An Invariant Manifold Approach to Modal Analysis of Nonlinear Structural Systems

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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.***