A new formulation and free vibration analysis of a nonlinear geometrically exact spinning Timoshenko beam

摘要

In this paper, a set of generalized nonlinear equations of motion for Timoshenko spinning beams (shafts) is derived by using the concept of geometrically exact theory. After simplification of equations by use of in-extensionality assumption, the partial differential equations of motion are discretized by Galerkin method and free vibration of a simply supported spinning shaft with movable end and a fixed-free spinning shaft are investigated using multiple scale method. The linear parts of equations of motion verified by other spinning Timoshenko models but since here, the transverse displacements and bending rotations were considered independently, the nonlinear parts of the proposed formulation are more accurate and complete than other Timoshenko models. The effect of linear damping, rotary inertia, gyroscopic effects and shear deformation, and also slenderness ratio are studied. It is shown that the effect of geometric nonlinearities is much greater than inertial nonlinearities. In addition, it will be seen that geometric nonlinearities are directly related to the slenderness ratio and separation of generalized transverse and flexural coordinates causes more accurate prediction of nonlinear model behavior.