初始纵横向荷载效应对薄板自振频率的影响分析The analysis for the effect of both initial in-plane and transverse loads on the natural frequency of plates

首先基于能量变分原理，给出了同时考虑初始纵向、横向荷载效应情况下板的应变能表达、动力平衡微分方程及单元刚度矩阵。再以动力平衡微分方程为基础，运用伽辽金法解得四种典型板同时考虑初始纵向、横向荷载效应的基频近似解，并运用瑞利法解得简支矩形板考虑初始纵向、横向荷载效应的前三阶频率近似解。然后相互验证了考虑初始纵向、横向荷载效应的板单元刚度矩阵和频率近似解的正确性，并进一步分析了初始纵向、横向荷载及相关因素对板自振频率的影响。结果表明，考虑初始纵向、横向荷载效应后，板的自振频率主要受初始纵向荷载、初始横向荷载、板的厚度及边界条件等因素的影响；初始纵向、横向荷载效应对板的基频影响明显于高阶频率；初始纵向、横向荷载对板的自振频率影响分别呈线性和抛物线规律。

The analysis for the effect of both initial in-plane and transverse loads on the natural frequency of plates

Firstly,the strain energy expression,the dynamic equilibrium differential equation and the element stiffness matrix are presented for plates considering the effect of both initial in-plane and transverse loads via the energy variation principle.Then,the approximate solutions of fundamental frequencies for four typical plates,considering the effect of both initial in-plane and transverse loads,are derived by the Galerkin method,based on the dynamic equilibrium differential equation.And the approximate solutions of the first three frequencies for a simply supported rectangular plate,considering the effect of both initial in-plane and transverse loads,are solved by the Rayleigh method.The frequencies obtained by the finite plate element agree well with the approximate solutions,the reliability and efficiency of the presented finite plate element and those frequency approximate solutions are verified.Finally,the frequencies of plates influenced by the effect of both initial in-plane and transverse loads and related factors are analyzed by numerical examples,result shows that the key physical factors governing the effect of initial loads on the frequencies of plates are the initial loads (both in-plane and transverse),thickness and boundary condition etc.The effect of both initial in-plane and transverse loads on fundamental frequency is more remarkable than that on high-order frequencies.The frequencies influenced by initial in-plane loads and initial transverse loads respectively follow the linear pattern and the parabolic pattern.