Abstract:

The dynamic stiffness matrix for a uniform and straight thinwalled Timoshenko beam element is derived by directly solving the governing differential equations of coupled bendingtorsional vibration of beam element. The effects of bendingtorsion coupling, warping stiffness, shear deformation and rotary inertia are taken into account in the present formulations. An illustrative example on the application of the dynamic finite element method is given for a bendingtorsion coupled channel section thinwalled beam In order to obtain the natural frequencies and normal modes, the automated Muller root search method and the bisection method ombined with frequency sweep method are used to solve the frequency characteristics equation. The effects of warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the particular thinwalled beam are discussed. Numerical results demonstrate the accuracy and effectiveness of the proposed method and show the warping stiffness, shear deformation and rotary inertia can change the natural frequencies and mode shapes of the open thinwalled beam substantially.

Key words: vibration and wave, thinwalled Timoshenko beam, warping stiffness, bendingtorsion coupled vibration, dynamic finite element method

摘要：

通过直接求解单对称均匀薄壁Timoshenko梁单元弯扭耦合振动的运动微分方程，推导了其精确的动态刚度矩阵。在本文研究中考虑了弯扭耦合、翘曲刚度、转动惯量和剪切变形的影响。针对某弯扭耦合的薄壁梁算例，应用本文推导的动态刚度矩阵，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，计算了该薄壁梁的固有特性，并讨论了翘曲刚度、剪切变形和转动惯量对该弯扭耦合薄壁梁的固有频率和模态形状的影响。数值结果验证了本文方法的精确性和有效性，并指出随着模态阶次的增加，剪切变形、转动惯量和翘曲刚度对薄壁梁的固有特性的影响更加显著。

关键词: 振动与波, 薄壁Timoshenko梁, 翘曲刚度, 弯扭耦合振动, 动态有限元法