Modified Fourier Approach for Vibration Analysis of Spinning Beam with Elastic Restraints

This paper presents a unified method for analyzing the dynamic behavior of spinning beams under elastic constraints. Based on the Timoshenko beam theory, a dynamic model of a spinning beam with elastic constraints is established. The displacement and bending angle are represented by a modified Fourier series. Compared with the traditional Fourier series, the improved Fourier series eliminates the discontinuity of the derivative at the boundary by introducing auxiliary polynomials, making it more suitable for elastic constraints. The governing equations and boundary conditions are coupled together using the energy method to form a set of standard linear equations. By solving this equation, the modes of the spinning beam structure under elastic constraints can be concisely and quickly obtained. Finally, by comparing with other methods, it is proved that the method has good convergence and practicability. Then, the effects of spinning speed, boundary stiffness and slenderness ratio on the whirling characteristics are analyzed. The results show that the linear spring has a more pronounced effect on the whirl frequency than the torsion spring. Different boundary constraints produce different elastic intervals. Mode exchange was found with increasing spinning speed. In the case of elastic constraints, the mode exchange occurs at lower spinning speed. This method has a certain universal applicability to the dynamic analysis of spinning beams under elastic constraints, and the research results can provide theoretical reference for subsequent research.