插值矩阵法分析轴向受载的Euler-Bernoulli梁双向弯曲与扭转耦合振动

Analysis of bi-axes bending-torsional coupled vibration of Euler-Bernoulli beams under axial load by interpolating matrix method

文章运用插值矩阵法研究了轴向受载的Euler-Bernoulli梁的双向弯曲扭转耦合自由振动问题。选择梁横截面的剪切中心作为坐标原点,坐标轴平行于梁截面的几何轴,振动微分方程中有关梁截面几何特性的参数均采用相对于几何轴的参数。轴向受载的Euler-Bernoulli梁的双向弯曲扭转耦合自由振动频率的计算转化为一组非线性常微分方程特征值问题,运用插值矩阵法求解,获得了3种边界条件下梁弯扭耦合振动的固有频率及其相应振型函数的计算结果,将数值计算结果与已有结果比较表明,文中方法具有很高的精度和效率。

The bi-axes bending-torsional coupled free vibrations of axially loaded Euler-Bernoulli beams with bi-asymmetric cross sections are studied by the interpolating matrix method. The reference point coincident with shear center of the beam is chosen and the reference axes are parallel to the geometric axes of cross sections, The geometric parameters of the cross sections of the beam are determined with respect to geometric axes. Then the governing equations of the free vibrations of axially loaded Euler- Bernoulli beams are transformed into a set of nonlinear characteristic ordinary differential equations （ODEs） with the natural frequencies orders. The interpolating matrix method is introduced to solve the derivative ODEs with three kinds of boundary conditions. Then the natural frequencies and the as- sociated mode shape of the beam are obtained. The numerical results show that the method is efficient and has very high accur^icy while comparing with the existing solutions.