Hamilton体系下中厚板静力弯曲和自由弯曲振动问题的一类模型及通解

对于中厚板的静力弯曲和自由弯曲振动问题,引入两个辅助函数,采用胡海昌在Reissner板理论基础上提出的中厚板微分方程及边界条件,将两类问题的控制方程引入Hamilton体系,分别得到Hamilton体系下中厚板静力弯曲和自由振动问题的微分方程组模型. 比较后得到了Hamilton体系下中厚板静力和振动问题的统一模型,其特点是: 微分方程组模型的统一形式中Hamilton矩阵在对角线位置有2个零子块矩阵. 对于中厚板静力和振动问题,比较了所得齐次微分方程组的特征根,给出齐次微分方程组的通解并进行了比较,从而使问题的求解更理性化和合理化,求解过程遵循一套统一的方法论,便于把这类解法推广到其它问题.

A type of model and ordinary solutions for static bending and free vibration problems of moderate thick plate in Hamilton system

For static bending and free vibration problems of moderately thick plate, by introducing two auxiliary functions and using the differential equations and the boundary conditions based on Reisser plate theory presented by Hu Hai-chang, the governing equations were introduced into Hamilton system, and the differential equation set model of static bending and free vibration problems of moderately thick plate were presented respectively. The unitary model for the two problems was given after comparison. In the presented united forms, the Hamilton matrix H includes two zero block matrix in the diagonal place. For the homogeneous differential equation set, the characteristic roots were compared for the static bending and free vibration problems of moderately thick plate. Then the ordinary solutions of the homogeneous equations were obtained and compared. Thus the solution finding is rational and reasonable. The solving procedure follows a set of methodology,and the solving procedure can be extended to other problems.