基于比例边界有限元的复合梁自由振动频率计算

Natural Vibration Frequencies of Laminated Composite Beams Based on the Scaled Boundary Finite Element Method

将比例边界有限元方法(SBFEM)拓展用于计算复合梁的自由振动频率.该方法将梁简化为一维模型,并且仅选用x和z方向的弹性线位移作为基本未知量.从弹性力学基本方程出发,通过比例边界坐标、虚功原理和对偶变量技术推导得到了复合梁的一阶常微分比例边界有限元动力控制方程,其通解为解析的矩阵指数函数.利用Padé级数求解矩阵指数函数可得各个梁层的动力刚度矩阵,根据自由度匹配原则组装得到复合梁的整体刚度和质量矩阵.求解特征值方程,最终可得复合梁的自由振动频率.该方法对复合梁的层数和边界条件均无限制,具有广泛的适用性.将该文的解与三层、四层和十层复合梁振动频率的数值参考解以及阶梯型悬臂梁固有频率的实验实测值进行对比,验证了比例边界有限元算法的准确性、高效性和快速收敛性.

The scaled boundary finite element method(SBFEM)was extended to calculate the natural frequen-cies of laminated composite beams.With this method,the beam was simplified as a 1D model.Only the dis-placement components along the x and z directions were selected as the fundamental unknowns.Based on the fundamental equations of elasticity and the scaled boundary coordinates,under the principle of virtual work and with the dual vector technique,the 1st-order ordinary differential scaled boundary finite element dynamic equa-tion for composite beams was obtained,with its general solution in the form of the analytical matrix exponen-tial function.The Padéexpansion was utilized to solve the matrix exponential function and the dynamic stiffness matrix for each beam layer was acquired.According to the principle of matching degrees of freedom,the global stiffness and mass matrices of the laminated beam were gained.The eigenvalue equation was solved to give the natural vibration frequencies of the laminated composite beam.The results show that,the proposed method is widely applicable without limitation on the layer number and boundary conditions.Comparisons between the numerical natural frequencies and the experiment results of 3-,4-and 10-layered step-shaped cantilever beams,validate the accuracy,high efficiency and fast convergence of the SBFEM.