Dynamic Resonances in Nonlinear Mechanical Systems

这项工作的重点是机械系统，其中一个或多个振荡频率是动态变量，其演变取决于系统的状态。随着时间的演进，这些系统可以慢慢地进出共振状态。该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，考虑了这些系统中存在的组件之间耦合的全部复杂性。在这项工作中，两个主要的研究课题将被解决，i)共振对直接耦合的动态行为的影响，多自由度系统，和ii)在一个或多个组件的周期-振幅依赖存在的非线性系统中的共振效应。这些包括失谐系统，其中具有动态频率的组件耦合到所有剩余的谐振模式，以及存在额外内部谐振的示例。采用奇异摄动方法对多自由度系统进行简化，并通过解析技术和数值研究分析共振区域的相空间输运。最后，分析预测将在现有的实验装置上进行验证，该实验装置将用于更全面地解释物理系统的复杂性。动力系统中的共振提供了一种机制，通过这种机制，复杂的耦合系统可以在其动力行为中经历意想不到的质变，这项工作将导致对这种行为的更完整的理解。这些主题代表了一类与复杂、相互连接的机械系统相关的重要问题，并将确定参数和耦合的特征，这些参数和特征在确定相互连接的机械系统中共振的影响方面起着重要作用。

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.