The following notation is used throughout the chapter on Pulleys:
A =Area of rim (sq. in.). a = " arm ( " " ). b = Center of pulley to center of belt (inches; practically equal to R). C1 = Total centrifugal force of rim (lbs.). c - Distance from neutral axis to outer fiber (inches). D = Diameter of pulley (inches). D1 = " " hub ( " ).
d1 = " " bolt at root of thread (inches). d = Diameter of bolt holes (inches). g =Acceleration due to gravity (ft. per sec.). h = Width of arm at any section (inches). I = Moment of inertia. L = Length of arm, center of belt to hub (inches). L1 = Length of rim flange of split pulley (inches).
l = Length of hub (inches).
N = Number of arms.
n = " " rim bolts, each side.
P =Driving force of belt (lbs.).
P1 = Force at circumference of shaft (lbs.). P2 = Force at circumference of hub (lbs.), p = Stress in rim due to centrifugal force (lbs. per sq. in.). R = Radius of pulley (inches). 8 = Fiber stress (lbs. per sq. in.). • = Fiber stress in flange (lbs. per sq. in.). T = Thickness of web (inches). t = " " rim ( " ).
If a flexible band be wrapped completely about a pulley, and a heavy stress be put upon each end of the band, the rim of the pulley will tend to collapse just like a boiler tube with steam pressure on the outside of it. A compressive stress is induced which is very nearly evenly distributed over the cross-section of the rim, except at points where the arms are connected thereto. At these points the arms, acting like rigid posts, take this compressive stress. Now, a pulley never has a belt wrapped completely round it, the fraction of the circumference embraced by the belt being usually about ½, and seldom, even with a tightener pulley, reaching |. Assuming the wrap to be ½ the circumference, and that all the side pull of the belt comes on the rim, none being transmitted through the arms to the hub, we then have one-half of the rim pressed hard against the other half by a force equal to the resultant of the belt tensions, which, in this case, would be the sum of them. Dividing the pulley by a plane through its center and perpendicular to the belt, the cross-section of the rim cut by this plane has to take this compressive stress.
This analysis is satisfactory from an ideal standpoint only, for the intensity of stress due to the direct pull of the belt, with the usual practical proportions of rim, would be very small. Moreover, the element of speed has not been considered.
When the pulley is under speed, a set of conditions which complicates matters is introduced. The centrifugal force due to the weight of the rim and arms is no longer negligible, but has an important influence upon the design and material used. This centrifugal force acts against the effect of the belt wrap, tending to reduce the compressive stress, or, overcoming the latter entirely, sets up a tensional stress both in the rim and in the arms. It also tends to distort the rim from a true circle by bowing out the rim between the arms, thus producing a bending moment in the rim, maximum at the points where the rim joins each arm.
It can readily be imagined that the analysis in detail of these various stresses in the rim acting in conjunction with each other is quite complicated - far too much so in fact, to be introduced here. As in most cases of such design, however, one controlling influence can be separated out from the others, and the design based thereon with sufficient margin of strength to satisfy the more obscure conditions. This is rational treatment, and the "theory" will be studied accordingly.
The rim, being fastened to the ends of the arms, tends, when driving, to be sheared off, the resisting area being the areas of the cross-sections of the arms at their point of joining the rim. The force that produces this shearing tendency is the driving force of the belt, or the difference between the tensions of the tight and loose sides.
Again, at the point of connection of the arms to the hub, e shearing action takes place, so that, if this shearing tendency were carried to rupture, the hub would literally be torn out of the arms. Now, viewing the arms as beams loaded at the end with the driving force of the belt, and fixed at the hub, a heavy bending stress is set up, which is maximum at the point of connection to the hub. If the rim were stiff enough to distribute this driving force equally between the arms, each arm would take its proportional share of the load. The rim, however, is quite thin and flexible; and it is not safe to assume this perfect distribution. It is usual to consider that one-half the whole number of arms take the full driving force.