高维非线性系统同宿和异宿轨道的计算及混沌动力学

Lots of nonlinear vibration problem in the actual project need to be described by high dimensional nonlinear dynamic system, and the calculation of homoclinic and heteroclinic orbits is the key problems in the analysis of complex dynamics problems such as multi-pulse chaos problem, thus the development of relative theoretical method to study the high-dimensional nonlinear dynamics system is urgently needed. This project starts with the calculation of homoclinic and heteroclinic orbits in high dimensional nonlinear systems, and develops the theoretical method which appropriate for global bifurcation and chaotic dynamics in high-dimensional nonlinear system, and then the analysis results of the vibration control of composite material plate can provide theoretical support for further research. This project takes composite truss core sandwich arched plate as research object, and establish dynamic equation of elastic deformation; Based on Padé approximation theory to form a new method to directly solve the expression of homoclinic and heteroclinic orbits in high dimensional nonlinear systems, and learn the impact of complex dynamic behaviors which affected by the structure parameter and excitation parameters; By simplifying and promoting generalized Melnikov theorem, the multiple pulse chaotic motions in high-dimensional nonlinear system is analyzed. Theoretically, the research of this project can form an effective method to analyze the problem of multi-pulse chaotic dynamics in high-dimensional nonlinear systems, and then promotes the development of high-dimensional system dynamics theory. In the aspect of application, this project can achieve a better understanding of the complex vibration behavior of composite truss core sandwich plate, also provide technical reserves for design in aerospace and safe operation of aircraft.

工程中存在很多复杂振动问题需要用高维非线性动力系统进行描述，而其同异宿轨道计算是多脉冲混沌等复杂动力学问题分析中必须首先解决的关键问题，迫切需要发展相关理论方法进行研究。本项目从高维非线性系统同异宿轨道的计算入手，发展适用于高维系统全局分岔与混沌动力学的理论方法，应用分析结果为复合材料板的振动控制提供理论支持。本项目以复合材料点阵夹芯拱形板的复杂振动问题为研究对象，建立其弹性变形动力学方程；以Padé逼近理论为基础，形成一套直接求解高维非线性系统同异宿轨道表达式的新方法，研究结构和激励参数对复杂动力学行为的影响；简化并推广广义Melnikov定理，研究高维系统多脉冲混沌运动行为。本项目的研究，在理论上可形成有效的高维非线性系统多脉冲混沌动力学问题分析方法，推动高维系统动力学理论发展。应用上有助于深入理解复合材料点阵夹芯板的复杂振动行为，为其在航空航天中的应用设计和飞行器安全运行提供技术储备

本项目从高维非线性系统同异宿轨道的计算入手，发展适用于高维系统全局分岔与混沌动力学的理论方法，应用分析结果为复合材料板及其他结构的振动控制提供理论支持。(1)考虑了复合材料点阵夹芯拱形板的初始挠度，建立能够体现复合材料点阵夹芯拱形板实际弹性变形的高维非线性方程，讨论物理参数和激励参数对复合材料点阵夹芯拱形板复杂力学行为的影响，总结拱形板跳跃现象和振动规律的产生机理。(2)讨论物理参数和激励参数对复杂动力学行为的影响，运用改进的Padé逼近方法求解各类MEMS结构非线性动力系统中的复杂同异宿轨道的表达式，以此来获取更为精确的分岔参数范围，还研究了这些复杂系统中的分岔控制、振动模态和优化策略等问题。(3)将相关理论与微谐振器的实际问题结合应用，计入形态误差建立了静电驱动单极板和双极板微谐振器关键振动元件的连续体数学模型，结合有限元仿真结果采用适用于此类方程的离散简化分析体系，并将其应用到分析器件复杂振动时的频率跳跃现象及多次跳跃现象。. 此外，为进一步验证理论研究的有效性、挖掘新的科研问题，形成了一套分子动力学仿真建模流程，以实现一类碳纳米MEMS关键零部件的屈曲力学性能分析；搭建了一套具有非接触、高速和微米级测量精度特点的MEMS动态测量平台，可实现面内弹性体运动测量，以此设计了一类非线性四稳态能量采集器；还分析了多自由度生物细长杆的异宿轨道现象及其对应的三维形态问题，研究了生物卷须攀爬打结现象中的非线性行为机理。