Hidden Instabilities due to Coexistence Phenomenon in Autoparametric Excitation

当工程结构受到振动源（如电机）的载荷时，共振发生在法向模态振动附近。 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，则简正模态振动是周期性运动，称为非线性简正模态.本文研究二自由度系统中的非线性简正模态.这类系统受到一种称为自参数激励的现象的影响. 这涉及到能量在非线性正常模式之间以共振方式转移，非线性正常模式可能变得不稳定。 如果一个结构被设计成在一个特定的振动模式下工作，而这个模式被证明是不稳定的，那么这个设计就是失败的。确定一个非线性正常模式是否稳定的通常程序涉及到使用一个被称为Floquet理论的数学理论。 这提供了基于例如运动微分方程的数值积分的不稳定性预测。 在拟议的研究中，我们解决了一个问题，（称为共存），其中Floquet理论预测稳定性，但其中涉及不稳定性，如果模型稍有改变，将导致。由于没有工程模型完美地代表物理情况，虽然共存现象出现在各种物理系统的模型中，但很少有研究旨在了解与共存现象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），和2）检查所产生的不稳定性的性质。这个项目将资助一个博士水平的研究生，从而将研究和教育结合起来。 研究结果将通过在科学/工程会议上提出并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的设计麻烦，可能会造成的隐藏的不稳定性，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.