Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft

The nonlinear axial-lateral-torsional free vibration of the rotating shaft is analyzed by employing the Rayleigh beam theory. The effects of lateral, axial and torsional deformations, gyroscopic forces and rotary inertia are taken into account, but the shear deformations are neglected. In the new developed dynamic model, the nonlinearities are originated from the stretching of beam centerline, nonlinear curvature and twist and inertial terms which leads to the coupling between the axial, lateral and torsional deformations. The deformed configuration of the cross section of the beam is represented by the axial and lateral deformations, also the geometry of the beam in the deformed configuration is represented by Euler angles. A system of coupled nonlinear differential equations is obtained which is examined by the method of multiple scales and the nonlinear natural frequencies are determined. The accuracy of the solutions is inspected by comparing the free vibration response of the system with the numerical integration of the governing equations. The effect of the spin speed and radius-to-length ratio of the rotating shaft on the free vibrational behavior of the system is inspected. The study demonstrates the effect of axial-lateral-torsional coupling on the nonlinear free vibrations of the rotating shaft.

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