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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:—

Diameter  of Rope.	Smallest diameter of Pulley, which should be used with the Rope.	H.P. per Rope for Smallest Pulley at 100  revs. per minute.
in.	in.	in.
3/4	14	5/8
1	21	1
15/8	42	8
21/8	66	16

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＝πD3f /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＝CθπD4/32L,

(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(D4−d4)/16D,

(9)

and equation (8) becomes

T＝Cθπ(D4−d4)/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+(T2+M2) 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