双参数弹性地基上矩形薄板自由振动问题的Hamilton方法

Free Vibration Problems of Rectangular Thin Plates on Two-parameters Elastic Foundation by the Hamiltonian Method

运用矩阵多元多项式的带余除法把双参数弹性地基上矩形薄板的振动方程转化为Hamilton系统,利用分离变量法给出对应的Hamilton算子.通过计算得到对边简支问题所对应Hamilton算子的本征值和本征函数系,并证明了该本征函数系的辛正交性和在Cauchy主值意义下的完备性.根据本征函数系的完备性,得到对应Hamilton系统的通解,进而给出双参数弹性地基上对边简支矩形薄板问题振型函数的通解.此外,通过两个例子说明此方法可以计算出自由振动问题的频率和振型函数.

The free vibration equations of rectangular thin plates are transformed into infinite dimensional Hamiltonian systems using pseudo-division algorithm for matrix multi-variable polynomial.Hamiltonian operators are obtained by means of separation variable method.By calculating the Hamiltonian operators for the rectangular plates with two opposites simply supported,the eigenvalues and eigenvectors of the Hamiltonian operators are derived.Then,the symplectic orthogonality and the completeness of the eigenvectors（in the sense of Cauchy＇s principal value）are proved.Based on the completeness of the eigenvectors,the general solutions of the Hamiltonian systems are obtained and the general solutions of the plate problem on elastic foundation of two parameters with two opposites simply supported can be obtained.Furthermore,two examples are given to illustrate that the frequency and deflection of the free vibration problem are directly solved by the present method.