Free vibration analysis of a nonlinear slender rotating shaft with simply support conditions

•The free vibration of a nonlinear rotating shaft is considered.•The effect of forward and backward mode on the free vibration is investigated.•The effect of rotary inertia and rotational speed on free frequencies is studied.•The nonlinear free frequencies for a rotating shaft are determined.

The free vibration analysis of a nonlinear slender rotating shaft with simply support conditions is studied. In the slender rotating shaft the effect of shear deformation is negligible. Accordingly, in the modeling of the system rotary inertia and gyroscopic effect are considered. The equations of motion are derived by the extended Hamilton principle. The nonlinear system analyzed utilizing multiple scales method. The forward and backward nonlinear frequencies of the slender rotating shaft are obtained. It is seen that for natural vibration of a slender rotating shaft, backward and forward modes are involved.