POWER TRANSMISSION, MECHANICAL Mechanical transmission of power is effected generally by means of belts or ropes, by shafts or by wheel gearing and chains. Each individual method may be used separately or in combination. The problems involved in the design arrangement of the
nisms for the mechanical distribution of power are conveniently approached by the consideration of the way in which the me chanical energy made available by an engine is distributed to the several machines in the factory. By a belt on the fly-wheel of the prime mover the power is transmitted to the line shaft, and pulleys suitably placed along the line shaft by means of other belts trans mit power, first, to small countershafts carrying fast and loose pulleys and striking gear for starting or stopping each engine at will, and then to the driving pulleys (q.v.) of the several machines.
Quantitative Estimation of the Power
dealing with the matter quantitatively the engine crank-shaft may be taken as the starting point of the transmission, and the first motion-shaft of the machine as the end of the transmission so far as that particular machine is concerned.
T be the mean torque or turning effort in ft.lb. which the engine exerts continuously on the crank shaft when it is making N revolutions per second. It is more convenient to express the revolutions per second in terms of the angular velocity w, that is, in radians per second. The relation between these quantities is w =27r.N. Then the rate at which work is done by the engine crank shaft is To.) foot-pounds per second, equivalent to Tw/55o horse power. This is now distributed to the several machines in varying proportions. Assuming for the sake of simplicity that the whole of the power is absorbed by one machine, let
be. the torque on the first motion-shaft of the machine, and let
be its angular velocity, then the rate at which the machine is ab sorbing energy is
foot-pounds per second. A certain quan tity of energy is absorbed by the transmitting mechanism itself for the purpose of overcoming frictional and other resistances, otherwise the rate of absorption of energy by the machine would exactly equal the rate at which it was produced by the prime mover assuming steady conditions of working. Actually there fore
would be less than Tco so that Ticol = nTo.), where n is called the efficiency of the transmission. Considering now the general problem of a multiple machine transmission, if
T2, CO2, T3, CO3, • • • are the several torques and angular velocities of the respective first motion shafts of the machines, ( Tice' ± T2c02-1- T3c034- • • = nno (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, n
is the over-all efficiency of the distributing mechanism. We now consider the separate parts of the transmitting mechanism.
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 in pounds 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)n.o foot-pounds per second. If the horse power required to drive the machine be represented by h.p., then (1—u)rw= 55o 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 0, in radians (fig. 2), the coefficient of frictionµ between the belt and the pul ley, the mass of the belt and the speed of the belt. Consider an element of the belt (fig. 2) subtending an angle de 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 dO, 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
where v is the linear velocity of the belt in feet per second. Hence the frictional resistance of the element to sliding is (t—Wv1/2)00, and this must be equal to the dif ference of tensions dt when the element is on the point of slipping, so that
The solution of this equation is t— WIP/g = e", (4) u—
where t 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 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 propor tioned so that the stress per square inch shall not exceed a certain safe limit determined from practice.