spindle by a separate band. The dimensions of the pulleys are generally so arranged that the return motion of the lathe spindle is faster than the forward motion. An alternative arrangement consists in providing two loose pulleys on the counter-shaft, driven by open and crossed belts respectively, and arranging two clutches on the shaft, so that by the movement of a sliding block, controlled by hand, one or other of the clutches can be put in gear. A u The proportions of cone pulleys for open or crossed belts may be determined by considering the expression for the half length (I) of a belt wrapping round pulleys of radius rl and V2 respectively, and with centres distant c apart. The value of l may be easily shown to be (11-l-r1)1r/2~|-(r1ir2)a+c cos o., where the positive sign is to be taken for a crossed belt and the negative sign for an o en belt. In determining the dimensions of corresponding drums of) cone pulleys it is evident that for a crossed belt the sum of the radii of each pair remains a constant, since the angle o. is constant, while for an open belt a. is variable and the values of the radii are then obtained by solving the equations ri = l/1r- c(m sin a. -I-cos a) + so sin a, r, = l/1r- c(u. sin a +cos a.) - ic sin a. The value of o. is in general small, and an approximate solution may be obtained by substituting two or three terms of the expansions for sin a and cos a.. This, however, leads to a troublesome numerical solution. An accurate geometrical solution by C. Culmann gives

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Fig. 2.

the linear equivalents of the above equations in the following manner. A rectangle ABCD (fig. 2), with side AB=1r c/2 and AD=c, is constructed, and the quadrant AEF is drawn with centre D and radius DA. F B is the evolute of this circle, and for any radius DE at an angle a and corresponding tangent EG terminated by the evolute, the perpendicular distance of G from the line AD is c(cos a.+a. sin a). If now a line be drawn from A to the bisector H of the side BC, it will meet the vertical through G in and l]=c(cos o.-l-a. sin a)/vr. A circular are, centre D and radius c/2, meets D E in K, and the perpendicular KL gives so sin a. This distance is marked off from the point I in each direction. whereby the points M and N are obtained, the distance apart of which represents the value rl-r2. If now the value l/-rr=O] be marked off, and a horizontal line be drawn through the point O, the line OM represents n-I-12. Repeating this construction for all values of a. between 0° and 90°, we obtain a curve BPC, which can be used for determining the ratios of corresponding drums of cone pulleys or of conical drums for open belts. The curve BPC is generally used with the abscissae spaced more conveniently for practical applications, and a modification of the diagram by ]. F. Klein (fourn. Franklin Inst., vol. lxxix.) is often used instead.,

FIG. 3.

Ki

5

When pulleys are mounted on

shafts which are parallel to one

another, the band will retain its

position, provided that its central

line advances towards each

pulley in the diametral plane of this latter. This condition is fulfilled in the example shown by ng. 3, in

which the central planes of each

pulley pass through the points of

delivery of the other pulley for the

given direction of motion. If the

motion is reversed the condition is

no longer satisfied and the belt will leave the pulleys. In more complicated cases guide pulleys must be used. In the most general case for inclined pulleys, any two points may be chosen on the line of intersection of the diametral planes, and tangents drawn to the pitch circles of the pulleys. Guide pulleys are set with their diametral planes in the planes containing corresponding pairs of tangents, and a continuous belt wrapped round these pulleys in due order can then be run in either direction. The rims of pulleys for hemp or other ropes or cords are grooved, and the sides are usually either inclined at 4;5° or curved to give a sharper angle at the outside than at the bottom of the groove; in the latter case, as the rope wears it engages in a groove of greater angle and less effective grip. Wire ropes are injured by the lateral crushing of the material, and in this case the grooves are wide enough to allow the rope to rest on the rounded bottom, which is lined with leather or wood to diminish the wear and increase the friction. In English practice there are as many separate endless ropes as there are pairs of grooves in the two pulleys to be connected, but in cases of American practice the rope is continuously wound round the two pulleys, and the free end passes over a pulley mounted on a movable weighted carriage to adjust the tension. It is of considerable importance that the effective radius of action of the rope remain constant throughout each pulley, otherwise the wear on the rope becomes very great and its life is diminished. The grooves must be turned exactly alike, and the rope must be of the same diameter throughout to diminish slip. Pulleys may be detach ably connected to a shaft by friction clutches, so that they may be thrown in and out of engagement at will. The section, fig. 4, shows

a clutch for a rope-driven pulley

A, which runs freely on a bush A

B on the shaft, and is provided C

with an enlarged cylindrical ' ' 'o

nave or clutch box C. A split ' 'jéng m ring D, carried by the clutch B, s& . 'i'=§ 31.il I eww A

I !

v l@ F

i

124

)),)) .

lp' c.

? iii!

and turning with it, can be 5 “

thrust against the clutch box f

by right- and left-handed screws

E, so that a sufficient grip is

obtained to cause the clutch and

the pulley to turn as one piece.

The engagement of the pulley

rn » " 1." Ef'l|

and clutch is determined by a

hand-controlled block F sliding

on the shaft, the movement of which is communicated to the right- and left-handed screw shafts by links G connected to the levers H.

FIG. 4.

The resistance to slipping of a flat belt on a pulley may be obtained by considering the equilibrium of a small arc of the pulley surface subtending an angle 116 at the centre, and having tensions T and T+dT at its extremities. Neglecting quantities of the second order, the pressure on the pulley is Td0, and the friction is;.¢T¢l0 where it is the coefficient of friction between the belt and the pulley. We have therefore dT=pTd0 and dT/T=nd0. Integrating the expression for an angle of wrapping 0, we obtain the relation log e T1/T2 = 110, where T, and T2 are the end tensions. For leather belts on cast-iron pulleys the value of n may be taken as o-4, giving a ratio of the tensions on the tight and slack sides of T1/T2=3-514, when the angle of wrapping is. i8o°. For ropes in the grooves of cast-iron pulleys, where ¢ is the inclination of the sides of the grooves, the value of the normal pressure is increased in the ratio of cosec éqh =I. A usual value of;.c for hemp ropes on cast-iron pulleys is 0-3, and the exponential log ratio is therefore O'3'lI' cosec %¢ when 0 =1r. At high speeds the centrifugal tension of the belt or rope, of amount wa”/g, may be considerable, and must be subtracted from the end tensions.

Pulley Blocks.-Frames or blocks containing pulleys or sheaves are used in combination for lifting heavy weights. There are usually two blocks, of which one A (fig. 5) is fixed, and the other B is movable, and a rope or chain, with one end secured to one of the blocks at C, passes round the sheaves in a continuous coil, leaving a free end D at which the effort is applied. In the arrangement shown there are three equal sheaves in each block, and each set turns on a pin secured in the framing. The load, supported by the lower hook, is raised by hauling on the free end and, neglecting any slight Obliquity of the plies of rope, the free end moves six times as fast