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International audience; The severe dynamic environment of take-off and flight of launchers presents risks of damage and life cycle limitation of the payload. To improve the dynamic comfort of the payload, damping treatments are employed at the final stage of the launcher assemble. One well-known solution is the limitation of the vibration amplitudes by dissipating the energy at the bolted joints between the stages. To our knowledge, there is not a simple and effective method that allows an accurate damping prediction at the full structure level from the knowledge of localized dissipation between the components. Indeed, to model the damping of an assembled structure made up of several components, one carries out a 3D modeling of the interfaces [1] to have a good representation of the dissipation. The result is a damping model identified by interface, which is difficult to exploit in the prediction of the dynamic behavior of the full model. Moreover, for uniformly distributed dissipation, it is common to use the Rayleigh damping assumption to represent the damping matrix in the equivalent viscous model. In general, common methods [2] identify this matrix at a single interface. In this work, we propose a method of modeling the multi-interface Rayleigh damping which consists of the following steps: (i) calculation of the modal damping of each interface taken separately by using the modal effort method [3] or the modal displacement method developed in PERMAS finite element software; (ii) identification of the proportional damping matrix of each interface (proportionality coefficients of stiffness and mass matrices); (iii) assembly of the identified damping matrices; (iv) prediction of the dynamic response of the complete structure from the global model. The proposed method was validated on a reduced model of the ARIANE 5 launcher to calculate the dynamic responses taking into account the different types of dissipation in the interfaces for the asymmetric load case and successfully compared to the reference results. The main advantages of the proposed method are the simplicity of the numerical implementation, the reduction of the CPU, the quality of prediction and the perspective of integration in an optimization procedure of interfaces design with high damping performances.

International audience; Developing techniques for vibration energy harvesting (VEH) based on various energy conversion mechanisms is being a focus of interest. In this context, a multimodal approach in a quasiperiodic system for vibration energy harvesting is investigated . The basic idea to enhance the system performances consists in introducing imperfections. Thus, mistuning is achieved by varying the mass of few cells . These imperfections will lead to the vibration energy localization in regions close to the imperfections which will be exploited to maximize the harvested energy.

International audience; It is known that, when the mechanical coupling between the components is weak, small imperfections in a periodic structure can induce vibration localization. Stochastic analysis of near-periodic coupled pendulums chain is discussed in this paper. Perfect periodicity of the system is disturbed by varying randomly the length of one of the pendulums which is considered as an uncertain parameter. Its randomness is modeled in a probabilistic framework by a random variable according to a given range of dispersion level. Stochastic effects on vibration localization in mistuned four coupled pendulums chain is investigated through statistical evaluations. To do so, the propagation of uncertainties is performed using the Latin Hypercube Sampling method.

International audience; Quasi periodicity and variability effects on structures are spread in all the engineering branches. The inherent research works are carried out with both numerical and experimental activities. A more specific goal is the analysis of the influence of such effects on the vibroacoustic response. The modern tools for investigating the most significant impacts of imperfections and irregularities on the vibrational and acoustical response of a given structure are codes based on combination of wave and finite element, spectral finite elements, transfer matrix methods and the adoption of stochastic variables. In view of the foreseen work inside the VIPER project, this paper refers to the analysis of the most recent literature and possible initial strategy.

International audience; In this paper, the potential benefit of mistuning on the harvested energy from a nonlinear periodic weakly coupled system is investigated. The nonlinearities take into account the Duffing nonlinear spring due to an electromagnetic set-up and the cubic electrical damping, which is used in the electrical circuit as nonlinear resistive load to harvest energy. For simulation, a two degree-of-freedom (dof) spring-mass-damper system is considered. The mistuning is achieved by a slight variation of one of the masses. An electromagnetic energy harvester is used to harvestenergy from the motion of the mistuned mass. The response and the harvested power of the mistuned system is obtained using harmonic balance method as a function of the mistuning parameter. The effects of the mistuning, damping and excitation level on the harvested power are investigated. The performance (harvested power) of the linear and nonlinear electrical damper is also compared for different values of mistuning and excitation levels. The system can easily be generalized to n DOF to represent a quasi-periodic structure.

International audience; The aim of this paper is to present the nonlinear dynamic responses of damaged laminated composite structure involving large deflection where a novel unilateral damage model is introduced. Only the matrix cracking is retained as damage form and the investigated structures are made of unidirectional layers of polymer matrix reinforced with long fibers. The damage is described by a single scalar variable representing the degradation of the transverse Young’s modulus. The structure behavior depends on the cracks state. Hence, in order to analyze the cracks opening/closure effect, the normal stress component to the fiber direction is chosen as a parameter to control the state of the cracks. If it is positive, the cracks are supposed to be opened and the damage effect is applied. If the sign of the normal stress component is negative, the cracks are closed and the virgin material behavior is recovered. Based on the firstorder shear deformation theory and the Von Karman’s large deflection plate theory, a Serendib finite element with five degree of freedom incorporating a nonlinear strain displacement relation due to large deflection is developed using the updated Lagrangian approach. Since the damage induces nonlinearity, the resulting nonlinear system of equations, combining the geometric and material nonlinearities, is solved in time domain using Newmark method and Newton-Raphson incremental-iterative method. Numerical examples of composite structures are considered and the influence ofgeometrical nonlinearity on the dynamic responses and the resulting damage is demonstrated. The stiffness degradation due to impact damage is also investigated. In addition, the proposed model permits the detection and the localization of the damage in laminated structures in large deflection. Thereby, this method can be used in life-time estimations and monitoring strategies of complex composite structures.

International audience; In many engineering area, quite often vibration is not desirable and the interest lies in reducing it by dissipation of energy or damping. Damping presents one of the most important physical aspects to model and estimate, since it plays a large role in determining the performance of a dynamic system and the amplitude of vibrations. The structural damping is well known and is typically responsible for a light level of damping compared to the joint damping. Damping in jointed interfaces is still not well mastered because of the complexity of the dissipation by friction, which includes nonlinear and localization effects. So, when eigenmodes are coupled with damping due to the nonlinear behaviour and the localization effects, as in the case of the most assembled structures, the Rayleigh assumption of proportional damping can lead to significant errors. The aim of this paper is the investigation of the representation assessment of the damping in a modal framework, in the case of localized dissipation. In order to quantify the effects of the nonlinear and the modal coupling on the dynamic responses, a quantification/correction method based on perturbation approach using asymptotic expansion of the frequency responses is proposed. The limitations of the proportional damping are then inspected and the correction of error estimations of modal damping due to the neglectof coupling terms is illustrated. A double-beam structure with a nonlinear interface is proposed to illustrate the validity and effectiveness of the proposed method.

International audience; We study wave propagation in a damped driven discrete nonlinear two-dimensional (2D) periodic structure. The considered system is composed of identical spherical particles and subjected to harmonic horizontal base excitation, where its collective nonlinear dynamics is caused by the Hertzian contact law and the applied compressive load. The coupled equations governing the nonlinear vibrations of the considered system have been solved using an analytical-numerical solving procedure, which allows a detailed study of complex model interactions of the periodic structure. The multiple scales method coupled with standing wave decompositionwere used to transform this nonlinear system into a set of complex differential equations, which has been numerically solved. Numerical simulations have been performed in order to analyze the stability, the modal interactions and the bifurcation topologies resulting from the collective dynamics of particles in contact, in a 2D configuration.

International audience; To achieve robust design, in presence of uncertainty, nonlinearity and structural periodicity, a metamodel combining the Latin Hypercube Sampling (LHS) method for uncertainty propagation and an enriched Craig-Bampton Component Mode Synthesis approach (CB-CMS) for model reduction is proposed. Its application to predict the time responses of a stochastic periodic nonlinear micro-system proves its efficiency in terms of accuracy and reduction of computational cost.

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International audience; The collective nonlinear dynamics of a periodic structure of coupled pendulums is investigated under primary resonance. The coupled equations governing the nonlinear vibrations of the considered system are normalized and transformed into a set of coupled complexalgebraic equations using the multiple scales method coupled with standing wave decomposition. The resulting nonlinear system is numerically solved using the asymptotic numerical continuation technique. It is shown that the number of additional multi-mode solutions is large even for an array of few pendulums and the stability of the corresponding branches can be tuned with respect to the design parameters. In practice, the nonlinear stiffness can be functionalized in order to enhance the performances of the system in term of energy transfer.

International audience; Among probabilistic uncertainty propagation methods, the generalized Polynomial Chaos Expansion (gPCE) has recently shown a growing emphasis. The numerical cost of the non-intrusive regression method used to compute the gPCE coefficient depends on the successive Latin Hypercube Sampling (LHS) evaluations, especially for large size FE models, large number of uncertain parameters, presence of nonlinearities and when using iterative techniques to compute the dynamic responses. To overcome this issue, the regression technique is coupled with a robust condensation method adapted to the Craig-Bampton component mode synthesis approach leading to computational cost reduction without significant loss of accuracy. The performance of the proposed method and its comparison to the LHS simulation are illustrated by computing the time response of a structure composed of several coupled-beams containing localized nonlinearities and stochastic design parameters.

Colloque avec actes et comité de lecture. Internationale.; International audience; La problématique de l'amortissement dans les structures assemblées présente un réel challenge. L'amortissement est généralement modélisé sous l'hypothèse de Rayleigh ou de Caughey même si la dissipation n'est pas répartie de façon uniforme sur toute la structure. L'amortissement proportionnel ou bien de Rayleigh, condition nécessaire, est exprimé comme étant une combinaison linéaire des matrices de masse et de raideur Caughey a introduit une condition nécessaire et suffisante qui conduit à une matrice d'amortissement généralisée diagonale. Cette hypothèse d'amortissement n'est pas toujours justifiée mais conduit à une matrice d'amortissement généralisée diagonale permettant de faire des calculs de prédiction à faible cout numérique. Parmi les méthodes de calcul des amortissements modaux, on utilise la méthode de l'espace d'état et la méthode énergétique. Lorsque la matrice d'amortissement généralisé n'est pas diagonale (cas des dissipations localisées), ces méthodes ne permettent pas de calculer ou d'estimer les termes d'amortissement généralisé extra-diagonaux. La problématique posée est alors la quantification de l'erreur commise lorsque l'on néglige les termes extra-diagonaux dans le calcul des niveaux vibratoires ; avec un bon ratio précision / coût de calcul. Le but de ce travail est d'investiguer la validité de l'hypothèse de diagonalité et de proposer une méthode très peu couteuse de quantification a posteriori de l'erreur d'estimation de l'amortissement modal par la méthode des perturbations.

International audience; This paper focuses on the prediction of the coupled vibroacoustic behaviour of a structurecavity system: the recent works in the field of smart lightweight structures to reduce the structural vibrations and the radiated noise require to be able to precisely predict this vibroacoustic behaviour for an adequate design of control laws. A classical way to solve such a problem is to use the finite element method commonly adopted for industrial problems.

International audience; In order to investigate the collective nonlinear dynamics of an array of coupled pendulums under simultaneous external and parametric excitations, a computational model is developed while considering the main sources of nonlinearities. The equations of motion are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). Numerical simulations are performed in the case of two coupled pendulums in order to investigate the complexity of the frequency responses in terms of bifurcation topologies and energy transfer.

International audience; Collective dynamics in periodic lattices of coupled nonlinear Duffing-Van Der Pol oscillators is modeled and investigated under simultaneous external and parametric resonances. The resonators are coupled with linear and nonlinear springs. Numerical simulations have been performed in the case of two coupled oscillators for which the basins of attraction have been analyzed in the multistability domain in order to check the efficiency of the multimode branches.

International audience; The vibroacoustic modeling using the finite element method (FEM) still remains costly in numerical computing. This is notably due to the formulation that can lead to a non-symmetric problem, and/or to a very large number of degrees of freedom to deal with [1]. The non-linear behavior commonly found in the vibration of thin structures, complicates the numerical solving. The reduced order method is used to reduce the numerical computation costs.The classical reduced order method is known to be unsuitable for non-linear problems due to the modal superposition principle that is not valid anymore.The combined approximation method [2] is a reduced order method which has been first developed for optimization and robustness studies via reanalysis [3]. It consists in building a reduced order basis issued from the modal basis of the initial problem enriched by a function taking into account the changes in the problem. This function is developed in a Taylor expansion to avoid the inversion of the current stiffness matrix.In this paper, this method is adapted to geometrical nonlinearities by considering the non-linear problem as a perturbation of the initial linear model. For thin structures [4] a coupling between in-plane and bending movements appears for large displacements. The initial basis should then take into account this effect.By considering a light fluid as air, vibroacoustic coupling effect can modify the acoustic behavior. A structural reduced order basis is built on by using the combined approximation method. The acoustic basis that includes eigenvectors of the conservative acoustic model is enriched by the static response of the fluid due to the presence of the structure. Therefore the fluid-structure reduced-order computational model is constructed using an extended Ritz basis which allows the nonlinear dynamical response of the vibroacoustic problems to be predicted with a good accuracy.The temporal response of a thin plate supported on a closed cavity is presented to illustrate the efficiency of the method. Results show a high precision level, and the error produced by the reduced order model is explicitly related to the level of non-linearity.

International audience; Complex mechanical structures usually encountered in the automotive or aerospace sectors are composed of an assembly of several components, exhibiting different mechanical properties and joined at their interfaces by different junction types. Hence, their various dynamic behaviours and the applied external loading may generate important efforts on the main structure, resulting in high acceleration responses of the on-board equipments, finally affecting their performance reliability and security.

International audience; In this paper, an experimental characterisation of a particle enclosure under harmonic excitation is performed. The developed procedure allows the identification of the intrinsic properties of the microparticles versus frequency. Some parameters influencing the measurement of dissipated power are considered.; Dans ce papier une caractérisation expérimentale d'un amortisseur de vibration par microparticules sous excitation harmonique a été effectuée. La procédure développée permet d'identifier les propriétés intrinsèques des microparticules en fonction de la fréquence. Certains paramètres qui influent sur la mesure de la puissance dissipée sont pris en compte dans ce papier.

National audience; In this paper, an experimental characterisation of a particle enclosure under harmonic excitation is performed. The developed procedure allows the identification of the intrinsic properties of the microparticles versus frequency. Some parameters influencing the measurement of dissipated power are considered.

International audience; The concept of energy pumping is an innovative dynamic phenomenon; it gives rise to new generation of dynamic absorbers. Theoretical studies and feasibility tests are necessary for better understanding of their dynamic behaviour and to be applied on real structures or machines. In this paper, numerical evidence is firstly given for the passive and broadband targeted energy transfer in the case of a linear system under shock excitation with Multiple Dynamic Absorbers or Nonlinear Energy Sink (NES). Secondly, it is shown that many NES absorb shock energy in only way and dissipate this energy locally, without "spreading" it returns to the linear system. The numerical results of optimisation in the case of NES linked to a linear beam are compared to Tuned Mass Dampers (TMD) linked to the same beam.

International audience; This paper presents a reduced order method dedicated to nonlinear vibroacoustic problems. Both structural and acoustics behavior are nonlinear. The structural nonlinearity is due to large displacements while the acoustic nonlinearity is due to the high intensity level in the fluid. The Kuznetsov equation is used to formulate the nonlinear acoustic problem. The reduced order model is based on the Ritz bases of the uncoupled linear problem. Nonlinear behavior is considered as a perturbation of the linear model so that the resolution is compared to a reanalysis problem. The Combined Approximation method dedicated to reanalysis problems is used to enriche the Ritz reduced basis.

International audience; a hybridization of the Wave Finite Element Method (WFEM) with the Generalised Polynomial Chaos Expansion (GCE). The WFE is a spectral method dealing with wave propagation in periodic structure. This method proved its efficiency in different domains; structural vibration, non-destructive testing, etc. However, the WFE is limited to deterministic media. Knowing that uncertainties affect dynamic behavior in Mid- and High frequencies, the combination of WFE and GCE is used to predict the effect of uncertainties on the dynamic response of periodic media. The uses of the GCE is based on the iso-probabilistic transformations for usuel distributions to Gaussian one to use the Hermite-Chaos expansion.The presented approach is validated for two periodic waveguides connected through a junction with uncertain parameters. The obtained results are verified vs Monte Carlo simulations.

International audience; L'objectif de ce travail est de montrer qu'il est possible de mener l'optimisation multiobjectifs en utilisant un algorithme heuristique aussi performant qu'un Algorithme Génétique (AG), qui est actuellement le plus utilisé dans les solveurs d'optimisation. Il s'agit de l'algorithme d'optimisation basé sur l'approche Pareto (Pareto Ant Colony Optimization : P-ACO). Dans ce papier on montrera à travers l'étude d'une plaque raidie en composite multicouches que l'algorithme P-ACO est aussi performant qu'un AG mais a l'avantage d'être plus aisé à mettre en oeuvre numériquement. La modélisation de la structure composite est réalisée par le code de calcul commercial ANSYS®.

International audience; Dynamic analysis of composite structures : HSDT formulation combined with reduction method

International audience; This paper presents a reduced order method adapted to a nonlinear dynamic problem. Geometrical non-linearity with large displacements behaviour is considered. The reduction tool is based on the combined approximation method that consists in expansing a binomial series. This basis is issued from the nonli-near behaviour considered as a modification of the linear model. The numerical simulation of the time response of a thin plate modeled by the finite element method (FEM) shows the applicability of the proposed method.; Cet article présente une méthode de réduction adaptée aux problèmes de dynamique non-linéaire. La non-linéarité étudiée est de type géométrique en grands déplacements. La réduction est basée sur la méthode des approximations combinées qui consiste à construire une base de série binomiale actualisée. Cette base est issue du comportement non-linéaire considéré comme une modification du modèle linéaire. La simulation numérique de la réponse temporelle d'une plaque mince modélisée en éléments finis montre l'intérêt de la méthode proposée.

International audience; This paper presents a reduced order method adapted to a nonlinear dynamic problem. Geometrical non-linearity with large displacements behaviour is considered. The reduction tool is based on the combined approximation method that consists in expansing a binomial series. This basis is issued from the nonli-near behaviour considered as a modification of the linear model. The numerical simulation of the time response of a thin plate modeled by the finite element method (FEM) shows the applicability of the proposed method.; Cet article présente une méthode de réduction adaptée aux problèmes de dynamique non-linéaire. La non-linéarité étudiée est de type géométrique en grands déplacements. La réduction est basée sur la méthode des approximations combinées qui consiste à construire une base de série binomiale actualisée. Cette base est issue du comportement non-linéaire considéré comme une modification du modèle linéaire. La simulation numérique de la réponse temporelle d'une plaque mince modélisée en éléments finis montre l'intérêt de la méthode proposée.

International audience; A procedure based on a power flow mode theory is presented to optimize the power flow dissipated at the junctions between substructures. These modes are derived from an eigenproblem associated to the minimization of an energy operator requiring the computation of the dynamic flexibility matrix, which remains prohibitive in a robust design context. A genetic algorithm is thus coupled to an approximate reanalysis method to determine junction parameter values which are both optimal and robust with regard to uncertainties.; Cette étude propose une procédure d'optimisation de la puissance dissipée aux jonctions entre différentes sous-structures par une méthode de modes de flux de puissance. Ceux-ci sont issus d'un problème aux valeurs propres associé à la minimisation d'un opérateur énergétique, nécessitant le calcul de la matrice de flexibilité dynamique, qui se révèle coûteux dans un contexte de conception robuste. On propose donc de coupler un algorithme génétique à une procédure de réanalyse approchée, afin de déterminer des valeurs des paramètres de jonctions à la fois optimales et robustes vis-à-vis des incertitudes.

International audience; Accurate determination of values of resonant frequencies and vibration characteristics of railway motor stators is an important issue for designers. The presence of heterogeneous laminated core, windings and assembling complexity of stators represent its main modelling difficulties. The aim of this paper consists on developing and validating, with a step by step approach, a model of a typical design stator by confronting it to experimental results. This approach leads finally to characterize mean behaviour of a typical design stator and to establish generic modelling rules for such structures.; La maîtrise du calcul dynamique des stators de moteurs ferroviaires représente un enjeu majeur pour le concepteur. Parmi les difficultés de modélisation de ces derniers figurent le comportement de leur cœur hétérogène feuilleté et bobiné ainsi que la complexité de leur assemblage. L'objectif de ce travail consiste à construire et valider, par une approche allant des composants à la structure complète, le modèle d'un stator d'architecture type en effectuant des corrélations calculs-essais. Ceci permet à terme de caractériser son comportement moyen et d'établir des règles de modélisation transposables à d'autres architectures.

International audience; The purpose of this study is to propose a design procedure to optimize the power flow dissipated at the interface between the components of a complex structure. At first, the underlying power flow mode theory is presented. It aims at dynamically characterizing the structural junction forces by projecting them onto a principal subspace of the power flow pattern at the interface. The obtained projection coefficients are then used to build a cost function, which is optimized with regard to the interface parameters, by means of a nonlinear programming algorithm. It is further verified that the optimal solutions truly minimize the optimization problem. Finally, an a posteriori robustness analysis is performed using an info-gap approach to determine which solutions are both optimal and robust with regard to the considered uncertainties. The method is illustrated by numerical simulations performed on a simple coupled structure.

International audience; Synthèse modale non-linéaire par une méthode d'approximations combinées généralisée

International audience; Prediction of the dynamic behavior of a structure treated by particle damper

The effective design of viscoelastic dampers as applied to real-world complex engineering structures can be conveniently carried out by using modern multiobjective numerical optimization techniques. The large number of evaluations of the cost functions normally combined with the typically high dimensions of finite element models of industrial structures makes multiobjective optimization very costly, sometimes unfeasible. Those difficulties motivate the study reported in this paper, in which a strategy is proposed consisting in the use of evolutionary algorithms specially adapted to multiobjective optimization of viscoelastic systems, combined with robust condensation and metamodeling. After the discussion of various theoretical aspects, a numerical application is presented to illustrate the use and demonstrate the effectiveness of the methodology proposed for the optimal design of viscoelastic constrained layers.

International audience; Complex structure optimization is an important part of the design process where the initial state of the structure is modified, sometimes in a significant way, in order to reach the best possible performance, for a given set of constraints.By exploring as much as possible the design parameter space, optimal solutions (displacements, constraints, eigenmodes, etc.) are then calculated. These successive multiple analysis or reanalysis thus imply a large computation cost which often remains prohibitive. The modifications applied to the structure can be of various types: topological, acting on the form of the structure or its boundary conditions; parametric, acting on the physical parameters of the structure (mass, stiffness, thickness); in a global (whole structure level) or a local way (component level). Therefore, the aim of reanalysis methods [1] is to approximate the responses of a structure whose parameters have been perturbed or even modified without solving the new equilibrium equation system associated to the updated structure: only the initial solutions and the perturbed data are used. Moreover, when the problem is non-linear, the re-actualization of the tangent stiffness matrix at each time step of the Newton-Raphson integration algorithm implies many reanalysis leading to a high computational time. To mitigate these difficulties, one proposes a robust reduction method adapted to non-linear and large sized dynamic models. This study especially focuses on geometrical non-linearities, i.e. large displacements [2]. The presented reduction method is based on the combined approximations method introduced by Kirsch [3,4].

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International audience; Reliability optimization in structural dynamics using the heuristic methods