Hidden Instabilities due to Coexistence Phenomenon in Autoparametric Excitation

当工程结构由振动源加载时，共振发生在正常模式振动的附近。这些共振通常是不受欢迎的，导致巨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，则简正模的振动是周期运动，称为非线性简正模。这项工作涉及两自由度系统中的非线性简正模。这类系统受到一种称为自参激励的现象的影响。这涉及到能量在非线性简正模之间以共振的方式传递，其中一个非线性简正模可能会变得不稳定。如果一个结构被设计成在特定的振动模式下工作，如果该模式被证明是不稳定的，那么设计将是失败的。确定非线性简正模式是否稳定的通常程序涉及到使用一种被称为弗洛奎特理论的数学理论。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。在提出的研究中，我们讨论了一种情况(称为共存)，在这种情况下，Floquet理论预测了稳定性，但如果模型稍有改变，就会涉及不稳定性。由于没有任何工程模型完美地描述了物理情况，所以在共存的模型中，这种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，但很少有研究旨在了解相关的不稳定性。这个项目的目的是1)确定哪些系统表现出共存(从一般的系统类别)，以及2)检查由此产生的不稳定性的性质。这个项目将资助一名博士水平的研究生，从而将研究和教育结合起来。研究成果将通过在科学/工程会议上发表并在科学期刊上发表的方式广泛传播。通过这种方式，希望让工程师们意识到隐藏的不稳定因素可能会导致设计上的麻烦，这是本研究的主题。

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.