To facilitate the calculations in connection with equation (5), tables are constructed giving the ratio t/u for various values ofµ and 0. (See W. C. Unwin, Machine Design, 12th ed., p. 377.) The ratio should be calculated for the smaller pulley. If the belt is arranged as in fig. 1, that is, with the slack side uppermost, the drop of the belt tends to in crease 0 and hence the ratio t/u for both pulleys.
The following example illustrates the use of the equations for the design of a belt in the ordinary way. Find the width of a belt to transmit 20 h.p. from the flywheel of an engine to a shaft which runs at 180 revolutions per minute (equal to 18.84 radians per second), the pulley on the shaft being 3 ft. diameter. Assume the engine fly wheel to be of such diameter and at such a distance from the driven pulley that the arc of contact is I 20°, equal to 2.094 radians, and further assume that the coefficient of friction j. =0.3. Then from equation (5)
2.718
that is log
0.6282, from which t/u=1.87, and u=t/r.87. Using this in (3) we have t(1---1/1.87) 1-5 X18.84=550 X 20, from which t=838 lb. Allowing a working strength of 300 lb. per square inch, the area required is 2.8 sq.in., so that if the belt is a in. thick its width would be '1.2 in., or if -h. in. thick, 15 in. approximately.
The effect of the force constraining the circular motion in diminishing the horse power transmitted may now be ascertained by calculating the horse power which a belt of the size found will actually transmit when the maximum tension t is 838 lb. A belt of the area found above would weigh about 1.4 lb. per foot. The velocity of the belt, v.= w r=18.84 X 1.5=28.26 ft. per second. The term
therefore has the numerical value 34.7. Hence equation (2) becomes (t-34.7)/(u-34.7)=1.87, from which, inserting the value 838 for t, u=464.5 lb. Using this value of u in equation (I) Thus with the comparatively low belt speed of 28 ft. per second the horse power is only diminished by about 5%. As the velocity increases the transmitted horse power increases, but the loss from this cause rapidly increases, and there will be one speed for every belt at which the horse power transmitted is a maximum. An increase of speed above this results in a diminution of transmitted horse power.
If the weight of a belt per foot is given, the speed at which the maximum horse power is transmitted for an assigned value of the maximum ten sion t can be calculated from equations (3) and (4) as follows:— Let t be the given maximum tension with which a belt weigh ing W lb. per foot may be worked. Then solving equation (4) for u, subtracting I from each side, and changing the signs all through : t — u = (t — W
g) (1—
And the rate of working U, in foot-pounds per second, is Differentiating U with regard to v, equating to zero, and solving for v, we have v =
Utilizing the data of the previous example to illustrate this matter, t= 838 lb., W =1.4 lb. per foot, and consequently, from the above expression, v=8o ft. per second approximately. A lower speed than this should be adopted, how ever, because the above investigation does not include the loss incurred by the continual bending of the belt round the circum ference of the pulley. The loss from this cause increases with the velocity of the belt, and operates to make the velocity for maxi mum horse power considerably lower than that given above.
When a belt or rope is working power is ab sorbed in its continual bending round the pulleys, the amount depending upon the flexibility of the belt and the speed. If C is the couple required to bend the belt to the radius of the pulley, the rate at which work is done is Co., foot-pounds per second. The value for C for a given belt varies approximately inversely as the radius of the pulley, so that the loss of power from this cause will vary inversely as the radius of the pulley and directly as the speed of revolution. Hence thin flexible belts are to be preferred to thick stiff ones. Besides the loss of power in transmission due to this cause, the bending causes a stress in the belt which is to be added to the direct stress due to the tensions in the belt in order to find the maximum stress. In ordinary leather belts the bending stress is usually negligible; in ropes, however, especially wire rope, it assumes paramount importance, since it tends to overstrain the outermost strands and if these give way the life of the rope is soon determined.