Nonlinear modal analysis based on invariant manifolds - Application to rotating blade systems
Analyse modale non linéaire basée sur les variétés invariantes - Application à des systèmes aubagés
Résumé
In the design-analysis cycle of complex structural systems such as rotorcraft, aircraft, and ground vehicles, it is necessary to understand their vibratory response thoroughly. If the vibration of interest is restricted to small neighborhoods of the static equilibrium positions, then the assumption of a linear system can be made. The corresponding analysis procedure is then greatly simplified, through the use of modern tools such as Finite Element Analysis and Modal Analysis. In contrast, when the amplitudes of oscillations are large, beyond the scale of linearization, or when a system behaves inherently nonlinearly with respect to its equilibrium configurations, then nonlinear equations of motion must be used in the model.
It is well known that nonlinear systems exhibit much richer and more complex behavior than their linear counterparts (i.e., bifurcations, internal resonances, sensitivity to initial conditions, etc.). Moreover, for a nonlinear system linear superposition is no longer valid, and the internal nonlinear coupling present between the linear normal modes of the system may necessitate the use of models with a relatively large number of degrees of freedom (DOF) in order to capture the system dynamics accurately. As a result, studies of nonlinear systems often sacrifice either time (through a large, expensive computer model) or accuracy (through the elimination of possibly significant mechanisms).
This research is aimed at the development and implementation of model reduction methods for certain classes of nonlinear structural systems, based on the invariant manifold approach initially developed by Shaw and Pierre [1?4], and further developed and implemented by Boivin [5, 6] and Pesheck [7]. The primary goal of this dissertation is to extend the nonlinear modal analysis methodology to large-scale structural systems (particularly
those modeled with the finite element method) with various types of nonlinearities (e.g., polynomial and piecewise linear), including systems subject to external excitation and those with internal resonances. Another objective of this work is to apply the invariant manifold approach to an industrial structure with a complex, intrinsic nonlinearity. The nonlinear dynamics of an important class of engineering rotating structures are investigated, namely rotorcraft blades.
It is well known that nonlinear systems exhibit much richer and more complex behavior than their linear counterparts (i.e., bifurcations, internal resonances, sensitivity to initial conditions, etc.). Moreover, for a nonlinear system linear superposition is no longer valid, and the internal nonlinear coupling present between the linear normal modes of the system may necessitate the use of models with a relatively large number of degrees of freedom (DOF) in order to capture the system dynamics accurately. As a result, studies of nonlinear systems often sacrifice either time (through a large, expensive computer model) or accuracy (through the elimination of possibly significant mechanisms).
This research is aimed at the development and implementation of model reduction methods for certain classes of nonlinear structural systems, based on the invariant manifold approach initially developed by Shaw and Pierre [1?4], and further developed and implemented by Boivin [5, 6] and Pesheck [7]. The primary goal of this dissertation is to extend the nonlinear modal analysis methodology to large-scale structural systems (particularly
those modeled with the finite element method) with various types of nonlinearities (e.g., polynomial and piecewise linear), including systems subject to external excitation and those with internal resonances. Another objective of this work is to apply the invariant manifold approach to an industrial structure with a complex, intrinsic nonlinearity. The nonlinear dynamics of an important class of engineering rotating structures are investigated, namely rotorcraft blades.
Lors de la phase de conception de structures complexes telles que véhicules, avions ou turbo machines, il est essentiel de connaître leur réponse vibratoire. Dans le cadre des petites perturbations, l'étude des vibrations linéaires est suffisant. Par contre, lorsque les amplitudes augmentent, au delà du seuil de linéarisation ou même lorsque les structures ont un comportement non linéaire intrinsèque, les équations du mouvement non linéaires ne peuvent être simplifiées et doivent être analysées telles quelles, engendrant une augmentation sensible des temps de calcul
Il est connu que les systèmes non linéaires sont à l'origine de manifestations complexes dans un espace plus vaste que leur homonymes linéaires. Le principe de superposition largement utilisé n'est plus valable et comprendre les paramètres importants du système est peu recommandé en intégration temporelle directe.
Ce travail de recherche se concentre sur le développement et la programmation de méthodes de réduction de systèmes non linéaires dans le cadre des variétés invariantes. Plus précisément, il s'agit de généraliser, au domaine non linéaire, les approches modales utilisées quotidiennement dans le domaine linéaire.
Il est connu que les systèmes non linéaires sont à l'origine de manifestations complexes dans un espace plus vaste que leur homonymes linéaires. Le principe de superposition largement utilisé n'est plus valable et comprendre les paramètres importants du système est peu recommandé en intégration temporelle directe.
Ce travail de recherche se concentre sur le développement et la programmation de méthodes de réduction de systèmes non linéaires dans le cadre des variétés invariantes. Plus précisément, il s'agit de généraliser, au domaine non linéaire, les approches modales utilisées quotidiennement dans le domaine linéaire.
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