I ·i PART THE FIRST. ' ' CHAPTER I. ON TRAINS OF MECHANISM IN GENERAL. 18. MECHANISM may be defined to be a combination of parts, connecting two or more pieces, so that the motion of one compels the motion of the others, according to a law of connexion. depending on the nature of the combination. The motion of elementary combinations are aingle or ag­ gregate. Aggregate motions are produced by combining in a peculiar manner two or more single combinaiions, as will hereafter appear in Part II. All that follows in this Part relates to the single combinations alone. 19. The motion of every piece in a machine being defined, as in the Introduction, by path, direction and velo­ city, it will be found, that its path is assigned to it by its connexion with the frame-work of the machine; hut its direction and velocity are determined by its connexion with �me other moving piece in the train. Thus a wheel de­ scribes circles, because its axis is supported by holes in the frame; but it describes them swiftly or slowly, backwards or forwards, by virtue of its connexion with the next wheel in the train, which lies between it and the moving power. This connexion afef' cts the ratio of the velocities, and the Telative direction of motion of the two pieces in ques­ tion, but its action is independent of the actual velocities or · -··- ·----- - THA,INS OF MECHANISK. 15 directions of eithet piece*, as in the familiar example already quoted of the two hands of a clock, where the connexion by wheel-work is so contrived, that while one hand revolves uniformly in an hour, the Qtber shall revolve uniformly in twelve. But this connexion has this 1nore general property, that it will also compel the latter to revolve with an angular velocity of one twelfth of the former, whatever be the actual velocity communicated to either; as, for example, when we set the clock by moving the minute-hand rapidly to a new place on the dial, and similar!y with respect to direction, the two hands will always revolve the same way, whether we turn one of them backwards or forwards. Since Mechanism is a connexion between two or more bodies, governing their proportio11al velocities and relative directions, and not affected by their actual velocities or directions; it follows that a systematic arrangement of the principles of mechanism must �e based upon the proportions and relations between the velocities and directions of the pieces, and not upon their actual and separate motions. 20. Proportional velocities may be divided into those in which the ratio is constant, and those in which it varies. Let Y and 11 be the velocities of two bodies, then y is V the velocity ratio; and if the velocities are uniform, let S, a T; y s h� the spaces described in the same time ••• - a:::: - V 8 a constant ratio; consequently between uniform velocities the velocity ratio is constant, which indeed is sufficiently obvious. If however the velocities be not uniform, and yet the Vel_ocity ratio constant, let the bodieg in any successive · • W• aull &g4 • Aw CQDtriv.ncea in which thia ia not atrictl1 uu.e with respect to the direction, t but they are not of a nature to vitiate the generality of he priftdpJ.. 16 TJl.A.INS OF llECHANISM. intervals of time T, T,, T,, ... move with velocities V, V,, V,, .•. and v, v,, v,, respectively, of any different magni­ tudes, but so that the two velocities at the same instant always preserve the same ratio; V V V .• . -�-, =-,,, &c. ... =c. v, " II Hence if S, S,, S1o1 ••• and a, a,, s,, be the spaces de­ scribed with these velocities by the two bodies in the inter­ vals T, T , T respectively, we have I II S - S-, - S- S - - + S,+ So11+ ... C 11 ...... - --------'-'--- . 8 s, s,, s+s,+s,,+ ••. And as this is true whatever be the magnitude oi the intervals of time, it is also true when they are taken so small that the changes of velocity become continuous, and therefore when the 1Jelooity ratio is constant it is obtained by comparing the entire spaces described in the same in­ terval of time, whatever changes the actual velocities of the bodies may have undergone during that tim-e. And in the same manner it may be shewn that in re­ volving bodies the angular velocity ratio, if constant, is equal to the ratio of the synchronal rotations, notwithstand­ ing the velocities of rotation may vary, and also to the inverse ratio of the periods if the angular velocities be uniform. When the velocity ratio varies, the relations of motion between two pieces may often be more simply defined by means of the law of their corresponding positions than by the ratio of their velocities. 21. With respect to actual direction we have seen that it has only two values, but the relation of direction between two bodies moving in given paths may be con­ veniently divided into two classes. In the first, while one continues to move in the same direction, the other shall al110 1'RA1NS OF ?vlECHANISll\ . 17 persevere in its own �irection; but if one change the oth�r shall change. To this class he]ongs the clock-hands ; and 111 this instance both hands move the same way round the circle. But this is not necessary; it may· be that when one piece revolves to the right the other tnay revolve to the left, and vice versa, as in a pair of flatting rollers ; or again in the old simple mangle, so long as the handle is turned in one direction, the bed of the mangle will travel forwards, but when the motion of the handle is reversed, the bed of the mangle also returns. In all these _cases the directional relation is constant. In another class the· connexion is of this natu1·e, that while one body perseveres in the same direction, the other shall change its direction; as, for ex­ ample, in a saw-mill. The saw-frame moves up and down, changing its direction periodically, but the piece from which it derives this motion revolves continually in the same direction. In cases of this kind the directional relation changes. 22. We have thus two kinds of directional relation� and two of the velocity ratio, by means of which it will appear, that all the simple combinations of mechanism, for the modification of motion, may he distributed into three classes:- CLASS A. Directional relation and Velocity 'l'atio constant. CLAss B. Directional relation constant-Velocity ratio varyin_g. CLASS C. Directional relatwn changing periodical­ ly-Velocity ratio either constant or 'Va'ryi• ., ng. . ' This latter class might have been divided into two, by arranging the constant and variable velocities under 2 18 T&AINS OF,, MECHANISM, separate heads; but it will be found that the contrivances for effecting these two conditions are so much alike, that this division would only introduce needless complication. 23. In those classes of combinations in which either the velocity ratio or the directional relations change, it will generally happen, from the very nature of mechanism, that the changea will recur in cycles. But, since these changes are independent of the actual velocities of the bodies, the cycles cannot be periodic in time, but will recur with reference to the path of one of the moving bodies, the same velocity ratio and directional relation generally corresponding to the arrival of this body at the same point of its path, and so on in succession for the different phases. The true argument*, as it is called, of the change being in fact the path of one of the bodies, and not the time of its motion. 24. A train of mechanism is composed of a series of moveable pieces, each of which is so connected with the frame-work of the machine, that when in motion every point of it is constrained to move in a certain path, in which, however, if considered separate}y from the other pieces, it is at liberty to move in the two opposite directions, and with any velocity. Thus wheels, pullies, shafts, and revolving pieces generally, are so connected with the frame of the machine, that any given point is compelled when in motion to describe a circle round the axis. Sliding pieces are compelled by fixed �ides to describe straight lines, and so on. 25. These pieces are connected in successive order' either by contact or by intermediate pieces, so that when • Vide Whewell's Philoaophy ofthe Ind,utive Sciencei, Vol. 11• p. 542. TitAJNS OF MECHANISM. 19 the first piece in the series is moved from any external caus e, it compe ls the second· to move, which again gives motion to the third, and so on. 26. The act of giving motion to a piece is termed driving it, and that of receiving motion from a piece is termed following it. The piece or pa ,rt of a piece which · is appropriated to transmitting motion to the next is the driver, and the part which receives motion is the follower. 27. The law of motion of one piece in a train may differ in any ,vay from the law of motion of the next piece in the series, and the change is efef cted by the mode of connexion. The s.,y stematic examination of the different cases under which these changes may be arranged, consti- tutes the principles of mechanism. One piece may drive another either by immediate con­ tact or by an intermediate or connecting piece. The dif­ ferent modes of doing it will be best explained by taking an example of each in its most elementary and general shape. 28. Communication of Motion by l1ontact. Let .AC, BD be two successive pieces of a train of mechanism, moving . on centers .A and B respectively, and let 1 BD be the driver, and .AC the follo,ver, the curved edge of the first touching that of the second. If the driver be C moved into a new position 11ear the first, as shewn by dotted lines, its edge will press that of the follower, and move it also into a new position. Let m be the point of contact in the first position, and let n and p be the respective points of the edges that come into contact in the second position as at r. Now, during the motion every 2-2 20 'l'BAINS 01'' MKCHANISM. point between p and m in one curve has been successively in contact with some other point between n and m in ·the other ; and if fron1 the nature of the curves nm is not equal to pm, sliding must have taken place between the edges through a space equal to the difference. But if nm be equal to pm no sliding will have happened. In the first case the communication of motion is said to be by sliding contact, and in the second by rolling contact. 1'his mode of action supposes either that the curves are both convex ; or should the curvature lie in the same direction, that the convex edge has a greater curvature than that of the concave edge at the point of contact. If this be not the case, successive contacts may take place at discontinuous point:-;. 29. Communication o.f Motion by Interrnediate Pieces. Let AP, BQ be a driver and follower, moving on centers at A and B respectively, and 2 A let a rod or link, PQ, be jointed at its extremities to the driver and follower at P and Q. Then, if the driver be moved into a new position A p, it will by means of the. link place the follower in a position B q. If the driver Q push the follower before it, the link mmt be rigid, but if the driver drag the follower after it, the link may be flexible, the principle of linkwork only requiring that the connexion between the link and its pieces shall be at constant points, and the distance between the two points of attachment invariable. Let A CE be a driver, BDF a follower whose centerF of motion are A and R, and whose f f#JN,e1 s. II . n ;\ . .. �,, ;· · ,·(" ' ·, -� ".· ',, 1 •· ) CoR. 2. Produce AP and BQ to meet in K, and drop KL perpendicular to PQ, then pm : Pm :: PL : KL, and qn : Q n :: KL : QL ; whence, compounding, pm : Qn :: PL : QL, which shews that L is the intersection of the two positions of the link. Coa. 3. If the path of the pieces be rectilinear, or any other curve than a circle, let Pp, Qq be the elements of the paths ; . then since Pm- = qn,, Pp . cos p Pm = Qq . cos Qqn ; . -Pp - c-os- Q-qn . . Qq cospPm , where the angles are those made by the link with the re­ spective directions of motion ; and hence The linear velocities are to each other inversely as the cosines of the angles which the link makes with the respec­ tive paths. 33. To find the Velocity Ratio in Cootact Motions. Let .A, B be the centers o•f mo- 7 tion of two pieces connected by the contact of curved edges, and let M be the point of contact in • • a given pos1t •1 on. Let P, Q be the respective centers of curvature of the edges, corresponding to the point of con- ,_ tact M ; join PQ; therefore this line will pass through the TltAINS OF MECHA. NISM. 25 Joint of contact M. Now in considering the communication �f motion through a small angle, the circles of curvature roav be substituted for the curved edges. But the line PQ being thus equal to the sun1 of the radii of two circles, will be constant during that small motion, and hence the motion be the same if a pair of rods .AP, BQ, connected by a link PQ, be substituted. Join .AB, meeting PQ in T, then by the last proposition, the angular motions of the arms .AP, BQ are to each other as the segments BT, .AT, and PQ is the common normal to the two curves ; whence in the com­ munication of motion by contact, the angular motions of the pieces a1·e inve1'sely as the segments into which the common normal divides the line of centers. 34. To find the quantity of sliding in Contact Motions. Let A and B be the two centers, M the point of contact, MD the 8 common normal ; then, Suppose the curves to move '.D into the new positions, shewn by the dotted lines, and very near .•····- ······:P.:--+��-·� the first, and let m be the new point of contact, and p and •n the ••' • new positions of the points which ••• • were in contact at M. . .. Now since every point of .. mn ._ must have necessarily touched . some point or other of mp, during the change from the first to the :4. second position, a sliding or shifting of the surfaces must have taken place equal to the difference between mp and mn. Join pn, which will ultimately represent this dif­ ference, and become a right . line perpendicular to the 26 TRAINS OF MECHANISM. normal MD. Also .1.Hp, Mn are ultimately perpendicular to· AM, BM. In the small triangle Mpn, the sides Mp, Mn, pn are respectively perpendicular to AM, BM, MD, and conse­ quently make mutually the same angles with each other as these latter lines ; pn sin therefore -- = --pM-n = sin B1HA pM sin pnM sin , DMB pn in which expression is the ratio of the sliding to the pM elementary quantity of motion of the point of contact in one of the pieces, DMB is the angle between the normal and the radius of contact of the other piece, and sin BMA = sin (BAM + ABM) = the sine of the sum of the angular distances of the radii of contact fron1 the line of centers. . . pn sin BM..dS1m1larly, nM = sin DMA . 35. From these expressions it appears that in the small triangle pnM, pn can only vanish with respect to nJlf or pM when sin BMA vanishes; that is, when the radii of contact coincide with the line of centers. But when pn vanishes the sliding vanishes, and the contact becomes Toll­ ing contact. Hence it appears that in rolling contact the curves must he so formed, that the point of contact shall always lie on the line of centers. Also the common normal .will cut the line of centers at the point T, (Fig. 7,) which will be now the point of contact, and therefore in rolling contact, the angular velocities are inversely as the segments into which the point of contact divides the line of centers. 36. Let the curves be a pair of involutes of circles, and let BD be a perpendicular from B upon MD. But this perpendicular is constant in the involute ; . . 1'RAI�S OF lIJi:CHA �lSM. BD I therefore sin DMB = BJlf cc BJ.lf ; .. . pn o: BM sin BJ.lf A vari es as the perpendicular upon x p M AM produced. B t if the curves be an epicycloid turning on the center A, in :ontact with a radial line ,vhich turns round B ; then DMB is a right angle, • n and p cc sin BJ.tl .A . pM 37, Tofind the Velocity Ratio in wrapping connea1ions. .. . .,1 9 s ..... q __ ,. .- · ,r ' . .• ,' ' t { •�' } ..• '• . .. • • •. . • '' • ' . '' ' ' ' ..' •' I ' . • ' . '.., \ . . ....,.;; ' . .'. . . . .. .. ..... --�-e··· •, Let A, B be the centers of motion, PQ the ,vrapping con­ nector touching the curves at P and Q, and let the point P be moved to p very near to its first position, then ,vill Q be drawn to q, and the connector will touch the curves in t,vo new points of contact, which may be 1· and s respectively. Now, in the action of wrapping or unwrapping, the con­ nector touches the curves in a series of consecutive points · between q and S or p and r, and ultimately q coincides with S and p with r. The extre1nities of the connector may therefore be considered at any given 1noment as if jointed to the two curves at the points of contact, and turning upon these points in the manner of a link. The relative velocities of the curves arc therefore n1on1entarilv ,; 28 TRAINS 01'' MECHANISM. the same as if AP, BQ were a pair of rods connected by a link PQ. Hence the angular velocities of the pieces are to each other inversely as the segm-ents into which the connector divides the line of cente·rs. 38. If the line of direction of the link in link-work, of the con1mon normal to the curves in contact motion, and of the connector in wrapping motion, be severally termed the line of action, we can express the separate propositions which relate to the Velocity Ratio, by saying that the angular velocities of the two pieces are to each other in­ versely as the segments into which the line of action divides the line of centers, or inversely as the perpendiculars from the centers of motion upon the line of action. I have confined these investigations, for the present, to motions in the same plane. The cases of motions in dif­ ferent planes, are more simply examined as the individual cornbinations which require them occur. 39. It has been shewn that the principal pieces which constitute a train of mechanism are compelled to move each in a given path. Now it is generally sufficient to consider this path as a circle, for in fact the pieces always either revolve or move in right lines ; and the contrivances by which motion is communicated in a rectilinear path, are the same as those by which it is given to a revolving piece, and derived from the latter by that familiar geometrical artifice by which a right line is considered as the arc of a circle whose radius is infinite. I t will presently appear that, in this way, much complication of classes will be avoided. Thus, for example, a pinion driving a rack is plainly the same contrivance as a pinion driving a toothed wheel, the rack being considered as a portion of a toothed wheel whose radius is infinite. 'fRA INS oF !I.I F.CHA NJSM. . t ru e, that J)ieces in a train may be fo und ,vhich I t JS . . . descr1" b o ther paths ' such as elliptical, ep1cy clo1dal, or s1nu- e · . d by corn b1.n1n ous bnes ; b ut the se are always produce g . . together various c1rcu lar motion�s, and therefore th e motion . . of the piece is act ually produced by pieces that travel 111 . ctrcu1 a r pat h s. Cases of this kind fall under the head of . Aggregate Mo ttons, to which a separate Part of tlu . s ,vo1. !-.. has been assigned. 40. The path of a revolving piece may be consider(•d as unlimited in extent in either direction, since the piece· may go on perfot·tniug any nu1nber of rotations in the sanH· direction. But a piece that travels in a right line is neces­ sarily limited in its motion either ,vay, to the length of that line. Again, the n1et hod by \.vhich motion is com1n unicatr•cl from one piece to another, may be of such a nature as to limit the motion of these pieces, although they n1ay be capable of unlimited motion, considered apart fron1 thi� connexion. For example, if the driver and follo\.ver he revolving cylinders, . and therefore capable of unlirnited motion, the communication of motion may be effected by a string whose ends are fixed one to each cylinder, and coiled round it, so that when the driver revolves it sha]l com1nu­ nicate motion to the follower by coiling the string round itself and uncoiling it from the follo\rer ; in ,v hich case the rotations of each cylinder are limited to the number of coils which its circumference contains ,vhen the other is ernpty. It appears, then, that the motion of a pair p ofie c coes n nm eca ty e d be limited either by the figure of one or b t oh te h oi r paths, f or by the nature of th l ei im r i ed c onnexion ; � c a o n an d nexion may be fo rmed betw vi ec ee n v ue rr ts lo a imr , ited paths, but if either the l i pm a.i tt he s d, o r the cot nh ne e xim ono he tion of the pieces \Vil] be lirnited. TRAINS OF l\1F:CHANli-�1. In classifying the con1munication of rnotion, ho,vcver, the union of unJi1nited connexions ,vith liniited paths, ,vill require but little attention, as the modifications to ,vhich they lead are in general sufficiently obvious ; but the dis­ tinction between limited ancl unlimited n1ethods of com1nu­ nication is of 1nore in1portance. -�----.. :-·- -· ---