Fig. 23.

35. Two plain cones, having their axes parallel, as shown in Fig. 23, will obviously answer the same purpose as the ordinary form of speed pulleys. The slant faces of the cones may be formed by any continuous curve; but with this condition-that the sum of the diameters at every position of the band shall be a constant.

Fig. 24.

Guide Pulleys.

36. By the intervention of guide pulleys the direction of cords may be changed into any other direction. Thus, by means of the guide pulleys B and C, the motion of the chord in the direction C D is changed into the direction a B.

D

The cords D C and C B should be in the plane of the pulley c; and the chords C B and B A should be in the plane of the pulley B.

37. Two guide pulleys, Е and н, may be employed to transmit

Fig. 25.

F

motion from the wheel A to the wheel B, when the axes of these wheels have any given direction.

Let EH be the line where the planes, passing through the two wheels, intersect each other. In this line assume any two convenient points E and H; in the plane of the wheel a draw the tangents E C and HD; and in the plane of the wheel B draw the tangents E F and H G ; then CE F G H D will be the path of the endless cord, which will be kept

in this path by a guide pulley at E, in the plane of c E F, and another guide pulley at н, in the plane of D H G.

The relative velocities of the axes A and B depend entirely upon the ratio of the radii, A D and B G, of the two wheels. See Art. 30.

To prevent Wrapping Connectors from Slipping.

38. The slip of the band on the wheel, when it is not excessive, is in many cases rather an advantage than otherwise; but when motion is to be transmitted from one wheel to another according to some given exact ratio, gearing chains of various forms are employed as the wrapping connectors.

39. In some cases the links of the gearing chain lay hold of pins or teeth formed upon the wheel, as shown in fig. 27. In other cases, the links of the gearing are joined together, someFig. 27.

Fig. 26.

thing like a watch chain, and carry teeth which pass into certain notches made at corresponding distances on the edge of the wheel, as shown in fig. 26.

40. When a belt moves a conical wheel, it always happens that the belt gradually moves towards the broad end of the

Fig. 28.

Fig. 29.

Fig. 30.

wheel: this is owing to the belt being more stretched on that side than it is on the other.

41. This property enables us to construct a wheel so that a belt shall not shift on its edge; this is simply effected by making the edge to swell a little in the middle, as shown in fig. 29.

D

42. When two rollers have to make only a limited number

cord is uncoiled from the other extremity, as shown in fig. 30. 43. By a similar arrangement of cords on the cylinder E F (see fig. 31), a reciprocating motion of this cylinder will produce a back and forward motion of the carriage A B.

Systems of Pulleys.

44. A system of pulleys must at least contain one movable pulley. When a wheel, forming a part of a system of wheels connected together by cords, has a progressive motion, it materially affects the velocity ratio of the receiver and the operator of the mechanism. There are a great many different systems of pulleys, but they all depend upon the different combinations of movable and fixed pulleys, and the different modes of reduplication of a cord.

45. In this system of pulleys there is one movable block and

Fig. 32.

a single continuous cord with three duplications, so that whilst the moving force P acts by one cord, the movable block with its load is suspended by six cords: if w ascend one foot, each of these cords will be shortened one foot, and therefore the cord P will be lengthened six feet; that is to say, the velocity of P will be six times that of w.

46. In the system of pulleys represented in fig. 33 there are two distinct cords and two movable pulleys A and B, making two duplications of cord; then if A ascends one foot, в must ascend two

Р

feet, and the cord at p must be lengthened four feet; that is, the velocity of p will be four times the velocity of w.

Generally if there are n movable pulleys in such a system, then,

47. The system of pulleys represented in fig. 34, contains two movable pulleys, one fixed pulley, and two single cords. In this case the velocity ratio of P to w is as four to one.

48. Fig. 35 represents a similar system of pulleys, in which the velocity ratio of p to w is as five to one.

In all these systems of pulleys the velocity ratios are con

stant.

Fig. 36.

49. In the compound wheel and axle, represented in fig. 36, the axle is made of different thicknesses as at A and B, and a continuous cord coils round these parts in different directions, and passes round the wheel of the movable pulley D. In one revolution of the wheel cp the space moved over by

the pulley D is equal to half the difference of the circumferences of the

A

W

axles A and B. Putting R, for the radius of the wheel c r, R for

the radius of the axle A, and r for the radius of the axle B; then we have for the velocity ratio

This piece of mechanism belongs to a class which produces what has been called differential motions, their object being to produce a slow and definite motion in a body by the most simple and practicable means.

TO PRODUCE A VARYING VELOCITY RATIO BY MEANS OF

WRAPPING CONNECTORS.

50. To find the ratio of the angular velocities of two eccentric wheels, moved by a cord wrapping over each.

Fig. 37.

K

Let D C be a cord wrapping round the wheels, whose axes of motion are A and B; their line C D will be a tangent to the two curves forming the edges of the wheels. On D C produced let fall the perpendiculars a Q and B K; then the velocity of the cord, in this position of the wheels, will be equal to the velocity of the point Q, and

at the same time it will also be equal to the velocity of the point K hence we find,

angular velocity A C

Fig. 38.

D

angular velocity B D

C

that is to say, the angular velocities are inversely as the perpendiculars let fall upon the cord from the axes of motion. 51. Let в be a movable pulley suspended from the continuous cord P A B C, passing over a fixed pulley A, and attached to a point c in the same horizontal line with A. Let fall B D perpendicular to a c; then B C will always be equal to B A, and B will move in the vertical line B D. Hence we find

B

W