浅正弦波纹圆板大挠度问题的摄动解法

Perturbation Method of Large Deflection Problem of Circular Plate with Shallow Sinusoidal Corrugation

基于扁壳的非线性大挠度理论,用摄动法和幂级数方法求解了波纹圆板的大挠度方程.选取无量纲中心挠度作为摄动参数,将描述波纹圆板的非线性微分方程组化为一系列线性微分方程组.对于中心平台部分,描述各阶摄动的线性微分方程组成为通常的欧拉方程,可以得到精确解;对于波纹部分,不能直接得到各阶摄动的精确解,采用幂级数方法求解.再根据边界条件、连续条件和摄动条件,将摄动问题化为线性代数方程组进行求解,得到了具有中心平台的浅正弦波纹圆板在各种荷载作用下的具有中心挠度二次项的弹性特征.

Based on the nonlinear theory of large deflection of shallow shells,the large deflection equations of corrugated circular plate were solved by using the perturbation and power series method.Selecting dimensionless deflection at the center as parameter of perturbation,the nonlinear differential equations describing corrugated circular plate were reduced to a series of linear differential equations.For the plane central region,the linear differential equations describing the perturbation of different orders were reduced to Euler equations which can be solved analytically.For the corrugated parts,the accurate solution for the perturbation of different orders can not be obtained.Therefore,power series method was used to solve the linear differential equations.Then,according to boundary conditions,continuous conditions and conditions of perturbation,the perturbation problems were reduced to the solution of the linear algebraic equations.The elastic characteristics of circular plate with shallow sinusoidal corrugation and a plane central region with the second order term of deflection at the center under different load were obtained.