面内变刚度矩形薄板自由振动问题的辛弹性分析

Symplectic elasticity approach for free vibration of a rectangular plate with in-plane material inhomogeneity

摘要:

基于辛弹性的方法分析了变刚度矩形薄板的自由振动问题。假设矩形板的弯曲刚度沿板的长度方向呈指数函数变化而泊松比为常数，利用变分原理将其导入辛体系，并应用分离变量法和本征值展开给出了求解面内变刚度矩形薄板自振频率的一种解析方法。这种方法不同于传统的逆解法或者半逆解法，它不需要提前假设试函数，是一种更为理性的正向的求解方法。通过这种方法可以得到变刚度板自由振动的频率方程，数值算例表明该方法计算简便、结果精确，可以得到变刚度板的各阶自振频率。在此基础上，详细研究了不同边界条件下，梯度指数、泊松比以及长宽比对变刚度板自振频率的影响。

Abstract:

A symplectic elasticity approach for the analysis of the free vibration problems of rectangular plates with in-plane variable stiffness is presented in this paper. Employing the Hamiltonian variational principle, the problem is formulated within the framework of state space and solved using the method of separation of variables along with the eigenfunction expansion technique based on the assumption that the flexural stiffness of plate varies exponentially with the length coordinate and the Poisson ratio is constant. Unlike the classi¬cal semi-inverse methods where a trial shape function is required to satisfy the geometric boundary conditions, this symplectic approach proceeds without any shape functions and it is a more rational and forward solution method. Exact frequency equations of a Kirchhoff rectangular plate with in-plane variable stiffness are derived. Numerical results are given and the effects of different boundary conditions, the gradient index, Poisson's ratio and aspect ratio on natural frequency are studied through numerical examples.