扁薄锥壳的非线性振动

本文首先用最小作用量原理推导出扁薄锥壳大振幅的变分方程。假设薄膜张力由两项组成,将协调方程化为两个独立的方程,选取扁锥壳中心最大振幅为摄动参数,采用我们提出的摄动变分法,将积分方程和微分方程线性化,使近似求解成为可能。我们对周边夹紧固定的圆底扁锥壳大振幅问题进行了求解。一次近似得到了扁锥壳线性固有频率,二次近似得到了频率比和中心最大振幅一次特征关系式,三次近似得到了频率比和中心最大振幅二次特征关系式。根据本文提供的特征关系式可进行工程设计。

Non-linear Vibration of Conical Shallow Thin Shells

In this paper, large amplitude variational equation of conical shallow thin shells is derived according to the principle of minimal action quantity. Assuming that the membrance tension is comprised of two parts, modifying the compatible equation into two parts, and taking the maximum central amplitude of the shallow conical thin shell as the perturbation parameter, the integral equation and the differential equation are linearized by the perturbation variation method 1] suggested by the authors, and thus it becomes possible to solve the problem approximately. A large amplitude problem of the circular conical shallow shell under the clamped edges boundary conditions is solved in this paper. By the first-order approximation, the linear natural frequency is obtained, first. By the second-order approximation, the first-order characteristic equation relating the ratio of the frequencies with the central maximum amplitude is obtained, then. By the third-order approximation, the second-order characteristic equation relating the ratio of frequencies and tne central maximum amplitude is obtained, finally. The characteristic equations obtained in this paper can be a reference for engineering designs.