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：(a) 能够简单地处理非光滑周期荷载（如阶跃或冲击周期荷载），以及(b)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。
Abstract
Nonlinear phenomena widely exist in structural analysis. Obtaining the periodic response of nonlinear systems is crucial to analyzing the frequency response characteristics, bifurcation and stability characteristics of structures. Therefore, a method for solving the periodic response of nonlinear systems based on time finite element method is proposed. On the basis of Galerkin time finite element method, in this method, periodic boundary conditions are introduced and combined with Newton iterative method to solve problem. The advantages of this method are: (a) it can simply deal with nonsmooth periodic loads (such as step or impact periodic loads), and (b) it can directly calculate the transfer matrix and Floquet multiplier according to the system matrix of the time finite element, and then determine the stability of the periodic solution. Finally, the correctness, convergence and accuracy of the proposed time finite element method in the periodic response solution and stability analysis of nonlinear systems are verified by numerical examples.
关键词
非线性系统 /
时间有限元 /
非光滑周期荷载 /
稳定性分析 /
稳态响应。
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Key words
nonlinear systems /
time finite method /
Non-smooth periodic loads /
analysis of stability /
Steady-state response.
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