Dynamic Resonances in Nonlinear Mechanical Systems

这项工作集中在机械系统中，其中一个或多个振荡频率是动力学变量，其演化依赖于系统的状态。随着时间的推移，这些系统可以慢慢地进入和离开共振状态。这一建议研究了这些缓慢变化的系统在共振附近的动力学行为，解释了这些系统中存在的组件之间耦合的全部复杂性。本文主要研究两个问题，一是共振对直接耦合多自由度系统动力学行为的影响，二是非线性系统中存在一个或多个分量的周期-振幅依赖性时的共振效应。这包括失谐系统，其中具有动态频率的分量耦合到所有剩余的共振模式，以及存在附加内部共振的例子。利用奇异摄动法对多自由度系统进行约化，并通过解析技术和数值研究分析相空间通过共振区的传输。最后，分析预测将在现有的实验设备上得到验证，该设备将用于更全面地解释物理系统的复杂性。动力学系统中的共振提供了一种机制，通过这种机制，复杂耦合系统的动力学行为可以发生意想不到的质的变化，这一工作将导致对这一行为的更全面的理解。这些主题代表了与复杂的、相互关联的机械系统相关的一大类问题，并将确定在确定相互关联的机械系统中的共振影响方面发挥重要作用的参数和耦合特性。

This work focuses on mechanical systems in which one or more of the frequencies of oscillation are dynamical variables, whose evolutions are dependent on the state of the system. As time evolves these systems can slowly move in and out of a state of resonance. This proposal investigates the dynamical behavior of these slowly-varying systems near resonance, accounting for the full complexity of the coupling between components that exists in these systems. Two primary research topics are to be addressed in this work, i) the effects of resonances on the dynamical behavior of directly coupled, multi-degree-of-freedom systems, and ii) resonant effects in nonlinear systems in the presence of period-amplitude dependence of one or more of the components. These include mistuned systems where the components with dynamical frequencies are coupled to all the remaining resonant modes as well as examples in the presence of additional internal resonances. The multi-degree-of-freedom systems will be reduced using singular perturbation methods and the transport of phase space through the resonance region will be analyzed through analytical techniques and numerical investigations. Finally, the analytical predictions will be verified on an existing experimental apparatus that will be used to more fully account for the complexities of physical systems. Resonances in dynamical systems provide a mechanism by which complex coupled systems can undergo unexpected qualitative changes in their dynamical behavior, and this work will lead to a more complete understanding of this behavior. These topics represent a significant class of problems with relevance to complex, interconnected mechanical systems and will identify parameters and the characteristics of coupling which play an important role in determining the impact of resonances in interconnected mechanical systems.