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在Hamilton辛对偶力学体系下,给出了求解一对边简支平面自由振动问题精确解的一般方法,并用该方法求得了一对边简支另一对边固支的矩形平面自由振动问题的精确解.首先用空间变量分离方法,求解矩形域平面自由振动问题的Hamilton正则方程,得到两个坐标方向的本征值关系;再利用Hamilton算子矩阵本征向量之间的共轭辛正交关系,得到广义振型函数向量的一般表达式;最后引入边界条件确定了两个空间本征值、频率方程和广义振型函数向量;讨论了固有振动频率与空间本征值的对应关系.把辛对偶方法和经典方法进行了比较,结果说明了本文方法的正确性和普适性.
In Hamilton symplectic dual mechanics system, a general method for solving exact plane free vibration problem of simply supported plane is given. By using this method, the free-vibration problem of rectangular plane with one pair of other simply supported edges is obtained Exact solution.Firstly, by using the method of spatial variable separation, the Hamilton regular equations of free vibration of the rectangular domain are solved to obtain the eigenvalues of the two coordinate directions. Then, the conjugate symplectic orthonormal between eigenvectors of Hamiltonian matrix The general expressions of generalized mode function vectors are obtained. Finally, the boundary conditions are used to determine two eigenvalues of space, frequency equation and generalized mode function vector. Corresponding relations of natural vibration frequency and space eigenvalue are discussed. The symplectic and classical methods are compared with each other, and the results show that the method in this paper is correct and universally applicable.