广义辛方法结构非线性轴向振动与控制的理论、数值与实验

Structures with axial motion are commonly found not only as the key dynamic components in large engineering structures but also as one of the major elements, such as sensors and actuators, in many micro-electro-mechanical systems. Due to varying degrees of external excitation during dynamic motion, axial nonlinear vibration and stability problems are normally inevitable. Such nonlinear dynamics responses will cause either structural failures or significantly reduce the performance and serviceable life of the designs. As a result, a systematic research into the nonlinear axial vibration and stability of such structures, and active control of the dynamic responses, is not only research enriched but also possesses great potential for engineering applications...This primary objective of this research project is to investigate with solutions the various engineering issues of structures with axial motion. Using nonlinear axial vibration of a structure with uniform axial motion as a breakthrough, a generalized Hamiltonian system with symplectic solution methodology will be established. Subsequently, the model and solution technique will be generalized for structures with axial acceleration where an augmented conservative quantity will be derived and an augmented symplectic conservative system will be developed. Based on the theory of symplectic geometry and a unique symplectic computational technique, the bifurcation and chaos nonlinear problems of the structures will be further theoretically analyzed and numerically simulated. The influences and consequences of symplectic coupled variables will be analytically enunciated, and the transformation relations and rules between the coupled variables will be theoretically revealed. Using sensors and actuators as design elements, a coupled and active control approach will be developed. The success of this project will not only become the basis of further research and design considerations for engineering structures with axial motion, but also will provide scientific knowledge and information for the active control of in-service structures with nonlinear axial vibration and stability problems.

轴向运动结构不只是许多大型工程动力装置中的关键结构，也是微机电动力系统中的主要元件之一（如传感器、制动器等）。由于实际运行中不同程度的扰动影响，往往会产生非线性轴向振动和稳定性问题。该振动会引发安全事故或严重降低设备使用性能及寿命，因此研究轴向运动结构及非线性动力行为、稳定性及其控制具有重要的工程意义和广阔的应用前景。本项目将针对轴向运动结构及其关键科学问题展开研究。以匀速轴向运动结构非线性振动作为突破口，建立一种广义哈密顿体系的辛求解方法。以此为基础，利用加速轴向运动结构非线性振动的近似守恒量，发展一种近似保辛体系。进一步采用辛空间理论与辛计算方法，对该类结构的分岔、混沌等非线性行为进行理论研究和数值仿真。阐明对偶变量之间的作用和影响，揭示控制变量的存在的转换关系和规律，通过设置传感器和制动器形成一种耦合主动控制方法，为含该类结构的工程装置前期研发和设计以及后期使用中的控制提供科学依据。

轴向运动结构不只是许多大型工程动力装置和传送系统中的关键结构，也是微机电动力系统中传感器和制动器等的主要元件之一。在复杂环境下，微小扰动往往会产生非线性轴向和横向振动和稳定性问题。该振动会引发安全事故或严重降低设备使用性能及寿命，因此研究轴向运动结构及非线性动力行为、稳定性及其控制具有重要的工程意义和广阔的应用前景。本项目将针对轴向运动结构及其关键科学问题展开了研究。以匀速轴向运动结构非线性振动作为突破口。考虑轴向运动的经典结构梁、弦和板等基本模型，构造状态空间下的原变量及其对偶变量。通过变分原理和守恒性，导出广义哈密顿正则方程以及对应的初、边值条件。从而建立一种广义哈密顿体系和辛求解方法。将固有频率和振动模态归结为哈密顿体系本征值和本征解问题，以及将非线性振动问题归结为本征解展开问题和代数方程问题。形成一种辛方法和解析方法。以空间坐标模拟时间和以时间模拟空间坐标，在时域构造广义非线性哈密顿体系。在该体系下，分析了轴向运动结构非线性振动的稳定性问题，研究其分岔、混沌等非线性行为和超谐共振问题等。考虑时间的尺度，提出一种研究超谐共振问题的哈密顿体系多尺度方法。建立一套轴向运动结构非线性振动问题的数值仿真系统和数值方法。该方法可从本质上揭示超谐共振现象和机理。研究结果表明，超谐共振现象是引起系统失稳和破坏的主要因素之一。为解决该问题，通过搭建轴向运动基本结构的实验平台，开展了外激励载荷引发的各阶振动模态的研究，分析轴向运动结构的非线性力学行为，并验证频率与本征值、振型与本征解之间的联系。通过调整运行速度以及支承条件，实现规避共振和超谐共振，达到主动控制目的。对于具体问题提出具有鲁棒性和自适应性的控制方案。该系统将为具有该类结构的新型装置研发、设计以及后续使用提供可靠依据，为工程应用中该类传送系统设备的设计方案提供了控制技术。同时，为解决非线性问题提供一条路径和方法。