复杂非线性连续系统新型全局离散和高效非线性动力学分析方法研究

A new global discretization method and an efficient nonlinear dynamic analysis method are developed for complex nonlinear continuous systems. Traditionally, complex boundary conditions are very hard to be completely satisfied, and there is no direct approach to analyze nonlinear dynamic behaviors of continuous systems. A method for decomposing the dependent variable and complex and coupled boundary conditions will be developed. Guidelines for selecting trial functions will be established for one-dimensional continuous systems, and a new global discretization method with all the boundary conditions satisfied will be developed. A new spatial and temporal discretization method will be developed for one-dimensional continuous systems, and harmonically balanced equations will be derived using tensor operators. An efficient incremental harmonic balanced method will be used to analyze nonlinear dynamic behaviors and stability. An inverted elastic pendulum experiment will be conducted to study its nonlinear dynamic phenomena under combined parametric and forced excitations. A mapping method from a complex domain where a two-dimensional problem needs to be solved to a simple domain will be developed. The new global discretization method will be extended to two-dimensional continuous systems and the dynamic behavior of a rectangular plate with complex boundary conditions will be studied as an example.

本项目以复杂非线性连续系统为研究对象，以发展精确的全局离散方法和高效的非线性特性分析手段为研究目标，针对连续系统复杂边界条件难以全部精确满足和无法直接对连续系统进行非线性特性分析的问题，开展复杂非线性连续系统全局离散与非线性特性分析研究。掌握一维连续系统复杂边界与耦合边界下的应变量和边界条件的分解方法，确立全局离散试探函数选取准则，发展精确满足所有复杂边界条件的一维连续系统新型全局离散方法。研究一维连续系统时间空间同时离散方法，建立张量算子表达的谐波方程，掌握用于一维连续系统的高效增量谐波平衡方法，探究其非线性振动规律。开展非线性动力学实验研究，揭示参数激振和强迫振动共同作用下柔性摆的非线性动力学现象,以验证理论分析结果。掌握二维连续系统复杂问题域到简单域的映射方法，拓展新型全局离散方法到二维连续系统，以二维矩形板在复杂边界下的振动模型为例，分析其动力学特性，验证新方法的有效性。

本项目着眼于研究新型全局离散方法和高效非线性动力学分析方法。在全局离散方法研究方面，提出一种采用转角和轴向及剪切应变对梁进行描述的新型全局离散方法，用于复杂未定边界条件问题，可精确得到连续系统的应力；在高效非线性动力学分析方法方面，建立一种基于快速傅里叶变换和张量缩并的高效Galerkin平均-增量谐波平衡法，实现了高效求解。由于引入独立采样过程，实现了谐波平衡过程与待求解问题的解耦，可通用化求解微分方程的稳态周期响应，该算法可作为通用微分方程半解析求解分析器使用；提出适用于指数-3微分代数方程的高效Galerkin平均-增量谐波平衡法及稳定性分析方法，用于对多体动力学系统进行非线性分析。可求得高维多体动力学系统的稳态周期解、确定稳定区间、分析分岔等复杂动力学行为。新算法高效、通用性强，有潜力发展为通用化的多体动力学求解分析软件；基于并行计算技术，提出了使用于模态离散连续系统的高性能谐波平衡法，大幅提高了计算速度，提升了对高维连续系统的非线性分析能力。