非对称Bernoulli-Euler薄壁梁的弯扭耦合振动

COUPLED BENDING-TORSIONAL VIBRATIONS OF ASYMMETRICAL BERNOULLI-EULER THIN-WALLED BEAM

通过直接求解均匀Bernoulli-Euler薄壁梁单元自由振动的控制运动微分方程,推导了其精确的动态传递矩阵。采用Bernoulli-Euler弯扭耦合梁理论,假定梁横截面没有任何对称性,考虑了薄壁梁在两个方向的弯曲振动及翘曲刚度的影响。动态传递矩阵可以用于计算非对称薄壁梁及其集合体的精确固有频率和模态形状。针对具体的算例,给出了各种边界条件下固有频率的数值结果并与文献中已有的结果进行了比较,还讨论了翘曲刚度对固有频率和模态形状的影响,结果表明如果忽略翘曲刚度的影响,可能得到毫无意义的结果。

The exact dynamic transfer matrix is derived for a straight and uniform Bernoulli-Euler thin-walled beam element whose elastic and inertial axes are not coincident by directly solving the governing differential equations of the motion. The Bernoulli-Euler bending-torsion coupled beam theory is used, and the cross-section is asymmetric. The bending vibrations in two perpendicular directions are coupled with the torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix can be used to calculate exact natural frequencies and mode shapes for asymmetrical thin-walled beams and its assemblages. Numerical results are given for the example of thin-walled beam with a variety of boundary conditions, and numerical solutions of natural frequencies are tabulated for comparison. The effect of warping stiffness on natural frequencies and mode shapes is also discussed. The results show that when the warping effect is neglected, the associated errors become increasingly large as the frequency order increases.