基于耦合振子系统等效的人工周期结构振动带隙研究

Artificial periodic structures are used a lot in the field of civil engineering and machine. Excessive vibrations are easily produced under the effect of external excitations in those structures. How to reduce vibration by adjusting the parameters of periodic structures has always been a hot academic topic cared in engineering mechanics and material physics. The current thinking on this issue mostly focus on the wave angle, and researches are mainly based on elastic wave motion theory and lattice energy band theory. This project plans to transfer the artificial periodic structures to coupled oscillators systems. Consequently, the research of vibration band gap can be equivalent to the research of amplitude death. Specific research plan is presented as follows: first, the typical artificial periodic structures will be transformed into appropriate coupled oscillators systems by means of experiments and numerical simulations. Second, suitable experiments will be designed to make sure the equivalence between the coupled oscillators systems and the original periodic structures. Third, researches on the mechanism and evolution process of amplitude death in the equivalent systems will be conducted. Finally, the phenomenon of amplitude death in large-scale coupld oscillators systems will be discussed based on the synchronization theory of complex network. The purpose of this study is to reduce the difficulty and scale of computations in the research of vibration ban gap in artificial periodic structures with the aid of the bifurcation and control theory of nonlinear dynamical systems and the synchronization theory of complex network. Thus, new insights will be provided to the analysis and design of the space arrangement of large-scale complex artificial periodic structures from the angle of vibration.

土木、机械等领域大量使用的人工周期结构极易在外激励作用下产生过大振动，如何通过调节周期结构自身参数来达到减振目的一直以来都是工程力学、材料物理等领域关心的热点学术问题。目前对这一问题的思考大都着眼于波动角度，研究主要基于弹性波动理论和晶格能带理论。本项目计划将人工周期结构等效成耦合振子系统，从而将周期结构中振动带隙的研究转化为耦合振子系统中振幅死区的研究，具体研究计划如下：（1）通过实验和数值模拟将典型人工周期结构转化成耦合振子系统（2）设计合适的实验确保耦合振子系统和原周期结构具有等效性（3）研究等效耦合振子系统中振幅死区现象出现的机理和演化过程（4）结合复杂网络同步理论讨论大规模耦合振子系统的振幅死区现象。研究目的是借助非线性动力系统分岔、控制理论和复杂网络同步理论来降低人工周期结构振动带隙研究的计算难度和规模，从振动角度为分析和设计大规模复杂人工周期结构空间排布提供一条新思路。

本文针对周期结构的带隙问题，从非线性动力学的角度进行了机理研究，内容包括四个方面一、将典型人工周期结构转化成耦合振子系统；二、对已有的耦合振子系统设计合适的控制策略来引发或抑制振幅死区并与实测对照确保耦合振子系统和原结构具有等效性；三、研究等效耦合振子系统中振幅死区现象出现的机理和演化过程；四、结合复杂网络同步讨论大规模耦合振子系统的振幅死区的对应关系。本项目对上述内容的研究结论对应如下：一、通过分析周期单元的动力学特性和结构中振动传播规律找到合理转化的依据。二、通过实际观测结合已有研究结论确保将原始周期结构转化为耦合振子系统的合理性和可靠性。三、通过对典型耦合振子系统的动力学特性分析，阐述了振幅死区的出现机理，并对参数诱发振幅死区做了详尽的讨论和研究。四、将振幅死区的出现和振子之间的同步联系起来。通过对振子同步的研究来探讨振幅死区的出现。本项目研究从新的角度理解周期结构中带隙出现，有助于寻找新的减振、隔振方法。