考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，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，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。

COUPLING BENDING-TORSIONAL VIBRATION OF BERNOULLI-EULER THIN-WALLED BEAM INCLUDING WARPING EFFECT

The dynamic transfer matrix is developed for a straight and uniform slender thin-walled beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion for coupled bending-torsional vibration of the beam element. The simple Bernoulli-Euler beam theory is employed, and the cross section of the thin-walled beam is assumed to be monosymmetric. The bending-torsional coupling and the effect of warping stiffness are included in the present formulations. But the effects of shear deformation and rotary inertia are considered to be negligible and not included in this study. The derived dynamic transfer matrix can be used for calculation of the exact natural frequencies and mode shapes for thin-walled beams and structures approximated as composed of thin-walled beams. The automated Muller root search method and the bisection method combined with frequency sweep method are used to solve frequency characteristics equation. Two illustrative examples on the application of the presented theory are given for bending-torsional coupled beams with thin-walled semi-circular and channel cross sections. The effect of warping stiffness on natural frequency is discussed. Compared with those available in the literature, numerical results demonstrate the accuracy and effectiveness of the presented method and show the warping stiffness can change the natural frequency of open thin-walled beam substantially.