An Invariant Manifold Approach to Modal Analysis of Nonlinear Structural Systems

这项研究的基本目标是发展一种非线性结构的模态分析方法，该方法利用新引入的非线性简正振型公式。这一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式的非线性结构的模态分析可能需要不切实际的数量的此类项，以便产生令人满意的模式收敛。这是因为，即使对于线性化的结构，线性模型也产生非耦合的模态动力学，从而产生最优的模态收敛，在非线性情况下，这些线性模式被非线性项耦合，导致模态收敛的恶化。所提出的非线性模态分析是基于不变流形的非线性运动简正模的定义和表述。这种方法是通用的，既不限于线性系统，也不限于保守系统，也不限于有限维系统。此外，该公式对于弱非线性系统是有建设性的，并提供了非线性模式的物理性质、相关的模式动力学以及非线性模式坐标与物理坐标的关系。提出的方法可以解决许多重要的研究问题，包括如何为非线性动力系统的模态分析选择最优振型的悬而未决的问题。为了解决这些问题，选择了综合策略，包括分析、计算和实验三个部分。

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.***