Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia

In this paper the free vibrations of an in-extensional simply supported rotating shaft with nonlinear curvature and inertia are considered. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. To analyze the free vibrations of the shaft, the method of multiple scales is used. This method is applied to the discretized equations, and directly to the partial differential equations of motion, which demonstrates the same results. An expression is derived which describes the nonlinear free vibrations of the rotating shaft in two transverse planes. It is found that in this case, both forward and backward nonlinear natural frequencies are being excited. The results of perturbation method are validated with numerical simulations.