For inside gearing, if r1 be the less radius and r2 the greater, 1/r1 − 1/r2 is to be substituted for 1/r1 + 1/r2.
§ 103. Friction of Cords and Belts.—A flexible band, such as a cord, rope, belt or strap, may be used either to exert an effort or a resistance upon a pulley round which it wraps. In either case the tangential force, whether effort or resistance, exerted between the band and the pulley is their mutual friction, caused by and proportional to the normal pressure between them.
Let T1 be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it; T2 the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; θ the ratio of the length of that arc to the radius of the pulley; dθ the ratio of an indefinitely small element of that arc to the radius; F = T1 − T2 the total friction between the band and the pulley; dF the elementary portion of that friction due to the elementary arc dθ; ƒ the coefficient of friction between the materials of the band and pulley.
Then, according to a well-known principle in statics, the normal pressure at the elementary arc dθ is T dθ, T being the mean tension of the band at that elementary arc; consequently the friction on that arc is dF = ƒT dθ. Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc, or dT = dF = ƒT dθ; which equation, being integrated throughout the entire arc of contact, gives the following formulae:—
| hyp log. | T1 | = ƒθ |
| T2 |
| T1 | = eƒθ |
| T2 |
F = T1 − T2 = T1 (1 − e − ƒθ) = T2 (eƒθ − 1)
(65)
When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest, and when the pulleys are next set in motion, so that one of them drives the other by means of the belt, it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged. Its value is given by this formula—
| T1 + T2 | = | eƒθ + 1 |
| 2 | 2 (eƒθ − 1) |