Modal Analysis of Structures in Time-Periodic Elastic State

Abstract : Time-periodic systems are an important niche in dynamical systems, analysing these systems profoundly opens up new possibilities for engineering. The goal of this thesis is to provide a comprehensive guide on computing Floquet forms and application of modal analysis of structures in time-periodic elastic state. The method is illustrated by fundamental examples. Modal analysis consists of computing the mode shape and the natural frequency of the equations of motion, this method is limited to linear time-invariant structures. Linear time-periodic systems have attracted attention over the past decades. Floquet theory has been applied to analyse stability. Using the same theory it is possible to compute periodic eigenvectors, known as Floquet forms, which are time-periodically equivalent to modes. Thus allowing the generalisation of modal analysis to linear time-periodic structures. Floquet forms are computed as eigensolutions of Hill’s matrix. The eigenspectrum requires treatment to find linearly independent Floquet forms. By using the archetypical case of a Ziegler column eigenvector sorting is shown to be the most efficient method to sort out Floquet forms. Projecting the physical equations of motion on Floquet forms results in a system of uncoupled equations with time independent coefficients. Truncating the number of Floquet forms results in a reduced order model. Computing Floquet forms is simplified by first projecting the equations of motion on classic modes. A finite element model of a periodically prestressed beam shows convergence of the solution with increasing number of Floquet forms. Thus proving the concept of time-periodic modal analysis.
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Submitted on : Tuesday, July 30, 2019 - 9:51:37 PM
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Barend Julius Bentvelsen. Modal Analysis of Structures in Time-Periodic Elastic State. Engineering Sciences [physics]. Sorbonne Universites, UPMC University of Paris 6, 2019. English. ⟨tel-02167604⟩

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