Multiple scales solution for free vibrations of a rotating shaft with stretching nonlinearity

In this paper, free vibration of a simply supported rotating shaft with stretching nonlinearityis investigated. rotary inertia and gyroscopic effects are included, but shear deformation is neglected.the equations of motion are derived with the aid of the hamilton principle and then transformed to thecomplex form. to analyze the free vibration, the method of multiple scales is directly applied to the partialdifferential equation of motion. an analytical expression, as a function of system parameters, is derived,which describes the nonlinear free vibration of the rotating shaft in two transverse planes. the effectsof rotary inertia, external damping and rotating speed on the forward and backward nonlinear naturalfrequencies are considered. it is shown that both forward and backward nonlinear natural frequencies arebeing excited. to validate the perturbation results, we use numerical simulation.