Fig. 2677.
In the accompanying illustration, [Fig. 2677], a is the driving and b the driven pulley, rotating as denoted by the arrows; hence c is the driving and d the slack side of the belt. Now let us examine how this slackness is induced. It is obvious that pulley a rotates pulley b through the medium of the side c only of the belt, and from the resistance offered by the load on b, the belt stretches on the side c. The elongation of the belt due to this stretch, pulley a takes up and transfers to side d, relieving it of tension and inducing its slackness. The belt therefore meets pulley b at the point of first contact, e, slack and unstretched, and leaves it at f, under the maximum of tension due to driving b. While, therefore, a point in the belt is travelling from e to f, it passes from a state of minimum to one of maximum tension. This tension proceeds by a regular increment, whose amount at any given point upon b is governed by the distance of that point from e. The increase of tension is, of course, accompanied by a corresponding degree of belt stretch, and therefore of belt length; and as a result, the velocity of that part of the belt on pulley b is greater than the velocity of any part on the slack side of the belt; hence the velocity of the pulley is also greater than that of the slack side of the belt. In the case of pulley a the belt meets it at g under a maximum of tension, and therefore of stretch, but leaves it at h under a minimum of tension and stretch, so that while passing from g to h the belt contracts, creeping or slipping back on the pulley, and therefore effecting a reduction of belt velocity below that of the pulley. To summarize, then, the velocity of the part of the belt enveloping a is less than that of a to the amount of the creep; hence the velocity of the slack side of the belt is that of a minus the belt creep on a. The velocity of the part of the belt on b is equal to that of the slack side of the belt plus the stretch of the belt while passing over b; and it follows that if the belt or slip creep on one pulley is equal in amount to the belt stretch on the other, the velocities of the two pulleys will be equal.
Fig. 2678.
Now (supposing the elasticity of the belt to remain constant, so that no permanent stretch takes place) it is obvious that the belt-shortening which accompanies its release from tension can only equal the amount of elongation which occurs from the tension; hence, no matter what the size of the pulleys, the creep is always equal in amount to the stretch, and the velocity ratio of the driven pulley will (after the increase of belt length due to the stretch is once transferred to the slack side of the belt) always be equal to that of the driving pulley, no matter what the relative diameters of the pulleys may be. In [Fig. 2678], for example, are two pulleys, a and b, the circumference of a being 10 inches, while that of b is 20; and suppose that the stretch of the belt is an inch in a revolution of a (a being the driving pulley). Suppose the revolutions of a to be one per minute, then the velocity of the belt where it envelops a and b, and on the sides c and d, will be as respectively marked.
Thus the creep being an inch per revolution of a, the belt velocity on the side c will be nine inches per minute, and its stretch on b being an inch, the velocity of b will be ten inches per minute, which is equal to the velocity of the driving pulley.
It is to be observed, however, that since a receives its motion independently of the belt, its motion is independent of the creep, which affects the belt velocity only: but in the case of b, which receives its motion from the belt, it remains to be seen if stretch is uniform in amount from the moment it meets this pulley until it leaves it, for unless this be the case, the belt will be moving faster than the pulley at some part of the arc of contact.
