of the arm will then set all the wheels in motion except A, which is to be made an immovable or dead wheel, and we shall have D and A moving relatively to each other just as before, that is to say, D will turn very slowly over A at rest. FIG. 253. DID Al 11 ' I 1 ' i i Miiirn'.niiiyi!! i-i'lil'liMU'Ml " I 1.1 I uri lil'i'i'l 'I'll'' 1 ' l-..lK..MMrn:.:".r l''l li AL an easy example, take wheels of the following numbers viz., A = 60, B = 45, C = 40, D = 65. AC 60 x 40 12 - -- If we now rotate the arm and carry round the train it will be found that D makes one revolution when the arm has been carried through a little more than 5^ revolutions, which is also evident from the formula upon observing that = 5*, which is a little greater than 5^. 242 Elements of Mechanism. So, again, taking the formula - = i , and substituting for the values given previously, we have in the respective examples, a 4000 a 10000 Hence the arm will make 4000 or 10,000 revolutions respec- tively while the wheel D turns round once. ART. 195. These examples lead us to compare the move- ment of any wheel in an epicyclic train with that in another train where the axes are fixed in space, and to regard the subject from a different point of view. Referring again to the fundamental case, viz., that of three equal wheels, A, B, and C, we have seen that if the arm be fixed, and A makes one turn, the wheel C will also turn once in the same direction. But if the arm re- volve round A fixed, the wheel C will apparently run round just as it did upon the last supposi- tion, and yet at the end of a revolution of the arm it will be found that the wheel C has not turned at all. The explanation is that the fixed train gives the absolute motion of C due to its connection with A, whereas the epicyclic train exhibits the relative motion of C with regard to A, which in this case is nothing, because A and C rotate with equal velocities in the same direction. The same thing is true with respect to any other wheel in the train, such as B. Thus, when the axes are fixed in space, A and B revolve in opposite directions, and the motion of B relatively to A is twice its absolute motion, and thus we account for the fact that in the epicyclic train B will rotate twice while the arm goes round once. So also in Art. 194 the fixed train gives the absolute motion of D, viz., fths of a revolution for each revolution of A, and the epicyclic train exhibits the relative motion of D as compared with thatpf A, viz., -^Vrrth of the movement of A in the fixed train. /ART. (96) Another illustration of aggregate motion is found -/in Equation clocks. In these nearly obsolete pieces of mechanism /> Equation Clock. 243 FIG. 255. the minute hand points to true solar time, and its motion therefore consists of the equable motion of the ordinary minute hand plus or minus the equation or difference between true and mean solar time. In clocks of this class the hand pointing to true solar time is fixed to the bevel wheel. The wheel A moves as the minute hand of an ordinary clock; the intermediate wheel B is fixed to a swinging arm, EB, as in Art. > 191, and the position of C will * be in advance of that of A when EB is caused to rotate a little in the same direction, and behind that of A when EB is moved in the opposite direction. Thus, as C goes round during each hour of the day, the hand attached to it may be a few minutes before or behind another showing mean time, and deriving its motion at once from A. The required motion of EB is obtained from a cam plate, Q, curved as in the diagram, and attached to a wheel which revolves once in a year. ART. 197. In the manufacture of rope the operation of FIG. 256. ' laying,' or twisting the strands effected by special machinery. into a perfect rope, has been 244 Elements of Mechanism. The Rev. Edmund Cartwright, the inventor of the power- loom, was also the first inventor of a machine for making rope. The general character of the contrivance will be understood from the sketch, which is taken from the specification of the invention. The machine itself is called a ' Cordelier,' and consists of a frame placed upon a horizontal shaft PQ, and terminating in a laying-block R, which serves the double purpose of directing the strands to the rollers at K, where they are twisted into rope, and of forming a support or bearing for one end of the shaft. Three spool frames carry the bobbins, or spools, which contain the supply of strands, and the strands, as they are unwound from the bobbins, pass through delivery rollers at D, E, and F, and thence onward to the laying top. All this is simple enough, and might be the invention of any- one ; but there is yet a difficulty to be overcome, which we pro- ceed to explain. Upon examining a rope it will be found that the twist of the rope is always in the opposite direction to that of the strands, and it follows that it the bobbins were absolutely fixed to the rotating frame the strands themselves would be untwisting during the whole operation. This untwisting is provided against in a rope- walk by the use of two machines, one at each end of the walk. The strands are attached to hooks on one of the machines, and these hooks are made to rotate with a velocity which exactly neu- tralises the twist of the machine which is forming the strands into a finished rope. Tn the Cordelier the difficulty is at once removed by the intro- duction of an epicyclic train. A dead wheel A, so fitted that it remains stationary while the shaft PQ rotates within it, gears with a second wheel B, and this latter with a third wheel C, equal to A, whose axis terminates in one of the spool frames. Now we have just proved that in such a train C will run round A without rotating at all upon its own axis, and hence the bobbin may be carried round without in the slightest degree untwisting the strand. In order to make this matter still more apparent we refer the student to fig. 257, which is intended to show three positions of a spool when rotating in a frame without the intervention of an epicyclic train. It is quite evident that the spool has made one The Cordelier, 245 rotation round an imaginary axis through its centre while rotating once round the centre of the frame. In fig. 258, on the other hand, where an epicyclic train, with C equal to A, is interposed, the bobbin will take the positions C, C', C", during a revolution, and the rotation just referred to will be exactly neutralised. FIG. 257. FIG. 258. ART. 198. We have stated that the twist of a rope is always in the opposite direction to that of the strands, and it may be asked, Why is this, and what is the reason that a rope does not untwist itself? The answer is that any single strand or cord, when twisted up, will always tend to untwist in virtue of the elasticity of its fibres, and that each separate strand in a rope exerts this tendency throughout its whole length ; but since the twist of the rope is in the opposite direction, the aggregate of all these comparatively feeble forces is felt as a powerful force restraining the whole rope from becoming untwisted. It follows, therefore, that by putting a little extra twist upon the strands of a rope in the process of laying, the rope itself will become harder or more tightly twisted. If anyone will try and make a small piece of cord out of three pieces of string he may at once satisfy himself of the correctness of what has been stated. Take three pieces of string, or fine sash line, thread them through holes in a small plate or disc, to keep them separate, and fasten them together at one end, leaving the other ends free. Upon twisting the knotted end and slowly advancing the disc, a cord will be made which will untwist as soon as it is handled. 246 Elements of Mechanism. Whereas by continually twisting each individual strand, and allowing the knotted end to turn in the opposite direction to that in which the strands are being twisted, a hard piece of cord may be made which will have no tendency whatever to untwist. There is a model in the collection belonging to the School of Mines which shows this experiment in a striking manner. (Fig. 259.) The driving apparatus consists of an arrangement for rotating at the same time three hooks. Each hook, /, is formed of a bent piece of wire terminating in an upright portion, a, which is threaded into a flat disc A. There is a detached sketch of one of these bent wires. Upon carrying A round in. a circle, it will be found that each hook rotates on its own axis. This motion has been explained in Art. 96. Take now three pieces of braided sash line, which have no twist, and suspend a weight W to each of them. Make a small loop at the opposite end of each line, and hang them all up on one of the hooks. A small conical block B, having a handle H, and grooved as in the sketch, is held in such a manner as to receive each line and to direct its motion. The operator now rotates the hook /, and allows the block B to descend slowly while the cord is being twisted. But on looking closely at the sash lines it will be found that each weight VV is turning on its axis during the whole operation. In truth, the weights are made in the form of long cylindrical bars, in order to permit this movement. 1 he result is that there is no twist what- ever remaining on the individual sash lines or strands. Now remove the block, when it will be found that the cord, which appears to the eye well made and perfect, will at once un- twist, and is, in fact, of no use whatever. It is defective in not having any power of retaining the twist which is essential to the hardness and durability of a rope or cord. The apparatus can, however, be so arranged as to put a strong twist upon each individual strand, in such a manner that the twist shall be retained in the finished cord, and shall act always to twist the same more tightly. For this purpose the weights are tied together by a piece of string at D, and can no longer ro ate separately. Each strand is Twist of a Rope. 247 hung on a separate hook, and the respective hooks /, /, /, rotate together by the carrying round of the disc A. The block is held differently, being placed as near as possible to D, and it is moved slowly upwards while the cord is being made. This is exactly the operation performed in a rope-walk, except that the strands are carried along in a horizontal line. FIG. 259. a ? a There is no difficulty about twisting the cord, for the surplus twist put upon the strands causes the weights W to go round to- gether underneath the block B, and a well-formed cord is made as the block rises. When completed, the string at D may be un- tied, the strands may be taken from the hooks, but there is no untwisting. On the contrary, the cord will bear handling, and is quite hard and durable. ART. 199. Many years ago Captain Huddart incorporated the invention of the Cordelier into some useful machinery for manufacturing rope, and he employed the same epicyclic train, but made the wheel C smaller than A in the proportion of 13 to 248 Elements of Mechanism. 14, as in the case of the wheel G in Ferguson's paradox. The result was that a slight additional twist, or forehard, as it is termed, was given to the strands of the rope. Among the apparatus belonging to the School of Mines is a hand machine for manufacturing fine hard cord, resembling whip- cord. It is, in fact, a miniature Cordelier, and instead of the epi- cyclic train of wheels for keeping the bobbins parallel during the rotation, there is a single dead wheel or grooved pulley, A, and FlG 26a three strong india-rubber cords, connecting A with the separate axes C, C', C", on vhich the bobbins are placed. Each of the grooved pulleys, C, C', C", is of smaller diameter than the wheel A, and therefore turns slowly backward in the op- posite direction to that in which it is carried. This fact is made clear in the sketch, for the arrow indicates the direction of rotation of the frame carrying the bobbins, and the dark, lines, aa, bb, cc, show the manner in which the respective bobbins rotate back- ward. That is, when C arrives at C', the line aa will have turned into the position bb, and when it arrives at C", the same line will have turned into the position cc. Whereas, if the pulleys C, C', C" were each equal to A, the line aa would have remained parallel to itself throughout the motion. In Mr. Smith's wire-rope machine, which is described in the papers of the Institute of Mechanical Engineers for the year 1862, the bobbins are placed one behind the other in the axis of a re- volving fiame, and have simply a slow unwinding motion on their axes as the wire strands are run off ; the important result being that the rate of manufacture is greatly increased. There is no question as to the superiority of this arrangement in a mechanical point of view, for the process of laying has no tendency to twist the strands when the bobbins themselves lie in the axis of rotation of the frame which surrounds them. Drilling Machines. 249 ART. 200. A further illustration of aggregate motion occurs in machinery for drilling and boring. In a drilling machine the spindle which carries the cutting tool revolves rapidly, and at the same time advances slowly in the direction of its length. The movement is obtained upon an 'obvious principle, which may be stated as follows : Conceive a nut, N, to be placed upon a screw-bolt, FG, and to be so held in a ring or collar that it can rotate freely without being capable of any other motion. If the nut be fixed, and FG be turned in the direction of the arrow, it is clear that the bolt must ad- vance through the nut. If, again, the screw be prevented from turning, and the nut be made to ro- tate in the same direction as before, the bolt will come back again. And, finally, if by any contrivance different amounts of rotation be impressed at the same time upon the nut and the screw, the bolt will receive the two longitudinal movements simultaneously, and the aggregate motion will be the sum or difference of these component parts. ART. 201. Suppose the wheels D and C to be attached to the bolt and nut respec- tively, and to be driven by the pinions A and B, which are fixed upon the same spindle ; and let A, B, C, D represent the numbers of teeth upon the respec- tive wheels. If (a) be the number of rotations made by either A or B while the nut fixed to C makes m rotations, and the wheel D makes n rotations, we shall have- = ^, and - = 4 a C a D 2 5O Elements of Mechanism. Therefore (a) rotations of A will cause a travel of the bolt FG through a space fa n) x pitch of the screw = a ^? _ M x pitch of the screw. ART. 202. We shall proceed to examine the construction of FIG. 263. FlG - 2fi 4- a small Drilling Machine, which may be worked either by hand or by steam-power, but is not self-acting. Drilling Machine. 25 1 The general arrangement of the machine is shown in fig. 264. The power is applied to turn the bevel wheel D, which again drives C, and causes the case or pipe containing the drill spindle to rotate. This provides for one part of the motion, viz., the rotating of the drill spindle, and the hand wheel K drives the spur wheels M and N, and advances the drill into the work in a manner which we shall endeavour to make clear. The drill spindle is formed in two pieces, as shown in fig. 263, and the upper or screwed portion does not rotate with the lower cylindrical portion which carries the drill, but simply moves it up and down by means of a collar without interfering with its rota- tion. The screwed piece works in a nut forming the boss of the wheel N, and is prevented from rotating by a feather sliding in a groove or slot which runs along the whole length of the screw, and which cannot be seen in the view given in the drawing, the feather itself being fixed in a stop-collar at N. Hence the rotation of the wheel N, by reason of its connection with the hand wheel K, will raise or depress the whole spindle as required. The rotation of the drill spindle is provided for by cutting a groove m n in the lower part of it, and attaching a corresponding projection or feather to the inside of the pipe AB. This allows the spindle to move lengthways in the pipe, and ensures its rotation just as if it were a part of the tube in which it is held. A machine of this construction might easily be made self- acting, as in a very useful form manufactured by Messrs. Smith, Beacock, and Tannett. Here the screwed spindle is not em- ployed, but a rack and pinion is substituted for it, and the pinion is slowly raised or depressed by an endless screw and worm wheel set in motion by a hand wheel similar to K. The self-acting portion consists of a small cone pulley, which draws off a motion of rotation from the driving shaft, and the axis of this pulley is fitted with a second endless screw and worm wheel placed just over the hand wheel, and which can be slid into gear so as to produce the self-acting motion. Thus the same slow rotation may be given to the driving pinion on the axis of the hand wheel, by the steam-power, which is other- wise given to it directly by the workman ; the cone pulley of course 252 Elements of MecJidnism. providing for varying amounts of feed according to the require- ments of the work. ART. 203. A Drilling Machine by Mr. Bodmer, of Man- chester, is made self-acting in the following manner : The drill spindle (fig. 265) has a screw-thread traced upon it. A groove is cut longitudinally along the spindle, and a projection upon the interior of the boss of the wheel D fits accurately into the groove. Thus the spindle can traverse through the wheel D, although the spindle and wheel must turn together. FIG. 265. FIG. 266. A nut H, in the form of a pipe, and having a wheel, C, at the bottom of it, receives the spindle. This wheel and pipe are shown separately in sec- tion. If a pinion, A, turning in the di rection shown by the arrow, engage the wheel D, it will screw the spindle rapidly out of the pipe H, and bring it down towards the work. Suppose a second pinion, B, turning in the same direction as A, to act upon C, it will move the nut instead of the screw, and the drill spindle will rise rapidly so long as it is prevented from rotating. (Fig. 266.) Thus far we have provided for bringing the spindle down to its work, and for raising it up again. It remains to apply the principle of aggregate motion, and to cause the drill spindle to become the recipient of these two movements in a nearly equal Drilling Machine. 253 degree, and thereby to ensure the slow descent accompanied by a rapid rotation, which is required in process of drilling. - The result of the combination is shown in fig. 267, where the wheels A and B are moved together : the wheel A tends to depress the spin- dle, the wheel B tends to raise it, and, since A is greater than B, the spindle descends by the difference of these motions, having further the motion of rotation given by the wheel A. The motions of A and B are ob- tained from the driving pulleys I, N, and L. I is an idle pulley, N drives A, and L drives B. When the strap is on N the drill descends to the work, when the strap is on L it ascends from the work, and when the strap is partly on N and partly on L the drilling pro- ceeds. The practical objection to this movement is that the rate of feed is invariable so long as the train of wheels remains the same. It may be thought better to control the feed by means of a cone pulley, where the strap can be readily shifted so as to change the advance of the cutter. ART. 204. A Boring Machine would be employed to give an accurate cylindrical form to the interior surface of a steam cylinder. In the annexed example the boring FIG. 268. cutters are attached to a frame which rides upon a massive cast-iron shaft or boring bar, and rotates with it : this frame is fur- ther the recipient of a slow longitudinal movement given by a screw. An annular wheel, A, shaped as in the diagram, rides loose upon the bar, and drives a pinion, P, at the end of the feed- ing screw which advances the cutters, the boring bar being recessed in order to receive the screw. 254 Elements of Mechanism. FIG. 269. It is quite apparent that as long as the rotation of the wheel A is identical with that of the boring bar, the pinion P will not turn at all ; and, further, that a slow motion will be impressed upon P if the rotation of A be made to lag a little behind that of the bar. A spur wheel, B, is keyed to the bar, a small shaft fixed at the side carries the wheels C and D, and thus motion is imparted to A, the driver of the feeding screw. Let the numbers of teeth upon B, C, D be 64, 36, 35, and let the wheel A have 64 teeth, both upon the outside and the inside of its circumference, the pitch of the screw being \ an inch, and the number of teeth upon the pinion being 16. 5 - > . C x A 36 x 64 36 That is, A loses ^ tn f a revolution for every complete rotation of the boring bar. At the same time the pinion P moves through J$ x T | or \ of a revolution, and the cutter advances through \ ^ x \ an inch or through y^th of an inch. ART. 205. This slow rotation of the screw which advances the boring head may be obtained in a more simple manner by a combination which virtually embodies the sun and planet wheels of Watt. Conceive that two wheels, A and B, of 40 and 80 teeth re- spectively, are attached to the bar CAB, which has a centre of motion at C. If the bar be carried round C, and A be made a dead wheel, the effect of depriving A of the rotation due to its connection with the arm will be to cause B to rotate relatively to the arm just as if the axes of both wheels were fixed in space. The movement is shown in the diagram, where A has turned Feed Motion. 255 through half a right angle from its first position relatively to tht arm, while the arm itself has been carried through a right angle. The student will distinguish between the absolute and relative rotations of B ; the absolute amount of the rotation of B is one right angle and a half. This also appears from the formula, viz. e = ~^. Substituting the values #/=o, e = , we have- But (n a} represents the number of rotations of the wheel B relatively to the arm while the latter is making (a) revolutions, and the analysis therefore shows that the angular velocity of B relatively to the arm is half that of the arm itself, and also that both rotations take place in the same direction. Further, it must be noted that the position of C makes no difference in the result, which will be the same if the point C be somewhere between A and B. 1 In the application of this movement to the boring machine, the centre of motion is between the axes of the wheels, in the line marked C in the diagram, and the numerical value of e is less than ^, probably about ^. The wheel B is placed upon the axis of the screw which advances the boring cutters, the rotating arm being now a part of the solid end of the boring bar ; the wheel A rides upon a separate stud, and is attached to a bar AD of some convenient 2 5 6 Elements of Median ism. length which passes through and rests upon a fork in an indepen- dent upright support placed at some little distance from the machine. As the wheel A is carried round the axis of the boring bar this rod slides a little to and fro in the fork, and controls the wheel A so as to render it impossible for it to rotate, or, in other words, to make it a dead wheel. The wheel B will now turn slowly under the action of A so far as its position relatively to the boring bar is concerned, and upon our supposition, the screw will advance the boring cutters by a space equal to its pitch in five complete revolutions. This would give a feed dependent upon the pitch of the screw, which could of course be varied at once by changing the wheels A and B. r. 206. Sir J. Whitworth's Friction Drilling Machine is an application of the principle of aggregate motion. AD is the drill spindle, which is driven in the usual manner by the bevel wheel B. E and F are two worm wheels embracing the screwed portion of the spindle upon opposite sides. They are of peculiar construction, being hollowed out so as to fit against the small screwed spindle, and they work with a V-threaded screw upon AD. If E and F be prevented from turning, they will form a nut through which the spindle will screw itself rapidly. If E and F be allowed to turn quite freely, the drill spindle will set them in motion, and the nut will be virtually eliminated. The drill spindle may then be regarded as the recipient of two equal and opposite motions : it is depressed by screwing through the nut, it is elevated by the turning of the wheels. If the rotation of the wheels be in any degree checked by the application of friction, the equality is destroyed, and the drill spindle descends to a corresponding extent. FIG. 271. Watt's Indicator. 257 FIG. 272. A friction brake, regulated by a screw, restrains the motion of E and F, and gives a perfect command over the working of the machine. When B is at rest the worm wheels act upon the screwed part of the spindle just as a pinion does upon a rack, and the drill can be rapidly brought down to the work. This method of converting a screw and worm wheel into a rack and pinion is quite worthy of attentive consideration : it is em- ployed in the well-known lathes by the same firm. ART. 207. Waffs Indicator is an instrument used to ascer- tain the actual horse-power of a working steam-engine. The principle upon which it is constructed is the following : A pencil oscillates through the space of a few inches in a horizontal line, with a velocity which always bears a fixed ratio to that of the piston, whereby its motion is an exact counterpart upon a very reduced scale of the actual motion of the piston in the steam cylinder ; and at the same time it is the subject of a second movement in a vertical line, which is caused by the pressure of the steam or uncondensed vapour in the cylinder, and occurs whenever the pressure of the steam or va- pour upon one and the same side of the piston of the engine be- comes greater or less than that of the atmosphere. Under the influence of these independent motions the aggre- gate path of the pencil will be a curve which is capable of inter- pretation, and which affords a wonderful insight into actions which are taking place in the interior of the cylinder. S 2 5 8 Elements of Mechanism. An excellent early form of the apparatus is known as McNaught's indicator, and consists of a small cylinder, A, fitted with a steam-tight piston, B. The piston rod, BD, is attached to a spiral steel spring, which is capable of extension and compression within definite limits, and is enclosed in the upper part of a tube which carries the cylinder A. The pencil is attached to a point in the rod BD, and traces the indicator diagram upon a piece of paper wrapped round a second cylinder by the side of the first. The cylinder, A, is freely open to the atmosphere at the top, and a stopcock admits the steam from below when required. The indicator is usually fixed upon the cover at one end of the steam cylinder of the engine. When the stopcock is opened and the lower side of B is in free communication with the interior of the cylinder, the pressure of the steam will be usually greater or less than that of the atmosphere : if it be greater, B will rise against the pressure of the spring, and if it be less, the pressure of the atmosphere upon the upper surface of B will overcome the resistance of the spring and cause the pencil to descend. FIG. 273. At the same time, the cylinder which carries the paper is made to turn with a motion derived at once from that of the piston in the engine, but much less in degree, and thus a curve is traced out somewhat of the character represented above. The Indicator. 259 Here PQ is the atmospheric line, and is the path of the pencil when the pressure of the steam is equal to that of the atmosphere, or when the spiral spring is neither extended nor compressed. As the steam enters the cylinder, the piston may be supposed to be descending, and the pencil to be describing the upper por- tion of the curve : when the piston returns, the pencil moves to the left through DEA, and thus the diagram is traced out. We may examine this matter with more particularity as follows : the steam is admitted when the piston reaches the top of its stroke, and the pencil rises with a rapid motion from A to B ; the full pressure of the steam is then maintained while the pencil, recording a portion of the travel of the piston, moves from B to C ; at C the steam is cut off, and the pencil falls gradually as the steam expands with a diminishing pressure ; at D the steam pours into the con- denser, and the fall becomes sudden ; from E to A the cylinder is in full communication with the condenser, and the pencil describes a line somewhat inclined to the line PQ, the position and form of which depend upon the perfection of the vacuum in the condenser. The strength of the spiral spring being ascertained, the curve tells us exactly the number of pounds by which the pressure of the steam urges the piston onward during every inch of its path in one direction, and the amount of resistance which the uncon- densed vapour or gases existing in the condenser oppose to its passage in the other direction. The area of the curve, therefore, affords an estimate of the work done in the engine during one complete stroke, and is a graphic representation of the same. The engineer estimates this area by simple measurement in the most direct manner which occurs to him, and the actual indicated horse-power is obtained by multiplying the work done in one stroke by the number of strokes made in a minute, and then dividing by 33,000, the number of foot-pounds which form the measure of rate of work called a horse-power. ART. 208. The object of the indicator being to ascertain the xact pressure of the steam or vapour in the cylinder at each point of the stroke of the piston, it has been found to be a great advantage to diminish as much as possible the play of the spring >vhich controls the pencil. In this way the vibration and irregu- 260 Elements of Mechanism. larity of motion of the pencil is greatly reduced. But the play given to the spring determines the height of the diagram, and we do not wish to reduce this, but rather the contrary. It is not easy to reconcile these contradictory requirements, but, nevertheless, a form of indicator has been invented by Mr. Richards which solves the difficulty, and has become most deservedly popular. It is an ingenious application of the combination of two bars and a link forming a parallel motion, and will be understood at once from the drawing, which is taken from a small model repre- senting very closely the essential parts of an actual instrument. The parallel motion bars AB and CD carry the pencil, which traces out upon a drum a copy of the vertical movement of the piston E of the indicator, but magnified by reason of the attach- ment of the piston to a point S near the fulcrum of the bar CD. FIG. 274. The principle of the apparatus is precisely the same as that which we have already explained, and the only difference consists in the application of the parallel motion bars to enlarge the diagram. The Indicator. 261 The drum derives its motion from any part of the engine whose movement is coincident with that of the piston, and the spiral spring can be changed so as to suit different engines. The connecting link is not set perpendicularly to the bars AB, CD, but makes an angle with them as shown, an artifice which causes the pencil to describe a line free from any sensible curvature. The parallel motion is set out in a separate diagram, in order that it may be thoroughly understood. FIG. 275. The link RS is parallel to BD when the motion begins, and it remains parallel throughout, for R and P are both constrained to describe vertical straight lines. Hence we have the pantograph in a disguised form. Also, travel of P : travel of R = CD : CS. In the indicator as constructed the movement of R is mag- nified about four times. It should be understood that the frame carrying the motion bars is attached to a collar which can be rotated on the cylinder, whereby the pencil is readily brought up to the paper or removed from it. ART. 209. There is a curious movement derived from the employment of a dead wheel in a train which has been applied by Mr. Goodall in a machine used for grinding glass into powder by the action of a pestle and mortar. The pestle is made to sweep round in a series of nearly circular curves contracting to nothing, and then expanding again so as to command the whole surface of the mortar. We shall show that the contrivance is merely a solution of the problem of obtaining an expanding and contracting crank. Let CQE be a crank whose centre of motion is D; conceive 262 Elements of MecJianism. that D is a dead wheel on the same axis, and that A is a larger wheel riding upon one end C of the crank arm. FIG. 276. Suppose, further, that a piece Q, capable of sliding along CE, is attached by a link SQ to a point S in the circle A, which is not its centre ; and, finally, that a pencil at P is connected with Q by a link PRQ constrained always to pass through a fixed point R. As the circle A and the crank CQ travel together round the dead wheel D, it has been proved in Art. 195 that the wheel A will turn relatively to the arm just as it would do in an ordinary train with fixed axes. Hence the point S travels slowly round in the dotted circle, thereby causing the point Q to move to and fro along CQ. It may be arranged that Q shall start from D, and it will travil along CE through a space equal to twice CS. When Q is at D, the point P is motionless, whereas, while Q moves further from D, and continually sweeps round, by virtue of its being a point in a revolving crank, it is evident that P will trace out an expanding spiral, which will return again to nothing when Q is pulled back to D by the action of the wheel A. It now only remains for us to consider what would be the actual construction of the apparatus. The drawing is taken from a model, and not from the machine itself. The crank CDE is a bar whose parallel arms are connected by a vertical piece, and which carries the wheel C upon one arm and the sliding piece KQ upon the other arm. This crank is driven by the spur wheels L and M connected with the handle H; it Expanding Crank. 263 therefore rotates round the axis of the dead wheel D, and carries C and KQ upon opposite sides of the vertical axis through D. The link SQ connects the wheel C with the piece KQ, and this latter piece is again connected by QR with the pestle, it being provided that QR shall pass through a guide at some fixed point about half-way between Q and the mortar. The pestle is swung from a ball and socket joint at some convenient height above P. The rotation of the crank round the dead wheel causes C to turn slowly upon its own axis, the point S therefore travels slowly round C, hence the end Q of the connecting rod QRP is some- times at a distance from D, and at other times is exactly over it, and during the whole time Q is a part of the crank DQ, and sweeps rougd with the arm. Thus the required motion is provided for. ./ >^ART. 210. The oval chuck affords an instance of aggregate Y/ motion. It is based upon the following property of an ellipse, / which is taken advantage of in constructing elliptic compasses for drawing the curve. Let ACA', BCB, repre- sent two grooves at right angles to each other, and traced upon a plane surface ; PDE, a rod furnished with pins at D and E. If this rod be moved into every possible position which it can assume while the pins remain in the grooves, the point P will de- scribe an ellipse. 264 Elements of Mechanism. Draw PN perpendicular to AC, and PM perpendicular to CB'. Let CN = * PE=*, PM , y PN EM . * 2 , y^PM 2 + EM 2 = ' * a* 2 ~~ PE 2 which is the equation to an ellipse. In drawing an ellipse we should fix the paper and move the rod over it, but in turning an ellipse in a lathe we should fix the describing tool and move the piece of wood or metal underneath it ; thus the conditions of the problem become changed, and the construction is modified accordingly. An equivalent for the grooves ACA', BCB' may be arrived at as follows : Describe a circle about E of radius larger, than ED, and let two parallel bars, QR, ST, be connected by a perpendicular link HK, equal in length to the diameter of the circle, and thus form a rigid frame embracing the circle, and capable of moving round it. FIG. 279. FIG. 280. B As the frame moves round the circle we must provide that HK shall pass through D in every position as represented in the diagram. If we draw DC parallel to QR, and EC parallel to HK, it is easy to understand that the imaginary triangle DCE in fig. 280 is exactly the same as the triangle DCE in fig. 278 and exists The Oval Chuck, 265 throughout the motion ; and that whereas we formerly moved the bar EP over a fixed plane and described an ellipse, so now we have arranged to obtain the same motion with a fixed bar and a movable plane, and shall trace out precisely the same curve. This is a very good example of aggregate motion. The plane upon which the ellipse is traced is the subject of two simultaneous movements : by one of them a line, HK, in the plane is made to revolve round D as a centre, and by the other the same line re- ceives a sliding motion in alternate directions through D. Thus an oval, or more properly an ellipse, may be turned in the lathe. 266 Elements of Mechanism. CHAPTER VIII. ON TRUTH OF SURFACE AND THE POWER OF MEASUREMENT. ART. 211. The subject matter comprised under the title of this chapter is so large that it cannot be discussed fully, and all that can be done is to present a brief sketch of some important facts connected with it. We have to describe a method of mechanical measurement, founded upon truth of surface, which is probably the reverse of that which most persons would form for themselves. The idea of measurement is commonly associated with optical con- trivances, whereby, if we desired to measure some minute interval of length, say in ten-thousandths of an inch, we should naturally proceed to the task armed with powerful lenses or microscopes, and relying mainly on the sense of sight. It would be something quite novel and unexpected to discover that the sense of touch would do more for us than the eye, and that, in mechanical measurement, it is more easy to feel minute differences of size by the aid of surfaces properly prepared and adjusted, than it is to recognise and compare such differences in the field of view of a microscope. We shall presently refer to a measuring machine, used in the construction of difference gauges, wherewith a workman can readily test gradations of size differing by t uuooth of an inch, and may in special cases carry on the operation as far as 4uth f an mcn > a graduation in the rim of the large wheel advances by ^^-^ths of an inch, which is rather more than ^th of an inch, an interval that can be readily observed without assisting the eye by lenses. In this machine there is space for measuring circular gauges up to 6 inches in diameter, and bars up to 12 inches in length. But for ordinary purposes, and for the construction of small gauges, a much smaller instrument will suffice, the working parts, however, being the same. ART. 218. We have now to speak of the construction of case-hardened cylindrical gauges, or standards, for comparison in the workshop, and the drawing shows the two principal forms. 1. There is an external gauge A, cylindrical in form, with a handle, and of varying diameters from T \j- inch up to 2 inches. 2. There is an internal gauge B, also cylindrical, and exactly fitting upon A. These gauges are made in pairs, and serve to test the diameter 280 Elements of Mechanism. of any solid cylinder, or of any cylindrical opening having the same diameter. In the collection of the School of Mines are two of these ex- ternal or plug gauges, and one internal or collar gauge, whereof the respective diameters are one inch, one inch less T ^o^o tn mcn > and one inch. Calling them in the order as above stated by the letters A, A', and B, we find these results : 1. Wipe A and B, which are exact i-inch gauges, with a clean dry cloth, and endeavour to pass one over the other. We cannot do it, for the surfaces at once bite together, and we should say that the plug was too large for the collar. Now rub a very little of the finest oil upon the surfaces of A and B, and it will be found quite easy to pass B upon A. If A be held in a vertical position, B will slowly sink from the top to the bottom of the cylinder. The smoothness and yet tightness of the fit is most remarkable, the oil preventing the adhesion and jamming together of the metallic surfaces. 2. We next proceed to test A' and B, remembering that the diameter of A' is y^g-uth of an inch less than the internal dia- meter of B, and we begin by wiping the surfaces so that all oil is removed, and they are perfectly dry and clean. It will now be found that B fits quite loosely on A'. If the gauge A' be held in a vertical position, B will fall freely from the top to the bottom. Again rub some oil upon the surfaces, which will fill up the vacant space, and A' will pass through B very smoothly, but some slight resistance will be felt. By handling these gauges the difference of fit due to a difference of TTf^rth of an inch becomes very apparent. The practical value of difference gauges is well understood. For instance, the cylinder of the moving headstock of a lathe requires as good a fit as possible, but that means that a true and proper allowance should be made in the size of the parts working together, and Sir J. Whitworth states that in the case of an ordinary lathe the hole in the headstock should be ^oW 11 of an mch larger than the cylinder. In like manner gauges would prove to be of great service in Millionth Measuring Mac/tine. 281 carrying out the manufacture of an axle, the journal being made to a standard gauge, and the bearing being bored out so as to fit a difference gauge somewhat larger in size, and of the precise difference in diameter, which experience has shown to be neces- sary, regard being had to the conditions under which the axle is to work. It may therefore be conceded that every manufacturer should be in a condition to produce difference gauges for use in his workshop. ART. 219. Sir J. Whitworth has constructed a measuring ma- chine with rectangular plane bars in place of the sliding cylinders, and with a higher mechanical multiplier, whereby he has measured U P tOTTToUoo of an inch. We have not space to describe the apparatus fully, but may state that the sliding bars are rectangular, with faces made truly plane, and that they move in right-angled V-g rooves > whose sur- faces are also true planes. The ends of the measuring bars are circular planes, each about njths of an inch in diameter, and the utmost care is taken to ensure that these plane ends shall be truly perpendicular to the respective axes of the bars. We have described the method of securing this result. Of the two measuring bars one is advanced by a screw having twenty threads to the inch, and terminating in a graduated hand- wheel with 250 divisions on its rim. This gives a quick motion of yffVfith of an inch for each graduation. The other measuring bar is actuated by a combination of screw-gearing. There is first a screw with twenty threads to the inch, which terminates in a worm-wheel having 200 teeth upon its rim. Then the worm wheel is itself driven by a tangent screw carrying a hand-wheel with a micrometer graduation of 250 divisions upon its circumference. It follows that the rotation of the tangent screw through one division will advance the bed screw of twenty threads by a space equal to _ x -- x of an inch, 250 200 20 or TOOT, oTToth of an inch. 282 Elements of Mechanism. Results of this character could be extended as far as we pleased in theory, but not in practice. The accuracy and truth of the pieces upon which we rely are so severely tested that the power of human execution soon fails, and hence we can appreciate the interest which this apparatus has awakened. From the dimensions of the wheels in this machine it has been found that a motion of -oooooi inch in the measuring plane is equivalent to a motion of '04 inch on the rim of the graduated hand-wheel, whence it follows that the machine magnifies the motion 40,000 times, or that the eye observes a graduation to traverse over a space 40,000 times as great as that which is being measured. In order to estimate the degree of tightness between the plane face of a measuring bar and the corresponding plane surface of an object under measurement, a so-called feeling piece or gravity piece has been employed. The gravity piece consists of a small plate of steel with parallel plane sides, and having slender arms, one for its partial support, and the other for resting on the finger of the observer. One arm of the piece rests on a part of the bed of the machine and the other arm is tilted up by the fore-finger of the operator. The plane surfaces are then brought together, one on each side of the feeling piece, until the pressure of contact is sufficient to hold it supported just as it remained when one end rested on the finger. This degree of tightness is perfectly definite, and depends on the weight of the gravity piece but not on the estimation of the observer. In this way the expansion due to heat when a 36-inch bar has been touched for an instant with the finger-nail may be de- tected. Also the movement of 'ooooor inch has been indicated by the gravity piece becoming suspended instead of falling, and the piece has fallen again on reversing the tangent screw through two graduations, representing -000002 ; showing the almost infinite- simal amount of play in the bearings of the screws. For the coarser measuring machine the workman relies upon the sense of touch for feeling the size of the body which is being passed between the measuring planes. Standards of Length. 283 ART. 220. The national standard of length is a rectangular bronze bar, 38 inches long, and i square inch in transverse section. A cylindrical hole, f inch in diameter, is sunk near each end to the depth of \ an inch ; a second small hole is then bored at the bottom of the larger one for the reception of a gold plug, forming a table T V inch in diameter, on which three fine lines are engraved at intervals of r ^th of an inch in directions transverse to the length of the bar. The distance between the two middle lines is the length of the standard, and the object of excavating the holes is to obtain a measurement along the axial line of the bar. This is an example of what is termed line measure, and line measure bars are compared by the aid of fixed microscopes, with optical contrivances for reading to s^^th f an mcn In the year 1834 Sir J. Whitworth obtained two standard yards in the form of line measure bars, and by the aid of microscopes transferred the mean distance between each pair of engraved lines to a rectangular end measure bar, as nearly as he could accomplish the task. At the same time he constructed a millionth measuring machine for the reception of the bar. It now became comparatively easy to subdivide the yard into feet, and for this purpose three bars were prepared, each a little longer than one foot. A temporary abutment was then raised in the bed of the measuring machine, and the bars were reduced and tested until (i) they became respectively of the same length, and (2) they filled up the length of the standard yard when placed end to end. Further subdivisions of the foot were made, and finally a standard inch was arrived at Again, standard end measure bars gave birth to standard cylin- drical gauges, and thus the mechanical measures adopted in the workshop have been originated and have been reproduced by the employment of end measure bars and a good measuring machine. 284 Elements of Mechanism. CHAPTER IX. MISCELLANEOUS CONTRIVANCES. WE propose to examine in our concluding chapter various miscel- laneous pieces of mechanism, and certain special contrivances which are of frequent occurrence in machinery, and with which a student of applied mechanics ought to render himself familiar. ART. 221. The invention of counting wheels is due to the celebrated Cavendish, who constructed a piece of apparatus for registering the number of revolutions of his carriage wheel. . This apparatus is deposited in the Museum of George III. at King's College. There is but one guiding principle in this branch of mechanism, however varied may be the details of the separate parts. Each wheel of a series, A, B, C, &c., possesses ten pins or teeth, and it is contrived that one tooth only of C shall be suffi- ciently long to reach those of B ; similarly B is provided witli one long tooth which is capable of driving A. Counting Wheels. 285 Thus C goes round ten times while B makes one revolution, and so on for the other wheels ; in this way the series is adapted for counting units, tens, hundreds, thousands, &c. In fig. 289 the arm EF imparts rotation to the first, or ratchet wheel, by means of the paul HD ; the number now registered is 988 ; after two vibrations of the arm the zero of C will reach the highest point, the tooth P will drive B through the space of one tooth ; and the number registered will be 990 ; after ten more vibrations of the arm, P will again advance B, and at the same instant Q will move A, and will bring its zero up to the highest point : the three wheels will now register ooo, having passed the number 999, which is the last they can give us. The wheels are retained in position by the rollers R, R, R, mounted upon springs. As each roller is forced in between two pins, it not only acts as a detent, but also adjusts the wheel in its right position. Mr. Babbage employed this contrivance in his calculating machine. / ART. 222. A small counting apparatus is attached to every gas-meter used in houses, and registers the number of cubic feet of gas consumed ; here, however, the step by step motion is not employed, the dial plates are fixed, and a separate pointer travels round each dial respectively. The pointers are placed upon the successive axes of a train of wheels, composed of a pinion and wheel upon each axis, the number of teeth on any wheel being ten times that upon the pinion which drives it. Suppose, for example, that the pointer on the plate registering thousands completes a revolution and adds ten thousand to the score, its neighbour on the left will have moved over one division on the dial registering tens of thousands, and thus an inspection of the pointers throughout the series will at once indicate the consumption of the gas. These index-fingers move alternately in opposite directions, being attached to the successive axes of a train of wheels ; the figures upon the counting wheels are also placed in the reverse order on every alternate wheel. As we are only concerned with the counting apparatus, it is not necessary to explain the manner in which the flow of gas through the meter sets the train of wheels in motion, but we may 286 " Elements of Mechanism. point out that there is no ratchet wheel employed, and that the flow of gas keeps up a constant rotation in an endless screw, which starts the train and maintains it in action. A reliable counting apparatus which will record the exact number of impressions made by a printing machine is indispensable in some public departments, and it is found that the best result is arrived at by combining a ratchet wheel having a few deep well- cut teeth with the train of wheels used in the gas-meter. The practice is to place upon the axis of the first or ratchet wheel carrying the units a pinion of 10 teeth gearing with a wheel of 100 teeth, then another pinion of 10 drives a wheel of 100 teeth, and so on, as far as we please. The train of wheels cannot fail to record the hundreds, thousands, &c., accurately ; the only possi- bility of a mistake occurs with the units, but if the paul be car- ried well over the teeth of the ratchet, and if the wheel itself be driven at each advance a little beyond the point necessary to give another unit, if, in other words, the movement should be a little over-pronounced, the register will be perfectly exact. In order to avoid the objection that the successive wheels turn in opposite directions, an idle wheel is interposed between each alternate pinion of 10 and its wheel of 100. All the pointers then Devolve in the same direction as the hands of a clock. ART. 223. Where it is intended to print the figures regis- (/tered, as in the numbering of bank-notes, the step by step motion is essential, and, further, each wheel must carry the letters upon its edge, and not upon the face ; the apparatus employed is the same in principle as that of Cavendish, but the construction differs, the wheels being placed side by side and close together. In order to present a fair idea of the construction of a num- bering machine, that is of a machine designed for printing con- secutive numbers, we refer in the first instance to a rough model belonging to the School of Mines, and shall afterwards give some description of a complete apparatus. In the model, the wheels are flat cylindrical discs, having the numerals, i, 2, 3, 4, 5, 6, 7, 8, 9, o, painted upon their edges. On one side of each disc a ratchet wheel with 10 teeth is carved out, while on the opposite face of the disc only one nick or cut rn is formed. The cut rn is adjacent to the numeral o, and is in- Numbering Machine. 28; tended to serve the same purpose as the projecting tooth in the previous arrangement. The drawing shows the complete ratchet in side elevation, and the position of the teeth with reference to the numbers on the disc. Also it will be noticed that the first driver DH is a slender bar which encounters the teeth of the ratchet on which it works, whereas the second driver NL is twice as broad as DH, and en- counters both the ratchet on the second numbering wheel, and also that slice of the rim of the first disc on which the nick rn is situated. FIG. 290. It is apparent that so long as NL is resting on the rim of the first disc at any part except where the cut rn is formed, it will be held above the teeth of the second ratchet, and will be inoperative, but that \vhen NL falls into rn it can drive the second wheel. 288 Elements of Mechanism. Thus let the unit wheel mark 9, and the second wheel mark o, the number read on the first two wheels being 09, meaning 9. At the next stroke NL falls both into rn opposite the numeral o, and into the second ratchet at the part opposite the numeral i, whereby NL advances both wheels by the space of one tooth, and the number 10 takes the place of 09. In like manner there is a second cut or nick pm adjacent to the numeral o on the second wheel, and on the opposite side to the complete ratchet. This determines the period when the next driving paul MT shall advance the third numbering wheel, and thus the series is continued. It follows that the consecutive advance of the respective wheels may be provided for by the employment of ratchets, having alternately ten teeth and one tooth respectively, and placed in regular order upon the numbering wheels, as shown in the diagram. /V^RT. 224. We can now explain with more particularity the //construction of a numbering machine, a considerable portion of which is set out in the annexed diagrams. The mechanism is automatic or self-acting, the operator grasps the handle H, and moves it to and fro as far as it will go ; each time that he does so the type is inked, the numbering wheels are adjusted, and an impression is taken. In order to comprehend the operation we may point out that the principal working parts are the following : 1. The handle H, centred at C, and provided with an arm carrying the printing wheels. The central axis of these wheels is at P, and since P is rigidly attached to the handle and C is also a fixed point, it follows that CP is of constant length. 2. The two 'cranks, CP, BQ, with the connecting rod PQ, whereof CP is an imaginary line, but BQ and PQ control the inking apparatus. It has already been stated that advantage may be taken of the different; positions of PQ in the above combi- nation, and an example is here afforded, as will be seen imme- diately. 3. The numbering wheels lie side by side, and the projecting portions, marked^,/, are the successive numerals, o, i, 2 . . . 9. 4. The inking rollers, marked I, I, work in slots in the arm Numbering Machine. 289 PM, and arc pulled towards P by' elastic bands, not shown in the drawing. Connected with the inking rollers is the circular table FIG. 291. RR, upon which a supply of printing ink is spread out. In the working of the machine, the rollers, I, I, run upon the table, RR, and receive from it a supply of ink ; they then pass on to the u 290 Elements of Mechanism. faces of the printing type, and supply them with sufficient ink for an impression. 5. There is an impression table, EE, on which the operation of printing is performed. 6. There are two ratchet wheels in combination, being an ordinary and masking ratchet wheel respectively, which advance the numerals according to some required rate of progression. For a description of such wheels we refer the student to Art. 138 ; they are not inserted in the present diagrams. FIG. 292. By comparing the annexed sketch with that given previously, the operation of the machine will be clearly understood. When the handle H is depressed to the full extent, the num- bering wheels are brought down to the printing table, EE, and an impression is taken. At the same time the inking rollers run back upon RR, and take up a supply of ink. During the time that the handle is being raised, the ratchet wheels do their work, and advance certain numbers as may be required. The inking rollers, in their turn, run from the table, RR, to the type, and supply the numerals with sufficient ink for the next impression, and thus the process goes on with a degree of ease and certainty which it is one of the triumphs of mechanical art to accomplish. Numbering Machine. 291 ART. 225. As regards the operation of the ratchet wheels, it will be remembered that an ordinary masked ratchet suffices to suspend the operation of advancing the unit wheel until two strokes have been made, and thus it becomes easy to print each number in a series, such as 101, 102, 103, &c., twice over. Also after the unit ratchet has done its work, some method embodying the principle set forth in previous articles will be employed for carrying on the motion to the wheels printing tens, hundreds, and so on. But there is yet something more to be provided for, inasmuch as it is often an advantage to print the odd numbers, as 101, 103, 105, &c., in one column, and the even numbers, as 102, 104, 106, &c., in another column. Or the same machine may be required to print consecutive numbers. The arrangement for effecting this double purpose will be understood by referring back to Art. 138, and it will be there seen that the numbering wheels are carried by an arm in a circular sweep from the inking apparatus to the printing table ; in tra- velling along they encounter the paul, which is fixed to the frame- work, and if the circle should simply graze, as it were, against the paul during its travel, one tooth only would be taken up ; whereas by setting the paul so that it meets the circle at a point nearer its centre, and strikes it more directly, two teeth may be taken up, and thus either one or two units may be advanced at each impression. i //ART. 226. The ninth chapter was devoted to illustrations of the importance of truth of surface, and we may now refer to a lecture diagram of a complete machine, which is an ordinary example of a combination of elementary surfaces, viz., the true plane, the screw, and the cylinder. The machine is in use at Woolwich for turning out rapidly the bosses or naves of wheels, and is also a direct application of the copying principle. The block of wood, intended for the nave, is supported be- tween centres, as in a lathe, and is rotated rapidly by a belt passing over a driving pulley E. The adjacent pulley I rides loose on the shaft, and is an idle pulley. 292 Elements of Mechanism. The cutter c is carried by a slide rest, which resembles the ordi- nary slide rest of a lathe. There is a true plane surface DD, with inclined sides, supporting the saddle/^. The handle H operates a screw which traverses this saddle to and fro along the line parallel to the line of centres of the lathe. The form of the nave is determined by the cam-groove in the plate AB in which a roller r runs as shown in the diagram. A slide carrying the cutter c is attached to r, and it follows that the traversing of the saddle fe along DD will cause the cutter c to advance or recede in exact accordance with the outline of the cam-groove. FIG. 293. When the block is first put in the lathe it is rough and un- even, and the cutter c should be advanced slowly and cautiously, in order that it may commence by paring off the principal in- equalities. The cutter is therefore brought in or out by a screw actuated by a hand wheel A, and at the same time is traversed longitudinally by the handle H, the definite position of the cutter on the saddle being, of course, quite independent of its motion as due to the cam-groove. In this way the machine does its work rapidly and effectually, the cutter runs to and fro along the outline of the block, and Escapements of Watches. 293 removes the material while copying and preserving the outline of the guiding curve. The amount removed is also under the control of the operator, and is regulated by the hand wheel h. AX'ART. 227. We have reserved for the concluding chapter an account of the principal conditions which obtain in the construc- tion of the escapements of watches, and have to show that the principle of the ^ dead beat' is recognised in the three forms which are in common use. And here we may remark that the pendulum of a clock appears as the balance wheel in a watch. A wheel, pivoted on very small steel pivots, and working in jewelled supports, is attached to a flat spiral steel spring in pocket watches, or to a more powerful helical steel spring in marine chronometers. This wheel vibrates under the action of the pull of the spring just as a pendulum would do under the pull of the earth, but under better conditions theoretically, for the force of the spring increases with the angle through which the balance wheel swings, and in direct proportion to that angle ; the result, there- fore, is that the swing of the watch pendulum is always performed in very nearly equal times whether the arc of swing be increased or diminished. We have what is technically called an isochronous pendulum in the balance wheel, and this is important, because the time is not affected by small changes in the arc of swing. Further, it should be noted that the balance wheel swings through an angle which is enormous as contrasted with the swing of the pendulum, being more than a whole revolution in the case of the chronometer or lever watch. Consider now the construction of the chronometer escapement, which fulfils our conditions with an exactness that may well sur- prise us, and which exhibits in its arrangement a marvellous amount of mechanical skill and forethought. The detent, which corresponds to the anchor pallets, consists of four principal parts : 1. The locking-stone, D, a piece of ruby, upon which the tooth of the escape wheel rests. 2. The discharging spring, Ar, which is a very fine strip of hammered gold. 294 Elements of Mechanism. 3. A screw at A to fix Ar to the stem of the piece SD. 4. The shank of the detent, consisting of a projecting arm Dy, the part DA, and the portion at S, which is cut away to form a spring which may bend and act as a pivot on which the whole detent can be moved a little. The small circle on the left hand has a projecting piece which keeps the escapement in action, and it is a part of the stem of the balance wheel. FIG. 294. As the balance wheel swings to and fro, this roller also vibrates, and when passing downwards it encounters the spring Ar, and pushes it aside without any perceptible effort, because the spring bends from the distant point A. On its return the projection finds the spring to be capable of bending, not from the distant point A, but only from the point g against which it rests. The roller therefore takes the spring and the whole detent with it and raises the locking stone D from the point of the escape wheel, the escape wheel at once flies round, and before it can be caught upon the next tooth by the return of the detent to its normal position, is enabled to give an impulse to the balance wheel by striking against the point d in the manner shown in fig. 295. The whole arrangement can now be studied from the drawings, and is complete with the exception of a banking screw which supports the detent when coming back to the position of rest. It will be seen that the large circle F is fixed to the smaller one, and that the projection marked d is quite clear of the escape wheel while a tooth is resting against the detent. The advance of the escape wheel is so instantaneous that it is not seen to move : it appears to tremble a very little, but it comes to The Chronometer Escapement. 295 rest again so quickly that the eye cannot follow and can scarcely detect the motion. It is, of course, made evident by watching the spokes of the wheel. FIG. 295- What, then, has been the action ? In the first swing of the balance the only obstacle has been the bending aside of the spring Ar, which is no more than bending a light feather. In the second swing the pendulum or balance wheel has had to lift the detent : this is a momentary and very small action against it, but as quick as thought the action is compensated and the balance receives its impulse through equal distances on each side of the middle of its swing, according to the principle of the dead-beat escapement. Here, then, theory and practice are in exact accord. It should be noted that the impulse is given at every alternate swing of the balance, and not with every swing as in the case of the clock pendulum. / ART. 228. The Lever Escapement comes next in order, and here we return to the anchor pallets. The escape wheel is locked 2c)6 Elements of Mediants in, by these pallets, and gives its impulse upon their oblique edges in the manner described in Art. 55. The balance wheel is free during the greater part of the swing (hence the name of Detached Lever), and oscillates through a considerable angle. The unlocking occupies an angle of about 3, and the impulse is given through about 9. These are just the conditions which prepare us for the principle of the dead beat. FIG. 296. The pallets, mp, qn, are jewels inlaid into the arms, the light steel bar DH is the lever movable about C as a centre. An open jaw at one end is capable of receiving a ruby pin, P, attached to the roller, which is on the axis of the balance wheel and moves with it. There are also banking pins, and a small guard pin to prevent the lever from falling out of position. As the balance vibrates the pin P swings to and fro with it : in doing so it enters the open cut at the end of the lever, and removes the locking portion of the pallet from the point of the escape wheel. Instantly the escape wheel flies forward, and by pressing against one oblique edge suddenly pushes on P, and the lever is no longer moved by the balance wheel but imparts an impulse to it. Very soon, that is after the nine degrees of the swing are consumed, the ruby leaves the lever behind, and the wheel goes on detached and unchecked in its swing. On its return the pin finds the lever where it had left it, carries Escapements of WatcJies. 297 it forward, unlocks the escape wheel, receives its impulse, leaves the lever behind, and the balance is free for the rest of the swing. The only action against the balance is that of unlocking the escape wheel so as to enable it to give the impulse. This is very brief in duration as compared with the whole swing, and the watchmaker takes care that it shall be as little as possible. The impulse is given just at the middle of the vibration, and the con- struction follows out the theory very closely. ART. 229. Lastly, we may refer to the escapement of the -'so-called Geneva Watches, which is Graham's cylinder movement. Here the balance is attached to a very thin cylinder centred at o, and the point of a tooth rests upon either the outside or the inside of this cylinder during a part of the swing. In this respect the action corresponds exactly to the friction of the escape tooth against the circular part of the pallets in the dead beat. As the cylinder vibrates round its centre o, the tooth pn comes under the edge at r, and pushes the cylinder onward : this gives an impulse. The tooth soon passes r, flies into the cylinder, and is stopped by the concave surface near s ; the cylinder now vibrates in the opposite direction, pn escapes, and in doing so gives another impulse at s to the cylinder in the opposite direction, and thus the action goes on. The impulse would not be given in the middle of the swing, but through small arcs equally distant from the middle point, and equal in length to each other. Hence this combination is nearly identical with the dead-beat es- capement, although inferior to it in this latter particular. The manner in which the effects of the expansion and con- traction of the material of the pendulum rod and the balance wheel, due to changes of temperature, are rendered innocuous, forms a separate branch of the subject. FIG. 297. 298 Elements of Mechanism. ART. 230. The fusee is adopted in chronometers, and in most English watches, in order to maintain a uniform force upon the train of wheels, and to compensate for the decreasing power of the spring. The spring is enclosed in a cylindrical barrel, and sets the wheels in motion by the aid of a cord or chain wound partly upon the barrel and partly upon a sort of tapering drum called a fusee. As the spring uncoils in the barrel, the pull of the cord de- creases in intensity ; at the same time, however, the cord unwinds FIG. 298. itself from the fusee, and continually exerts its strain at a greater distance from the axis, that is, with a greater leverage, and with more effect. The theoretical form of the fusee is a hyperbola, being the section of a right cone made by a plane parallel to the axis of the cone. To prove this statement we must first recognise the law ac- cording to which an elastic body under extension or compres- sion exerts a force of restitution whereby it tends to recover its original form. This law was stated by Dr. Hooke as being contained in the maxim ut tensio sic w's, by which it is intended to convey that when a body is extended or compressed in a degree less than that which produces a permanent derangement of form, the force necessary to keep it extended or compressed is proportional to such extension or compression. Take a spiral steel spring balance, for example ; hang upon it successive weights of i, (2), (2 + 1), (3 + 1) Ibs., the index point The Fusee. 299 will descend through equal spaces for each additional pound weight, and will rise by equal spaces as each pound is successively removed. Assuming the law to hold exactly when the spiral spring is subject to a force of torsion instead of one of direct extension, we shall have the pull of the spring proportional to the angle through which the barrel has been made to turn. Let DPBA represent one-half of the sec- tion of a fusee, DPB being the curve whose equation is to be found. Draw DE, PN, BA perpendiculars on EA ; take ER, QN, SA to represent the pull of the spring when the chain is at the points D, P, and B, respectively. According to Hooke's law, the force of the spring will decrease uniformly as the chain passes from D to B, therefore RQS must be a straight line inclined to EA. Produce RQS to meet EA in C. Then ^rr = ;pr.> which is a constant ratio, by reason of the V^.N VX.A. law of elasticity. Assume that this ratio is represented by m, .-. QN = m . CN. In order that the fusee may accomplish its object, the product of the pull of the spring into the arm NP must remain constant for every position of P. Hence, calling CN = x, NP = y, we have (pull of spring) x NP = ;// . CN x NP = mxy. But this product is not to vary, /. mxy = a constant quantity, or xy = a constant, which is the equation to a hyperbola. In practice, where great accuracy is required, the strength of the spring is tested by fixing a light lever to the winding square of the fusee, and observing whether the pull of the spring is 300 Elements of MecJianism. balanced in every position by the same weight hung at the end of the lever. The fusee would be cut away a little where it was _4^eSsary to do so. jr ART. 231. In mechanism the fusee is frequently employed to transmit motion instead of to equalise force, and enables us to derive a continually increasing or decreasing circular motion from the uniform rotation of a driving shaft. The groove of the fusee may be traced upon a cone or other tapering surface, or it may be compressed into a flat spiral curve : in all cases the effect produced will be that due to a succession of arms which radiate in perpendicular directions from a fixed axis, and continually increase or decrease in length. The fusee can of course only make a limited number of turns in one direction. A flat spiral fusee occurs in spinning machinery, and serves to regulate the velocity of the spindles, and to ensure the due winding of the thread in a succession of conical layers upon a bobbin or cop. The formation of the cop is a problem upon which a vast amount of mechanical ingenuity has been expended ; and without FIG. 300 FIG. 301. entering too much into details, we may ob- serve that there are two distinct stages in the process of winding the yarn upon a spindle so as to produce a finished cop. The copbottom (fig. 300) is first formed upon a bare spindle by superposing a series of conical layers with a continually increasing vertical angle. The body of the cop is then built up by winding the yarn in a series of equal conical layers. (Fig. 301.) The winding-on of the yarn begins at the base of the cone and proceeds upwards to the vertex ; the spindles are driven by a drum which rotates under the pull of a chain, and they may be made to revolve with in- creasing rapidity by placing a fusee upon the driving shaft and causing the chain to coil upon it. The Fusee. 301 FIG. 302. Such an arrangement as shown in fig. 302 will be adapted to the winding of a uniform supply of thread upon a conical surface ; and we can easily compre- hend that a fusee of fixed dimensions will do veiy well for building up the body of the cop after the foundation is made. The main difficulty occurs in producing the copbot- tom, where the series of conical layers of con- tinually increasing ver- tical angle demands a fusee whose dimensions shall gradually contract towards the centre. The method of contracting the form explained as follows : Fig. 303 represents portions of two flat discs having axes at A and B, and upon which are cut radial and curved grooves in the FIG. 303. of the fusee may be manner indicated ; it being arranged that when one plate is placed upon the other, the pins P and Q shall travel in both sets of grooves at the same time. We can easily see that the blocks which carry the pins will 302 Elements of Mechanism. move along the radial grooves as the disc B turns relatively to A, and that by this combination we can obtain a spiral fusee of any required form, and can contract or enlarge its dimensions at pleasure. ART. ^^. If two cords be wound in opposite directions round a drum, A, and the ends of the cords be fastened to a movable carriage, it is evi- dent that the rotation of A in alternate di- rections will cause a reciprocating movement in the carriage. This is a mangle in its simplest form, and the objection that the handle must be con- tinually turned in opposite directions may be obviated by the use of the mangle wheel. It is clear that if the drum were divided into two portions, and that each half instead of being cylindrical were formed into a fusee, the motion of the piece driven by the rope would be no longer uniform but would vary with the dimensions of the fusee. Hence the drum A has been replaced by a spiral fusee in the self-acting mule of Mr. Ro- berts, and thus the motion of the carriage is gradually acce- lerated until it has reached the middle of its path, and then decreases to the end of the movement. It must be understood that the cord fastened at A goes off at C, while that fastened at B passes on to D. FIG. 306. A helical screw of a varying pitch traced upon a cylinder would produce a similar variable motion of the mule-carriage, and has Winding-on Motion. 303 FIG. 307. been applied in a machine constructed upon a different principle, in order to obtain a continually decreasing motion of the carriage. It replaces the fusee. ART. 233. Mr. Robertas winding-on motion reposes upon the principle of the fusee, though in a modified form. Let one end of a rope which is coiled round a drum be at- tached to a point, P, in the movable arm CP ; it is evident that the rotation of CP about the centre of motion C will cause some portion of the rope to be unwound from the barrel (fig 307). Draw CS perpendicular to 7 the direction of the rope ; then, at any instant of the motion/' / the arrangement supplied ^^ the jointed rods, CP, BQ, \"<\ mentioned in Art. 93, and it is manifest that the rate at which the string is ^- j wound will vary as the per- pendicular CS. *\ This ratek is greater t when CP\is perpendicular to PQ\ bu\ decreases to nothingVvhen x GS vanishes, and here\ therefore, the varying arm of the fusee exists in a latent form. \\ Next conceive that the conditions are changed, and that the drum B moves to the right hand through a moderate space, while CP remains fixed. The cord will unwind from the drum with a nearly uniform velocity. If, finally, the arm CP be not fixed, but be made to move from a position a little to the left of the vertical into one nearly hori- zontal during one journey of the drum, it is abundantly clear that we shall subtract from the uniform motion of unwinding that amount which is due to the action of a fusee, and that if the spindles derive their motion from the rotation of the drum they will continually accelerate as the drum recedes from CP. In this way we can make up the " body of the cop. To form the cop- bottom, it is necessary that the winding on should begin more rapidly, and should gradually diminish. This character of motion 304 Elements of Mechanism. is produced by causing the nut P to traverse CD in successive steps during each journey of the drum. As soon as the cop has FIG. 308. attained its full diameter the nut ceases to travel along CD, and the thread is wound in uniform conical layers. ART. 2^.. Harrison's going fusee is employed in watches or clocks having a fusee for the purpose of keeping the timepiece going while the spring is being wound up. The principle on which it is constructed will be readily under- stood. Referring to the small sketch in fig. 309, conceive that a force applied in direction of the arrow at x is to be communicated to y. Let x and y be connected by a spring S, and suppose further that some resistance to motion is felt at y, then the force at x would compress the spring until y began to yield, after which x and y would move together as if they were one piece. Next suppose that x becomes fixed, then it is apparent that the elasticity of the compressed spring S will cause a push to be felt at j, and that for a short interval the spring would urge y onward just as if the driving pressure on x had been maintained unimpaired. In other words, if we interpose a spring between the driver and the thing driven, it is possible to obtain some amount of working pressure from the spring after the driving force has ceased to act. We pass on to explain the drawing, which is taken from a model arranged for making the contrivance intelligible to a learner. If the construction adopted in a watch were more closely followed it would be difficult to show the working parts. A weight W, hanging upon a string wound round the disc A, The Going Fusee. 305 supplies the pull of the chain on the fusee, and the disc is pro- vided with a ratchet, called the winding ratchet, having an ordi- nary detent at R. On the same axis as A is a circular plate B, having a ratchet, called the going ratchet, which is prevented from recoiling by a detent P. Connected with A, and forming part of it, is the great wheel D, which is shown in the model as a pitch circle without teeth, and which drives the pinion E. FIG. 309. The object is to keep E rotating in the direction of the arrow, and from what has been stated it is apparent that so long as W acts upon A it will act also on D and will drive E. The difficulty occurs when the weight is being wound up, in which case A is turned in the reverse direction, and D would be powerless to drive E, unless some new agent were called into play. In the model this agent is supplied by the strong indiarubber cord ST, and in a watch it takes the form of a curved strip of steel, whose ends are brought near together. A circular slot marked by a dark thick line terminating in T determines the amount through which the recoil of the spring can act, and it should be noted that ST is attached at S to a pin on the plate B, and at T to a pin in the plate D. The first action of W is to stretch ST, and when the spring is sufficiently stretched D begins to move and travels round, the going ratchet slipping tooth by tooth under the detent P. When 306 Elements of Mechanism. the winding-up takes place and the pressure of W is taken off, the action of the spring begins, and there is a pull of S towards T, and of T towards S. The detent P, acting on the going ratchet, prevents any motion in the direction ST, and therefore T ap- proaches towards S, but in doing so ST pulls D in the same direction as that in which W previously moved it, and the result is that the motion of E continues, notwithstanding that the weight W has ceased to act. ART. a|2> The mechanism of a keyless watch, so far as the winding of the spring and the setting of the hands are concerned, may be of interest to the student. There are three separate things to be provided for : 1. The spring is to be wound up without opening the case or inserting a key. 2. The same button or handle which turns the spring should be available for setting the hands. 3. No injury should happen if the winding button be turned in the wrong direction. FIG. 310. Taking number 3 first, the contrivance adopted is a very old Keyless Watch. 307 one in mechanism, being a ratchet cylindrical coupling, whereof the two parts are held together by a spring. It is shown at R, in figs. 310 and 311, the spring being marked Q, and terminating in a tail, which holds the coupling by the groove S. When the spring is in action the two halves of the coupling are locked together, but upon depressing the stud P the tail S is forced down, and the halves of the coupling are separated. The button H, which is rotated for winding the watch-spring, or for setting the. hands, is connected directly with RC, the lower half of the coupling, and when the spring is in action the turning of RC causes L also to rotate. Thus, when the button is rotated in the direction of the arrow, the lower half of the coupling drives the upper half, but when the button is rotated in the opposite direction, the ratchet teeth on the lower half slip over those in the upper half, and all that happens is that RC vibrates up and down by the depth of a tooth. The movement is harmless, for no part of the mechanism which does any work is affected by it. The winding square in an ordinary watch is replaced by the spur wheel D, having a detent r, actuated by a spring, whereby the toothed wheel serves the same purpose as a ratchet wheel with saw-cut teeth. This contrivance is borrowed from the engineers, as will be remembered. The spur wheel D gears with another spur wheel B, on the face of which is a flat bevel wheel which engages with another bevel wheel L attached to the upper half of the ratchet coupling. The rotation of H in the direction of the arrow drives RC, which again drives L, and so causes B and D to rotate, and thereby to wind up the watch. The setting of the hands is accomplished by a wheel A which operates on the minute hand. The lower half of the coupling RC terminates in a cylindrical toothed wheel, technically known as a crown or contrate wheel, which is brought into gear with the spur wheel A when the stud P is depressed. It will be apparent that when RC is lowered the winding-up stops, and at that time the hands can be set. Whereas when RC is allowed to rise the crown wheel is thrown out of gear with A, and the winding-up Can begin. Each action is shown separately in the sketches. x 2 308 Elements of Mechanism- ART. 236. The snail is chiefly found in the striking part of repeating clocks. It is a species of fusee, and is used to define Flo I2 the amount of angular deviation of a bent lever ABD, furnished at the end A with a pin which is pressed against thk - curve of the snail by a spring, and\is attached at the other end to a curv&d rack, whose position determines the number of blows which will be struck upon the bell. In order to form the snail, a circle is- divided into a number of equal parts (twelve, for ^exiample), and a series of steps are formed by cutting away the plate and leaving a circular boundary in each position. As the snail revolves, ABf) jpasses by jerks into twelve different positions, and the clocklstrikes the successive hours. Since the point A Describes a circle about B, it is clear that the depth of each step must vary in order to obtain a constant amount of angular motioii in the arm B A during each progressive movement. It will be seehjhat the circular arc described by the end of the small lever has its tangent at A, when in the position sketched, parallel to the vertical diameter which divides the snail into two equal parts, and this reduces the inequality between the steps. ART. 2. The disc and roller is equivalent to the fusee, and is now but nttle used, on account of the probability that the roller will occasionally slip. This arrangement consists of a disc A, revolving round an axis perpendicular to its plane, and giving motion to a rolling plate B, fixed upon an axis which intersects the axis of the disc A at right angles. Supposing the rotation of the disc to be uniform, that of the roller B will continually decrease as it is shifted towards the centre of A, and conversely. This is precisely the effect produced by a fusee. The roller may be a wheel furnished with teeth, and may roll upon a spiral rack, as shown in the diagram. The Disc and Roller. 309 As the disc revolves the pinion P slides upon the square shaft, and is kept upon the rack by the action of a guide-roller, R, which travels along the spiral shaded groove. FIG. 3I3 . This example is by no means put forward as a good mechanical contrivance, for indeed the disc and roller possesses an inherent defect which should be diminished as much as possible in practice and not exaggerated. The bounding circles of the roller run with the same linear velocity, whereas the circular paths upon which they are both respectively supposed to roll move with different linear velocities, by reason of their being concentric circles of un- equal size traced upon a plate which rotates with a uniform angular velocity about an axis through the common centre. It is geometrically impossible that the bounding circles of the roller can both roll together, or the combination will fail in exact- ness as a piece of pure mechanism, and thus two rollers running round upon a flat surface or bed-stone in the manner suggested form an excellent pulverising or grinding apparatus. These rollers are called edge-runners : they are of large size and very weighty, and are placed near to the vertical axis about which they run. It is apparent, without any proof, that the disc B will change the direction of its rotation as soon as it has passed over the centre of the plate A. This follows from the nature of circular motion, and, as we have indicated, the disc B will come to a standstill when its circumference reaches the centre of the driving plate A. In like manner, if the axis of B be carried along in a straight 3 1 o Elemen is of Median ism. line passing over the centre of A, the student will understand thai if A rotates in one direction while B is approaching its centre, anc in the opposite direction, as soon as B has crossed the centre, ii necessarily follows that the reciprocation of A will impart a con- tinuous rotation in one direction to the roller B. FIG. 314. Such a result is exhibited in the sketch on the left-hand side of fig. 314, where the arrow shows that A is rotating in one direc- tion when B is above the centre C, and in the opposite direction when B is below the same centre ; while it becomes evident that the direction of rotation of B is correctly indicated by the arrow at P, and remains invariable. ART. 238. This movement has been applied in the construc- tion of a continuous indicator. It has been shown that Watts indicator gives the amount of work done in any stroke of a steam engine, and it follows that the general performance of the engine would be inferred from a comparison of several indicator cards taken at intervals. The continuous indicator by Messrs. Ashton and Storey will, however, furnish a complete register of the work done in a_given time. In this apparatus the piston of the indicator is attached tc the roller B, and the plate A takes up, on a din)rmshed scale, the reciprocating motion of the piston of the engine. Continuous Indicator. 311 A general idea of the mechanism of the instrument may be gathered from the diagram. A grooved wheel, M, is connected directly with the disc A, and receives a reciprocating motion from the piston of the engine just as if it were the barrel of an ordinary indicator. The roller B is connected on the side marked P with the indicator piston, and travels up and down in the vertical line PE, the motion of B being subject to the opposing forces of a' steel spring and the steam pressure, just as if it were the pencil in an ordinary indicator. The axis EP is provided with an elongated pinion, as shown, which gears into a wheel H in every position, the intent being that the rotation of B shall be carried by H to a recording or counting apparatus at the top of the instrument, and in the line ba produced. The principle relied on is that the work done by the engine in a given time will be directly proportioned to the number of re- volutions made by H in the same time. Taking a condensing engine as an example, it is apparent that when trW indicator piston is in the position which accords with the tra/ing of ~an atmospheric line in an ordinary diagram, the disc BHies exactly over the centre of the plate A and does not rotate. In subh a state of things the wheel H also remains at rest and no work is done. Whereas any increase of the steam pres- sure carries B above the centre of A, and causes H to rotate with a velocity which increases as the pressure rises. Or, if the vacuum be improved, B sinks to a greater distance below the centre of A, and H rotates more rapidly during the return stroke, the result being t register work done in both cases. Also, if at any time there-should be a subtraction of area in an ordinary indicator diagram, there will be a like subtraction here by reason that B wlrV- cross over the centre before the direction of rotation of A is changed, whereupon H will begin to rotate in the reverse direction, and some of the work previously scored up will be, as it were, rubbed out. Hence, the instrument sums up, or inte- grates, as it is termed, the actual performance of the engine in any given interval of time. ART. fi^ Step wheels constitute a modification of toothed wheels ; they are due to Dr. Hooke, and are used to ensure a smooth action in certain combinations of wheel-work. 312 Elements of Mechanism. It is evident that the action of two wheels upon each other becomes more even and perfect when the number of teeth is in- creased, but that the teeth at the same time become weaker and less able to transmit great force. Dr. Hooke's invention overcomes the difficulty, and virtually increases the number of teeth without diminishing their strength. Several plates or wheels are laid upon one another so as to form one wheel, and the teeth of each succeeding plate are set a FIG. 315. little on one side of the preceding one, it being provided that the last tooth of one group shall cor- respond within one step to the first tooth of the next group. The principal part of the action of two teeth occurs just as they pass the line of centres, and there are now three steps instead of one from the tooth A to the tooth B. A single oblique line might replace the succes- sion of steps, but we should then introduce a very objectionable endlong pressure upon the bearings of the wheels. Pinions of this construction are to be met with in planing ma- chines, and are employed to drive the rack which is underneath the table ; so again step wheels are used in marine steam engines where the screw shaft is driven from an axis considerably above it. They are valuable where strength and smoothness of action are to be combined. ART. 240. It has been shown that in the case of ordinary bevel wheels the pitch cones have a common vertex, and there- fore of necessity the axes of rotation meet in a point. In some cases, however, it may be convenient that the axes of bevel wheels should pass close to each other without intersecting ; the teeth have then a twisted form, and the wheels are known as skew bevels. A general idea of the principle of construction adopted in wheels of this kind may be obtained by a simple experiment. Fasten a light rod of wood or a bright steel wire, such as a knitting pin, obliquely to the face-plate of a lathe, in such a manner that the direction of the rod intersects at an angle the axis or line of the lathe. Set the plate in rapid rotation, when the rod will appear to be replaced by a shadowy double cone. The effect is Skciv Bevels. 313 due to the fact that any impression made on the retina of the eye endures for an appreciable time. Now fix the rod in such a manner that it no longer cuts the line of centres, but passes obliquely on one side thereof, and set the face-plate in rapid rotation as before. The double cone will be replaced by a surface resembling a dicebox, which is curved in every part, and yet is generated by a straight line. This sur- face is known to mathematicians as a 'hyperboloid of revolution.' From the mode of generation it is clear that the surface in question is symmetrically disposed about the line of centres of the lathe, and we may regard that line as its axis. If two such hyperboloids be brought into contact along a pair of generating lines they will roll together, and will serve to com- municate a motion of rotation from the axis of one to that of the other ; that is, they will enable us to communicate motion between two axes which are not parallel and do not meet, the directions of the generating lines in each surface determining the general direction of inclined teeth which might lie on the re- spective surfaces. Thus two hyperboloid surfaces replace, in the case of skew bevels, the pitch cones of ordinary bevelled wheels. ART. 241. Root's Blower, a mechanical equivalent for a fan, is a special contrivance which has been modified and varied by the ingenuity of patentees, and has been extensively used. FIG. 316. FIG. 317. The sketch fig. 316 is from a model in the collection of the 3 1 4 Elements of Mechanism. School of Mines, and shows the principle of the action of this form of air-compressing machine. There are two rotating pistons, B and D, centred on axes at C and F, and driven by a pair of equal spur wheels in opposite directions with equal velocities. A circular case surrounds the pistons, and it is provided that there shall be no actual contact of the surfaces of the rotating pistons, either with each other or with the casing, but that the working parts shall approach as closely as may be without any rubbing action or internal friction. In the diagram the arrows show the manner in which the air enters the casing at A, and is swept forward to the exit at E, the two small arrows on the pistons being employed to indicate their relative directions of rotation. In fig. 317 the blower is converted into a ventilating appa- ratus for a colliery, and is in use at a pit belonging to the South Durham Coal Company ; it consists of two pistons, each 25 feet in diameter and 13 feet wide, driven by engines having a pair of 28-inch cylinders with 4-feet stroke. The clearance between the periphery of one of the pistons and the central circle of the other is inch, and it is stated that when the ventilator runs at 21 revolutions per minute the amount of air passing through the machine is about 118,000 cubic feet per minute. ART. 242. The Governor of a steam engine usually appears under the form invented by Watt, and has proved of the greatest possible value in steam machinery. The diagram shows this well-known piece of apparatus, and the principle of its action may be described very briefly as follows : The engine imparts rotation to the balls of a heavy conical pendulum, and maintains them at a certain inclination to the vertical ; if the velocity of the engine be increased, the balls open out more widely ; if it be diminished they collapse, and in doing so they set in motion a system of levers which is connected with a throttle valve, and thereby regulate the supply of steam to the cylinder i. A common method of constructing the governor has been that shown in fig. 318. The balls are suspended at the points E The Governor of a Steam Engine. 315 TO VALVE and H, a little on either side of the central vertical spindle CB. Each arm, as HD, is connected by a link to a sliding block ST. As the rate of rotation increases the balls fly out, ST rises, and in doing so actuates a lever which controls a steam valve and diminishes the supply of steam. The arm DH is produced to meet the vertical axis in C, and DB is drawn perpendicularly to CB, whence the balls and suspending arms lie upon the surface of a cone whose axis is CB. The chief point to notice is that the number of re- volutions made per minute by the balls depends upon the height of the cone, viz., CB. The effect of placing E and H at a little distance from the axis CB is to cause the variation in the height of the cone to become greater for any given rise of the balls, and thereby to render the governor less sensitive. Thus the heights of the cone in the two positions shown are CB and cb respectively, the vari- ation being equal to Cc + B& Special methods of construction have been originated for reducing the amount of this variation ; as to which the reader is referred to the author's text book on the Steam Engine. An engineer can easily arrange that the variation in speed admitted by the governor shall not exceed one-tenth of the mean velocity, but it is of the essence of the invention that some change in the speed should be admissible : the balls cannot alter their position unless the time of a revolution changes, and they cannot accumulate such additional momentum as may be sufficient to move the valve until the rate of the engine has sensibly altered. In some cases, as where the engine drives machinery for very fine spinning, it may be desirable to obtain an almost absolute uniformity of motion ; or, again, it may be an object to avoid the fluctuations in speed to which the common governor is liable when any sudden change occurs in the load upon the engine. 316 Elements of Mechanism. ART. 243. In order to control the engine with almost theo- retical exactness, and to provide against the objections to which Watt's governor is exposed in certain extreme cases, Mr. Siemens has put forward a remarkable adaptation of epicyclic trains to the conical pendulum ; and we shall proceed to an examination of his invention. The original construction of this governor is exhibited in the diagram, and is better adapted for the purposes of explanation than a more recent arrangement (fig. 319). An epicyclic train of three equal wheels, A, B, C, is placed between the driving power and the conical pendulum ; of these, A is driven by the engine, C is connected with the pendulum, and B is capable of running round A and C to a small extent defined by stops, the joints at F and B being so constructed as to permit of such a motion. FIG. 319. JW>^ A The wheel, B, is also con- nected by the system of levers to a weight, K, and shuts the steam valve when its motion has lifted K through a certain space. The valve spindle passes through the centre of motion, E, and is turned by the arm FE. A conical pendulum, DP, is suspended by a ball-and-socket joint at S, and the extremity D moves in a circular groove, DH. T/ie Chronometric Governor. 317 In this way the rotation of C is communicated directly to the pendulum. It will be seen that a certain amount of energy is absorbed in preserving the pendulum at a constant angle with the vertical, and it is a part of the contrivance to increase artificially the friction which opposes the motion of the pendulum, and thus finally to make the pressure exerted by the weight, K, an actual measure of the amount of the maintaining force. The governor is at work when the velocity of the engine is just sufficient to keep K raised through a small space. In order to understand the peculiar action introduced by the epicyclic train we should remember that one of these two things will happen : either A and C will turn at the same rate, or else B will shift its position and run round the axis AH ; there can be no departure from the rigid exactness of this statement. Now, the wheel C is connected with the pendulum, and its rotation cannot be maintained without a constant expenditure of energy ; in other words, the tendency of C is to lag behind A, and to cause B to run round the axis AH. This indisposition in C to accept the full velocity of A is artificially increased by the friction until B shifts its position and raises the weight K permanently, and then of course it follows that the pull of K evidences itself as a constant pressure tending to drive the wheel C. The pendulum being in this manner retained in permanent rotation, suppose that any increase were to occur in the velocity of A : the wheel C is in connection with a heavy revolving body, and can only change its velocity gradually, but K is already lifted, in the sense of being counterpoised, and the smallest increase of lifting power can therefore raise it higher ; thus the tendency to an increase in the velocity of A will at once cause B to change its position, and will control the steam valve. So sensitive is this form of governor to fluctuations in speed, that an alteration of ^th of a revolution may suffice to close the throttle valve altogether. It is in its power to move the valve, as well as in its sensitiveness, that this arrangement presents so re- markable a contrast to Watt's governor, where the moving force on the valve spindle is only the difference between the momenta 3 1 8 Elements of Mechanism. stored up in the two positions of the balls. In this form of governor the power is only limited by the strength of the rods and levers ; for it is apparent that the whole momentum stored up in the revolving pendulum would in an instant be brought to bear upon the valve spindle if any sudden alteration were to occur in the velocity of the wheel A. In the method which has been adopted at Greenwich for registering the times of transits of the stars by completing a gal- vanic circuit at the instant of observation, a drum carrying a sheet of paper is made to revolve once in two minutes. A pricker actuated by an electro-magnet, and moving slowly in a lateral direction, is set in motion at the end of each beat of the seconds pendulum of a clock, and thereby makes a succession of punctures in a spiral thread running round the drum. The observer touches a spring at the estimated instant of the time of transit of a star across a wire of the telescope, and, producing a puncture inter- mediate to those caused by the pendulum, does in fact record the exact period of the observation. The regularity of motion in the drum is a matter of vital importance, and was at one time ensured by the employment of a clock train moving under the control of this pendulum of Mr. Siemens. But, inasmuch as the tendency to simplify mechanical move- ments is always leading to new results, it has been found advan- tageous to replace the more complicated governor, with its train of wheels, by a conical pendulum controlled by a so-called dipper. The pendulum is driven by an ordinary clock train, and the dipper is carried round with the ball, and is merely a small flat plate dipping partially into a bath of glycerine and water. When the pendulum is accelerated the dipper enters the liquid more deeply, and the drag is greater, whereas when the velocity of the pendulum diminishes the dipper rises, and the motion is less retarded. It is said that this apparatus fulfils its purpose ex- tremely well. ART. 244. The Double Eccentric for Reversing an Engine. When the piston is near the middle of its stroke in a locomotive or marine engine, the slide-valve will have moved over the steam- ports in the manner pointed out in fig. 320. The slide-valve is connected with a point in the circumfer- TJic Double Eccentric. 319 ence of a small circle which represents the path of the centre of the eccentric pulley, and the piston is connected with a point in a larger circle, representing the path of the centre of the crank-pin. The piston and valve are shown as separated in the drawing, but the small circle is repeated in the position which it actually occupies, and the method of reversal is the following : In the upper diagram the piston is supposed to be moving to the right, and the valve to the left, the piston having travelled so far in its stroke that the valve is returning to cut off the steam : in order, therefore, to change the motion, we must drive the piston back by admitting the steam upon the opposite side, and by letting out that portion of the steam which is urging the piston forward. Hence we must move the valve into the position shown in the lower diagram, and shift the centre of the eccentric pulley from A to B : the piston will then return before it reaches the end of the cylinder, and the movement of the engine will be reversed. In examining the diagram it should be understood that the crank which works the slide-rod is inclined at an angle somewhat greater than 90 to the crank which is attached to the piston ; and also that the crank of the slide-rod is always in advance of the larger one in its journey round. The engine would not work if 320 Elements of Mechanism. the larger crank were to turn in the opposite way to that shown in the sketch. This explanation shows that in reversing an engine we must either shift the eccentric from the one position into the other, or else we must employ two eccentrics, and provide some means of connecting each of them in turn with the slide-valve. V ART. 245. The link motion commonly appears under three (\ "forms, (i) There is the shifting link, having its concave side * towards the axle or crank shaft : this arrangement was introduced by the celebrated Stephenson, and is known as Stephenson's Link Motion. (2) There is the stationary link, where the curvature is in the opposite direction. (3) There is the straight link, which is derived from a combination or moulding together of the two former contrivances. i. Stephenson's link motion is shown in fig. 321. AB is the starting lever, under the control of the engine-driver, and is represented as being pushed forward in the direction in which the engine is moving ; CD is the link, provided with a groove, along which a pin can travel ; a short lever, centred at R, is connected at one end, Q, with the slide-valve, and at the other end with the pin which moves in the link. It is clear that so long as the pin remains near the point D, the lever centred at R will be caused to oscillate just as if the pin were attached to the extremity of the outer eccentric bar, and that the outer eccentric alone will be concerned in the motion of the valve. If now the engine-driver wishes to reverse his engine, he pulls back the lever AB, and by doing so he raises the link CD until the pin comes opposite to the end of the inner eccentric bar. The raising of the link is caused by the motion imparted to the bell- crank lever, GEF, which is centred at the point E. A counter- poise to the weight of the link is attached to the axis passing through E at some little distance behind the bell-crank FEG, so as to be out of the way of the moving parts, and the object of this counterpoise is to enable the engine-driver to raise the heavy link and bars easily. The inner eccentric bar now alone comes into play, and the two eccentrics being fastened to the crank axle at the angles Reversing an Engine. 321 322 Elements of MecJianisin. indicated in the first part of the article, it is apparent that the valve will be shifted, and that the action of the engine will be reversed. 1fhe stationary link shown in fig. 322 was invented by Mr. :h, of the Great Western Railway. It will be seen that the 'link RS is here suspended by an arm 1>C, so as to be stationary so far as any up-and-down movement is concerned, and that it is circular in form, being struck by a radius equal to DR, whereby also its concavity lies towards the cylinder and away from the axle or shaft of the engine. In the sketch the forward eccentric is in operation, and the motion is readily traced from the axle to the slide, which is shown as having partly uncovered the steam port marked A. On pulling the rod H, which is in connection with the starting lever, or its equivalent, the bell crank K 32<5 ' at an angle a to SA, anc draw A^r perpendiculai to Sjf, xy perpendiculai to SA, yz perpendiculai to S.r, zv perpendiculai to SA, and so on. Then SA x S_y = S* 2 , by similar triangles, also, Sz; x Sy = Sz 2 , and so on. .'. S^, S_y, S^, &c., are the respective values of r\, r 2 , r 3) &c. whence the curve can be set out. The curve has the property that the tangent at every point i: inclined at the same angle to the radius vector at the point COP' sidered. e Taking the equation r = aem, we have .'. r - = m, a constant quantity. dr But r is the tangent of the angle which the curve makes dr with the radius vector at any point. Let this angle be , therefore tan = a constant, or is invariable. , dy di) Also r -- , = r m, dr' dr whence the second rolling curve is identical with the first spiral. Rolling Ellipses, 327 In like manner we could prove that two equal ellipses centred upon- opposite foci would roll together. 248. In practice rolling curves must be provided with upon the retreating edge, other- wise the driver would leave the follower, and the revolution would not be com- pleted (fig. 327). As is usual in all cases where seg- mental wheels are employed, a guide must direct the teeth to the exact point where they commence to engage each other. The guide may be dispensed with by carrying the teeth all round the curves : this construction is usually adopted in practice, although, strictly speaking, it destroys the rolling action entirely. A quick return of the table in small planing machines has been effected by the aid of rolling ellipses. The table is driven by a crank and connecting rod, and the crank exists under the form of a flat circular plate, centred on one of the foci, and having a groove radiating from the axis as a line of attachment for one end of the connecting rod. As the plate may be set in any position upon the elliptical wheel, we propose to inquire what will be the effect of a change of direction in the groove or crank. Let the ellipses have the position shown in fig. 328, S and H being the centres of motion, and SPHQ being perpendicular to Aa, the axis of one ellipse. Draw PR perpendicular to Dd, and let the ellipse DRdH? be the driver, ro- tating as shown. While P^/R is rolling upon PrtQ, the ellipse Aa makes half a revolution ; and while RDP is rolling upon QAP, it makes the remaining half-revolution. FIG. 328. 328 Elements of Mechanism. Suppose T)d to revolve uniformly, then the times of a half re- volution of Aa will be in the same proportion to each other as the angle PSR to the angle 360 PSR. The quick half revolution occurs when the shaded segments are rolling upon each other. If, therefore, the table be made to move in the line HS produced, and the crank be placed in a direction perpendicular to Aa, we shall obtain the greatest possible difference between the periods of advance and return. The practical difficulty with rolling wheels exists during that part of the revolution where the driver tends to leave the follower, and it can only be obviated by making the teeth unusually deep ; also the wheels should work in a horizontal plane. ART. 249. An instance of rolling curves is exhibited in the sketch, and occurred in one of the many attempts made to improve the printing press before the invention of Mr. Cowper enabled the newspapers to commence a real and vigorous existence. The type was placed upon each of the four flat sides of a rect- angular prism, to which the wheel B corresponded in shape, and the paper was passed on to a platten corresponding in form and size with the pitch-line of the wheel A. The prism and platten being in the same relative position as the wheels B and A, we can understand that the type would be in the act of impressing the paper while the convex edge of the wheel A rolled upon the flat side of B, and that in this way we should obtain four impressions for each revo- lution of the wheels. By this construction, the patentees, Messrs. Bacon and Donkin, intended to introduce the principle of continuous rotation as opposed to the reciprocating movement in a common press ; and the object of imitating exactly upon the wheels A and B the form of the print- ing prism and of the platten, was to FIG. 329. Hookcs Joint. 329 ensure that the paper and type should roll upon one another with exactly equal velocities at their opposing surfaces, and that no slipping or inequality of motion should destroy the sharpness of the impression. ART. 250. Hooke's Joint is a method of connecting two axes, whose directions meet in a point, in such a manner that the rotation of one axis shall be communicated to the other. Here AB and CD represent two axes whose directions meet in the point O ; the extremities of AB and CD terminate in two semicircular arms which carry a cross, PQSR ; the arms of this cross are perfectly equal, and the joints P, Q, S, and R permit the necessary free- dom of motion. As the axis AB revolves, the points P and Q describe a circle whose plane is perpendicular to AB, and at the same time the points S and R describe another circle whose plane is perpendicular to CD. These two circles are inclined at the same angle as the axes, and are represented in fig. 331 ; thus the arm OP starts from P, and moves in the circle PP'L, while the arm OR starts from R, and de- scribes the circle RR'Q inclined to the former. Let OP', OR' be corresponding positions of the two arms, then P'R' is constant, but changes its inclination at every instant, and as a consequence the relative angular velocities of OP' and OR' are continually changing. To find the relative angular velocities of the axes AB and CD, we proceed as follows : Let the circle prq (fig. 332) represent the path of P, ptq being the projection upon this circle of the path of R, and suppose a to be the angle between AB and CD ; then the dimensions of the curve ptq, which vill be an ellipse, can be at once deduced from the equation O/ Or cos a." Draw Rm perpendicular to Or, then Rm will be the actual 330 Elements of Mechanism. vertical space through which OR has descended while OP de- scribes the angle /K)P. But the path of R is really a circle, and only appears to be an ellipse by reason of its being projected upon FIG. 332. FIG. 334. a plane inclined to its own plane. In order, therefore, to estimate the actual angular space through which OR has moved, we must refer this motion to the circle which R really describes (fig. 333), and thus by making R'm' = Rm, we can infer that w'OR' will be the angle which OR describes while OP moves through the angle /OP. But the angle /OP = angle wOR, and hence we can represent the motion of both axes upon one diagram by combining the ellipse /R^ with the circle prR.'g (fig. 334). This being done, we may draw R'RN perpendicular to/O^, and join OR, OR' ; it will at once appear that the angles ROr, R'Or are those described in the same time by the axes AB and CD. Hence the axes AB and CD revolve together, but unequally, and the angles which they describe in the same time can always be found by construction. First draw the circle prq in a plane perpendicular to one axis, and having O for its centre, next construct the ellipse whose major axis is the diameter pQg equal to POQ, and whose minor axis is the product of POQ x the cosine of the angle between the axes. Then take OR' any position of OP, draw R'RN perpendicular to pQq, join OR' and OR. It now appears that R'Or and ROr will represent the angles described by the axes AB and CD in the given time. Furthermore, OR and OR' coincide when R is at the end of an axis of the ellipse /R^, an event which must happen four Double Hookes Joint. 331 times as the cross goes round once ; and there is therefore this curious result, that however unequal may be the rate at which the axes are at any time revolving, they will coincide in relative posi- tion four times in one revolution. The single joint may often be very useful in light machinery which is required to be movable, and the parts of which do not admit of very accurate adjustment ; but it will be understood that the friction, and especially that irregularity which we have just proved to exist, would render it necessary to confine the angle between the shafts within narrow limits in actual practice. ART. 251. Now that the general character of the movement is understood, we shall be in a position to comprehend the change which is effected by interposing a double joint between the axes. FIG. 335. 'D i. Take the case where AB and CD are parallel axes con- nected by an intermediate piece BC, having a Hooke's joint at both the points B and C. Conceive that the arms of the crosses at B and C are placed in the manner shown in the sketch, or let each vertical arm be con- nected with the forks at B and C. If AB revolves uniformly, BC will also revolve with a varying velocity dependent upon the angle ABC, but the variable velocity which BC receives from AB is precisely the same as that which it would receive from DC if the latter axis were the driver and were to revolve uniformly. 332 Elements of Mechanism. It follows therefore that the motion which AB imparts to CD will be a uniform velocity of rotation exactly equal to that of AB. Hence a double Hooke's joint may be used to communicate uniform motion between two parallel axes whose directions nearly coincide. If, however, the construction were varied and the vertical arm PQ of the first cross were connected with E, while the horizontal arms, s, r, were connected with F, we should communicate no doubt a motion of rotation between the axes, but it would no longer be uniform but variable, by reason that we could not return by the same course reversed under like conditions. The devi- ations from uniform rotation would no longer oppose and correct. each other, but they would act together and increase the in- equality. This is seen at once upon constructing the diagrams which represent the relative rotation between each pair of axes. 2. Let AB and CD be inclined to each other, and be con- nected by the piece BC jointed at B and C, and so placed that the angle ABC is equal to the angle BCD. As in the former case we must be careful to connect B and C with the corresponding arms of the crosses, and we have seen that the inequality produced by DC in the motion of CB depends both upon the angle BCD and the position of the cross ; it is therefore the same whether CD lies in the direction shown, or in the dotted line CH parallel to AB. In both cases the angle between the axes and the position of the cross will respectively coincide. But we have seen that when the parallel axes AB and CH are connected by joints at B and C in the manner stated, the axes AB and CH will rotate with equal uniform velocities, and we conclude, therefore, that they will also rotate in a similar manner when placed in the position ABCD. Hence a double Hooke's joint may be employed to com- municate a uniform rotation between two axes inclined at a given angle. ART. 252. It will be found that well-known propositions of Euclid obtain a new significance when applied to movable com- binations of the lines of figure. Parallel Axes. 333 FIG. 336. Referring again to the triangle from which we deduced the law of motion of the crank and connecting rod : i. Let APB represent such a triangle, A and B being fixed points, and the angle APB being a rigid angle. Also, let the sides PA, PB, be produced indefinitely in order that the dimensions of / c ^^ the triangle may vary by the ' sliding of PA, PB through the points A and B respectively. Let PAB = 0, PBA = 0, then + = 180 APB = a constant. .*. $9 + cty = o, or ?0 = c, whence the angular velocity of PA about A is equal and opposite to the angular velocity of PB about B. Also since APB is a rigid angle, and since the angles in the same segment of a circle are equal to one another, we infer that the point P lies always in a circular arc passing through A and B. It will simplify the result if we take APB 90, as we can then apply the property that the angle in a semicircle is a right angle, also AB will in that case be the diameter of the circle traced out by the point P. In fig. 337 take A and B to represent two fixed axes, and let DEFH be a rigid rectangular cross whose arms can slide through the points A and B. The motion will only be possible so long as the arms of the cross have perfect liberty to slide through the points A and B as well as to rotate about them. Let this be arranged, and join AB ; then we have PAB + PBA=9o in every position of the cross. Hence if the angle PAB increase by the rotation of PA, the angle PBA must diminish equally by the rotation of PB, or ED and FH must revolve with equal angular velocities. Fro. 337. 334 Elements of Mechanism. Also P, the angle of the cross, will describe a circle whose diameter is AB, and our proposition follows directly from Euclid, for if P move on to any point Q, the angles QAP, QBP are angles in the same segment of a circle, and are therefore equal to each other. This movement was put into a practical shape by Mr. Oldham, and used in machinery at the Bank of England. The student may easily construct a model after the manner of Hooke's Joint, when the centre of the cross will be seen to describe a circle whose diameter is the perpendicular distance between the axes while the arms of the cross slide to and fro through holes in the forked arms that spring from the axes and support them. 2. Bisect AB in C, and take P and C as fixed centres of motion. As before, let APB be a right angle, then the rotation of AB about C will set up a rotation in both PA and PB, whereby each of the latter lines will tend to rotate with half the angular velocity of AB. In order to make the motion continuous, the lines AP, BP must be produced so as to form a rectangular cross, and they may be conveniently formed as straight grooves in a plain board whose axis passes through P. The bar AB will then be provided with pins working in the grooves. Describe a circle with centre C and radius equal to CA or CB ; draw any fixed diameter A'CB', and join A'P. FIG. 33 8. Then it is proved in Euclid that the angle at the centre of a circle is double the angle at the circumfer- ence when both angles stand upon the same arc, that is angle ACA'=2 angle APA', or the angular velocity of CA is twice that of the cross. As the driver ACB revolves the pins A and B will oscillate to and fro along their respective grooves and will traverse through the centre P. ART. 253. The differential worm wheel and tangent screiv is a combination- which will be understood without any drawing. The Geneva Stop. 335 Here two worm wheels, differing by one tooth in the number which they carry, are placed side by side and close together, so as to engage with an endless screw. As the wheels are so very nearly alike the endless screw can drive them both at the same time, and it is evident that one wheel will turn relatively to the other, through the space of the extra tooth, in a complete revo- lution, and that a very slow relative motion will thus be set up. In this way, if one wheel carries a dial plate, and the other a hand, we may obtain the record of a very large number of revolu- tions of the tangent screw. ART. 254. Where a train of wheels is set in motion by a spring enclosed in a barrel it becomes of consequence not to overwind the spring. The Geneva stop has been contrived with the view of preventing such an occurrence, and will be found in all watches which have not a fusee. Here a disc A, furnished with one projecting tooth, P, is fixed upon the axis of the barrel containing the mainspring, and is turned by the key of the watch. Another disc, B, shaped as in the drawing, is also fitted to the cover of the barrel, and is turned onward in one direction through a definite angle every time that the tooth P FIG. 339. passes through one of its openings, being locked or prevented from moving at other times by the action of the convex surface of the disc A. In this manner each rotation of A will advance B through a certain space, and the motion will continue until the convex surface of A meets the convex portion E, which is allowed to remain upon the disc B, in order to stop the winding up. The winding action having ceased, the discs will return to their normal positions as the mechanism runs down. Instead of supposing A to make complete revolutions let it oscillate to and fro through somewhat more than a right angle ; then B will oscillate in like manner and will be held firmly by the opposition of the convex to the concave surface except during the time that P is moving in the notch. 336 Elements of Mechanism. ART. 255. The Geneva stop has been applied by Sir J. Whitworth in his planing machine, in order to give a definite vibration to a piece from which the feed motion is derived. FIG. 340. The drawing shows a lever centred at A, and having a pin P at one end. The other end of the lever is weighted at W, the object of the weight being to cause the lever to fall over suddenly, and with sufficient power to carry the driving belt from one working pulley to the next in order. A pinion on a shaft terminating at A gears into a rack formed on a traversing rod which is moved longitudinally in alternate directions by tappets on the table of the machine. The rod, in its turn, actuates a bell crank lever which is connected with a fork employed for passing the driving belt from one pulley to another. As far as this explanation has gone it would appear that the weighted lever was designed simply for controlling the driving belt, but it will be seen that a portion of a Geneva stop is super- added, and this extra piece enables the lever to actuate also the feed motion. The axis A of the lever is surrounded by a circular plate, corresponding with A in fig. 339, and the piece BED has two concave circular cheeks at E and D exactly fitting the plate A. Also the pin P works in the open jaws as shown. The result is that the lever PA carries BED as it swings over, and locks it in either position, whether to the right or the left In the diagram the pin P is shown (i) in a vertical posi- The Star Wheel. 337 tion, while in the act of driving BED, and also, (2) after having fallen over to the right, at which time BED is securely locked. The extreme positions of the weighted lever are shown by the dotted lines pa, qb, and it only remains to point out that the feed motion is taken from the axis B by means of a grooved pulley and a catgut band. This band runs round another pulley which has already been described in Art. 132, and is there marked as F, and by a comparison with the previous description the general arrangement will be readily understood. The locking of BED is essential in order to prevent any motion of the cutter, before the completion of the cut. ART. 256. The star wheel is used in cotton-spinning machi- nery, and is analogous to the Geneva stop. If the convex portion E were removed, so as not to interfere with the rotation of A, we should virtually possess a star wheel in the disc B. See Art. 254, and fig. 339. In that case each rotation of A would advance B by the space of one tooth, or we should convert a continuous circular motion into one of an inter- mittent character. The usual form of the star wheel is given in the sketch, where the revolving arm encounters and carries forward a tooth at each revolution. The action is the same as if a wheel with one tooth were to drive another with several teeth. ART. 257. It is well known to mathematicians that the an- gular velocity of a rigid body about an axis may be properly repre- sented by a straight line in the direction of that axis, whence it follows that angular velocities may be combined according to the law which gives us the parallelogram of linear velocities or the parallelogram of forces. A remarkable illustration of the compounding of angular velo- cities has been afforded in the construction of the so-called Plimpton or roller skate for use on artificial ice. This skate runs like a wagon upon four wheels, but instead of the perch-pin being vertical, as in an ordinary wagon, it inclines z 338 Elements of Mechanism. inwards, and dips towards the centre of the skate. Indeed, each pair of wheels is provided with an inclined axis, and it will be presently seen that the skate will not serve its purpose unless the respective axes converge downwards to a point underneath the centre of the footboard. Everyone is aware of the manner in which curves are described by a skater upon ice. For example, when tracing out a circular sweep on, say, the right foot, technically distinguished as 'an outside edge,' the plan is to keep the leg and body quite straight and to lean over a little towards the centre of the circle. With a roller skate the same movement of the body produces the same result, but in a very different manner. The act of tilting over the footboard causes the fore wheels to deviate a little to- wards the right, and the hind w.heels to deviate a little towards the left, the respective axles of the front and hind pair meeting, as they ought to meet, in the centre of the circular path described by the skater, and our object is to exhibit a method of construction which necessarily produces this result. In order to simplify the explanation it will be better to confine our attention to the front pair of wheels, and the drawing shows a model which may illustrate the movement of the wheels as conse- quent upon the tilting of the footboard. The right-hand figure gives a perspective view of one half of the footboard AB, together with the inclined axis ae, and a pair of rollers attached to an axle standing at right angles to a pipe or hollow tube, which is threaded upon the immovable inclined axis. The next sketch shows the footboard resting on its wheels near the edge of a horizontal table, and it is apparent that if the Roller Skates. 339 skate were pushed forward a little it would advance in a line per- pendicular to the edge of the table. Now raise one edge of the footboard without in any way altering the direction in which its central dotted line points, and let it be noted that the direction of that line is at right angles to the edge of the table. In the drawing the footboard is shown as tilted through an angle xvy, and the immediate result is that the axle of the rollers turns to the right in the manner indicated by the arrow, and that if the skate were pushed forward it would immediately move in a curved line pointing to the left hand. It is extremely easy to construct the model, and the move- ment may then be studied with advantage. In applying general reasoning we say that there are three axes of rotation before us, and it will be better to take the case where the board in the model is held in a horizontal position parallel to the plane of the table. In exhibiting the model it is more convenient to hold it in this way, but the drawing is clearer when the board is allowed to drop with one end on the table. The three axes of rotation are : 1. A horizontal axis through the horizontal footboard AB. 2. The inclined axis ae. 3. A vertical axis through a. Here two simultaneous rotations about the vertical and hori- zontal axes may give a resultant rotation about the inclined axis, just as horizontal and vertical forces acting on a point have an in clined resultant. But again, in the case of forces, the combination of the in- clined resultant with the horizontal component would give the other vertical component ; so here, the combination of the rotation about the horizontal axis AB with another rotation about the inclined axis ae, gives a rotation about an imaginary vertical axis, viz., that passing through a, and hence the axle of the rollers does in effect rotate about a vertical axis, just as if it wefe the axle of an ordinary carriage provided with a vertical perch-pin. Such a rotation causes the rollers to deviate on one side of the normal direction. The specification of Mr. Plimpton's patent, granted to A. V. 34 Elements of Mechanism. Newton and numbered 2190 of the series for 1865, contains draw- ings of the skate, and shows the foot stand running upon four wheels, two on each side of the roller axle. The respective axles are supported in frames capable of turning to a small extent limited by stops, and are directed downwards to a point half-way between the heel and toe of the skate. These ledges perform the function of inclined axes. The invention is described as relating to an improvement in attaching rollers to the foot stand of a skate, whereby the rollers are made to turn by the rocking of the foot stand so as to cause the skates to run in a curved line either to the right or left. It has been stated by experts on the subject that the arrange- ment of two inclined axes for causing the roller axles to converge towards the centre of curvature of the path of the skater was a completely new invention at the date of the patent, and that no machine existed at that time in which a like motion had been arrived at in so simple a manner. APPENDIX INTRODUCTORY TO REULEAUX'S SYSTEM OF TEACHING MECHANISM In the following Appendix the writer proposes to make some observations upon a distinct method of analysing combinations of mechanism, as originated a few years ago by Professor Reuleaux, of Berlin, in a work entitled the ' Kinematics of Machinery,' which has been translated into English by Mr. Kennedy. It is scarcely possible to preface the inquiry by a general state- ment of the method adopted, inasmuch as the technical terms employed would be unintelligible without explanation, and should be carefully defined in the first instance. Accordingly, we com- mence with some general definitions, and shall allow the subject- matter to develop itself gradually. Art. L Def. : When a body is constrained to move in a definite manner by means of an envelope the combination is termed a pair. The envelope forms one element of the pair, and the enclosed body forms the other element. Such pairs are of two kinds, viz. higher pairs and lower pairs, the combinations which are formed being distinguished by the general terms, higher pairing and loiver pairing. It is essential at the outset to appreciate and understand this distinction, which runs through the whole subject- matter. Def. : When a body and its envelope are in surface contact, and so constructed that every point of the body is constrained to describe a straight line, all such straight lines being parallel to each other, the two pieces form a sliding pair. Such a pair is described and shown at p. 17 of this book. Def. : When a body and its envelope are in surface contact, 342 Elements of Meclianism. and so constructed that every point in the body is constrained to describe a circle, all such circles being in parallel planes and having a common axis, the two pieces form a turning pair. Such a pair is described and shown (ante) at pp. 17 and 18. Def. : When a body and its envelope are in surface contact, and so constructed that every point in the body is constrained to describe a screw thread of uniform pitch and having a common axis, the two pieces form a screw pair. An example is afforded by any ordinary and well-made screw and nut. It appears that there are only three kinds of lower pairs, namely, those before mentioned, which provide for the following move- ments : 1. Simple straight-line motion. 2. Circular motion. 3. The combination of rectilinear and circular motion in a given fixed ratio. Def. : Where one element completely surrounds the other so that no motion is possible except that which the combination is intended to produce, the pair is said to be dosed. In order to make these definitions clear, take the case of a horizontal shaft revolving in an ordinary circular bearing made in two halves and bolted together. If there are shoulders upon the shaft just outside the bearing, so as to prevent endlong motion, the pair will be complete, or will be a closed pair. If the bolts be taken out of the bearing and the upper half lifted off, it may be that the shaft will go on rotating just as before, being held in position by its weight or otherwise. The motion is, therefore, that of a turning pair, and there is artificial closure, because the desired movement is arrived at but the combination no longer forms a closed pair. It is, in fact, evident that the shaft may be lifted off its bearing while the rotation is going on, which would make an end of the turning pair. This result is shortly expressed by saying that the pair is not a closed pair. It will presently be necessary to refer to other analogous use of the word closure. Pairs and Chains. 343 2. There are, as we have stated, only three lower pairs, but the number of higher pairs cannot be estimated. There is higher pairing when two toothed wheels are brought into gear, inasmuch as there is definite constrained motion, but with line and not surface contact between two teeth, and it is hardly necessary to remark that one tooth is not an envelope of the other. Also, there is both rolling and sliding at the parts where the teeth are in contact. Reuleaux gives, as a fundamental example of higher pairing, a rectangular block sliding in a groove of varying curvature, where the motion may be constrained and definite, but where surface contact cannot take place. It is a property of all pairs, whether of the higher or lower kind, that, if one element be fixed, a definite motion can be given to the other element. It is also a fundamental property of a pair of elements that, if one element be fixed, every motion of the second element, except the particular motion required, is prevented. 3. The term chain, or kinematic chain, is applied to any com- bination of pairs wherein motion of the several parts is admissible. But a chain is made up of links, and accordingly, when two elements of different pairs in a chain are connected together, the elements so connected form a link in the chain. While the pairs are loose and unconnected there is no chain, but, as soon as the links are formed all through, the chain is ready for the final step which makes it an operative instrument. The final step is the fixing of one link, whereby the chain is supported and can exhibit its properties. Def. : If a chain be so constructed that, when one link is fixed, each other link can only accept one definite and determinate motion, the chain is said to be closed. Def. : A closed kinematic chain, in which one link is fixed, is called a mechanism. Def. : A mechanism, when set in motion by a mechanical force applied to one of its links, is called a machine. From what has preceded, it is clear that the order of arrange- ment of the links in a chain is material, and that changing the order in which the pairs are arranged may alter the properties of the chain in the communication of motion. Since a mechanism is derived by fixing a link in a chain it 344 Elements of Mechanism follows that the number of mechanisms which can be formed from a chain is equal to the number of links in the chain. Whether the mechanisms so formed are the same or different will depend upon other considerations. 4. The word ' chain ' having a technical meaning, and being used in a sense quite different from that ordinarily attributed to it, we propose to examine certain well-known combinations from the new point of view. For this purpose we shall commence with instances of lower pairing, by taking chains formed with four pairs of simple elements. For simplicity, let s represent a sliding pair, and let T re- present a turning pair. Also let the selected chains be : F.G. I. where the first diagram represents a simple combination of four turning pairs, after which one or two sliding pairs replace the corresponding turning pairs. It appears that the four-bar motion described (ante) page no is a combination of four turning pairs, the elemei/s of which are united, so as to constitute four links, namely, c p, P Q, Q B, B c. When one link (such as c B) is fixed, the arrangement exhibits the properties of a chain or mechanism, and the necessary calculations as to the relative motions of the separate parts have already been worked out. Also, since one T is the same as another, the combina- p ^^ \\ tion will only furnish one chain and one distinct mechanism, and, although there are practical differences with ( o & reference to the fact that sometimes B Q oscillates, and at other times per- forms complete revolutions while c P revolves, there is only one Four-Bar Motion. 345 general proposition proved at pages in and 112, which gives the relative velocities of c P and B Q, or of c p and P Q. 5. Taking, however, the second combination as made up of the pairs x T, T s, it appears that we have one chain from which we can deduce two distinct mechanisms, namely, those given by fixing the positions of the links derived from T s and T T. The chain itself is called a slider-crank chain. Also, we restrict the investigation to cases where the direction of sliding passes always through the centre of one turning pair. 6. Taking the direct acting horizontal engine, with a crank and connecting rod, as shown in the diagram, there is manifestly a chain made up of three turning pairs and one sliding pair. The turning pairs are marked c, p, Q in the drawing, and the sliding pair is the block at Q moving in the slides H K, L M. Inasmuch as the position of c relatively to the slides is fixed by attaching the bearings of the shaft at c, together with the slides, to the FIG. 3. framework of the engine, it is clear that the fixed link is derived from T s by uniting one element of T with one element of s. The first mechanism deducible from the slider- crank chain is therefore to be found in any ordinary locomotive engine. 7. Another mechanism is obtained by deriving the fixed link from T T, that is, by uniting an element of one turning pair with an element of another turning pair. This is done in the Whitworth shaping machine described and figured (ante} at pages 102 and 103. 346 Elements of Mechanism. It appears that in the Whitworth machine a small shaft is in- serted into a bearing B, bored out in the main shaft, which is sufficiently large for the purpose. Hence the centres of two turning pairs are fixed, namely, the centre B, and the main shaft which carries the wheel F, and a new mechanism results, but the chain itself differs in no respect from that which exists in the ordinary crank and connecting rod. That the chain cannot vary is seen from an inspection of Fig. i, which shows that wherever s is inserted the three T's must always follow in order, and hence that the relative motion of the separate links will be the same. The slider-crank chain is therefore always one and the same chain, but two distinct mechanisms can be derived from it, namely, those enumerated above. 8. It is part of our subject to consider the manner in which by varying the details of construction any given mechanism may appear in a disguised form. FIG. 5. The slider-crank chain being taken as an example, we shall deal first with the mechanism where T s is the fixed link, as shown in the diagram. Slider- Crank Cliain. 347 1. The crank CP may be replaced by a pin p attached to the face of a plate riding upon a shaft c, as in the Whitworth shaping machine. 2. The pin at p may be enlarged so as to embrace the centre c, and the envelope of the pin may take the form of a hoop attached by a rod to the sliding block Q. This is the construction of an ordinary eccentric and rod as employed for actuating a slide valve (see page 51). If we regard the eccentric in this manner, it becomes un- necessary to prove that the combination is an equivalent for the crank and connecting rod, inasmuch as either arrangement gives one and the same mechanism. 3. The next step is to enlarge the pin at Q so as to make it embrace the shaft c with its eccentric circle, and to surround FIG. 7- this pin by a hoop furnished with sliding guides which pass through bearings at A, B, as in the diagram. It is apparent that when the shaft c revolves carrying the eccentric circle with it, the enlarged pin R will oscillate about the centre Q, and we shall have the point Q reciprocating in the 348 Elements of Mechanism. line C Q, just as in the ordinary crank and connecting-rod arrange- ment. 4. If in the normal diagram the pin at Q be united to the envelope of the slider at Q we have the fixed link s T as it stood originally. But P Q will now oscillate, and c must travel to and fro in the line Q c in order that P c may perform complete revolutions about P. The reciprocation of c has been taken advantage of in Stannah's pump. 5. We pass on to other variations of the mechanism where T T is a fixed link. The Whitworth shaping machine being the fundamental in- stance of this mechanism, it appears That if in the diagram of the Whitworth machine the crank c P be made less than CQ, the slotted bar being QP, there will be oscillation of Q P instead of rotation. This is the mechanism of a well-known quick-return shaping machine, as described at p. 104, which may take the form of an oscillating cylinder engine, by increasing the distance c Q, and by converting the slotted link into a cylinder oscillating upon trunnions at Q. 6. The mechanism of the oscillating engine is also apparent from the normal diagram. If a fixed link connects the turning pairs at P and Q, and the sliding guides H K, L M be converted into a cylinder oscillating on trunnions at E, and provided with a piston Q, it becomes apparent that the arrangement is the same as that of the shaping machine just referred to, but in a disguised form. Double Slider- Crank Chain. 349 9. The remaining combination in Fig. i is that of two turning and two sliding pairs, which is technically known as The double slider-crank chain. Here the order may be s s T T or s T s T at pleasure, and the result is that we can form two distinct chains. These chains will be expressed in a diagram as shown. FIG. 9. Fig. (i) will give three distinct mechanisms namely, those obtained by fixing the links derived respectively from ss, TT, ST; and the following examples of these several mechanisms are sug- gested. i. The mechanism derived from fixing ss is to be found in the elliptic compasses. The construction is the following : In two grooves A B, c D, at right angles to each other, are fitted two rectangular blocks bored for the reception of the pins E, F. A rod E F p, carrying a pencil at p, is attached to the pins, whereby the pencil P is competent to describe an ellipse on the 350 Elements of Mechanism. plane A B c D. The sliding pairs are the blocks and grooves, while the turning pairs are the pins E, F, with their envelopes. 2. The mechanism derived from fixing XT is to be found in Oldham's coupling, where two fixed shafts, whose directions are parallel and lie close together, are to be united by a coupling so that either may drive the other. Here each shaft terminates in a disc with a rectilinear slot or groove, and a flat-faced block having a corresponding rib or pro- jection on each side engages with and couples together the t'vo shafts, the directions of the ribs when projected on a plane parallel to either face of the block being at right angles to each other. The block will slide up and down in the grooves as the shafts revolve, and will communicate the necessary rotation. 3. An example of the mechanism where s T supplies the fixed link is to be found in the bullet machine described (ante) at P a e 55- Here the centre of motion of the shaft c is fixed, and the guides in which the shaft ABDE slides are also fixed. The first turning pair is the shaft c with its bearings, the second turning pair is the eccentric disc P working in the square block. The sliding pairs are the block and frame, together with the reciprocating shaft and its bearings. Also, the fixed link is obtained by connecting the bearing of the shaft c with the guides in which the shaft ABDE reciprocates. The mechanism is that of a crank with an infinite link giving a simple harmonic motion. Fig. (2) will give only one mechanism- namely, that derived by fixing the position of Double Slider-Crank CJiain. 351 The mechanism so obtained is not commonly met with, but may be seen in the annexed form of steering gear by J. Rapson (patent No. 8214 of 1839). By this device the turning moment on the tiller remains con- stant in every position thereof. To effect this object a block Q R slides in guides D E, F H and carries a pin which runs through another sliding block p, which can move along the tiller c B as shown, the axis of the tiller being at c. The direction of the pull on the tiller ropes is shown by the arrows, and it is obvious that the pull on the tiller caused by turning the steering wheel remains constant at any inclination of the tiller to the dotted line. Here the combination is s T s' T', where s is the sliding pair Q R with its guides, T is the turning pair at c, s' is the sliding pair made up of the block and c B. T' is the turning pair at the pin p. Also the fixed link is ST. Note. It is important that one fundamental property of a chain should be clearly understood, namely, that in the same chain, whatever may be the mechanisms formed from it, the velocity- ratio of like parts is always the same. Thus, for example, when the student has worked out the velocity ratios for the Whitworth shaping machine, he can apply his results to the oscillating engine 10. In the three elementary pairs mentioned in Art. I, the reciprocal restraint of the elements is complete. That is why the term ' closed ' is introduced. 35 2 Elements of Mechanism. If the reciprocal restraint be incomplete, some kind of closure will become necessary. Thus, if two spur wheels, as A and K, be brought into gear, they form, as we have said, a higher pair, but unless they are sup- ported and held at the right distance, they will not work properly. The reciprocal restraint which is indispensable is here supposed to be obtained by carefully holding the wheels and allowing them to turn on their axes without separating. It is obvious that this could not be done effectively, and the above method, which would be an example of force closure, is not attempted in practice, but in place thereof it is common to mount the wheels A and B upon pins at their centres and to support the pins in a bar or frame which is rigidly supported. It will be apparent that two simple turning pairs have thus been introduced, giving a chain with two turning pairs and one higher pair. This is an example of pair closure. A couple of spur wheels in gear, when supported on a fixed frame and turning freely, is to be regarded therefore as a chain with three pairs. If the position of the wheel A be fixed, and the bar carrying B be allowed to rotate about the axis of A we arrive at an epicyclic train, giving the so-called sun and planet motion. But the chain in the case of the spur wheels with fixed axes is precisely the same as in the epicyclic train, and it follows that a proposition to the effect that the relative motion of the wheel B to the wheel A in the epicyclic train is the same as its relative motion to the wheel A in the train with fixed axes is a direct consequence of the fact that the chain is the same in both cases. This example is worth consideration because it leads to the Chain Closure. 353 inference that there may be other things to be thought of beyond the grouping of pairs into chains and mechanisms. It is well known to mechanics, and is in part apparent from the results set forth in the chapter on Aggregate Motion, that an epicyclic train exhibits properties of practical value and is com- petent to furnish results which would never have been arrived at by the use of trains of wheels with fixed axes. The method now before us gives but small assistance in inter- preting the peculiarities of an epicyclic train, or in showing to a student what may be done with it. 11. Hitherto the possible defect of closure in a pair has been pointed out, and it remains to explain what is meant by defect of chain closure. An instance of defect of closure in a chain is to be found in the jointed parallelogram c P B Q. If c P be the driver, the position of c B being fixed, it is apparent that when P is crossing the line of centres, namely c B, the arm Q B may continue to rotate in the same direction as c P, or may begin to move backward. Either motion is equally possible, the chain is unclosed, but QB may be constrained to rotate in the same direction as c P by introducing a new chain CP'Q'B as already explained (ante p. 115). FIG. 14. C B \ A This is therefore an example of the closure of one chain by the addition of another chain, and such a device is technically known as chain closure. In constructing models to exhibit any special movement, it is a common thing to employ some kind of closure for getting over difficulties which may arise. Thus (ante) at page 141 it is shown that when c P = P Q in the slider crank chain there is a failure of motion when Q arrives at c, and Mr. Booth, in a patent, No. 9824 354 Elements of Mechanism. of 1843, has shown how to deal with it. The method referred to is a case of pair closure. Thus Q p is produced to R, making p R = c P, and a pin fastened at R is caused to engage with a forked or pronged opening in the right position on c D, so as to introduce a supplemental turning pair just as Q arrives at c, whereby the motion is carried on, and the throw of Q is four times the length of the crank c P, or twice as great as in the ordinary move- ment. If the student has grasped the meaning which underlies these short notes, he will comprehend the leading idea which dominates throughout the German method of studying mechanism. By regarding combinations as chains of moving parts, he may be enabled to consolidate and group together a number of possible mechanisms which in practice assume forms apparently unlike, but in one sense identical, and after the system has been more fully worked out and applied it may become of value and im- portance as an additional means of generalising results. INDEX, AGG A GGREGATE motion. 215-65 * Alarum clock, 68 Annular wheel, 21 Arbor, axis, axle, 17 Archimedean drill, 166 Axis, instantaneous, 32 BALANCE wheel of watch, 293 Bell crank levers, 40-41 Belts or bands, 25 transfer of motion by, 25 how kept on pulleys, 26 open or crossed, 26 with axes at right angles, 28 Bevel wheels, 22 teeth of, 1 88 skew, 312 . Blower, by Root, 313 Bobbin motion, by Houldsworth, 233 model to illustrate, 235 application of, 238 Bodmer, drilling machine by, 252 Boring machine, 253 feed motion of, 254 use of epicyclic train in, 255 Brace, ratchet, 152 CALLIPERS, use of, 275 ^ Cam, definition of, 68 use of, in conversion of motion, 69 analysis of curve in simple cases, 69, 70 as a heart wheel, 71 for imitating handwriting, 72 in sewing machine, 73 altered form of, 74 in striking mechanism of a clock, 74 in a lever punching machine, 75 in printing machine, 75 in a rifling machine, 76 CRA Cam described on cylinder, 77, 78 example of, 79 other examples, from printing ma- chinery, 80-2 double, 82 expansion, 83 for multiplied oscillations, 84, 128 used in carpet weaving, 128 Cartwright's Cordelier, 244 Centrode, meaning of term, 32 Change wheels, 208 Chinese windlass, 220 Chronometer escapement, 2935 Chronometric governor, by Siemens, 316 Circles, angular velocity ratio of, in roll- ing, 23 Circular, into reciprocating motion, 42 Circular motion, of a point, 8, 9 transfer of, 15 relation of angles described in, 16 transmission of, 15, 16, 20 converted into reciprocating, 42 same by wheclwork, 85 Circumduction, motion of, 1 16 Clock train, 201 Clutch, 91 Combination of motions of translation and rotation, 33 of two and three spur wheels, 86 example of same, in screwing ma- chine, 87 in planing machine, 88 of two cranks and link, numerous ex- amples, 110-27 Cones, rolling of, 23 Conical pulleys, 200 Copying machinery, examples of, 92, 195, 206, 292 Cordelier, by Cartwright, 244 Counting wheels, 284 Crab, lilting, 203 356 Index. CRA Crane, wheel-work of, 204 Crank, and connecting rod, 42 analysis of motion of, 45-8 throw of, 45 contrivance for doubling throw of, 141 same by wheelwork, 216 expanding, 263 variable, rotation of, 100 Cranks, two, with link, 110-27 one oscillating, 113 applied in wool combing, 118 and in ventilating machine, 119 and in sewing machine, 120 and in shearing machine, 121 and in Stanhope levers, 123 multiple rotating, 115-16 example of, 247 Crown wheel, 21 escapement, 59 Curvature, circle of, 171 Cycloid, definition of, 180 Cylindrical gauges, 279 "pVEAD points, 149 4-/ Difference gauges, 280 Differential pulley, for carriages, by Sax ton, 217 screw, 218 pulley, by Weston, 221 motion, for cotton spinning, 236-40 Disc and roller, 309 applied for continuous indicator, 310 Drilling machine, principle of, 249 Drill spindle, motion of, 249 example of, 250 by Bodmer, 252 by Sir J. Whitworth, 256 Driver, meaning of term, 20 ECCENTRIC circle, properties of, 49 Eccentric, throw of, 50 construction in steam engine, 51 example of, 52 use of in drilling machines, 53 equivalent for crank and connecting rod, 50 same for crank and infinite link, 54 example of, 55 End measure, standard bars, 283 conversion into line measure, 275 machine for, by Sir J. Whitworth, 276 Epicloid, definition of, 173 Epicyclic tram, 222 theory of, 224 Epicyclic train, for straight line motion, 229 model to illustrate, 230 for astronomical models, 231 with bevel wheels, 232 in spinning machinery, 233 for slow motion, 240 compared with ordinary train, 242 in rope-making, 245 in Cordelier, 243-8 Equation clock, 243 Escapement, simple form, 58 crown wheel, 59 anchor, 62 with recoil, 62 dead beat, 66 examples of recoil, 68 pin wheel, 68 chronometer, 293-5 detached lever, 296 horizontal, by Graham, 297 FEATHERING paddle wheel, 117 * Feed motion, 155 of rifling machine, 155-6 by Sir J. Whitworth, 157 silent, 159 by Worssam, 160 of boring machine, 254 same by epicyclic train, 255 Ferguson's paradox, 227 Follower, meaning of term, io Foot-second, definition of, 5 Four-bar motion, 110-15 Fusee, 298 theory of, 299 flat spiral, used in cotton spinning ma- chinery, 301 other examples, 302 principle of, winding-on motion, 303 going, by Harrison, 305 GAUGES, standard cylindrical, 279 difference, 280 Gear, meaning of term, 22 Geneva stop, 335 applied in planing machine, 336 Glass-grinding machine, 262 Governor, of steam engine, by Watt, 314 chronometric, by Siemens, 316 Graham's cylinder "escapement, 297 Grasshopper engine, parallel motion, 142 Guide pulleys, 30-1 i H ARMONIC motion, simple, 9 amplitude of, 10 Index. 357 HAR Harmonic motion, period of, 10 phase of, i to analysis of, n composition of, 12 model to illustrate, 14, 42 application of, 39 Harrison's going fusee, 305 Heart wheel, 71 modified form of, 74 Hooke's joint, 329-32 Horizontal escapement, 297 Hunting cog, 214 Hypocycloid, definition of, 173 'property of, 176 T OLE wheel, 194 * Indicator, invented by Watt, 257 by McXaught, 258 by Richards, 260 diagram, 258 application of parallel motion in, 261 continuous, by Ashton and Storey, 310 Instantaneous axis, 32 Intermittent motion, example of, 57 theory of same, 58 by segmental wheels, 108-9 Involute teeth, 183-5 TACK, lifting, 38 J Joint, toggle, 125 applied in printing machinery, 126 in carpet weaving, 127 for multiplied vibrations, 128 Hooke's, theory of, 329 double Hooke's, 331 J EYLESS watch, 306-7 LATHE, Blanchard, 195 screw cutting, 206 Lazy tongs, 215 Lemie'le's ventilator, 119 Lever, bell crank, 40 nipping, 158 escapement, 296 Levers, Stanhope, 122 theory of same, 124-5 of Lagarousse, 162 modified form of same, 163 Line and end measure, 275 Link-work, meaning of term, no Link and two cranks, 1 10 theory of same, 111-13 one crank oscillating, 113-14 Link and two cranks, both cranks rota- ting, 115 applied in wool combing, 118 and for shuttle motion in sew- ing machine, 129 also in machine for cutting metals, 121 also in Stanhope levers, 122 Link motion, by Stephenson, 320 by Gooch, 322 Logarithmic spiral, setting out curve, 326 MACHINE for shaping the naves of wheels, 291 shaping, with quick return, 103 another form, 104 for drilling, 250 for measuring. 276 Mandril of a lathe, 18 Mangle rack, 105 for Cowper's printing machine, 106 double, with segmental wheel, 108 single, with segmental wheels, 108 Mangle wheel, 105 with quick return, 106 Marlborough wheel, 196 Masked ratchet, use of, 291 Measuring bars, preparation of, 273 Measuring machine, by Sir J . Whitworth, 276 millionth, 281 Mitre wheels, 23 Motion of rotation, 33 of translation, 33 of hour hand, 202 AJIPPING lever, 158 L^ Numbering machine, 286-291 OSCILLATION, centre of, 61 model to illustrate, 62 multiplied, 127 Oval chuck, theory of, 263-5 PADDLE wheel, feathering, 117 Pairs, elementary, 17 examples of, 17 cylindric, four parallel, no Pallets, 58 Pantograph, 137 in parallel motion of beam engine, 137 Paradox, Ferguson's, 227 Parallel axes, transfer of motion between, 323-4 358 Index. Parallel motion, by Watt, 129 theory of, 130-4 of beam engine, 137 of marine engine, 138 application of, 139 in grashopper engines, 142 in compound engines, 146 - in Richards' indicator, 261 Paul, meaning of term, 150 action of, 151 Pauls, of unequal length, 153 application of, 154 Peaucellier's straight line motion, 143 Pendulum, simple, bo law of oscillation, 60 centre of oscillation of rigid, 61 model to illustrate same, 62 mechanical action of, 65 Pin wheels, theory of, 179 Pinion, meaning of term, 22 lantern, 179 Pitch, diametral, 168 circular, 169 Photometer, Wheatstone's, 177 Planing machine, reversing motion, 89 feed motion, 157 Plimpton's roller skate, 338 Point, motion of a, 4 circular motion of, 9 harmonic motion of, 9 Power, telodynamic transmission of, 29 Pulley, convex rim of, 26 single movable, 219 Pulleys, fast and loose, 27 with inclined axes, 28 guide, 30 speed, 196 theory of same, 197 applied in lathe, 199 conical, 200 RACK and pinion, forms of teeth, 182 teeth derived from involute of circle, 188 Rack, mangle, 105 -- double, 108, 216 Ratchet wheel, 150 for driving in alternate directions, I 5 I used in lifting jack, 152 with click and hook, 162 masked, 163-5 compared with lifting pump, 167 Ratchet brace, 152 Reciprocating into circular motion, 148 examples, 166 Rectangular bars, 273 Reversing motion, by spur wheels, 85 with quick return, 86, 88 example of, 87 by disc wheels, 90 by bevel wheels, 91 by clutch, 91 adopted by Sir J. Whitworth, 92 example in rifling machine, 94 by pulleys and belts, 96 by slit bar and crank, 97 theory of same, 97-101 by double eccentric and link motion. j Roberts's winding-on motion, 303 I Rolling curves, theory of, 324 examples, 325 for quick return, 327 in printing machinery, 328 Root's blower, 313 Rope, twist of, 245 model to illustrate same, 247 extra twist, apparatus for, 248 CAXTON'S differential pulley, 217 J Scraping tool, 269 Screw surface, definition of, 34 Screw, pitch of, 34 right or left handed, 34, 207 single or double threaded, 35 Screw threads, 34 mechanical properties of, 34-6 uniform system of, by Sir J. Whit- worth, 37 Screw coupling, 219 Screw cutting, theory of, 205 lathe for, 206-8 Screw and worm wheel, used as rack and pinion, 256 Sector, use of, 170 Segmental wheels, 108, 109 Shaping machine, with quick return, by Sir J. Whitworth, 102 for locomotive wheels, 104 Siemens's chronometric governor, 316 Silent feed, 159 Similar curves, 135 Skate, Plimpton's, 338 Skew bevels, 312 Slit bar motion, 97 theory of, 97-101 Slow motion, by epicyclic train, 240 Snail, 308 Speed pulleys, 196-7 Spiral logarithmic, 326 Standard gauges, 279 Standards of length, 283 Stanhope levers, 123 Index. 359 Star wheel, 337 Step wheels, 312 Straight line motion, by Scptt Russell, 140 theory of, 140 exact, by Peaucellier. 143 multiple, 146-7 by epicyclic train, 229 Sun and planet wheels, 226 Surface plate, 268 appearance of, 270 method of preparing, 271 adhesion of, 272 Swash plate, 55 theory of, 56 TEETH, involute, theory of, 183 action of same, 185 contact of, in wheelwork, 213 Teeth of wheels, theory of, 172 general solution, 174 first case, 177 with radial flanks, 178 second case, 178 with involute curves, 183-5 general considerations, 185-7 lor bevel wheels, 188 Throw of crank, 45 doubled, 141 Toggle joint, 125 Tooth, root or flank, 21 point of, 21 pitch of, 21 Trains of wheels for given purposes, 209-14 Trains, epicyclic, 223-48 True plane, meaning of term, 267 Truth of surface, importance of, 266 Twist of a rope, 247 \7 ELOCITY, how measured, 5 how represented, 6 angular, 18 measure of, 18 Velocities, parallelogram of, 7 triangle of, 8 WOR Velocities, diagram of, in harmonic mo- tion, 13 Velocity ratio, 20 with parallel axes, 24 with inclined axes, 24 between crank pin and piston in direct-acting engine, 47 diagram of same, 48 in cam motion, 77 also for oscillating engine, 99 between crank and slit bar, 99-100 in four-bar motion, in Vibrations, multiplied, 128* , keyless, 306-7 * v Watts' indicator, 257 parallel motion, 129 Weston's differential pulley, 2*1 Wheel, toothed, 20 pitch circle of, 20 spur, 21 crown, 21 annular, 21 bevel, 21 face, 21 worm, 37 segmental, 108-5 Marlborough, 196 purchase, 216 with racks for doubling throw of crank, 216 Wheels, segmental, 108-9 in trains, 190 examples of, 192-4 in clock train, 201 for motion of hour hand, 202 in lifting crab, 203 in crane, 204 step, 312 Whitworth, Sir J., measuring machine, refer to Chapter VIII. Windlass, Chinese, 220 Worm wheel, 37 Worm barrel, 8r with movable switch, 82 INDEX TO APPENDIX. DULLET machine, 350 f^HAIN, or kinematic chain, 343 ^ of four turning pairs, 344 of three turning and one sliding pair, 345 slider crank, 345 - property of, 351 of three pairs, 352 Closure, meaning of term, 342 defect of, 353 example of chain, 353 same of pair, 353 Compasses, elliptic, 349 Coupling, Oldham's, 350 TROUBLE slider crank chain, 349 elementary mechanisms, 349, 350, 351 "PCCENTRIC, example of slider " crank chain, 347 Epicyclic train, 352 is chain of three pairs, 353 "CORCE closure, 352 Four-bar motion, 344 only one mechanism, 344 T INK of chain, 343 MACHINE, 343 Mechanism, definition of, 343 QSCILLATING engine, 348 ^ belongs to slider cr; chain, 348 crank TDAIR, meaning of the term, 341 sliding, turning, screw, 341, 342 closed, 342 closure, 352 Pairing, lower, 344 higher, 352 QUICK RETURN shaping machine, 348 "DAPSON'S steering gear, 351 CLIDER crank chain, 345 *^ elementary mechanisms, 345, 246 other forms, 347, 348 without rods, 347 Stannah's pump, 34-8 WHITWORTH shaping machine, 348 what chain it belongs to, 348 same mechanism as oscillating engine, 348 tfottiswoode &> Co. Printers, New-street Square, London. UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. NOV 4 APR 2 8 AUG 3 AUGll RECTJ Form L9-lCOm-9,'52(A3105)444 UC SOUTHERN REGIONAL LIBRARY FACILITY Untj TJ 115 A 000316609 7 6"