变截面Timoshenko悬臂梁自由振动分析

Free Vibration of Non-uniform Timoshenko Cantilever Beams

为考虑剪切变形和转动惯量的影响,基于模态摄动法基本原理,提出了一种求解变截面Timoshenko悬臂梁自由振动问题的近似解法。这一方法是利用等截面Euler梁的特征值和模态,将变截面Timoshenko梁特征方程的偏微分方程组转化为代数方程组进行求解,从而得到变截面Timoshenko梁的特征值和模态。该方法适用于求解任意复杂截面型式梁的动力特性,无论梁的截面变化是否连续。随后对截面阶跃变化和线性变化2类变截面梁进行算例分析,数值分析结果表明,这一方法简单、实用,具有良好的精度。

In order to consider the effect of shearing deformation and rotating inertia, an approximate method, which is based on modal perturbation method, is proposed for the free vibration analysis of nonuniform Timoshenko cantilever beams. With the eigenvalues and eigenvectors of uniform Euler beams, a set of partial differential characteristic equations of non-uniform Timoshenko beams is transformed into a set of algebraic equations for solutions of non-uniform Timoshenko beams. The method can solve the dynamic characteristics of beams with complicated cross-section, whether the cross-section variation is continuous or discontinuous. At the end, two types of beams, namely （a） discontinuous variation of thickness and （b） continuous and linear variation, are analyzed and shown that the method is simple, practicable and owns good precision.