Analyse modale des structures dans des états élastiques périodiques

Modal analysis of structures in time-periodic elastic state

• Auteur / Autrice : Barend Julius Bentvelsen
• Direction : Corrado Maurini, Arnaud Lazarus
• Type : Thèse de doctorat
• Discipline(s) : Sciences mécaniques
• Date : Soutenance le 05/10/2018
• Etablissement(s) : Sorbonne université
• Ecole(s) doctorale(s) : École doctorale Sciences mécaniques, acoustique, électronique et robotique de Paris
• Partenaire(s) de recherche : Laboratoire : Institut Jean Le Rond d'Alembert (Paris ; 2006-....)
• Jury : Président / Présidente : Olivier Doaré
• Jury : Examinateurs / Examinatrices : Benoit Prabel, Sébastien Neukirch
• Jury : Rapporteurs / Rapporteuses : Fadi Dohnal, Olivier Thomas

Mots clés

Mots clés contrôlés

Vibrations

Stabilité

Analyse modale (ingénierie)

Floquet, Théorie de

Systèmes dynamiques

Mots clés libres

Analyse modale

Systèmes périodiques

Floquet théorie

Vibrations

Stabilité

Ingénierie des structures

Modal analysis

Time-periodic systems

Floquet theory

Vibrations

Stability

Structural dynamics

Résumé

Les systèmes périodiques sont une niche importante dans l’analyse des systèmes dynamiques. L'analyse modale ouvre de nouvelles opportunités pour l’ingénierie et la recherche. Le but de cette thèse est de montrer de manière exhaustive comment calculer une base modale pour des systèmes périodiques et comment appliquer ça pour réduire les équations de mouvement. Pour montrer la méthode marche des cas fondamentaux sont analysés.

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.