VIII. On the whirling and vibration of shafts

Abstract 1. It is well known that every shaft, however nearly balanced, when driven at a particular speed, bends, and, unless the amount of deflection be limited, might even break, although at higher speeds the shaft again runs true. The particular or “critical“ speed depends on the manner in which the shaft is supported, its size and modulus of elasticity, and the size, weight, and position of any pulleys it carries. The theory for the case of an unloaded shaft first received attention at the hands of Professor Rankine, who obtained numerical formulae for the cases of an unloaded shaft resting freely on a bearing at each end, and for an overhanging shaft working in a shoulder at one end.