扁球壳的非线性固有频率

本文用最小作用量原理推导出扁球壳大振幅时的变分方程,继之将协调方程化为两个方程,选取扁球壳中心最大振幅为摄动参数,采用摄动变分法,将非线性方程线性化,对周边固定的扁球壳进行了求解,一次近似得到了扁球壳线性固有频率,二次近似和三次近似得到了扁球壳的非线性固有频率。从本文特征关系式容易给出圆薄板的线性和非线性固有频率,给研究这方面的动力工程问题提供了有价值的参考。

Non-linear Natural Frequencies of Shallow Spherical Shells

In this paper, based on the principle of minimum action quantity, the large amplitude variational equation of the shallow spherical shell is derived, and the compatible equation is divided into two equations. Taking the central maximum amplitude of the shallow spherical shell as the perturbation parameter, the nonlinear equation is linearized by the perturbation variational method, and the problem is solved under the condition of clamped edges. In the first-order approximation, the linear natural frequency of the shallow spherical shell is obtained. In the second and the third approximations,the non-linear natural frequency is derived. Moreover, it is quite easy to derive the linear and the non-linear natural frequencies of circular thin plates from the characteristic equation obtained in this paper. This paper provides the engineers concerned with a valuable reference.