基于时间有限元的非线性系统周期响应求解及稳定性分析

Solving and stability analysis of periodic response of nonlinear system based on time finite element method

非线性现象广泛存在于结构分析之中,获取非线性系统的周期响应对分析结构的频响特征、分岔与稳定特性至关重要。为此,一种基于时间有限元的非线性系统周期响应求解方法被提出。该方法在伽辽金时间有限元法的基础上,引入周期边界条件,并结合牛顿迭代法进行求解。该方法的优势在于:能够简单地处理非光滑周期荷载(如阶跃或冲击周期荷载);能够直接根据时间有限元的系统矩阵计算传递矩阵和Floquet乘子,进而判定周期解的稳定性。最后,通过数值算例验证了所提时间有限元在非线性系统周期响应求解及稳定性分析中的正确性、收敛性以及精度。

Nonlinear phenomena widely exist in structural analysis,and to obtain periodic response of nonlinear system is crucial for analyzing frequency response characteristics,bifurcation and stability characteristics of structures.Here,a method for solving periodic response of nonlinear systems based on time finite element was proposed.Based on Galerkin time finite element method,the proposed method could introduce periodic boundary conditions,and combine with Newton iteration method to solve periodic response of nonlinear systems.It was shown that the proposed method can simply handle non-smooth periodic load,such as,step or impact periodic load;it can directly calculate transfer matrix and Floquet multiplier according to system matrix of time finite element to judge the stability of periodic solution;the correctness,convergence,and accuracy of the proposed time finite element method in solving periodic response and stability analysis of nonlinear systems are verified with numerical examples.