(27)(The obliquity which is found to answer best in practice is about 141â„2°; its cosine is about 31/22, and its sine about1â„4. These values though not absolutely exact, are near enough to the truth for practical purposes.)Fig. 102.Suppose the base-circles to be a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is P1IP2. As the line of connexion of those pulleys is the same as that of the proposed teeth, they will rotate with the required velocity ratio. Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of contact P1IP2, that point will trace on a plane rotating along with the wheel 1 part of the involute of the base-circle D1D1′, and on a plane rotating along with the wheel 2 part of the involute of the base-circle D2D2′; and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Principle I. of § 45.Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.All involute teeth of the same pitch work smoothly together.To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along P1IP2, is less than the pitch in the ratio of cos obliquity : I; and consequently that, if distances equal to the pitch be marked off either way from I towards P1and P2respectively, as the extremities of the path of contact, and if, according to Principle IV. of § 45, the addendum-circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice it is usual to make the path of contact somewhat longer, viz. about 2.4 times the pitch; and with this length of path, and the obliquity already mentioned of 141â„2°, the addendum is about 3.1 of the pitch.The teeth of arack, to work correctly with wheels having involute teeth, should have plane surfaces perpendicular to the line of connexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action.§ 47.Teeth for a given Path of Contact: Sang’s Method.—In the preceding section the form of the teeth is found by assuming a figure for the path of contact, viz. the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II. of § 45, and by equation (25). This method of finding the forms of the teeth of wheels forms the subject of an elaborate and most interesting treatise by Edward Sang.All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set.Fig. 103.§ 48.Teeth traced by Rolling Curves.—If any curve R (fig. 103) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from § 30, that the instantaneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-circle for the moment, and that consequently the line AT, traced by a tracing-point T, fixed to the rolling curve upon the plane of the wheel, will be everywhere perpendicular to the straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If thesame rolling curve R, with the same tracing-point T, be rolled on theoutsideof any other pitch-circle, it will have thefaceof a tooth suitable to work with theflankAT.In like manner, if either the same or any other rolling curve R′ be rolled the opposite way, on theoutsideof the pitch-circle BB, so that the tracing point T′ shall start from A, it will trace the face AT′ of a tooth suitable to work with aflanktraced by rolling the same curve R′ with the same tracing-point T′insideany other pitch-circle.The figure of thepath of contactis that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E′I′E′, as the case may be) at a fixed point I (or I′).If the same rolling curve and tracing-point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form aset. The teeth of arack, of the same set, are traced by rolling the rolling curve on both sides of a straight line.The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch-surfaces are in rolling contact.Fig. 104.§ 49.Epicycloidal Teeth.—The most convenient rolling curve is the circle. The path of contact which it traces is identical with itself; and the flanks of the teeth are internal and their faces external epicycloids for wheels, and both flanks and faces are cycloids for a rack.For a pitch-circle of twice the radius of the rolling ordescribingcircle (as it is called) the internal epicycloid is a straight line, being, in fact, a diameter of the pitch-circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. For a smaller pitch-circle the flanks would be convex andin-curvedorunder-cut, which would be inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the radius of the describing circle, so that the flanks may be either straight or concave.In fig. 104 let BB′ be part of the pitch-circle of a wheel with epicycloidal teeth; CIC′ the line of centres; I the pitch-point; EIE′ a straight tangent to the pitch-circle at that point; R the internal and R′ the equal external describing circles, so placed as to touch the pitch-circle and each other at I. Let DID′ be the path of contact, consisting of the arc of approach DI and the arc of recess ID′. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch.The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID = E′ID; and the mean obliquity is one-half of that angle.It appears from experience that the mean obliquity should not exceed 15°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID′ should each be one-sixth of a circumference; therefore the circumference of the describing circle should besix times the pitch.It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth.§ 50.Nearly Epicycloidal Teeth: Willis’s Method.—To facilitate the drawing of epicycloidal teeth in practice, Willis showed how to approximate to their figure by means of two circular arcs—one concave, for the flank, and the other convex, for the face—and each having for its radius themeanradius of curvature of the epicycloidal arc. Willis’s formulae are founded on the following properties of epicycloids:—Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by the normal TI to the epicycloid at a given point T, with a tangent to the circle at I—that is, the obliquity of the action at T.Then the radius of curvature of the epicycloid at T is—For an internal epicycloid, Ï = 4r sin θR − rR − 2rFor an external epicycloid, Ï′ = 4r sin θR + rR + 2r(28)Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c = 2r sin θ; and thereforeFor the flank, Ï âˆ’ c = 2r sin θRR − 2rFor the face, Ï′ − c = 2r sin θRR + 2r(29)For the proportions approved of by Willis, sin θ =1â„4nearly; r = p (the pitch) nearly; c =1â„2p nearly; and, if N be the number of teeth in the wheel, r/R = 6/N nearly; therefore, approximately,Ï âˆ’ c =p·N2N − 12Ï âˆ’ c =p·N2N + 12(30)Fig. 105.Hence the following construction (fig. 105). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab = ac −1â„2p. Draw radii bd, ce; draw fb, cg, making angles of 751â„2° with those radii. Make bf = p′ − c, cg = p − c. From f, with the radius fa, draw the circular arc ah; from g, with the radius ga, draw the circular arc ak. Then ah is the face and ak the flank of the tooth required.To facilitate the application of this rule, Willis published tables of Ï âˆ’ c and Ï′ − c, and invented an instrument called the “odontograph.â€Â§ 51.Trundles and Pin-Wheels.—If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels.§ 52.Backs of Teeth and Spaces.—Toothed wheels being in general intended to rotate either way, thebacksof the teeth are made similar to the fronts. Thespacebetween two teeth, measured on the pitch-circle, is made about1â„6th part wider than the thickness of the tooth on the pitch-circle—that is to say,Thickness of tooth=5â„11pitch;Width of space=6â„11pitch.The difference of1â„11of the pitch is called theback-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about one-tenth of the pitch.§ 53.Stepped and Helical Teeth.—R. J. Hooke invented the making of the fronts of teeth in a series of steps with a view to increase the smoothness of action. A wheel thus formed resembles in shape a series of equal and similar toothed disks placed side by side, with the teeth of each a little behind those of the preceding disk. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel-work of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the aftermost end; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next pair has commenced.Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and expensive workmanship than common teeth; and helical teeth are, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery.§ 54.Teeth of Bevel-Wheels.—The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex.Fig. 106.The operations of describing the exact figures of the teeth of bevel-wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, substitutingpolesforcentres, andgreat circlesforstraight lines.In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used:—Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I, OB2I their pitch cones; OC1, OC2their axes; OI their line of contact. Perpendicular to OI draw A1IA2, cutting the axes in A1, A2; make the outer rims of the patterns and of the wheelsportions of the cones A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about O for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs ID1, ID2, with the radii A1I, A2I; those arcs will be thedevelopmentsof arcs of the pitch-circles B1I, B2I, when the conical surfaces A1B1I, A2B2I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces.§ 55.Teeth of Skew-Bevel Wheels.—The crests of the teeth of a skew-bevel wheel are parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel whose pitch surface is a cone touching the hyperboloidal surface at the given circle.§ 56.Cams.—Acamis 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 pitch-surface, 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, calledthreads, winding round it. Convex and concave screws are distinguished technically by the respective names ofmaleandfemale; a short concave screw is called anut; and when ascrewis spoken of without qualification aconvexscrew is usually understood.The relation between theadvanceand therotation, 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. Thepitchoraxial pitchof 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 thesame thread. If, therefore, the screw has several equidistant threads, the true pitch is equal to thedivided 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 thisnormal helix, may be called thenormal pitch; and when the screw has more than one thread the normal pitch from thread to thread may be called thenormal divided pitch.The distance from thread to thread, measured on a circle described about the axis of the screw, called the pitch-circle, may be called thecircumferential pitch; for a screw of one thread it is one circumference; for a screw of n threads, (one circumference)/n.Let r denote the radius of the pitch circle;n the number of threads;θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;Pathe axialpitchPa/n = padivided pitch;Pnthe normalpitchPn/n = pndivided pitch;Pcthe circumferential pitch;thenpc= pacot θ = pncos θ =2Ï€r,npa= pnsec θ = pctan θ =2Ï€r tan θ,npn= pcsin θ = pacos θ =2Ï€r sin θ.n(31)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 calledendless 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 right-handed or both left-handed, the angle between the directions of their axes is the sum of their obliquities; if one is right-handed and the other left-handed, 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.III. The angular velocities of the screws are inversely as their numbers of threads.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 awheel, the namescrewbeing 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.Fig. 107.§ 58.Coupling of Parallel Axes—Oldham’s Coupling.—Acouplingis 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 bysliding contactwas invented by Oldham, and is represented in fig. 107. C1, C2are the axes of the two parallel shafts; D1, D2two disks facing each other, fixed on the ends of the two shafts respectively; E1E1a bar sliding in a diametral groove in the face of D1; E2E2a 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 C1C2as 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 C1C2diametrically 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 movingtowardsthe pulley.Cordsrequire either cylindrical drums with ledges or grooved pulleys.Chainsrequire 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.Fig. 108.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.Thepitch-lineof a pulley or drum is a curve to which the line of connexion is always a tangent—that is 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 communicating a constant velocity ratio are circular. Theeffective radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius added to half the thickness of the connector. Theangular velocities of a pair of connected circular pulleys or drums are inversely as the effective radii.Acrossedbelt, as in fig. 108, A, reverses the direction of the rotation communicated; anuncrossedbelt, as in fig. 108, B, preserves that direction.ThelengthL of an endless belt connecting a pair of pulleys whose effective radii are r1, 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 = 2 √ {c2− (r1+ r2)2} + (r1+ r2)(Ï€ − 2 sin−1r1+ r2);c(32 A)and for an uncrossed belt:—L = 2 √ {c2− (r1− r2)2} + Ï€ (r1+ r2+ 2 (r1− r2) sin−1r1− r2;c(32 B)in which r1is the greater radius, and r2the 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 isapproachingeach pulley shall be in the plane of the pulley.§ 60.Speed-Cones.—A pair of speed-cones (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 speed-cones 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.Fig. 109.In order that the belt may fit accurately in every possible position on a pair of speed-cones, the quantity L must be constant, in equations (32 A) or (32 B), according as the belt is crossed or uncrossed.For acrossedbelt, as in A and C, fig. 109, L depends solely on c and on r1+ r2. Now c is constant because the axes are parallel; therefore thesum of the radiiof the pitch-circles 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 followingapproximateequation:—L nearly = 2c + Ï€ (r1+ r2) + (r1− r2)2/ c.(33)The following is the most convenient practical rule for the application of this equation:—Let the speed-cones be equal and similar conoids, as in B, fig. 109, but with their large and small ends turned opposite ways. Let r1be the radius of the large end of each, r2that of the small end, r0that of the middle; and let v be thesagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, thebulgingof the conoids in the middle of their length. Thenv = r0− (r1+ r2) / 2 = (r1− r2)2/ 2Ï€c.(34)2Ï€ = 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.§ 61.Linkwork in General.—The pieces which are connected by linkwork, if they rotate or oscillate, are usually calledcranks,beamsand levers. Thelinkby 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 ascoupling-rod,connecting-rod,crank-rod,eccentric-rod, &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 isthe line of connexion, and its extremities may be called theconnected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is thearmorcrank-arm.The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connexion in 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 crank-arms coincides with the line of connexion, the common perpendicular of the line of connexion and the axis of that crank-arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank-arm in question at such an instant is called adead-point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point 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 dead-points, 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 coupling-rod of such a length that the line of connexion is equal to the distance between the axes. The cranks pass their dead-points 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 coupling-rod, 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 crank-arm 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 theinstantaneous axis of the linkat that instant; and thevelocities of the connected points are directly as their distances from that axis.Fig. 110.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. 110); 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 C1be the axis, and T1the connected point of the beam of a steam-engine; T1T2the connecting or crank-rod; T2the other connected point, and the centre of the crank-pin; C2the axis of the crank and its shaft. Let v1denote the velocity of T1at any given instant; v2that of T2. To find the ratio of these velocities, produce C1T1, C2T2till they intersect in K; K is the instantaneous axis of the connecting rod, and the velocity ratio isv1: v2:: KT1: KT2.(35)Should K be inconveniently far off, draw any triangle with its sides respectively parallel to C1T1, C2T2and T1T2; the ratio of the two sides first mentioned will be the velocity ratio required. For example, draw C2A parallel to C1T1, cutting T1T2in A; thenv1: 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 crank-pin 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 crank-arm.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—Dead-Points.—The distance between the extremities of the path of the connected point in a reciprocating piece (such as the piston of a steam-engine) is called thestrokeorlength of strokeof that piece. When it is connected with a continuously turning piece (such as the crank of a steam-engine) the ends of the stroke of the reciprocating piece correspond to thedead-pointsof the path of the connected point of the turning piece, where the line of connexion is continuous with or coincides with the crank-arm.Let S be the length of stroke of the reciprocating piece, L the length of the line of connexion, and R the crank-arm 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 θ be the supplement of the arc between the dead-points,S2= 2 (L2+ R2) − 2 (L2− R2) cos θ,cos θ =2L2+ 2R2− S2.2 (L2− R2)(38)Fig. 111.§ 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. 111). Let O be the point of intersection of the axes OC1, OC2, and θ their angle of inclination to each other. The pair of shafts C1, C2terminate in a pair of forks F1, F2in 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 F1, F2. At each instant each of those points moves at right angles to the central plane of its shaft and fork, therefore the line of intersection of the central planes of the two forks at any instant is the instantaneous axis of the cross, and thevelocity ratioof the points F1, F2(which, as the forks are equal, is also theangular velocity ratioof the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. Themeanvalue of that velocity ratio is that of equality, for each successivequarter-turnis made by both shafts in the same time; but its actual value fluctuates between the limits:—α2=1when F1is the plane of OC1C2α1cos θandα2= cos θ when F2is in that plane.α1(39)Its value at intermediate instants is given by the following equations: let φ1, φ2be the angles respectively made by the central planes of the forks and shafts with the plane OC1C2at a given instant; thencos θ = tan φ1tan φ2,α2= −dφ2=tan φ1+ cot φ1.α1dφ1tan φ2+ cot φ2
(27)
(The obliquity which is found to answer best in practice is about 141â„2°; its cosine is about 31/22, and its sine about1â„4. These values though not absolutely exact, are near enough to the truth for practical purposes.)
Suppose the base-circles to be a pair of circular pulleys connected by means of a cord whose course from pulley to pulley is P1IP2. As the line of connexion of those pulleys is the same as that of the proposed teeth, they will rotate with the required velocity ratio. Now, suppose a tracing point T to be fixed to the cord, so as to be carried along the path of contact P1IP2, that point will trace on a plane rotating along with the wheel 1 part of the involute of the base-circle D1D1′, and on a plane rotating along with the wheel 2 part of the involute of the base-circle D2D2′; and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Principle I. of § 45.
Consequently, one of the forms suitable for the teeth of wheels is the involute of a circle; and the obliquity of the action of such teeth is the angle whose cosine is the ratio of the radius of their base-circle to that of the pitch-circle of the wheel.
All involute teeth of the same pitch work smoothly together.
To find the length of the path of contact on either side of the pitch-point I, it is to be observed that the distance between the fronts of two successive teeth, as measured along P1IP2, is less than the pitch in the ratio of cos obliquity : I; and consequently that, if distances equal to the pitch be marked off either way from I towards P1and P2respectively, as the extremities of the path of contact, and if, according to Principle IV. of § 45, the addendum-circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice it is usual to make the path of contact somewhat longer, viz. about 2.4 times the pitch; and with this length of path, and the obliquity already mentioned of 141â„2°, the addendum is about 3.1 of the pitch.
The teeth of arack, to work correctly with wheels having involute teeth, should have plane surfaces perpendicular to the line of connexion, and consequently making with the direction of motion of the rack angles equal to the complement of the obliquity of action.
§ 47.Teeth for a given Path of Contact: Sang’s Method.—In the preceding section the form of the teeth is found by assuming a figure for the path of contact, viz. the straight line. Any other convenient figure may be assumed for the path of contact, and the corresponding forms of the teeth found by determining what curves a point T, moving along the assumed path of contact, will trace on two disks rotating round the centres of the wheels with angular velocities bearing that relation to the component velocity of T along TI, which is given by Principle II. of § 45, and by equation (25). This method of finding the forms of the teeth of wheels forms the subject of an elaborate and most interesting treatise by Edward Sang.
All wheels having teeth of the same pitch, traced from the same path of contact, work correctly together, and are said to belong to the same set.
§ 48.Teeth traced by Rolling Curves.—If any curve R (fig. 103) be rolled on the inside of the pitch-circle BB of a wheel, it appears, from § 30, that the instantaneous axis of the rolling curve at any instant will be at the point I, where it touches the pitch-circle for the moment, and that consequently the line AT, traced by a tracing-point T, fixed to the rolling curve upon the plane of the wheel, will be everywhere perpendicular to the straight line TI; so that the traced curve AT will be suitable for the flank of a tooth, in which T is the point of contact corresponding to the position I of the pitch-point. If thesame rolling curve R, with the same tracing-point T, be rolled on theoutsideof any other pitch-circle, it will have thefaceof a tooth suitable to work with theflankAT.
In like manner, if either the same or any other rolling curve R′ be rolled the opposite way, on theoutsideof the pitch-circle BB, so that the tracing point T′ shall start from A, it will trace the face AT′ of a tooth suitable to work with aflanktraced by rolling the same curve R′ with the same tracing-point T′insideany other pitch-circle.
The figure of thepath of contactis that traced on a fixed plane by the tracing-point, when the rolling curve is rotated in such a manner as always to touch a fixed straight line EIE (or E′I′E′, as the case may be) at a fixed point I (or I′).
If the same rolling curve and tracing-point be used to trace both the faces and the flanks of the teeth of a number of wheels of different sizes but of the same pitch, all those wheels will work correctly together, and will form aset. The teeth of arack, of the same set, are traced by rolling the rolling curve on both sides of a straight line.
The teeth of wheels of any figure, as well as of circular wheels, may be traced by rolling curves on their pitch-surfaces; and all teeth of the same pitch, traced by the same rolling curve with the same tracing-point, will work together correctly if their pitch-surfaces are in rolling contact.
§ 49.Epicycloidal Teeth.—The most convenient rolling curve is the circle. The path of contact which it traces is identical with itself; and the flanks of the teeth are internal and their faces external epicycloids for wheels, and both flanks and faces are cycloids for a rack.
For a pitch-circle of twice the radius of the rolling ordescribingcircle (as it is called) the internal epicycloid is a straight line, being, in fact, a diameter of the pitch-circle, so that the flanks of the teeth for such a pitch-circle are planes radiating from the axis. For a smaller pitch-circle the flanks would be convex andin-curvedorunder-cut, which would be inconvenient; therefore the smallest wheel of a set should have its pitch-circle of twice the radius of the describing circle, so that the flanks may be either straight or concave.
In fig. 104 let BB′ be part of the pitch-circle of a wheel with epicycloidal teeth; CIC′ the line of centres; I the pitch-point; EIE′ a straight tangent to the pitch-circle at that point; R the internal and R′ the equal external describing circles, so placed as to touch the pitch-circle and each other at I. Let DID′ be the path of contact, consisting of the arc of approach DI and the arc of recess ID′. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch.
The obliquity of the action in passing the line of centres is nothing; the maximum obliquity is the angle EID = E′ID; and the mean obliquity is one-half of that angle.
It appears from experience that the mean obliquity should not exceed 15°; therefore the maximum obliquity should be about 30°; therefore the equal arcs DI and ID′ should each be one-sixth of a circumference; therefore the circumference of the describing circle should besix times the pitch.
It follows that the smallest pinion of a set in which pinion the flanks are straight should have twelve teeth.
§ 50.Nearly Epicycloidal Teeth: Willis’s Method.—To facilitate the drawing of epicycloidal teeth in practice, Willis showed how to approximate to their figure by means of two circular arcs—one concave, for the flank, and the other convex, for the face—and each having for its radius themeanradius of curvature of the epicycloidal arc. Willis’s formulae are founded on the following properties of epicycloids:—
Let R be the radius of the pitch-circle; r that of the describing circle; θ the angle made by the normal TI to the epicycloid at a given point T, with a tangent to the circle at I—that is, the obliquity of the action at T.
Then the radius of curvature of the epicycloid at T is—
(28)
Also, to find the position of the centres of curvature relatively to the pitch-circle, we have, denoting the chord of the describing circle TI by c, c = 2r sin θ; and therefore
(29)
For the proportions approved of by Willis, sin θ =1â„4nearly; r = p (the pitch) nearly; c =1â„2p nearly; and, if N be the number of teeth in the wheel, r/R = 6/N nearly; therefore, approximately,
(30)
Hence the following construction (fig. 105). Let BB be part of the pitch-circle, and a the point where a tooth is to cross it. Set off ab = ac −1â„2p. Draw radii bd, ce; draw fb, cg, making angles of 751â„2° with those radii. Make bf = p′ − c, cg = p − c. From f, with the radius fa, draw the circular arc ah; from g, with the radius ga, draw the circular arc ak. Then ah is the face and ak the flank of the tooth required.
To facilitate the application of this rule, Willis published tables of Ï âˆ’ c and Ï′ − c, and invented an instrument called the “odontograph.â€
§ 51.Trundles and Pin-Wheels.—If a wheel or trundle have cylindrical pins or staves for teeth, the faces of the teeth of a wheel suitable for driving it are described by first tracing external epicycloids, by rolling the pitch-circle of the pin-wheel or trundle on the pitch-circle of the driving-wheel, with the centre of a stave for a tracing-point, and then drawing curves parallel to, and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels.
§ 52.Backs of Teeth and Spaces.—Toothed wheels being in general intended to rotate either way, thebacksof the teeth are made similar to the fronts. Thespacebetween two teeth, measured on the pitch-circle, is made about1â„6th part wider than the thickness of the tooth on the pitch-circle—that is to say,
The difference of1â„11of the pitch is called theback-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel is about one-tenth of the pitch.
§ 53.Stepped and Helical Teeth.—R. J. Hooke invented the making of the fronts of teeth in a series of steps with a view to increase the smoothness of action. A wheel thus formed resembles in shape a series of equal and similar toothed disks placed side by side, with the teeth of each a little behind those of the preceding disk. He also invented, with the same object, teeth whose fronts, instead of being parallel to the line of contact of the pitch-circles, cross it obliquely, so as to be of a screw-like or helical form. In wheel-work of this kind the contact of each pair of teeth commences at the foremost end of the helical front, and terminates at the aftermost end; and the helix is of such a pitch that the contact of one pair of teeth shall not terminate until that of the next pair has commenced.
Stepped and helical teeth have the desired effect of increasing the smoothness of motion, but they require more difficult and expensive workmanship than common teeth; and helical teeth are, besides, open to the objection that they exert a laterally oblique pressure, which tends to increase resistance, and unduly strain the machinery.
§ 54.Teeth of Bevel-Wheels.—The acting surfaces of the teeth of bevel-wheels are of the conical kind, generated by the motion of a line passing through the common apex of the pitch-cones, while its extremity is carried round the outlines of the cross section of the teeth made by a sphere described about that apex.
The operations of describing the exact figures of the teeth of bevel-wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels all those operations are to be performed on the surface of a sphere described about the apex instead of on a plane, substitutingpolesforcentres, andgreat circlesforstraight lines.
In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used:—
Let O (fig. 106) be the common apex of a pair of bevel-wheels; OB1I, OB2I their pitch cones; OC1, OC2their axes; OI their line of contact. Perpendicular to OI draw A1IA2, cutting the axes in A1, A2; make the outer rims of the patterns and of the wheelsportions of the cones A1B1I, A2B2I, of which the narrow zones occupied by the teeth will be sufficiently near to a spherical surface described about O for practical purposes. To find the figures of the teeth, draw on a flat surface circular arcs ID1, ID2, with the radii A1I, A2I; those arcs will be thedevelopmentsof arcs of the pitch-circles B1I, B2I, when the conical surfaces A1B1I, A2B2I are spread out flat. Describe the figures of teeth for the developed arcs as for a pair of spur-wheels; then wrap the developed arcs on the cones, so as to make them coincide with the pitch-circles, and trace the teeth on the conical surfaces.
§ 55.Teeth of Skew-Bevel Wheels.—The crests of the teeth of a skew-bevel wheel are parallel to the generating straight line of the hyperboloidal pitch-surface; and the transverse sections of the teeth at a given pitch-circle are similar to those of the teeth of a bevel-wheel whose pitch surface is a cone touching the hyperboloidal surface at the given circle.
§ 56.Cams.—Acamis 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 pitch-surface, 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, calledthreads, winding round it. Convex and concave screws are distinguished technically by the respective names ofmaleandfemale; a short concave screw is called anut; and when ascrewis spoken of without qualification aconvexscrew is usually understood.
The relation between theadvanceand therotation, 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. Thepitchoraxial pitchof 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 thesame thread. If, therefore, the screw has several equidistant threads, the true pitch is equal to thedivided 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 thisnormal helix, may be called thenormal pitch; and when the screw has more than one thread the normal pitch from thread to thread may be called thenormal divided pitch.
The distance from thread to thread, measured on a circle described about the axis of the screw, called the pitch-circle, may be called thecircumferential pitch; for a screw of one thread it is one circumference; for a screw of n threads, (one circumference)/n.
Let r denote the radius of the pitch circle;
n the number of threads;θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;
n the number of threads;
θ the obliquity of the threads to the pitch circle, and of the normal helix to the axis;
then
(31)
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 calledendless 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 right-handed or both left-handed, the angle between the directions of their axes is the sum of their obliquities; if one is right-handed and the other left-handed, 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.
III. The angular velocities of the screws are inversely as their numbers of threads.
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 awheel, the namescrewbeing 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.—Acouplingis 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 bysliding contactwas invented by Oldham, and is represented in fig. 107. C1, C2are the axes of the two parallel shafts; D1, D2two disks facing each other, fixed on the ends of the two shafts respectively; E1E1a bar sliding in a diametral groove in the face of D1; E2E2a 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 C1C2as 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 C1C2diametrically 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 movingtowardsthe pulley.
Cordsrequire either cylindrical drums with ledges or grooved pulleys.
Chainsrequire 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.
Thepitch-lineof a pulley or drum is a curve to which the line of connexion is always a tangent—that is 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 communicating a constant velocity ratio are circular. Theeffective radius, or radius of the pitch-circle of a circular pulley or drum, is equal to the real radius added to half the thickness of the connector. Theangular velocities of a pair of connected circular pulleys or drums are inversely as the effective radii.
Acrossedbelt, as in fig. 108, A, reverses the direction of the rotation communicated; anuncrossedbelt, as in fig. 108, B, preserves that direction.
ThelengthL of an endless belt connecting a pair of pulleys whose effective radii are r1, 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:—
(32 A)
and for an uncrossed belt:—
(32 B)
in which r1is the greater radius, and r2the 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 isapproachingeach pulley shall be in the plane of the pulley.
§ 60.Speed-Cones.—A pair of speed-cones (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 speed-cones 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 speed-cones, the quantity L must be constant, in equations (32 A) or (32 B), according as the belt is crossed or uncrossed.
For acrossedbelt, as in A and C, fig. 109, L depends solely on c and on r1+ r2. Now c is constant because the axes are parallel; therefore thesum of the radiiof the pitch-circles 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 followingapproximateequation:—
L nearly = 2c + π (r1+ r2) + (r1− r2)2/ c.
(33)
The following is the most convenient practical rule for the application of this equation:—
Let the speed-cones be equal and similar conoids, as in B, fig. 109, but with their large and small ends turned opposite ways. Let r1be the radius of the large end of each, r2that of the small end, r0that of the middle; and let v be thesagitta, measured perpendicular to the axes, of the arc by whose revolution each of the conoids is generated, or, in other words, thebulgingof the conoids in the middle of their length. Then
v = r0− (r1+ r2) / 2 = (r1− r2)2/ 2πc.
(34)
2Ï€ = 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.
§ 61.Linkwork in General.—The pieces which are connected by linkwork, if they rotate or oscillate, are usually calledcranks,beamsand levers. Thelinkby 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 ascoupling-rod,connecting-rod,crank-rod,eccentric-rod, &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 isthe line of connexion, and its extremities may be called theconnected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is thearmorcrank-arm.
The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connexion in 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 crank-arms coincides with the line of connexion, the common perpendicular of the line of connexion and the axis of that crank-arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank-arm in question at such an instant is called adead-point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead-point 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 dead-points, 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 coupling-rod of such a length that the line of connexion is equal to the distance between the axes. The cranks pass their dead-points 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 coupling-rod, 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 crank-arm 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 theinstantaneous axis of the linkat that instant; and thevelocities 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. 110); 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 C1be the axis, and T1the connected point of the beam of a steam-engine; T1T2the connecting or crank-rod; T2the other connected point, and the centre of the crank-pin; C2the axis of the crank and its shaft. Let v1denote the velocity of T1at any given instant; v2that of T2. To find the ratio of these velocities, produce C1T1, C2T2till they intersect in K; K is the instantaneous axis of the connecting rod, and the velocity ratio is
v1: v2:: KT1: KT2.
(35)
Should K be inconveniently far off, draw any triangle with its sides respectively parallel to C1T1, C2T2and T1T2; the ratio of the two sides first mentioned will be the velocity ratio required. For example, draw C2A parallel to C1T1, cutting T1T2in A; then
v1: 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 crank-pin 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 crank-arm.
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—Dead-Points.—The distance between the extremities of the path of the connected point in a reciprocating piece (such as the piston of a steam-engine) is called thestrokeorlength of strokeof that piece. When it is connected with a continuously turning piece (such as the crank of a steam-engine) the ends of the stroke of the reciprocating piece correspond to thedead-pointsof the path of the connected point of the turning piece, where the line of connexion is continuous with or coincides with the crank-arm.
Let S be the length of stroke of the reciprocating piece, L the length of the line of connexion, and R the crank-arm 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 θ be the supplement of the arc between the dead-points,
S2= 2 (L2+ R2) − 2 (L2− R2) cos θ,
(38)
§ 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. 111). Let O be the point of intersection of the axes OC1, OC2, and θ their angle of inclination to each other. The pair of shafts C1, C2terminate in a pair of forks F1, F2in 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 F1, F2. At each instant each of those points moves at right angles to the central plane of its shaft and fork, therefore the line of intersection of the central planes of the two forks at any instant is the instantaneous axis of the cross, and thevelocity ratioof the points F1, F2(which, as the forks are equal, is also theangular velocity ratioof the shafts) is equal to the ratio of the distances of those points from that instantaneous axis. Themeanvalue of that velocity ratio is that of equality, for each successivequarter-turnis made by both shafts in the same time; but its actual value fluctuates between the limits:—
(39)
Its value at intermediate instants is given by the following equations: let φ1, φ2be the angles respectively made by the central planes of the forks and shafts with the plane OC1C2at a given instant; then
cos θ = tan φ1tan φ2,