Vibration analysis of geometrically nonlinear spinning beams

• Vibrations of nonlinear spinning beam with general boundary conditions are studied.
• Nonlinear curvature and inertia are considered.
• Perturbation method is directly applied to partial differential equations of motion.

In this paper, free vibration and primary resonances of an inextensional spinning beam with six general boundary conditions are studied. The spinning beam has large amplitude vibrations, which lead to nonlinearities in curvature and inertia. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. To analyze the free vibrations and primary resonances, the method of multiple scales is directly applied to the partial differential equations of motion. We use the concept of forward and backward mode shapes in our analyses. Expressions are derived which describe the nonlinear free vibration and primary resonances of the spinning beam with six general boundary conditions. It is shown that unlike the primary resonances in which only the forward modes are excited, in the free vibration case both forward and backward modes are excited. Numerical examples are presented for hinged–hinged and clamped–free boundary conditions and the results are verified by numerical simulations.