弹性地基上矩形薄板自由振动的精确解

EXACT SOLUTIONS FOR FREE VIBRATIONS OF RECTANGULAR THIN PLATES ON THE WINKLER FOUNDATION

根据分离变量法得到了Winkler弹性地基上矩形薄板自由振动问题的精确解,分析了地基模量对频率的影响,其中边界条件为CCCC、SCCC和SSCC情况的精确解过去被认为是难以得到的。在分离变量方法中,不需要事先人为的选取满足某一组对边边界条件的挠度函数,而是直接利用控制方程本征根给出振型函数通解的一般解析形式,再利用边界条件得到振型函数系数和频率方程的精确形式。数值结果与有限元结果及文献结果吻合较好,验证了该文方法和结果的的正确性。

The exact solutions of free vibrations of rectangular thin plates on the Winkler foundation are solved by using the method of separation of variables, and the effects of foundation on frequencies are analyzed. It was considered that there were not the closed form exact solutions for the plates with boundary conditions CCCC, SCCC and SSCC(S denotes simple support, and C denotes clamp). The exact eigenfunctions in the method of saperation of variables are directly solved from the governing differential equation, this is different from that in the inverse method, then the coefficients of the exact eigenfunctions and the exact eigenvalue equations are determined using boundary contions.The numerical results agree with those in literatures and FEM, this validates the correctness of the present method.