OB21 OBII their pitch cones; FIG. 106. OC1, OC2 their axes; OI their
line of contact. Perpendicular to OI draw AIIA2, cutting the axes in Al, A2; make the outer rims of the patterns and of the wheels
(30)
z
portions of the cones A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about 0 for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs ID1, ID2, with the radii Ail, A2I ; those arcs will be the developments of arcs of the pitchcircles B1I, B21, when the conical surfaces A1B1I, A2B2I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spurwheels; then wrap the developed arcs on the cones, so 'as to make them coincide with the pitchcircles, and trace the teeth on the conical surfaces.
§ 55. Teeth of SkewBevel Wheels.—The crests of the teeth of a skewbevel wheel are parallel to the generating straight line of the hyperboloidal pitchsurface; and the transverse sections of the teeth at a given pitchcircle are similar to those of the teeth of a bevelwheel whose pitch surface is a cone touching the hyperboloidal surface at the given circle.
§ 56. Cams.—A cam is a single tooth, either rotating continuously or oscillating, and driving a sliding or turning piece either constantly or at intervals. All the principles which have been stated in § 45 as being applicable to teeth are applicable to cams; but in designing cams it is not usual to determine or take into consideration the form of the ideal pitchsurface, which would give the same comparative motion by rolling contact that the cam gives by sliding contact.
§ 57. Screws.—The figure of a screw is that of a convex or concave cylinder, with one or more helical projections, called threads, winding round it. Convex and concave screws are distinguished technically by the respective names of male and female; a short concave screw is called a nut ; and when a screw is spoken of without qualification a convex screw is usually understood.
The relation between the advance and the rotation, which compose the motion of a screw working in contact with a fixed screw or helical guide, has already been demonstrated in § 32 ; and the same relation exists between the magnitudes of the rotation of a screw about a fixed axis and the advance of a shifting nut in which it rotates. The advance of the nut takes place in the opposite direction to that of the advance of the screw in the case in which the nut is fixed. The pitch or axial pitch of a screw has the meaning assigned to it in that section, viz. the distance, measured parallel to the axis, between the corresponding points in two successive turns of the same thread. If, therefore, the screw has several equidistant threads, the true pitch is equal to the divided axial pitch, as measured between two adjacent threads, multiplied by the number of threads.
If a helix be described round the screw, crossing each turn of the thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal helix, may be called the normal pitch; and when the screw h'is more than one thread the normal pitch from thread to thread i lay be called the normal divided pitch.
The d stance from thread to thread, measured on a circle described about the axis of the screw, called the pitchcircle, may be called the circumferential pitch; for a screw of one thread it is one circum
ference; for a screw of n threads, one circumference
n
Let r denote the radius of the pitch circle;
n the number of threads;
0 the obliquity of the threads to the pitch, circle, and of the normal helix to the axis;
pitch,
Pa=pa  the axial {
n divided pitch;
P pitch,
Pn the normal
n — p" divided pitch; P, the circumferential pitch;
2 irr Pc =pa cote=pnCOSB= ,
pa=pasece=p~tane=2,rrtan0 n
pa =p~ sin B =pa cos o 2irr sin 0 n
If a screw rotates, the number of threads which pass a fixed point in one revolution is the number of threads in the screw.
A pair of convex screws, each rotating about its axis, are used as an elementary combination to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws their threads must be parallel; and their line of connexion is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles:
I. If the screws are both righthanded or both lefthanded, the angle between the directions of their axes is the sum of their obliquities; if one is righthanded and the other lefthanded, that angle is the difference of their obliquities.
II. The normal pitch for a screw of one thread, and the normal divided pitch for a screw of more than one thread, must be the same in each screw.
Hooke's wheels with oblique or helical teeth are in fact screws of many threads, and of large diameters as compared with their lengths.
The ordinary position of a pair of endless screws is with their axes at right angles to each other. When one is of considerably greater diameter than the other, the larger is commonly called in practice a wheel, the name screw being applied to the smaller only; but they are nevertheless both screws in fact.
To make the teeth of a pair of endless screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to form a cutting tool; the larger screw, or " wheel," is cast approximately of the required figure; the larger screw and the steel screw are fitted up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure.
§ 58. Coupling of Parallel Axes—Oldham's Coupling.—A coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction with the same mean angular velocity. If the axes of the shafts are in the same straight line, the coupling consists in so connecting their contiguous ends that they shall rotate as one piece; but if the axes are not in the same straight line combinations of mechanism are required. A coupling for parallel shafts which acts by sliding contact was invented by Oldham, and is represented in fig. 107. Ci, C2 are the axes of the two parallel shafts; Dl, D2 two disks facing each other, fixed on the ends of the two shafts FIG. I07.
respectively; EIEI a bar sliding in
a diametral groove in the face of D1; E2E2 a bar sliding in a diametral groove in the face of D2: those bars are fixed together at A, so as to form a rigid cross. The angular velocities of the two disks and. of the cross are all equal at every instant; the middle point of the cross, at A, revolves in the dotted circle described upon the line of centres C1C2 as a diameter twice for each turn of the disks and cross; the instantaneous axis of rotation of the cross at any instant is at I, the point in the circle C1C2 diametrically opposite to A.
Oldham's coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practibility or permanency of their exact continuity.
$ 59. Wrapping Connectors—Belts, Cords and Chains.—Flat belts of leather or of gutta percha, round cords of catgut, hemp or other material, and metal chains are used as wrapping connectors to transmit rotatory motion between pairs of pulleys and drums.
Belts (the most frequently used of all wrapping connectors) require nearly cylindrical pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on the pulley, unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one pulley on to another of equal size alongside of it, by pressing against that part of the belt which is moving towards the pulley.
Cords require either cylindrical drums with ledges or grooved pulleys.
Chains require pulleys or drums, grooved, notched and toothed, so as to fit the links of the chain.
Wrapping connectors for communicating continuous motion are endless.
Wrapping connectors for communicating reciprocating motion have usually their ends made fast to the pulleys or drums which they connect, and which in this case may be sectors.
The line of connexion of two pieces connected by a wrapping connector is the centre line of the belt, cord or chain ; and the comparative motions of the pieces are determined by the !principles of § 36 if both pieces turn, and of § 37 if one turns and the other shifts, in which latter case the motion must be reciprocating.
The pitchline of a pulley or drum is a curve to which the line of connexion is always a tangent—that 's to say, it is a curve parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the wrapping connector.
Pulleys and drums for communi
cating a constant velocity ratio are circular. The effective radius, or radius of the pitchcircle of a circular pulley or drum, is equal to the real radius added to half the thickness of the connector. The
then
(31)
angular velocities of a pair of connected circular pulleys or drums are inversely as the effective radii.
A crossed belt, as in fig. 1o8, A, reverses the direction of the rotation communicated ; an uncrossed belt, as in fig. Io8, B, preserves that direction.
The length L of an endless belt connecting a pair of pulleys whose effective radii are r2, with parallel axes whose distance apart is c, is given by the following formulae, in each of which the first term, containing the radical, expresses the length of the straight parts of the belt, and the remainder of the formula the length of the curved parts.
For a crossed belt:
L = 21tc2 — (ri + r2)2 + (ri + r2) 1 sin iri ` r2) ; (32 A)
and for an uncrossed belt:
(7—2 I
L = 2 %' fc2 — (ri — r2)9 + it (rl + r2 + 2 (rl — r2) sin 'r' c r2; (32 B)
in which ri is the greater radius, and r2 the less.
When the axes of a pair of pulleys are not parallel, the pulleys should be so placed that the part of the belt which is approaching each pulley shall be in the plane of the pulley.
§ 6o. Speed Cones.—A pair of speedcones (fig. 109) is a contrivance for varying and adjusting the velocity ratio communicated between a pair of parallel shafts by means of a belt. The speedcones are either continuous cones or conoids, as A, B, whose velocity ratio can be varied gradually while they are in motion by shifting the belt, or sets of pulleys whose radii vary by steps, as C, D, in which case the velocity ratio can be changed by shifting the belt from one pair of pulleys to another.
In order that the belt may fit accurately in every possible position on a pair of speedcones, the quantity L must be constant, in equations (32 A) or (32 B), according as the belt is crossed or uncrossed.
For a crossed belt, as in A and C, fig. 109, L depends solely on c and on rl + r2. Now c is constant because the axes are parallel ; therefore the sum of the radii of the pitchcircles connected in every position of the belt is to be constant. That condition is fulfilled by a pair of continuous cones generated by the revolution of two straight lines inclined opposite ways to their respective axes at equal angles.
For an uncrossed belt, the quantity L in equation (32 B)
is to be made constant. The exact fulfilment of this condition requires the solution of a transcendental equation; but it may be fulfilled with accuracy sufficient for practical purposes by using, instead of (32 B) the following approximate equation:
L nearly =2c +ir(ri+r2) + (rl— r2)2/c• (33)
The following is the most convenient practical rule for the application of this equation:
Let the speedcones be equal and similar conoids, as in B, fig. 109, but with their large and small ends turned opposite ways. Let ri be the radius of the large end of each, r2 that of the small end, ro that of the middle ; and let v be the sagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length. Then
v =1'2 — (ri +r2)/2 = (ri —r2)2/2irc.
22= 6'2832; but 6 may be used in most practical cases without sensible error.
The radii at the middle and end being thus determined, make the generating curve an arc either of a circle or of a parabola.
§ 62. Linkwork in General.—The pieces which are connected by linkwork, if they rotate or oscillate, are usually called cranks, beams and levers. The link by which they are connected is a rigid rod or bar, which may be straight or of any other figure; the straight figure being the most favourable to strength, is always used when there is no special reason to the contrary. The link is known by various names in various circumstances, such as couplingrod, connectingrod, crankrod, eccentricrod, &c. It is attached to the pieces which it connects by two pins, about which it is free to turn. The effect of the link is to maintain the distance between the axes of those pins invariable; hence the common perpendicular of the axes of the pins is the line of connexion, and its extremities may be called the connected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the arm or crankarm.
The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connexionin which case the angular velocity ratio at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the line of connexion upon the respective axes of rotation.
If at any instant the direction of one of the crankarms coincides with the line of connexion, the common perpendicular of the line of connexion and the axis of that crankarm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crankarm in question at such an instant is called a deadpoint. The velocity of the other connected point at such an instant is null, unless it also reaches a deadpoint at the same instant, so that the line of connexion is in the plane of the two axes of rotation, in which case the velocity ratio is indeterminate. Examples of deadpoints, and of the means of preventing the inconvenience which they tend to occasion, will appear in the sequel.
§ 62. Coupling of Parallel Axes.—Two or more parallel shafts (such as those of a locomotive engine, with two or more pairs of driving wheels) are made to rotate with constantly equal angular velocities by having equal cranks, which are maintained parallel by a couplingrod of such a length that the line of connexion is equal to the distance between the axes. The cranks pass their deadpoints simultaneously. To obviate the unsteadiness of motion which this tends to cause, the shafts are provided with a second set of cranks at right angles to the first, connected by means of a similar couplingrod, so that one set of cranks pass their dead points at the instant when the other set are farthest from theirs.
§ 63. Comparative Motion of Connected Points.—As the link is a rigid body, it is obvious that its action in communicating motion may be determined by finding the comparative motion of the connected points, and this is often the .most convenient method of proceeding.
If a connected point belongs to a turning piece, the direction of its motion at a given instant is perpendicular to the plane containing the axis and crankarm of the piece. If a connected point belongs to a shifting piece, the direction of its motion at any instant is given, and a plane can be drawn perpendicular to that direction.
The line of intersection of the planes perpendicular to the paths of the two connected points at a given instant is the instantaneous axis of the link at that instant; and the velocities of the connected points are directly as their distances from that axis.
In drawing on a plane surface, the two planes perpendicular to the paths of the connected points are represented by two lines (being their sections by a plane normal
to them), and the instantaneous axis by a point (fig. I Io) ; and, should the length of the two lines render it impracticable to produce them until they actually intersect, the velocity ratio of the connected points may be found by the principle that it is equal to the ratio of the segments which a line parallel to the line of connexion cuts off from any two lines drawn from a given point, perpendicular respectively to the paths of the connected points.
To illustrate this by one
example. Let Ci be the axis, and Ti the connected point of the beam of a steamengine; TiT2 the connecting or crankrod; T2 the other connected point, and the centre of the crankpin; C2 the axis of the crank and its shaft. Let vi denote the velocity of T1 at any given instant ; v2 that of T2. To find the ratio of these velocities, produce CiTi, C2T2 till they intersect in K; K is the instantaneous axis of the connecting rod, and the velocity ratio is
(34)
vl v2 : : KT1 : KT2.
(357
Should K be inconveniently far off, draw any triangle with its sides respectively parallel to CiTi, C2T2 and TiT2 ; the 'ratio of the two sides first mentioned will be the velocity ratio required. For example, draw C2A parallel to CiTi, cutting T1T2 in A; then
vl : v2 :: C2A : C2T2. (36)
§ 64. Eccentric.—An eccentric circular disk fixed on a shaft, and used to give a reciprocating motion to a rod, is in effect a crankpin of sufficiently large diameter to surround the shaft, and so to avoid the weakening of the shaft which would arise from bending it so as to form an ordinary crank. The centre of the eccentric is its connected point; and its eccentricity, or the distance from that centre to the axis of the shaft, is its crankarm.
An eccentric may be made capable of having its eccentricity altered by means of an adjusting screw, so as to vary the extent of the reciprocating motion which it communicates.
§ 65. Reciprocating Pieces—Stroke—DeadPoints.—The distance between the extremities of the path of the connected point in a reciprocating piece (such as the piston of a steamengine) is called the stroke or length of stroke of that piece. When it is connected with a continuously turning piece (such as the crank of a steamengine) the ends of the stroke of the reciprocating piece correspond to the
deadpoints of the path of the connected point of the turning piece, where the line of connexion is continuous with or coincides with the crankarm.
Let S be the length of stroke of the reciprocating piece, L the length of the line of connexion, and R the crankarm of the continuously turning piece. Then, if the two ends of the stroke be in one straight line with the axis of the crank,
S = 2R; (37) and if these ends be not in one straight line with that axis, then S, L — R, and L+R, are the three sides of a triangle, having the angle opposite S at that axis; so that, if 0 be the supplement of the arc between the deadpoints,
S2=2(L2+R2)—2(L2—R2) cos 0,
cos B = 2L2 + 2R2 — S2 (38) 2 (L2 — R2)
§ 66. Coupling of Intersecting Axes—Hooke's Universal Joint.—Intersecting axes are coupled by a contrivance of Hooke's, known as the " universal joint," which belongs to the class of linkwork (see fig. 11 I). Let 0 be the point of intersection of the axes OCR, OC2, and 0 their angle of inclination to each other. The pair of shafts C', C2 terminate in a pair of forks Fl, F2 in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross, having its centre at O. This cross is the link; the connected points are the centres of the bearings F', F2. At each instant each of those points moves at right angles to the central plane of its shaft and lork, therefore the line of intersection of the central planes of the two forks at any instant is the instantaneous axis of the cross, and the velocity ratio of the points Fa, F2 (which, as the forks are equal, is also the angular velocity ratio of the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. The mean value of that velocity ratio is that of equality, for each successive quarterturn is made by both shafts in the same time; but its actual value fluctuates between the limits:
a2 = r when Fl is the plane of OC'C2
al cos o (39)
and al = cos 0 when F2 is in that plane.
Its value at intermediate instants is given by the following equations: let 01, ¢2 be the angles respectively made by the central planes of the forks and shafts with the plane OC2C2 at a given instant ; then cos B=tan 4' tan ¢2,
a2 = _ d~2 =tan Oi+ cot 01,
al dot tan 02 + cot
§ 67. Intermittent Linkwork—Click and Ratchet.—A click acting upon a ratchetwheel or rack, which it pushes or pulls through a certain arc at each forward stroke and leaves at rest at each backward stroke, is an example of intermittent linkwork. During the forward stroke the action of the click is governed by the principles of linkwork; during the backward stroke that action ceases. A catch or pall, turning on a fixed axis, prevents the ratchetwheel or rack from reversing its motion.
Division 5.—Trains of Mechanism.
§ 68. General Principles. —A train of mechanism consists of a series of pieces each of which is follower to that which drives it and driver to that which follows it.
The comparative motion of the first driver 'and last follower is obtained by combining the proportions expressing by their terms the velocity ratios and by their signs the directional relations of the several elementary combinations of which the train consists.
§ 69. Trains of Wheelwork.—Let A', A2, A3, &c., A,,, 1, A,,, denote a series of axes, and a2, a3, &c., a,,, their angular velocities. Let the axis Al carry a wheel of N, teeth, driving a wheel of n2 teeth on the axis A2, which carries also a wheel of Na teeth, driving a wheel of na teeth on the axis A3, and so on; the numbers of teeth in drivers being denoted by N's, and in followers by n's, and the axes to which the wheels are fixed being denoted by numbers. Then the resulting velocity ratio is denoted by
a,,, 2 as &c. . . a,,, —N, . N2 . . &c.... (4r)
al al as am—' n2 . na . . &c. . . . n,„, that is to say, the velocity ratio of the last and first axes is the ratio
of the product of the numbers of teeth in the drivers to the product of the numbers of teeth in the followers.
Supposing all the wheels to be in outside gearing, then, as each elementary combination reverses the direction of rotation, and as the number of elementary combinations m—r is one less than thenumber of axes m, it is evident that if m is odd the direction of rotation is preserved, and if even reversed.
It is often a question of importance to determine the number of teeth in a train of wheels best suited for giving a determinate velocity ratio to two axes. It was shown by Young that, to do this with the least total number of teeth, the velocity ratio of each elementary combination should approximate as nearly as possible to 3.59• This would in many cases give too many axes; and, as a useful practical rule, it may be laid down that from 3 to 6 ought to be the limit of the velocity ratio of an elementary combination in wheelwork. The smallest number of teeth in a pinion for epicycloidal teeth ought to be twelve (see § 49)—but it is better, for smoothness of motion, not to go below fifteen; and for involute teeth the smallest number is about twentyfour.
Let B/C be the velocity ratio required, reduced to its least terms, and let B be greater than C. If B/C is not greater than 6, and C lies between the prescribed minimum number of teeth (which may be called t) and its double 2t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are, if possible, to be resolved into factors, and those factors (or if they are too small, multiples of them) used for the number of teeth. Should B or C, or both, be at once inconveniently large and prime, then, instead of the exact ratio B/C some ratio approximating to that ratio, and capable of resolution into convenient factors, is to be found by the method of continued fractions.
Should B/C be greater than 6, the best number of elementary combinations m—r will lie between
log B —log C log B —log C
log 6 and log 3
Then, if possible, B and ,C themselves are to be resolved each into m—r factors (counting r as a factor), which factors, or multiples of them, shall be not less than t nor greater than 6t; or if B and C contain inconveniently large prime factors, an approximate velocity ratio, found by the method of continued fractions, is to be substituted for B/C as before.
So far as the resultant velocity ratio is concerned, the order of the drivers N and of the followers n is immaterial: but to secure equable wear of the teeth, as explained in § 44, the wheels ought to be so arranged that, for each elementary combination, the greatest common divisor of N and n shall be either r, or as small as possible.
§ 70. Double Hooke's Coupling.—It has been shown in § 66 that the velocity ratio of a pair of shafts coupled by a universal joint fluctuates between the limits cos 0 and 1/cos 0. Hence one or both of the shafts must have a vibratory and unsteady motion, injurious to the mechanism and framework. To obviate this evil a short intermediate shaft is introduced, making equal angles with the first and last shaft, coupled with each of them by a Hooke's joint, and having its own two forks in the same plane. Let at, a2, as be the angular velocities of the first, intermediate, and last shaft in this train of two Hooke's couplings. Then, from the principles of § 6o it is evident that at each instant a2/a1= a2/aa, and consequently that as=a'; so that the fluctuations of angular velocity ratio caused by the first coupling are exactly neutralized by the second, and the first and last shafts have equal angular velocities at each instant.
§ 71. Converging and Diverging Trains of Mechanism.—Two or more trains of mechanism may converge into one—as when the two pistons of a pair of steamengines, each through its own connectingrod, act upon one crankshaft. One train of mechanism may diverge into two or more—as when a single shaft, driven by a prime mover, carries several pulleys, each of which drives a different machine. The principles of comparative motion in such converging and diverging trains are the same as in simple trains.
Division 6. Aggregate Combinations.
§ 72. General Principles.—Willis designated as "aggregate combinations " those assemblages of pieces of mechanism in which the motion of one follower is the resultant of component 'notions impressed on it by more than one driver. Two classes of aggregate combinations may be distinguished which, though not different in their actual nature, differ in the data which they present to the designer, and in the method of solution to be followed in questions respecting them.
Class I. comprises those cases in which a piece A is not carried directly by the frame C, but by another piece B. relatively to which the motion of A is given—the motion of the piece B relatively to the frame C being also given. Then the motion of A relatively to the frame C is the resultant of the motion of A relatively to B and of B relatively to C ; and that resultant is to be found by the principles already explained in Division 3 of this Chapter §§ 2732.
Class II. comprises those cases in which the motions of three points in one follower are determined by their connexions with two or with three different drivers.
This classification is founded on the kinds of problems arising from the combinations. Willis adopts another classification founded on the objects of the combinations, which objects he divides into two classes, viz. (r) to produce aggregate velocity, or a velocity which is the resultant of two or more components in the same path, and (2) to produce an aggregate path—that is, to make a given point
(40)
in a rigid body move in an assigned path by communicating certain motions to other points in that body.
It is seldc:n that one of these effects is produced without at the same time producing the other; but the classification of Willis depends upon which of those two effects, even supposing them to occur together, is the practical object of the mechanism.
§ 73. Differential Windlass.—The axis C (fig. 112) carries a larger barrel AE and a smaller barrel DB, rotating as one piece with the angular velocity al in the direction AE. The pulley or sheave FG has a weight W hung to its centre. A cord has one end made fast to and wrapped round the barrel AE; it passes from A under the sheave FG, and has the other end wrapped round and made fast to the barrel BD. Required the relation between the velocity of translation v2 of W and the angular velocity al of the differential barrel.
In this case v2 is an aggregate velocity, produced by the joint action of the two drivers AE and BD, transmitted by wrapping connectors to FG, and combined by that sheave so as to act on the fol
W lower W, whose motion is the same with that of
the centre of FG.
The velocity of the point F is al . AC, upward motion being considered positive. The velocity of the point G is — ai . CB, downward motion being negative. Hence the instantaneous axis of the sheave FG is in the diameter FG, at the distance
FG AC—BC
2 AC+BC
from the centre towards G; the angular velocity of the sheave is
AC + BC a2 _ —a' FG
and, consequently, the velocity of its centre is
FG AC—BC ai(AC—BC) (42)
vs=az 2 • ACtBC 2 or the mean between the velocities of the two vertical parts of the cord.
If the cord be fixed to the framework at the point B, instead of being wound on a barrel, the velocity of W is half that of AF.
A casecontaining several sheaves is called a block. A fallblock is attached to a fixed point; a runningblock is movable to and from a fallblock, with which it is connected by two or more plies of a rope. The whole combination constitutes a tackle or purchase. (See PULLEYS for practical applications of these principles.)
§ 74. Differential Screw.—On the same axis let there be two screws of the respective pitches pi and p2, made in one piece, and rotating with the angular velocity a. Let this piece be called B. Let the first screw turn in a fixed nut C, and the second in a sliding nut A. The velocity of advance of B relatively to C is (according to § 32) api, and of A relatively to B (according to § 57) —ap2; hence the velocity of A relatively to C is
a (PI P2), (46) being the same with the velocity of advance of a screw of the pitch pip2. This combination, called Hunter's or the differential screw, combines the strength of a large thread with the slowness of motion due to a small one.
§ 75. Epicyclic Trains.—The term epicyclic train is used by Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels or assist in driving them. The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in §§ 30, 31.
§ 76. Link Motion.—A slide valve operated by a link motion receives an aggregate motion from the mechanism driving it. (See STEAMENGINE for a description of this and other types of mechanism of this class.)
§ 77. Parallel Motions.A parallel motion is a combination of turning pieces in mechanism designed to guide the motion of a reciprocating piece either exactly or anproximately in a straight line, so as to avoid the friction which arises from the use of straight guides for that purpose.
Fig. 113 represents an exact
parallel motion, first proposed, it is
believed, by Scott Russell. The
arm CD turns on the axis C, and
is jointed at D to the middle of the
bar ADB, whose length is double
of that of CD, and one of whose
ends B is jointed to a slider, sliding
in straight guides along the line
CB. Draw BE perpendicular to
CB, cutting CD produced in E, then
E is the instantaneous axis of the bar ADB; and the direction of
motion of A is at every instant perpendicular to EA—that is, along
the straight line ACa. While the stroke of A is ACa, extending to equal distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by the sliding of B through a comparatively short stroke, and by rotatory motions at the joints C, D, B.
§ 78.* An example of an approximate straightline motion composed of three bars fixed to a frame is shown in fig. 114. It is due
to P. L. Tchebichev of St Petersburg. The links AB and CD are equal in length and are centred respectively at A and C. The ends D and B are joined by a link DB. If the respective lengths are made in the proportions AC: CD : DB =1:1'3:0'4 the middle point P of DB will describe an approximately straight line parallel to AC within limits of length about equal to AC. C. N. Peaucellier,
a French engineer officer, was the first, in 1864, to invent a linkwot k with which an exact straight line could be drawn. The linkwot k is shown in fig. 115, from which it will be seen that it consists of a rhombus of four equal bars ABCD, jointed at opposite corners with two equal bars BE and DE. The seventh link AF is equal in length to halt the distance EA when the mechanism is in its central position, The points E and F are fixed. It can be proved that the point C always moves in a straight line at right angles to the line EF. The more general property of the mechanism corresponding to proper. tions between the lengths FA and EF other than that of equality is that the curve described by the point C is the inverse of the curve described by A. There are other arrangements of bars giving straightline motions, and these arrangements together with the general properties of mechanisms of this kind are discussed in How to Draw a Straight Line by A. B. Kempe (London, 1877).
§ 79.* The Pantograph.—If a parallelogram of links (fig. ii6), be fixed at any one point a in any one of the links produced in either direction, and if any straight
line be drawn from this point to cut the links in the points
b and c, then the points a, b, c will be in a straight line for all positions of the mechanism, and if the point b be guided in any curve whatever, the point c will trace a similar curve to a scale enlarged
in the ratio ab : ac. This property of the parallelogram is utilized in the construction of the pantograph, an instrument used for obtaining a copy of a map or drawing on a different scale. Professor J. J. Sylvester discovered that this property of the parallelogram is not confined to points lying in one line with the fixed point. Thus if b (fig. 117) be C b
any point on the link CD, and if a  point c be taken on the link DE such that the triangles CbD and DcE are similar and similarly situated with regard to their respective links, then the ratio of the distances ab and
ac is constant, and the angle bac E
is constant for all positions of the FIG. 117. mechanism; so that, if b is guided in
any curve, the point c will describe a similar curve turned through an angle bac, the scales of the curves being in the ratio ab to ac. Sylvester called an instrument based on this property a plagiograph or a skew pantograph.
The combination of the parallelogram with a straightline motion, for guiding one of the points in a straight line, is illustrated in Watt's parallel motion for steamengines. (See STEAMENGINE.)
§ 80.* The Reuleaux System of Analysis.—If two pieces, A and B. (fig. 118) are jointed together by a pin, the pin being fixed, say, to A. the only relative motion possible between the pieces is one of turning about the axis of the pin. Whatever motion the pair of pieces may have as a whole each separate piece shares in common, and this common motion in no way affects the relative motion of A and B The motion of one piece is said to be completely constrained relatively to the other piece. Again, the pieces A and B (fig. 119) are paired together as a slide, and the only relative motion possible between them now is that of sliding, and therefore the motion of one relatively to the other is completely constrained. The pieces may be pairod
E
A
together as a screw and nut, in which case the relative motion is compounded of turning with sliding.
These combinations of pieces are known individually as kinematic pairs of elements, or briefly kinematic pairs. The three pairs mentioned above have each the peculiarity that contact between the two pieces forming the pair is distributed over a surface. Kinematic
pairs which have surface contact are classified as lower pairs. Kinematic pairs in which contact takes place along a line only are classified as higher pairs. A pair of spur wheels in gear is an example of a higher pair, because the wheels have contact between their teeth along lines only.
A kinematic link of the simplest form is made by joining up the halves of two kinematic pairs by means of a rigid link. Thus if AIBt represent a turning pair, and A2B2 a second turning pair, the rigid link formed by joining BI to B2 is a kinematic link. Four links of this kind are shown in fig. 120 joined up to form a closed kinematic chain.
In order that a kinematic chain may be made the basis of a mechanism, every point in any link of it must be completely constrained with regard to every other link. Thus in fig. 120 the motion of a point a in the link AIA2 is completely constrained with regard to the link BIB4 by the turning pair AIBI, and it can be proved that the motion of a relatively to the nonadjacent link A3A4 is completely constrained, and therefore the fourbar chain, as it is called, can be and is used as the basis of many mechanisms. Another way of considering the question of constraint is to imagine any one link of the chain fixed; then, however the chain be moved, the path of a point, as a, will always remain the same. In a fivebar chain, if a is a point in a link nonadjacent to a fixed link, its path is indeterminate. Still another way of stating the matter is to say that, if any one link in the chain be fixed, any point in the chain must have only one degree of freedom. In a fivebar chain a point, as a, in a link nonadjacent to the fixed link has two degrees of freedom and the chain cannot therefore be used for a mechanism. These principles may be applied to examine any possible combination of links forming a kinematic chain in order to test its suitability for use as a mechanism. Compound chains are formed by the superposition of two or more simple chains, and in these more complex chains links will be found carrying three, or even more, halves of kinematic pairs. The Joy valve gear mechanism is a good example of a compound kinematic chain.
A chain built up of three turning pairs and one sliding pair, and known as the slider crank chain, is shown in fig. 121. It will be seen that the piece Al can only slide relatively to the piece BI, and these two pieces therefore form the sliding pair. The piece Al carries the pin B4, which is one half of the turning pair A, B,. The piece AI together with the pin B4 therefore form a kinematic link AIB,. The other links of the chain are, BIAz, B2B3, A3A4. In order to convert a chain into a mechanism it is necessary to fix one link in it. Any one of the links may be fixed. It follows therefore that there are as many possible mechanisms as there are links in the chain. For example, there is a wellknown mechanism corresponding to the fixing of three of the four links of the slider crank chain (fig. I21). If the link d is fixed the chain at once becomes the mechanism of the ordinary steam engine; if the link e is fixed the mechanism obtained is that of the oscillating cylinder steam engine; if the link c is fixed the mechanism becomes either the Whitworth quickreturn motion or the slotbar motion, depending upon the proportion between the lengths of the links c and e. These different mechanisms are called
inversions of the slider crank chain. What was the fixed framework of the mechanism in one case becomes a moving link in an inversion.
The Reuleaux system, therefore, consists essentially of the analysis of every mechanism into a kinematic chain, and since each link of the chain may be the fixed frame of a mechanism quite diverse mechanisms are found to be merely inversions of the same kinematic chain. Franz Reuleaux's Kinematics of Machinery, translated by Sir A. B. W. Kennedy (London, 1876), is the book in which the system is set forth in all its completeness. In Mechanics of Machinery, by Sir A. B. W. Kennedy (London, 1886), the system was used for the first time in an English textbook, and now it has found its way into most modern textbooks relating to the subject of mechanism.
§ 81.* Centrodes, Instantaneous Centres, Velocity Image, Velocity Diagram.—Problems concerning the relative motion of the several parts of a kinematic chain may be considered in two ways, in addition to the way hitherto used in this article and based on the principle of § 34. The first is by the method of instantaneous centres, already exemplified in § 63, and rolling centroids, developed by Reuleaux in connexion with his method of analysis. The second is by means of Professor R. H. Smith's method already referred to in § 23.
Method i.—By reference to § 3o it will be seen that the motion of a cylinder rolling on a fixed cylinder is one of rotation about an instantaneous axis T, and that the velocity both as regards direction and magnitude is the same as if the rolling piece B were for the instant turning about a fixed axis coincident with the instantaneous axis. If the rolling cylinder B and its path A now be assumed to receive a common plane motion, what was before the velocity of the point P becomes the velocity of P relatively to the cylinder A, since the motion of B relatively to A still takes place about the instantaneous axis T. If B stops rolling, then the two cylinders continue to move as though they were parts of a rigid body. Notice that the shape of either rolling curve (fig. 91 or 92) may be found by considering each fixed in turn and then tracing out the locus of the instantaneous axis. These rolling cylinders are sometimes called axodes, and a section of an axode in a plane parallel to the plane of motion is called a centrode. The axode is hence the locus of the instantaneous axis, whilst the centrode is the locus of the instantaneous centre in any plane parallel to the plane of motion. There is no restriction on the shape of these rolling axodes; they may have any shape consistent with (oiling (that is, no slipping is permitted), and the relative velocity of a point P is still found by considering it with regard to the instantaneous centre.
Reuleaux has shown that the relative motion of any pair of nonadjacent links of a kinematic chain is determined by the rolling together of two ideal cylindrical surfaces (cylindrical being used here in the general sense), each of which may be assumed to be formed by the extension of the material of the link to which it corresponds. These surfaces have contact at the instantaneous axis, which is now called the instantaneous axis of the two links concerned. To find the form of these surfaces corresponding to a particular pair of nonadjacent links, consider each link of the pair fixed in turn, then the locus of the instantaneous axis is the axode corresponding to the fixed link, or, considering a plane of motion only, the locus of the instantaneous centre is the centrode corresponding to the fixed link.
To find the instantaneous centre for a particular link corresponding to any given configuration of the kinematic chain, it is only necessary to know the direction of motion of any two points in the link, since lines through these points respectively at right angles to their directions of motion intersect in the instantaneous centre.
To illustrate this principle, consider the fourbar chain shown in fig. 122 made up of the four links, a, b, c, d. Let a be the fixed link, and consider the link c. Its extremities are moving respectively in directions at right angles to the links b €Ind d; hence produce the links b and d to meet in the point One. This point is the instantaneous centre of the motion of the link c relatively to the fixed link a, a fact indicated by the suffix ac placed after the letter O. The process being repeated for different values of the angle 0 the curve through the several points Oar is the centroid which may be imagined as formed by an extension of the material of the link a. To find the corresponding centroid for the link c, fix c and repeat the process. Again, imagine d fixed, then the instantaneous centre Obd of b with regard to d is found by producing the links c and a to intersect in Obd, and the shapes of the centroids belonging respectively to the links b and d can be found as before. The axis about which a pair of adjacent links turn is a permanent axis, and is of course the axis
of the pin which forms the point. Adding the centres corresponding to these several axes to the figure, it will be seen that there are six centres in connexion with the fourbar chain of which four are permanent and two are instantaneous or virtual centres; and, further, that whatever be the configuration of the chain these centres group themselves into three sets of three, each set lying on a straight line. This peculiarity is not an accident or a special property of the fourbar chain, but is an illustration of a general law regarding the subject discovered by Aronhold and Sir A. B. W. Kennedy independently, which may be thus stated: If any three bodies, a, b, c, have plane motion their three virtual centres, Oab, Oba, Oaa, are three points on one straight line. A proof of this will be found in The Mechanics of Machinery quoted above. Having obtained the set of instantaneous centres for a chain, suppose a is the fixed link of the chain and c any other link; then Oa,is the instantaneous centre of the two links and may be considered for the instant as the trace of an axis fixed to an extension of the link a about which c is turning, and thus problems of instantaneous velocity concerning the link c are solved as though the link c were merely rotating for the instant about a fixed axis coincident with the instantaneous axis.
Method 2.—The second method is based upon the vector representation of velocity, and may be illustrated by applying it to the fourbar chain. Let AD (fig. 123) be the fixed link. Consider the link BC, and let it be required to find the velocity of the point B having given the velocity of the point C. The principle upon which c
the solution is based is that the only motion which B can have relatively to an axis through C fixed to the link CD is one of turning about C. Choose any pole 0 (fig. 124). From this pole set out Oc to represent the velocity of the point C. The direction of this must be at right angles to the line CD, because this is the only direction possible to the point C. If the link BC moves without turning, Oc will also represent the velocity of the point B; but, if the link is turning, B can only move about the axis C, and its direction of motion is therefore at right angles to the line CB. Hence set out the possible direction of B's motion in the velocity diagram, namely cbi, at right angles to CB. But the point B must also move at right angles to AB in the case under consideration. Hence draw a line through O in the velocity diagram at right angles to AB to cut cbi in b. Then Ob is the velocity of the point b in magnitude and direction, and cb is the tangential velocity of B relatively to C. Moreover, whatever be the actual magnitudes of the velocities, the instantaneous velocity ratio of the points C and B is given by the ratio Oc/Ob.
A most important property of the diagram (figs. 123 and 124) is the following : If points X and x are taken dividing the link BC and the tangential velocity cb, so that cx : x b = CX : XB, then Ox represents the velocity of the point X in magnitude and direction. The line cb has been called the velocity image of the rod, since it may be looked upon as a scale drawing of the rod turned through 90° from the actual rod. Or, put in another way, if the link CB is drawn to scale on the new length cb in the velocity diagram (fig. 124), then a vector drawn from 0 to any point on the new drawing of the rod will represent the velocity of that point of the actual rod in magnitude and direction. It will be understood that there is a new velocity diagram for every new configuration of the mechanism, and that in each new diagram the image of the rod will be different in scale. Following the method indicated above for a kinematic chain in general, there will be obtained a velocity diagram similar to that of fig. 124 for each configuration of the mechanism, a diagram in which the velocity of the several points in the chain utilized for drawing the diagram will appear to the same scale, all radiating from the pole O. The lines joining the ends of these several velocities are the several tangential velocities, each being the velocity image of a link in the chain. These several images are not to the same scale, so that although the images may be considered to form collectively an image of the chain itself, the several members of this chainimage are to different scales in any one velocity diagram, and thus the chainimage is distorted from the actual proportions of the mechanism which it represents.
§ 82.* Acceleration Diagram. Acceleration Image.—Although it is possible to obtain the acceleration of points in a kinematic chain with one link fixed by methods which utilize the instantaneous centres of the chain, the vector method more readily lends itself to this purpose. It should be understood that the instantaneous centre considered in the preceding paragraphs is available only for estimating relative velocities; it cannot be used in a similar mannerfor questions regarding acceleration. That is to say, although the instantaneous centre is a centre of no velocity for the instant, it is not a centre of no acceleration, and in fact the centre of no acceleration is in general a quite different point. The general principle on which the method of drawing an acceleration diagram depends is that if a link CB (fig. 125) have plane motion and the acceleration of any point C be given in magnitude and direction, the acceleration of any other point B is the vector sum of the acceleration of C, the radial acceleration of B about C and the tangential acceleration of B about C. Let A be any origin, and let Ac represent the acceleration of the point C, ct the radial acceleration of B about C which must be in a direction parallel to BC, and tb the tangential acceleration of B about C, which must of course be at right angles to ct; then the vector sum of
these three magnitudes is Ab, and this vector represents the acceleration of the point B. The directions of the radial and tangential accelerations of the point B are always known when the position of the link is assigned, since these are to be drawn respectively parallel to and at right angles to the link itself. The magnitude of the radial acceleration is given by the expression v2/BC, v being the velocity of the point B about the point C. This velocity can always be found from the velocity diagram of the chain of which the link forms a part. If dw/dt is the angular acceleration of the link, dw/dt X CB is the tangential acceleration of the point B about the point C. Generally this tangential acceleration is unknown in magnitude, and it becomes part of the problem to find it. An important property of the diagram is that if points X and x are taken dividing the link CB and the whole acceleration of B about C, namely, cb in the same ratio, then Ax represents the acceleratioi of the point X in magnitude and direction; cb is called the acceleration image of the rod. In applying this principle to the drawing of an acceleration diagram for a mechanism, the velocity diagram of the mechanism must be first drawn in order to afford the means of calculating the several radial accelerations of the links. Then assuming that the acceleration of one point of a particuar link of the mechanism is known together with the corresponding configuration of the mechanism, the two vectors Ac and ct can be drawn. The direction of lb, the third vector in the diagram, is also known, so that the problem is reduced to the condition that b is somewhere on the line tb. Then other conditions consequent upon the fact that the link forms part of a kinematic chain operate to enable b to be 'fixed. These methods are set forth and exemplified in Graphics, by R. H. Smith (London, 1889). Examples, completely worked out, of velocity and acceleration diagrams for the slider crank chain, the fourbar chain, and the mechanism of the Joy valve gear will be found in ch. ix. of Valves and Valve Gear Mechanism, by W. E. Dalby (London, 1906).
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