针对任意边界条件下中心开口矩形板的自由振动特性研究问题, 引入改进傅里叶级数方法, 用改进傅里叶级数形式表示开口矩形板的位移容许函数, 该级数形式具有收敛性好、精度高等特点, 采用沿边界均匀分布的线性弹簧模拟任意边界条件, 并结合位移连续条件和Rayleigh-Ritz能量泛函变分法, 对未知傅里叶展开系数求极值将问题转化为求解一个标准特征值方程问题, 通过求解方程可得到中心开口矩形板的固有频率及其对应振型;对不同边界组合不需重新推导公式, 只需改变模拟弹簧刚度值即可, 提高了效率, 最后通过数值算例与有限元方法的计算结果进行对比分析以验证文中方法的有效性和精确性。

This paper is based on the improved Fourier series method to establish a free vibration analysis model and calculate the natural frequency of a rectangular plate with a central opening. When considering the opening, only a quarter of the plate was studied by using the symmetry of the rectangular plate with central opening, which was divided into three regions. The admissible function displacement of each region was expressed by the improved Fourier series. By using the linear spring, which was uniform distribution along the border, arbitrary boundary conditions were simulated. And through the continuous conditions of displacements, the relationship of each regional plate was determined. According to the Rayleigh-Ritz energy functional and the variational method, the overall energy function can be obtained. The generalized eigenvalue matrix equation can be obtained by studying the extremum of the unknown improved Fourier series expansion coefficients. Solving the equation can lead to the natural frequencies and the corresponding vibration modes of the rectangular plate with central opening. Finally, according to the calculation of numerical examples, the results were compared with the finite element method to verify the accuracy and effectiveness of the method in this paper.