阶梯式Timoshenko梁自由振动的DCE解

本文基于微分容积法和区域叠加技术提出了微分容积单元法（Differential Cubature Element method，以下简称DCE方法），并用之求解阶梯式变截面Timoshenko梁的自由振动问题。根据梁的变截面情况将其划分为几个单元，在每个单元内应用微分容积法将梁的控制微分方程和边界约束方程离散成为一组关于该单元内配点位移的线性代数方程组，将这些方程组写在一起并在各单元之间应用连续性条件和平衡条件得到一组关于整个域内各点位移的齐次线性代数方程组，这是一广义特征值问题，由子空间迭代法求解该特征问题便可求得系统的自振动频率。数值算例表明，本方法能稳定收敛、并有较高的数值精度和计算效率。

Free Vibrations of Stepped Timoshenko Beams by Differential Cubature Element Method

Based on the differential cubature method and the domain superposition technique, the differential cubature element method was developed to solve the free vibration problems of stepped Timoshenko beams. The basic idea of the differential cubature element method is to divide the entire variable domain into several sub-domains(elements) according to the structure of a beam and to apply the differential cubature method for each element. The displacement compatibility and force equilibrium conditions should be employed for the conjunction nodes on the interface boundaries of the elements. Combining the discreted governing equilibrium equations, compatibility equations and boundary constraint equations, a set of linear homogeneous algebraic equations could be obtained. This is a typical eigenvalue problem, the eigenvalues of which could be calculated by sub-space iterative method. The simplicity, efficiency and applicability of this method are demonstrated by the convergence and numerical accuracy studies.