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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 solution to the free vibration problem of rectangular thin plate on Winkler elastic foundation is obtained based on the method of separating variables. The influence of the modulus of foundation on the frequency is analyzed. Exact solutions of the boundary conditions CCCC, SCCC and SSCC are considered to be difficult to obtain. In the method of separating variables, we do not need to select the deflection function of a certain set of boundary conditions in advance. Instead, we use the eigenvalue of the control equation to give the general analytical form of the general solution of the modal function, and then use the boundary conditions to get the vibration The exact form of the coefficient of function and frequency equation. The numerical results are in good agreement with the finite element results and the literature results, which verify the correctness of the method and the results.