厚圆板轴对称振动的弹性力学解

Elasticity Solution for Axisymmetric Vibration Problem of Thick Circular Plate

本文以轴对称三维弹性力学基本方程为基础,导出厚圆板强迫振动的状态方程式。利用Maclaurin级数和Sylvester定理,厚圆板的位移和应力可以用中面位移和应力的微分算子表示。通过载荷分解和圆板表面条件,可以得到厚圆板在对称载荷与反对称载荷作用下的振动控制方程。求解了厚圆板在周边固支和简支条件下的对称与反对称的自由振动问题。通过数值计算得到了这两类自由振动的固有频率。 本文的方法适用于求解厚圆板在任意厚跨比的强迫振动和自由振动的问题。

In this paper, based upon the basic equations of three dimensional theory of elasto - dynamics, the state e-quation of thick plate have been formulated. By using the Mclaurin series and Sylvester theorem, the dynamic response of stresses and displacements of thick plate can be expressed by the differential operator of the stresses and displacements in the neutral plane. By means of the load decompose and the surface load condition, the vibrational governing equations of thick plate subjected to the symmetric and unti - symmetric loads are derived. The symmetric and unti -symmetric free vibration problems of thick plate with the fixed and simple support edge conditions are solved. By the numerical calculation, the free vibration frequencies of thick plate are obtained. The method promoted in this paper is suit for solving the dynamic response and free vibration frequencies of thick plate in any value of 2h/a and any order vibrational mode.