Primary Resonance of Nonlinear Spinning Timoshenko Shaft Based on a Novel Third-order Approximation Model Derived from Geometrically Exact Nonlinear Model

In this paper, a set of generalized nonlinear equations of motion for spinning Timoshenko shafts is derived using the concept of a geometrically exact approach. In order to investigate the primary resonance of the shaft, the multiple scale method is applied to the discrete equations of motion. In this study, the effects of shear deformation, rotary inertia, gyroscopic terms, and linear damping were considered. To show the advantages of the Timoshenko models, a comparison is made between the results of Timoshenko and classical models. As a result, it can be seen that in the Timoshenko model, the amplitude of the vibration is directly related to the slenderness ratio of the shaft. Also, linear and nonlinear shear terms can affect the primary resonance of spinning shafts and their effects are more noticeable in higher vibrational modes.