where η is called the efficiency of the transmission. Considering
now the general problem of a multiple machine transmission, if
T1, ω1, T2, ω2, T3, ω3,. . . are the several torques and angular velocities
of the respective first motion shafts of the machines,
| (T1ω1+T2ω2+T3+ω3+ . . . .)＝ηTω | (2) |
expresses the relations which must exist at any instant of steady motion. This is not quite a complete statement of the actual conditions because some of the provided energy is always in course of being stored and unstored from instant to instant as kinetic energy in the moving parts of the mechanism. Here, η is the over-all efficiency of the distributing mechanism. We now consider the separate parts of the transmitting mechanism.
§ 3. Belts.—Let a pulley A (fig. 1) drive a pulley B by means of a leather belt, and let the direction of motion be as indicated by the arrows on the pulleys. When the pulleys are revolving uniformly, A transmitting power to B, one side of the belt will be tight and the other 'side will be slack, but both sides will be in a state of tension. Let t and u be the respective tensions on the tight and slack side; then the torque exerted by the belt on the pulley B is (t−u)r, where r is the radius of the pulley in feet, and the rate at which the belt does work on the pulley is (t−u)rω foot-pounds per second. If the horse-power required to drive the machine be represented by h.p., then
| (t−u)rω＝550 h.p., | (3) |
assuming the efficiency of the transmission to be unity. This equation contains two unknown tensions, and before either can be found another condition is necessary. This is supplied by the relation between the tensions, the arc of contact θ, in radians (fig. 2), the coefficient of friction ii between the belt and the pulley, the mass of the belt and the speed of the belt. Consider an element of the belt (fig. 2) subtending an angle dθ at the centre of the pulley, and let t be the tension on one side of the element and (t+dt) the tension of the other side. The tension tending to cause the element to slide bodily round the surface of the pulley is dt. The normal pressure between the element and the face of the pulley due to the tensions is t dθ, but this is diminished by the force necessary to constrain the element to move in the circular path determined by the curvature of the pulley. If W is the weight of the belt per foot, the constraining force required for this purpose is Wv2dθ/g, where v is the linear velocity of the belt in feet per second. Hence the frictional resistance of' the element to sliding is (t−Wv2/g)μdθ, and this must be equal to the difference of tensions dt when the element is on the point of slipping, so that (t−Wv2/g)μdθ＝dt. The solution of this equation is
| t−Wv2/gu−Wv-/g＝eμθ. | (4) |
where l is now the maximum tension and u the minimum tension, and e is the base of the Napierian system of logarithms, 2-718. Equations (3) and (4) supply the condition from which the power transmitted by a given belt at a given speed can be found. For ordinary work the term involving v may be neglected, so that (4) becomes
FIG. 2.
| t/u＝eμθ. | (5) |
Equations (3) and (5) are ordinarily used for the preliminary design of a belt to calculate t, the maximum tension in the belt necessary to transmit a stated horse power at a stated speed, and then the cross section is proportioned so that the stress per square inch shall not exceed a certain safe limit determined from practice.
To facilitate the calculations in connexion with equation (5), tables are constructed giving the ratio l/u for various values of μ and θ. (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 increase 0 and hence the ratio t/u for both pulleys.
§ 4. Example of Preliminary Design of a Belt.—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 flywheel to be of such diameter and at such a distance from the driven pulley that the arc of contact is 120°, equal to 2·094 radians, and further assume that the coefficient of friction μ=0·3. Then from equation (5) t/u＝e2.094×0.3＝2·7180.6282; that is loget/u=0·6282, from which t/u=1~87, and u=t/1·87. Using this in (3) we have t(1−1/1·87) 1·5 × 18·84 = 550 × 20, from which t= 838 ℔. Allowing a working strength of 300 ℔ per square inch, the area required is 2·8 sq. in., so that if the belt is 14 in. thick its width would be 11·2 in., or if 316 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 ℔. A belt of the area found above would weigh about 1·4 ℔. per foot. The velocity of the belt, v=wr=18·84×1·5=28-26 ft. per second. The term Wv2/g 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, 14=464'5 ℔. Using this value of u in equation (1)
H.P.＝(838−464·5)×18·84×1·5550＝19·15
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.
§ 5. Belt Velocity for Maximum 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 tension t can be calculated from equations (3) and (4) as follows:—
Let t be the given maximum tension with which a belt weighing W ℔ per foot may be worked. Then solving equation (4) for u, subtracting t from each side, and changing the signs all through: t−u＝(t−Wv2/g) (1−e−μθ). And the rate of working U, in foot-pounds per second, is
U = (t—u)v＝(tv−Wv3/g)(1−e−μθ).
Differentiating U with regard to v, equating to zero, and solving for v, we have v＝(tg/3W). Utilizing the data of the previous example to illustrate this matter, t=838 ℔ per square inch, W＝1·4 ℔ per foot, and consequently, from the above expression, v＝80 ft. per second approximately. A lower speed than this should be adopted, however, because the above investigation does not include the loss incurred by the continual bending of the belt round the circumference of the pulley. The loss from this cause increases with the velocity of the belt, and operates to make the velocity for maximum horse power considerably lower than that given above.
§ 6. Flexibility.—When a belt or rope is working power is absorbed 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 Cω foot-pounds per second. The value of 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 over strain the outermost strands and if these give way the life of the rope is soon determined.
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(From Abram Combe, Proc. Inst. M sch. Eng.) Fig. 3.-Rope driving; half-crossed rope drive, separate rope to each groove.
§ 7. Rope Driving.—About 1856 Tarnes Combe, of Belfast, introduced the practice of transmitting power by means of ropes running in grooves turned circumferentially in the rim of the pulley (fig. 3). The ropes may be led off in groups to the different floors of the factory to pulleys keyed 'to the distributing shafting. A groove was adopted having an angle of about 45°,
