The dynamic analysis of contact problems is related to great mathematical difficulties and, thus, unsurprisingly has to date not been solved completely. In this work, a semi-analytical method is proposed to evaluate some integrals of Green’s function used in the dynamic analysis of a rectangular plate resting on the surface of an elastic foundation of inertial properties (Lamb’s model). The great challenge herein is overcoming the singularity present in the study of Green’s function related to this problem. The proposed solution involves the discretization of the studied system (a rectangular plate resting on the surface of an elastic foundation of inertial properties), which leads to a numerical solution in matrix form. All the terms of the matrix are doubly indexed, and the singularity is present in the terms having the same indices. Therefore, special efforts are made to calculate the terms of the matrix having the same indices, in order to eliminate the singularity. This requires solving the integrals of the terms of the matrix with the same indices analytically and the integrals of the terms of the matrix of different indices by numerical methods. Finally, this study of Green’s function is used in the dynamic analysis of the above-defined system and was successfully accomplished with a semi-analytical method leading to determinate values of the Eigen-frequencies and the Eigen-shapes of the plate.

中文翻译：

---

用于接触问题的兰姆模型格林函数积分的评估

接触问题的动态分析与巨大的数学困难有关，因此，毫不奇怪，迄今为止尚未完全解决。在这项工作中，提出了一种半解析方法来评估格林函数的一些积分，这些积分用于对搁置在惯性特性弹性基础（兰姆模型）表面上的矩形板进行动力学分析。这里的巨大挑战是克服与此问题相关的格林函数研究中存在的奇异性。建议的解决方案涉及所研究系统的离散化（放置在惯性属性弹性基础表面上的矩形板），这导致矩阵形式的数值解。矩阵的所有项都被双重索引，并且奇异性出现在具有相同索引的项中。所以，为了消除奇异性，特别努力计算具有相同索引的矩阵的项。这需要解析求解具有相同索引的矩阵的项的积分，以及通过数值方法求解不同索引的矩阵的项的积分。最后，格林函数的这项研究用于上面定义的系统的动态分析，并通过半解析方法成功完成，导致确定板的特征频率和特征形状的值。这需要解析求解具有相同索引的矩阵的项的积分，以及通过数值方法求解不同索引的矩阵的项的积分。最后，格林函数的这项研究用于上面定义的系统的动态分析，并通过半解析方法成功完成，导致确定板的特征频率和特征形状的值。这需要解析求解具有相同索引的矩阵的项的积分，以及通过数值方法求解不同索引的矩阵的项的积分。最后，格林函数的这项研究用于上面定义的系统的动态分析，并通过半解析方法成功完成，导致确定板的特征频率和特征形状的值。