弹性支撑旋转Timoshenko梁动力学特性

摘要

为研究弹性支撑旋转梁动力学特性随转速及弹性支撑参数变化规律，考虑剪切效应、转动惯量和陀螺效应，采用Hamilton原理推导旋转Timoshenko梁动力学方程，应用Chebyshev谱方法获得系统涡动频率与模态振型数值解。结果表明，在高速转动状态下陀螺效应、支撑结构刚度对Timoshenko梁动力学特性有显著影响；各阶固有频率随着转速增加而分成正向涡动频率与反向涡动频率，高阶频率变化幅度更大；涡动频率随支撑结构直线刚度增加而呈阶梯状变化，当直线刚度增加到一定值后系统涡动频率将保持稳定；随着支撑结构转动刚度增加，涡动频率出现一个最小值与最大值，前者低于自由边界条件下频率值，后者高于固定边界条件下频率值。相关结果可用于各类旋转梁机构的设计与优化。%The dynamic characteristics of a spinning Timoshenko beam with elastic supports are investigated. The model of the beam is built and the rotatory inertia, shear effects and gyroscopic effects are considered. The governing equation is derived based on the extended Hamilton’s principle and is numerically solved by Chebyshev spectral method, and the natural frequencies and modals are obtained. It is shown that the gyroscopic effects and the support stiffness have a significant effect on the dynamic characteristics of the spinning Timoshenko beam. The natural frequencies are split into forward whirl speed and backward whirl speed, the distinction between the forward and backward whirl speeds becomes more obvious for higher order modes. The whirling speeds vary in step-wise with translational spring stiffness increasing and become stable when the stiffness exceeds a critical value. Two extremum values appear as the rotational stiffness increasing, the minimal value is smaller than the corresponding value under the free boundary condition and the maximum value is bigger than the corresponding value under the fixed boundary condition. The results can be used for designing and optimizing the mechanism of rotating beams.