International audienceThe definition of a non-linear normal mode (NNM) as an invariant manifold in phase space is used. In conservative cases, it is shown that normal form theory allows one to compute all NNMs, as well as the attendant dynamics onto the manifolds, in a single operation. Then, a single-mode motion is studied. The aim of the present work is to show that too severe truncature using a single linear mode can lead to erroneous results. Using single-non-linear mode motion predicts the correct behaviour. Hence, the nonlinear change of co-ordinates allowing one to pass from the linear modal variables to the normal ones, linked to the NNMs, defines a framework to properly truncate non-linear vibration PDEs. Two examples are studied: a discrete system (a mass connected to two springs) and a continuous one (a linear Euler-Bernoulli beam resting on a non-linear elastic foundation). For the latter, a comparison is given between the developed method and previously published results

International audienceIn order to build efficient reduced-order models (ROMs) for geometrically nonlinear vibrations of thin structures, a normal form procedure is computed for a general class of nonlinear oscillators with quadratic and cubic nonlinearities. The linear perturbation brought by considering a modal viscous damping term is especially addressed in the formulation. A special attention is focused on how all the linear modal damping terms are gathered together in order to define a precise decay of energy onto the invariant manifolds, also defined as nonlinear normal modes (NNMs). Then, this time-independent formulation is used to reduce the dynamics governing the oscillations of a structure excited by an external harmonic force. The validity of the proposed ROMs is systematically discussed and compared with other available methods. In particular, it is shown that large values of the modal damping of the slave modes may change the type of nonlinearity (hardening/softening behaviour) of the directly excited (master) mode. Two examples are used to illustrate the main features of the method. A two-degrees-of-freedom (dof) system allows presentation of the main results through a simple example. Then a water-filled circular cylindrical shell with external resonant forcing is considered, in order to show the ability of the method to substantially reduce the dynamics of a continuous structure. © 2006 Elsevier Ltd. All rights reserved

This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures dis-C. Touzé (B)

The nonlinear interaction of waves in a driven medium may lead to wave turbulence, a state such that energy is transferred from large to small length scales. Here, wave turbulence is observed in experiments on a vibrating plate. The frequency power spectra of the normal velocity of the plate may be rescaled on a single curve, with power-law behaviors that are incompatible with the weak turbulence theory of Düring et al. [Phys. Rev. Lett. 97, 025503 (2006)10.1103/PhysRevLett.97.025503]. Alternative scenarios are suggested to account for this discrepancy -- in particular the occurrence of wave breaking at high frequencies. Finally, the statistics of velocity increments do not display an intermittent behavior.

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von KármánÕs theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.

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In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von K" a arm" a an equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency o kn of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to o kn ; which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper. #

This article is dev oted to an experimental v alidati on of a theoretical model presented in an earlier contribution by the same authors. The non-linear forced v ibrations of circular plates, with the excitation frequency close to the natural frequency of an asymmetric mode, are investigated. The experimental set-up, which allows one to perform precise measurements of the v ibration amplitudes of the two preferential configurations, is presented. Experimental resonance curves showing the amplitude and the phase of each configuration as functions of the driv ing frequency are compared to the theoretical ones, leading to a quantitativ e v alidation of the predictions giv en by the model. Finally, all the approximations used are systematically discussed, in order to show the scope and relev ance of the approach.