760 MECHANICS [APPLIED MECHANICS. parallel to the axis, between the corresponding points in two suc cessive 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 has more than one thread the normal pitch from thread to thread may be called the normal 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 the circumferential pitch ; for a screw of one thread it is one circum- ,. . one circumference fereuce ; for a screw 01 n threads, - - . n Let r denote the radius of the pitch circle ; n the number of threads ; 6 the obliquity of the threads to the pitch circle, and of the normal helix to the axis ; Pa ) ( pitch, P_a = [ the axial I n ~~P a } ( divided pitch ; P,, the normal pitch, divided pitch ; P c the circumferential pitch ; then p e =p a cot 0= p tt cos 6= Pa=Pn sec 0= p e tan0 = p a COS0 2xrr ta.nO n 2irr sin . (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 con tact of their threads. Such screws are commonly called endless sarcivs. At the point of contact of the screws their threads must be parallel ; and their line of connexion is the common perpendi cular 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 obli quities ; 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 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. 68. 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 con necting their contiguous ends that they shall rotate as one piece ; but if the axes are not in the same straight line combinations of me chanism are required. A coupling for parallel shafts which acts by sliding contact was invented by Oldham, and is represented in fig. 21. C,, C. 2 are the axes of the two Fig. 21. AJL ^i> 2 uUv CWA.QO Ji I/IIG i/tw parallel shafts ; D 1( D 2 two disks facing each other, fixed on the ends of the two shafts respectively ; E^j a bar sliding in a diametral groove in the face of D,; E 2 E 2 a bar sliding in a diametral groove in the face of D 2 : 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 C^ 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 CjC,, 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 practicability or permanency of their exact continuity. 69. 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 a 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 sect. 46 if both pieces turn, and of sect. 47 if one turns and the other shifts, in which latter case the motion must be recipro cating. The pitch-line of 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. The effective 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. The angular velocities of a pair of connected circular pulleys or drums are inversely as the effec tive radii. A crossed belt, as in fig. 22, A, reverses the direction of the rota tion communicated ; an uncrossed belt, as in fig. 22, B, preserves that direction. The length L of an endless belt connecting a pair of pulleys whose effective radii are r,, r. 2 , with parallel axes whose distance apart is C, is given by the following formulie, 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 Ve 2 - (^ + rrf + K + r TT - 2 sin- 1 (32, A);
c /
and for an uncrossed belt, L-aVfc -K-r.,) 8 } + *(r 1 +rj + 2(r 1 -r.) sin-^p (32, B); in whuh r t is the greater radius, and r 2 the less. When tiie 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. 70. Speed-Cones (see fig. 23). A pair of speed-cones is a con trivance 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.