# 1911 Encyclopædia Britannica/Power Transmission/Mechanical

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I.—Mechanical
§ 1. Methods.—The mechanical transmission of power is effected in general 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 and arrangement of the mechanisms for the mechanical distribution of power are conveniently approached by the consideration of the way in which the mechanical 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 transmit power, first, to small counter shafts carrying fast and loose pulleys and striking gear for starting or stopping each engine at will, and then to the driving pulleys of the several machines. (See also Pulleys.)
§ 2. Quantitative Estimation of the Power Transmitted.—In 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.
Let T be the mean torque or turning effort 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 ω, that is, in radians per second. The relation between these quantities is ω＝2πN. Then the, rate at which work is done by the engine crank shaft is Tω foot-pounds per second, equivalent to Tω/550 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 T$_1$ be the torque on the first motion shaft of the machine, and let wl be its angular velocity, then the rate at which the machine is absorbing energy is T$_1$ω$_1$ foot-pounds per second. A certain quantity 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 therefore T$_1$ω$_1$ would be less than Tω so that
| T$_1$ω$_1$＝ηTω, | (1) |
​ where η is called the efficiency of the transmission. Considering now the general problem of a multiple machine transmission, if T$_1$, ω$_1$, T$_2$, ω$_2$, T$_3$, ω$_3$,. . . are the several torques and angular velocities of the respective first motion shafts of the machines,
| (T$_1$ω$_1$+T$_2$ω$_2$+T$_3$+ω$_3$+ . . . .)＝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 Wv$^2$dθ/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−Wv$^2$/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−Wv$^2$/g)μdθ＝dt. The solution of this equation is
| t−Wv$^2$/g/u−Wv-/g＝e$^{μθ}$. | (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＝e$^{μθ}$. | (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＝e$^{2.094×0.3}$＝2·718$^{0.6282}$; that is log$_e$t/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 1/4 in. thick its width would be 11·2 in., or if 3/16 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 Wv$^2$/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)×18·84×1·5/550＝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−Wv$^2$/g) (1−e$^{−μθ}$). And the rate of working U, in foot-pounds per second, is
U = (t—u)v＝(tv−Wv$^3$/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.
(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°, ​ and this is the angle now used in the practice of Messrs Combe, Barbour and Combe, of Belfast. A section of the rim of a rope driving wheel showing the shape of the groove for a rope 13/4 in. diameter is shown in fig. 4, and a rope driving pulley designed for six 13/4 in. ropes is shown in fig. 5. A rope is less flexible than a belt, and therefore care must be taken not to arrange rope drives with pulleys having too small a diameter relatively to the diameter of the rope. The principles of §§ 3, 4, 5 and 6, apply equally to ropes, but with the practical modification that the working stress in the rope is a much smaller fraction of the ultimate strength than in the case of belting and the ratio of the tensions is much greater.
Fig. 4.
Fig. 5.—Rope Pulley, 10 ft. diam., 6 grooves, 21/2 in. pitch, weight about 35 cwt. Constructed by Combe, Barbour & Combe, Ltd., Belfast.
The following table, based upon the experience of Messrs Combe, presents the practical possibilities in a convenient form:—
The speed originally adopted for the rope was 55 ft. per second. This speed has been exceeded, but, as indicated above, for any particular case there is one speed at which the maximum horse power is transmitted, and this speed is chosen with due regard to the effect of centrifugal tension and the loss due to the continual bending of the rope round the pulley. Instead of using one rope for each groove, a single continuous rope may be used, driving from one common pulley several shafts at different speeds. For further information see Abram Combe, Proc. Inst. Mech. Eng. (July 1896). Experiments to compare the efficiencies of rope and belt driving were carried out at Lille in 1894 by the Société Industrielle du Nord de la France, for an account of which see D. S. Capper, Proc. Inst. Mech. Eng. (October 1896). Cotton ropes are used extensively for transmitting power in factories, and though more expensive than Manila ropes, are more durable when worked under suitable conditions.
§ 8. Shafts.—When a shaft transmits power from a prime mover to a machine, every section of it sustains a turning couple or torque T, and if ω is the angular velocity of rotation in radians per second, the rate of transmission is Tω foot-pounds per second, and the relation between the horse power, torque and angular velocity is
| Tω＝550 H.P. | (6) |
The problem involved in the design of a shaft is so to proportion the size that the stress produced by the torque shall not exceed a certain limit, or that the relative angular displacement of two sections at right angles to the axis of the shaft at a given distance apart shall not exceed a certain angle, the particular features of the problem determining which condition shall operate in fixing the size. At a section of a solid round shaft where the diameter is D inches, the torque T inch-pounds, and the maximum shearing stress f pounds per square inch, the relation between the quantities is given by
| T＝πD$^3$f /16, | (7) |
and the relation between the torque T, the diameter D, the relative angular displacement θ of two sections L inches apart by
| T＝CθπD$^4$/32L, | (8) |
where C is the modulus of rigidity for the material of the shaft. Observe that θ is here measured in radians.- The ordinary problems of shaft transmission by solid round shafts subject to a uniform torque only can be solved by means of these equations.
Calculate the horse power which a shaft 4 in. diameter can transmit, revolving 120 times per minute (12·56 radians per second), when the maximum shearing stress f is limited to 11,000 ℔ per square inch. From equation (7) the maximum torque which may be applied to the shaft is T=138,400 inch-pounds. From (6) H.P.＝138,400×12·56/12×550＝264. The example may be continued to find how much the shaft will twist in a length of 10 ft. Substituting the value of the torque in inch-pounds in equation (8), and taking 11,500,000 for the value of C,
θ＝138,400×120×32/11,500,00×3·14×256＝0.·57 radians,
and this is equivalent to 3·3°.
In the case of hollow round shafts where D is the external diameter and d the internal diameter equation (7) becomes
| T＝πf(D$^4$−d$^4$)/16D, | (9) |
and equation (8) becomes
| T＝Cθπ(D$^4$−d$^4$)/32L. | (10) |
The assumption tacitly made hitherto that the torque T remains constant is rarely true in practice; it usually varies from instant to instant, often in a periodic manner, and an appropriate value of f must be taken to suit any particular case. Again it rarely happens that a shaft sustains a torque only. There is usually a bending moment associated with it. For a discussion of the proper values of f, to suit cases where the stress is variable, and the way a bending moment of known amount may be combined with a known torque, see Strength of Materials. It is sufficient to state here that if M is the bending moment in inch-pounds, and T the torque in inch-pounds, the magnitude of the greatest direct stress in the shaft due to the effect of the torque and twisting moment acting together is the same as would be produced by the application of a torque of
| M+(T$^2$+M$^2$) inch-pounds. | (11) |
It will be readily understood that in designing a shaft for the distribution of power to a factory where power is taken off at different places along the shaft, the diameter of the shaft near the engine must be proportioned to transmit the total power transmitted whilst the parts of the shaft more remote from the engine are made smaller, since the power transmitted there is smaller.
§ 9. Gearing Pitch Chains.—Gearing is used to transmit power from one shaft to another. The shafts may be parallel; or inclined to one another, so that if produced they would meet in a point; or inclined to one another so that if produced they would not meet in a point. In the first case the gear wheels are called spur wheels, sometimes cog wheels; in the second case bevel wheels, or, if the angle between the shafts is 90°, mitre wheels; and in the third case they are called skew bevels. In all cases the teeth should be so shaped that the velocity ratio between the shafts remains ​constant, although in very rare cases gearing is designed to work with a variable velocity ratio as part of some special machines. For the principles governing the shape of the teeth to fulfil the condition that the velocity ratio between the wheels shall be constant, see Mechanics, § Applied. The size of the teeth is determined by the torque the gearing is required to transmit.
Pitch chains are closely allied to gearing; a familiar example is in the driving chain of a bicycle. Pitch chains are used to a limited extent as a substitute for belts, and the teeth of the chains and the teeth of the wheels with which they work are shaped on the same principles as those governing the design of the teeth of wheels.
If a pair of wheels is required to transmit a certain maximum horse power, the angular velocity of the shaft being ω, the pressure P which the teeth must be designed to sustain at the pitch circle is 550 H.P./ωR, where R is the radius of the pitch circle of the wheel, whose angular velocity is ω.
§ 10. Velocity Ratio.—In the case of transmission either by belts, ropes, shafts or gearing, the operating principle is that the rate of working is constant, assuming that the efficiency of the transmission is unity, and that the product Tω is therefore constant, whether the shafts are connected by ropes or gearing. Considering therefore two shafts, T$_1$ω$_1$=T$_2$ω$_2$; that is ω$_1$/ω$_2$=T$_2$/T$_1$; i.e. the angular velocity ratio is inversely as the torque ratio. Hence the higher the speed at which a shaft runs, the smaller the torque for the transmission of a given horse power, and the smaller the tension on the belts or ropes for the transmission of a given horse power.
§ 11. Long Distance Transmission of Power.—C. F. Hirn originated the transmission of power by means of wire ropes at Colmar in Alsace in 18 50. Such a telodynamic transmission consists of a series of wire ropes running on wheels or pulleys supported on piers at spans varying from 300 to 500 ft. between the prime mover and the place where the power is utilized. The slack of the ropes is supported in some cases on guide pulleys distributed between the main piers. In this way 300 h.p. was transmitted over a distance of 6500 ft. at Freiberg by means of a series of wire ropes running at 62 ft. per second on pulleys 177 in. diameter. The individual ropes of the series, each transmitting 300 h.p., were each 1⋅08 in. diameter and contained 10 strands of 9 wires per strand, the wires being each 0⋅072 in. diameter. Similar installations existed at Schaffhausen, Oberursal, Bellegarde, Tortona and Zürich. For particulars of these transmissions with full details see W. C. Unwin’s Howard Lectures on the “Development and Transmission of Power from Central Stations” (Journ. Soc. Arts, 1893, published in book form 1894). The system of telodynamic transmission would no doubt have developed to a much greater extent than it has done but for the advent of electrical transmission, which made practicable the transmission of power to distances utterly beyond the possibilities of any mechanical system.
See W. J. M. Rankine, Treatise on Machinery and Millwork; and W. C. Unwin, Elements of Machine Design; and for telodynamic transmission see F. Reuleaux, Die Konstrukteur. (W. E. D.)