In this research, free and forced vibrations of the functionally graded material (FGM) sandwich beams are investigated by using the scaled boundary finite element method (SBFEM). The innovation of this method is that the discretization only exists in the axial direction of the beam, which reduces the spatial dimension of problem by one, the time-consuming mesh generation processes are greatly reduced. The homogeneous second-order differential equations of the beams are derived by the method of virtual work principle with the radial coordinates as independent variables, which realizes the characteristics of the FGM changing along the radial direction. By using the state space method and the Padé series expansion method, the governing equations are further solved analytically, and the mass and stiffness matrices of each layer of the FGM sandwich beams are obtained. The combination between layers of the FGM sandwich beams is realized according to continuity conditions of displacement. Therefore, the free and forced vibrations using Newmark time integration method of the FGM sandwich beams are further realized. A special attention is given to two different types of the FGM sandwich beams, one is the beam with FGM face sheets and the other is the beam with FGM core. Three different types of dynamic loadings are considered in the analysis of forced vibration. The convergence of the proposed method for solving the dynamic characteristics of the beams is verified by increasing the order of the higher order spectral elements. In order to prove the numerical stability of the current method for dynamic analysis of the FGM sandwich beams, the frequency of the FGM sandwich beams under various parameters and the transient vertical displacement of the beam loading points under three different loadings are solved. The numerical results of the current method are compared with the results obtained by using different theories in the other literature, which shows a good agreement. In addition, on the basis of the research, the doped FGM core in the uniform core is discussed by using the current method, which provides some guidance for the optimal design of the dynamic characteristics of the beams.

在这项研究中，功能梯度材料 (FGM) 夹层梁的自由振动和受迫振动通过使用缩放边界有限元方法 (SBFEM) 进行了研究。该方法的创新之处在于，离散化只存在于梁的轴向，将问题的空间维度减少了一个，大大减少了耗时的网格生成过程。利用虚功原理的方法，以径向坐标为自变量，推导出梁的齐次二阶微分方程，实现了FGM沿径向变化的特性。通过使用状态空间法和Padé级数展开法，进一步解析求解控制方程，得到每层FGM夹层梁的质量矩阵和刚度矩阵。FGM夹层梁的层间组合是根据位移的连续性条件实现的。因此，进一步实现了 FGM 夹层梁使用 Newmark 时间积分方法的自由振动和受迫振动。特别关注两种不同类型的 FGM 夹层梁，一种是带有 FGM 面板的梁，另一种是带有 FGM 芯的梁。在受迫振动分析中考虑了三种不同类型的动态载荷。通过增加高阶光谱元素的阶数，验证了所提出的用于求解梁的动态特性的方法的收敛性。为了证明当前FGM夹层梁动力分析方法的数值稳定性，求解了不同参数下FGM夹层梁的频率和三种不同载荷下梁加载点的瞬态竖向位移。将当前方法的数值结果与其他文献中使用不同理论得到的结果进行了比较，显示出良好的一致性。此外，在研究的基础上，利用现有方法对均匀纤芯中的掺杂FGM纤芯进行了探讨，为梁的动态特性优化设计提供了一定的指导。将当前方法的数值结果与其他文献中使用不同理论得到的结果进行了比较，显示出良好的一致性。此外，在研究的基础上，利用现有方法对均匀纤芯中的掺杂FGM纤芯进行了探讨，为梁的动态特性优化设计提供了一定的指导。将当前方法的数值结果与其他文献中使用不同理论得到的结果进行了比较，显示出良好的一致性。此外，在研究的基础上，利用现有方法对均匀纤芯中的掺杂FGM纤芯进行了探讨，为梁的动态特性优化设计提供了一定的指导。