Abstract A separation-of-variable method was proposed in 2009 by the present authors and co-workers (Xing YF, Liu B. New exact solutions for free vibrations of thin orthotropic rectangular plates. Compos Struct 2009;89:567–74), but this traditional separation-of-variable method is incapable for free boundary conditions. Hence, this work is to improve the traditional separation-of-variable method to make it capable of dealing with free boundary conditions. This new method is called the improved separation-of-variable method, in which the characteristic differential governing equation is solved directly, and the free boundary conditions are transformed into separable forms, which are achieved with the Rayleigh’s principle. In addition, the relationships between spatial eigenvalues and temporal eigenvalue are quite concise. For plates without free edges, the proposed method reduces to the traditional separation-of-variable method. Numerical experiments validate the proposed method. The present technique of handling free boundary conditions is a general methodology for eigenvalue problems such as free vibration and eigenbuckling.

中文翻译：

---

正交各向异性矩形薄板自由振动的改进变量分离法

摘要 本文作者和同事于 2009 年提出了一种变量分离方法（Xing YF, Liu B. New精确解法，用于薄正交各向异性矩形板的自由振动。Compos Struct 2009;89:567-74），但是这种传统的变量分离方法不能用于自由边界条件。因此，这项工作是为了改进传统的变量分离方法，使其能够处理自由边界条件。这种新方法称为改进的变量分离法，直接求解特征微分控制方程，将自由边界条件转化为可分离形式，利用瑞利原理实现。此外，空间特征值和时间特征值之间的关系非常简洁。对于没有自由边的板，所提出的方法简化为传统的变量分离方法。数值实验验证了所提出的方法。目前处理自由边界条件的技术是用于特征值问题（例如自由振动和特征屈曲）的通用方法。