高维非线性系统的周期解分岔和准周期运动的分析方法

This proposed research project focuses on innovative techniques and algorithms to build accurate and efficient dynamic analysis methods for bifurcations of periodic solutions and quasi-periodic motions of high-dimensional nonlinear systems in order to better explain distributed structural systems behavior, predict bifurcations and route to chaotic motions, and capture the characteristics of Hopf bifurcations, resulting in quasi-periodic motions. High- and infinite- dimensional nonlinear systems are an important part of today’s dynamic analysis in Engineering, providing the accurate models for most dynamics distributed structural systems. However, with the time-varying characteristics existing and the complexity and/or strong of nonlinearities increasing, high-dimensional nonlinear systems are facing the difficult task of ensuring rich nonlinear dynamic behaviors and bifurcations, such as steady-state responses under combined parametric and forced excitation, period-doubling bifurcations, internal resonances, quasi-periodic motions, and chaotic motions. To address the problem of predictive and replicable scientific discovery in Engineering Mechanics and beyond, this research project develops accurate computational methods for dynamic analysis of high-dimensional nonlinear systems.. The specific research objectives are four-fold. First, the work involves developing the incremental harmonic balance (IHB) method, implementing a fast Fourier transformation (FFT), to simplify the Galerkin average procedure. Integrating a prior with the Jacobian matrix is a key to improving prediction accuracy and efficiency for dynamic analysis by using the IHB method. Second, this project develops the IHB method combined with quasi-periodic functions and multiple time scales to accurately identify quasi-periodic motions of high-dimensional nonlinear systems. Third, this project develops a new precise Hsu’s method, computing the G. Floquet transition matrix based on the precise time integration algorithm, to identify stability regions, predict bifurcations and route to chaotic motions, and capture the characteristics of bifurcations of high-dimensional nonlinear systems with complicated and/or strong nonlinearities. Finally, these methods are in accurate analyzing dynamics of high-dimensional nonlinear systems, with direct applications in axially moving strings/beams, beams attached to a rotating hub, and curved beams under base harmonic excitation and experimentally validate them.

研究高维非线性系统的周期解分岔和准周期运动的分析方法，使之能够精确且高效地计算高维非线性系统的动力学响应，分析从出现分岔到出现混沌这一过渡阶段的复杂分岔，进一步研究Hopf分岔后导致的准周期运动的精确定量分析计算。高维非线性系统中往往含有时变参数，并受到强非线性和复杂非线性项的影响，使得系统出现分岔及分岔后的动力学问题更为复杂，亟待被深入研究和解决。. 本项目包含4个部分，1.增量谐波平衡法结合快速傅丽叶变换(FFT)，精确高效地分析高维非线性动力学响应；2.以准周期函数为谐波函数，引入多时间尺度，推导相应的增量谐波平衡法，精确分析高维非线性系统的准周期运动；3.精细积分法引进G. Floquet理论中的转移矩阵的Hsu法数值积分计算，准确确定分岔及分岔点的位置，分析各种产生混沌的分岔过程；4.结合三个实际工程例子-轴向运动绳索，旋转梁，基础激励作用下的弯曲梁的非线性动力学。

研究高维非线性系统的周期解分岔和准周期运动的分析方法，使之能够精确且高效地计算高维非线性系统的动力学响应，分析从出现分岔到出现混沌这一过渡阶段的复杂分岔，进一步研究Hopf分岔后导致的准周期运动的精确定量分析计算。高维非线性系统中往往含有时变参数，并受到强非线性和复杂非线性项的影响，使得系统出现分岔及分岔后的动力学问题更为复杂，亟待被深入研究和解决。. 本项目采用了理论分析方法、数值仿真与实验验证相结合的手段从机理上对高维非线性系统的周期解分岔和准周期运动进行研究，系统分析在各种复杂非线性因素影响下非线性系统准周期运动的定量的精确求解。本项目的研究取得了下列几个成果：1.对于高维非线性系统响应频谱含有等相距频率的边频带的准周期运动，其频谱含有两个基频，其中一个是载频(carrier frequency)，另一个是边频带中各相邻频率的等相距，提出了改进的两时间变量增量谐波平衡法，其中一个时间变量含有载频，另一个时间变量含有等相距频率，精确地计算出准周期运动，分析了能量在不同模态间不断转移的机理；2.为了提高计算效率，推广了快速傅丽叶变换(FFT)与增量谐波平衡法相结合的方法，应用于高维非线性系统周期振动分析中，推广了FFT和Broyden方法与增量谐波平衡法相结合的方法，应用于单自由度非线性系统的准周期运动中；3. 利用精细积分法引进G. Floquet理论中的转移矩阵的Hsu法数值积分计算，准确确定周期解的分岔类型及分岔点的位置；4.用上述的增量谐波平衡法和数值方法研究了实际工程中轴向运动梁、含有子系统的轴向运动梁、旋转梁和基础激励作用下的屈曲梁复杂的非线性动力学响应，包含有周解振动、准周期运动和混沌等。. 本项目的研究成果有助于理解高维非线性系统一些复杂的非线性动力学问题，为研究非线性动力系统的准周期运动提供新的研究途径，解决实际工程中高维非线性动力系统的振动问题。