Hidden Instabilities due to Coexistence Phenomenon in Autoparametric Excitation

当一个工程结构受到振动源（如电机）的载荷时，共振发生在正模态振动的邻域。这些共振通常是不希望的，导致大的变形和可能的破坏性影响。如果在工程模型中考虑非线性效应，则正常模态振动是称为非线性正常模态的周期性运动。本文研究两自由度系统的非线性正态模态。这样的系统受到一种称为自参数激励的现象的影响。这涉及到以共振方式在非线性正态模式之间传递能量，非线性正态模式可能变得不稳定。如果一个结构被设计成在特定的振动模式下运行，如果这种模式被证明是不稳定的，那么这个设计就是失败的。确定非线性正态模是否稳定的通常程序涉及使用称为Floquet理论的数学理论。这提供了不稳定性的预测，例如基于运动微分方程的数值积分。在提议的研究中，我们解决了一种情况（称为共存），在这种情况下，Floquet理论预测了稳定性，但如果模型稍微改变，就会导致不稳定性。由于没有一个工程模型能完美地代表一个物理情况，这种共存模型的不稳定性是不可避免的。尽管共存存在于各种物理系统模型中，但很少有研究旨在理解相关的不稳定性。这个项目的目的是1)确定哪些系统表现共存（从一般类型的系统），以及2)检查由此产生的不稳定性的性质。该项目将资助一名博士研究生，从而将研究和教育结合起来。研究结果将通过在科学/工程会议上展示和在科学杂志上发表而广泛传播。通过这种方式，希望工程师们能够意识到可能由隐藏的不稳定性引起的设计问题，这些不稳定性是本研究的主题。

When an engineering structure is loaded by a vibratory source such as amotor, resonances occur in the neighborhood of normal modal vibrations. These resonances areoften undesirable, resulting in large deformations and possible destructive effects. Ifnonlinear effects are taken into account in the engineering model, the normal mode vibrations areperiodic motions known as nonlinear normal modes.This work concerns nonlinear normal modes in two degree of freedom systems.Such systems are subject to a phenomenon known as autoparametric excitation. Thisinvolves energy being transferred between nonlinear normal modes in a resonant fashion, with thepossibility that a nonlinear normal mode may become unstable. If a structure were to be designed tooperate in a particular vibrational mode, and if that mode turned out to be unstable, the designwould be a failure.The usual procedure for determining whether a nonlinear normal mode isstable or not involves using a mathematical theory known as Floquet theory. This offerspredictions of instability based on for example numerical integration of the differential equations ofmotion. In the proposed research, we address a situation (called coexistence) in which Floquet theory predicts stability, but which involves instabilities which would result if the model was changed slightly.Since no engineering model perfectly represents a physical situation, such instabilities in models withcoexistence are inevitable.Although coexistence occurs in a variety of models of physical systems verylittle research has been aimed at understanding the associated instabilities. This project aims 1) todetermine which systems exhibit coexistence (from a general class of systems), and 2) to examine thenature of the resulting instability.This project will fund a graduate research student at the doctoral level andthus will integrate research and education. The results of the research will be broadly disseminated bybeing presented at a scientific/engineering meeting and published in a scientific journal. Inthis way it is hoped that engineers will be made aware of possible design trouble that could be caused by thehidden instabilities which are the subject of this research.