Critical Torsional Oscillations of a Rotating Accelerated Shaft

We assume that the forces applied to the shaft have a variable part which is a moment of constant amplitude M(x) per unit length, distributed along the shaft, and varying with a frequency proportional to the angular velocity. From the solution corresponding to the harmonic steady state vibration, we deduce, by using Heaviside's expansion, the motion due to a sudden application of the moment M(x). This enables us to compute the effect of the moment when applied with a linearly increasing frequency. In this analysis the damping will be neglected. In case of a viscous damping the linear character of the equations is not affected and the same method might be used. We did not carry this calculation for two reasons: 1. The exact result will be in general complicated, and involve viscous friction coefficients which will not be very accurately known. Besides, the effect of friction might be roughly taken into account by considering the steady state solution. 2. In most cases the damping is not viscous but due to the hysteresis or internal friction of the material. This is proved by the experimental fact that the energy absorbed in the vibration of elastic bodies is proportional to the frequency and not to its square. This effect might be taken roughly into account by energetic considerations