An Invariant Manifold Approach to Modal Analysis of Nonlinear Structural Systems

本研究的基本目的是开发一种非线性结构的模态分析方法，该方法利用了一种新引入的非线性正态振动公式。该主题的重要性源于这样一个事实，即依赖于相关线性化系统的模态的非线性结构的模态分析可能需要大量的此类项以产生令人满意的模态收敛。这是因为，即使对于线性化结构，线性模型也会产生不耦合的模态动力学，从而产生最优的模态收敛，在非线性情况下，这些线性模态被非线性项耦合，导致模态收敛性恶化。所提出的非线性模态分析是基于不变流形的非线性正态运动的定义和公式。这种方法是通用的，并不局限于线性系统，也不局限于保守系统，也不局限于有限维系统。此外，该公式对弱非线性系统是建设性的，并提供了非线性模态的物理性质，相关的模态动力学，以及非线性模态坐标与物理坐标相关的方式。所提出的方法将允许许多重要的研究问题得到解决，包括如何为非线性动力系统的模态分析选择最佳模态振型的突出问题。为了解决这些问题，我们选择了一种包括分析、计算和实验在内的综合策略

The basic objective of this research is the development of a modal analysis methodology for nonlinear structures, which utilizes a newly introduced formulation of nonlinear normal modes of vibration. The importance of the subject stems from the fact that a modal analysis of a nonlinear structure that relies on the modes of the associated linearized system may require an unpractical number of such terms in order to yield satisfactory modal convergence. This is because, even through for the linearized structure the linear models yield uncoupled modal dynamics and hence optimal modal convergence, in the nonlinear case these linear modes become coupled by the nonlinear terms, resulting in deterioration of the modal convergence. The proposed nonlinear modal analysis is based upon a definition and formulation of nonlinear normal modes of motion in terms of invariant manifolds. This approach is general and is not restricted to linear systems, nor to conservative systems, nor to finite-dimensional systems. In addition, the formulation is constructive for weakly nonlinear systems and provides the physical nature of the nonlinear modes, the associated modal dynamics, and the way in which the nonlinear modal coordinates are related to the physical coordinates. The proposed methodology will allow for many important research problems to be addressed, including the outstanding questions of how to choose optimal mode shapes for modal analysis of nonlinear dynamical systems. In order to attach these problems, and integrated strategy is chosen which involves analytical, computational, and experimental components.***