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International audience; An experimental demonstration of the broadband passive damping capacity of a vibro-impact acoustic black hole (VI-ABH) is reported. A VI-ABH is an adaptation of the classical ABH design consisting of a beam with a tapered edge of decreasing thickness creating an acoustic black hole (ABH), complemented by contact points on which the beam impacts during its vibration. The contact nonlinearity creates a rapid and efficient transfer of vibrational energy from the low-frequency range, where the ABH is known to be ineffective, to the high-frequency range, thus improving the global passive vibration mitigation characteristics. The optimal design of a VI-ABH follows the rule of locating the contact points at local maxima of the low-frequency modes. Experiments clearly demonstrate the gain in performance, both in forced and free vibrations.

International audience; A rectangular plate with a wedge profile creating an Acoustic Black Hole (ABH) termination is studied numerically. A particular emphasis is put on combining two different types of nonlinearity in order to improve the passive damping capacity of the ABH by transferring energy to the high-frequency range where it is more efficient. First, the addition of contact points to create a vibro-impact black hole (VI-ABH) is taken into account, following a previous study on beams. The contact nonlinearity allows for a rapid and efficient transfer of energy. Second, the large-amplitude vibrations of the plate in the ABH region where small thickness is reached, is also considered. The geometric nonlinearity is incorporated using a von Kármán plate model, and the regime of wave turbulence is shown to be triggered thus creating an energy flux from the low to the high frequencies. The linear characteristics of the ABH plate are first analyzed. Numerical results show the appearance of overdamped modes gathered in solution branches with constant number of half-waves in the transverse direction of the ABH, seen as a waveguide. The structure of the branches is shown to be more and more prominent when increasing the width of the plate, showing a transition from beam-like to full plate structure, with a fixed value for the fundamental cut-on frequency. The combination of both contact and geometric nonlinearities to improve the ABH effect is then reported. It is shown that the coexistence of both nonlinearities provides better passive damping efficacy.

International audience; Nonlinear flexural vibrations of slender beams holding both an Acoustic Black Hole termination and a contact non-linearity are numerically studied. The Acoustic Black Hole (ABH) effect is a passive vibration mitigation technique, which has shown attractive properties above a given cut-on frequency. In this contribution, a vibro-impact acoustic black hole (VI-ABH) is introduced, the contact nonlinearity being used as a mean to transfer energy from low to high frequencies. A numerical model of a VI-ABH is derived from an Euler-Bernoulli beam. The contact law is handled with a penalization approach, the visco-elastic layer with a Ross-Kerwin-Ungard model and the problem is solved with a modal approach combined with an energy-conserving time integration scheme. Numerical results show that the VI-ABH brings about important modifications, and changes the nature of more traditional black holes, by redistributing all the vibrational energy. It can lead to a strong decrease of the resonance magnitude at low frequencies. Under steady state noise excitation, parametric studies are realised in the cases of a single contact, a grid of contacts and bilateral contacts layouts, in order to find some optimal designs. Transient dynamics is also studied through the analysis of displacement signal envelope and energy decay time. All the numerical results constantly show that the combination the ABH effect and an energy transfer provided by contact nonlinearity leads to very attractive mitigation template including low frequencies.

International audience; Geometrically nonlinear vibrations of thin plates and shells with variable thickness are investigated numerically with the purpose of synthesizing the sound of cymbals. In cymbal making, taper refers to the gradual change in thickness from the centre to the rim, and is known to be a key feature that determines the tone of the instrument. It is generally used in conjunction with shape variations in order to enable the cymbal to play a bell-like sound when hit near its centre, or a crash sound when struck close to the edge. The von Kármán equations for thin plates with thickness and shape variations are derived, and a numerical method combining a Rayleigh-Ritz approach together with a Störmer-Verlet scheme for advancing the problem in time, is detailed. One main advantage of the method is its ability to implement easily any frequency-dependent loss mechanism which is a key property for sound synthesis. Also, the accuracy of the computation of the nonlinear restoring force is especially preserved. The method is employed to synthesize the sounds of cymbal-like instruments. The impact of taper is addressed and the relative effects of both thickness and shape variations, are contrasted.

International audience; A nonlinear magnetic vibration absorber is presented and used to control vibration of a three-storey structure. A distinctive feature of the absorber concerns its versatility for tuning the linear and nonlinear stiffness coefficients, depending on simple geometric design parameters such as the distance between fixed magnets and the moving one. In particular, the absorber can be set either as a nonlinear tuned vibration absorber, a nonlinear energy sink, or a bistable tuned vibration absorber, according to whether the linear stiffness term is positive, vanishing, or negative. The response of the primary structure and the vibration mitigation are investigated in the cases of impulsive shock, free vibration with imposed initial displacement, and single frequency excitation. Significant reductions of the primary structure vibrations are obtained for the three cases investigated, showing the ability of using a vibration absorber only relying on magnetic forces for passive control. The detailed comparisons of the absorbers performance show that, in this case study, no general guidelines can be easily deduced for selecting one of the three tunings for a nonlinear absorber. Depending on the excitation, the vibratory levels, and the frequency content of the excitation, the three configurations show advantages and drawbacks that are discussed.

International audience; The Acoustic Black Hole (ABH) effect refers to a special vibration damping technique adapted to thin-walled structures such as beams or plates. It usually consists of a local decrease of the structure thickness profile, associated to a thin viscoelastic coating placed in the area of minimum thickness. It has been shown that such structural design acts as an efficient vibration damper in the high frequency range, but not at low frequencies. This paper investigates how different types of vibration absorbers, linear and nonlinear, added to the primary system can improve the low frequency performance of a beam ABH termination. In particular, the conjugated effects of the Acoustic Black Hole effect and a Tuned Mass Damper (TMD), a Nonlinear Energy Sink (NES), a bi-stable NES (BNES), and a vibro-impact ABH (VI-ABH) are investigated. Forced response to random excitation are computed in the time domain using a modal approach combined with an energy conserving numerical scheme. Frequency indicators are defined to characterize and compare the performance of all solutions. The simulation results clearly show that all the proposed methods are able to damp efficiently the flexural vibrations in a broadband manner. The optimal tuning of each proposed solution is then investigated through a thorough parametric study, showing how to optimize the efficiency of each solution. In particular, TMD and VI-ABH show a slight dependence on vibration amplitude, while the performance of NES and BNES have a peak of efficiency for moderate amplitudes.

International audience; Collisions in musical string instruments play a fundamental role in explaining the sound production in various instruments such as sitars, tanpuras and electric basses. Contacts occuring during the vibration provide a nonlinear effect which shapes a specific tone due to energy transfers and enriches the hearing experience. As such, they must be carefully simulated for the purpose of physically-based sound synthesis. Most of the numerical methods presented in the literature rely on a compliant modeling of the contact force between the string and the obstacle. In this contribution, numerical methods from nonsmooth contact dynamics are used to integrate the problem in time. A Moreau-Jean time-stepping scheme is combined with an exact scheme for phases with no contact, thus controlling the numerical dispersion. Results for a two-point bridge mimicking a tanpura and an electric bass are presented, showing the ability of the method to deal efficiently with such problems while invoking, as compared to a compliant approach, less modelling parameters and a reduced computational burden.

International audience; This article is concerned with the vibration of a stiff linear string in the presence of a rigid obstacle. A numerical method for unilateral and arbitrary-shaped obstacles is developed, based on a modal approach in order to take into account the frequency dependence of losses in strings. The contact force of the barrier interaction is treated using a penalty approach, while a conservative scheme is derived for time integration, in order to ensure long-term numerical stability. In this way, the linear behaviour of the string when not in contact with the barrier can be controlled via a mode by mode fitting, so that the model is particularly well suited for comparisons with experiments. An experimental configuration is used with a point obstacle either centered or near an extremity of the string. In this latter case, such a pointwise obstruction approximates the end condition found in the tanpura, an Indian stringed instrument. The second polarisation of the string is also analysed and included in the model. Numerical results are compared against experiments, showing good accuracy over a long time scale.

International audience; Acoustic Black Hole effect (ABH) is a passive vibration damping technique without added mass based on flexural waves properties in thin structures with variable thickness. A common implementation is a plate edge where the thickness is locally reduced with a power law profile and covered with a viscoelastic layer. The plate displacement in the small thickness region is large and easily exceeds the plate thickness. This is the origin of geometric nonlinearity which can generate couplings between linear eigenmodes of the structure and induce energy transfer between low and high frequency regimes. This phenomenon may be used to increase the efficiency of the ABH treatment in the low frequency regime where it is usually inefficient. An experimental investigation evidenced that usual ABH implementation gives rise to measurable geometric nonlinearity and typical nonlinear phenomena. In particular, strongly nonlinear regime and wave turbulence are reported. The nonlinear ABH beam is then modeled as a von Kármán plate with variable thickness. The model is solved numerically by using a modal method combined with an energy-conserving time integration scheme. The effects of both the thickness profile and the damping layer are then investigated in order to improve the damping properties of an ABH beam. It is found that a compromise between the two effects can lead to an important gain of efficiency in the low frequency range.

The authors are grateful to Bertrand David (Telecom-ParisTech) for computing the code allowing the STFT filtering procedure used in Section 5.1. The filter has been designed in the framework of the PAFI project (Plateforme d’Aide la facture Instrumentale, www.pafi.fr) which is also thanked.; The vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics.; International audience; The vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics.

International audience; This paper presents a modal, time-domain scheme for the nonlinear vibrations of perfect and imperfect plates. The scheme can take into account a large number of degrees-of-freedom and is energy-conserving. The targeted application is the sound synthesis of cymbals and gong-like musical instruments, which are known for displaying a strongly nonlinear vibrating behaviour. This behaviour is typical of a wave turbulence regime, in which the wide-band spectrum of excited modes is observable in the form of an energy cascade. The modal method is selected for its versatility in handling complex damping laws that can be implemented easily by selecting appropriate damping values in each one of the modal equations. In the first part of the paper, the modal method is explained in its generality, and it will be seen that the method is valid for plates with arbitrary geometry and boundary conditions as long as the eigenmodes are known. Secondly, a time-integration, energy-conserving scheme for perfect and imperfect plates is presented, and implementation comments are given in order to treat efficiently the high-dimensionality of the resulting dynamical system. The scheme is run with appropriate parameters in order to produce sound samples. A simple impact law is considered for the exci-tation, whereas the flexibility of the method is highlighted by showing simulations for free-edge circular plates and simply-supported rectangular plates, together with various damping laws.

International audience; A passive control of aeroelastic instabilities on a two-degrees-of-freedom (dofs) system is considered here using shape memory alloys (SMA) springs in their pseudo-elastic regime. SMA present a solid-solid phase change that allow them to face strong deformations (∼10%) ; in the pseudo-elastic regime, an hysteresis loop appears in the stress-strain relationship which in turn gives rise to an important amount of dissipated energy. This property makes the SMA a natural candidate for mitigating undesired vibrations in a passive manner. A 2-dofs system is here used to model the classical flutter instability of a wing section in a uniform flow. The SMA spring is selected to act on the pitch in order to dissipate energy of the predominant motion. A simple phenomenological model for the SMA hysteresis loop is introduced, allowing for a quantitative study of the important parameters to optimize in view of an experimental design. Thanks to a simple phenomenological model for the SMA hysteresis loop, a quantitative numerical study is performed in order to exhibit the best tuning of the material parameters for controlling the flutter instability.

International audience; Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2 a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R → ∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified. © 2006 Elsevier Ltd. All rights reserved.

International audience; This paper summarizes some results obtained in the last few years for the modeling of nonlinear vibrating instruments such as gongs and cymbals. Linear, weakly nonlinear and chaotic regimes are successively examined. A theoretical mechanical model is presented, based on the nonlinear von Kármán equations for thin shallow spherical shells. Modal projection and Nonlinear Normal Mode (NNM) formulation leads to a subset of coupled nonlinear oscillators. Current developments are aimed at using this subset for sound synthesis purpose.

International audience; Non-linear normal modes (NNMs) are used in order to derive accurate reduced-order models for large amplitude vibrations of structural systems displaying geometrical non-linearities. This is achieved through real normal form theory, recovering the definition of a NNM as an invariant manifold in phase space, and allowing definition of new co-ordinates non-linearly related to the initial, modal ones. Two examples are studied: a linear beam resting on a non-linear elastic foundation, and a non-linear clamped-clamped beam. Throughout these examples, the main features of the NNM formulation will be illustrated: prediction of the correct trend of non-linearity for the amplitude-frequency relationship, as well as amplitude-dependent mode shapes. Comparisons between different models - using linear and non-linear modes, different number of degrees of freedom, increasing accuracy in the asymptotic developments - are also provided, in order to quantify the gain in using NNMs instead of linear modes.

International audience; This article is devoted to an experimental validation of a theoretical model presented in an earlier contribution by the same authors. The non-linear forced vibrations 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 vibration amplitudes of the two preferential configurations, is presented. Experimental resonance curves showing the amplitude and the phase of each configuration as functions of the driving frequency are compared to the theoretical ones, leading to a quantitative validation of the predictions given by the model. Finally, all the approximations used are systematically discussed, in order to show the scope and relevance of the approach. © 2003 Elsevier Science Ltd. All rights reserved.

International audience; A passive control of aeroelastic instabilities on a two-degrees-of-freedom (dofs) system is considered here using shape memory alloys (SMA) springs in their pseudo-elastic regime. SMA present a solid-solid phase change that allow them to face strong deformations (∼ 10%); in the pseudo-elastic regime, an hysteresis loop appears in the stress-strain relationship which in turn gives rise to an important amount of dissipated energy. This property makes the SMA a natural candidate for damping undesired vibrations in a passive manner. A 2-dofs system is here used to model the classical flutter instability of a wing section in an uniform flow. The SMA spring is selected on the pitch mode in order to dissipate energy of the predominant motion. A simple model for the SMA hysteresis loop is introduced, allowing for a quantitative study of the important parameters to optimize in view of an experimental design.

International audience; The non-linear dynamic thermomechanical behaviour of superelastic shape memory alloys is investigated. To this end, the Zaki-Moumni model, initially developed for quasi-static loading cases, is extended to simulate the uniaxial forced oscillations of a shape memory alloy device. First, the influence of loading rate is accounted for by considering the thermomechanical coupling in the behaviour of NiTi shape memory alloy. Comparisons between simulations and experimental results show good agreement. Then, the forced response of a shape memory alloy device is investigated at resonance. Both isothermal and non-isothermal conditions are studied, as well as non-symmetric tensile-compressive restoring force. In the case of large values of forcing amplitudes, simulation results show that the dynamic response is prone to jumps, bifurcations and chaotic solutions.

International audience; The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Kármán equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincaré maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.

International audience; A numerical study of the transition from periodic to chaotic motions in forced vibrations of circular plates, is proposed. A pointwise harmonic forcing of constant excitation frequency Ω and increasing values of the amplitude is considered. Perfect and imperfect circular plates with a free edge are studied within the von Kármán assumptions for large displacements (geometric non-linearity). The transition scenario is observed for different excitation frequencies in the range of the first eigenfrequencies of the plate. For perfect plate with no specific internal resonance relationships , a direct transition to chaos is at hand. For imperfect plate tuned so as to fulfill specific internal resonance relations, a coupling between internally resonant modes is first observed. The chaotic regime shows an attractor of large dimension, and thus is studied within the framework of wave turbulence.

International audience; A vibrating plate is set into a chaotic state of wave turbulence by a forcing having periodic and random components. Both components are weighted in order to explore continuously intermediate forcing from the periodic to the random one, but keeping constant its rms value. The transverse velocity of the plate is measured at the application point of the force. It is found that whatever the detail of the forcing is, the velocity spectra exhibit a universal cascade for frequencies larger than the forcing frequency range. In contrast, the velocity spectra strongly depend on the nature of the forcing within the range of forcing frequencies. The coherence function is used to extract the contribution of the velocity fluctuations that display a linear relationship with the forcing. The nonlinear contribution to the velocity fluctuations is found to be almost constant, about 55% of the total velocity fluctuations whatever the nature of the forcing from random to periodic. On the other hand, the nonlinear contribution to the fluctuations of the injected power depends on the nature of the forcing; it is significantly larger for the periodic forcing (60%) and decreases continuously as the randomness is increased, reaching a value of 40% for the pure random forcing. For all the cases of intermediate forcing from random to periodic, a simple model of the velocity response recovers in a fairly good agreement the probability density function of the injected power. The consequence of the existence of a linear-response component is discussed in the context of the fluctuation-dissipation theorem validation in experiments of out-of-equilibrium systems. © 2010 The American Physical Society.

Mots cles : Plaque stratifiee / non-linearites geometriques / flambage / vibrations / piezoelectrique Abstract - Buckling and non-linear vibrations of a piezoelectric stratified plate. Application to a MEMS mass sensor. This study proposes a model of a circular, non-symetrically laminated, piezoelectric plate, subjected to non-linear large amplitude vibrations. This model is built to simulate the behavior of a MEMS bio-sensor, designed to detect, automatically and autonomously, the presence of a given molecule in an aqueous solution. Because of both the laminated structure and the fabrication process, prestresses are observed, with different signs and intensities from one layer to another. This is responsible of a non-planar deformed geometry of the plate at rest. Moreover, experimental results show that a geometrically non-linear response is observed, with curved freqeuncy responses. A continuous non- linear model of the von-Karman type is used and discretized by an expansion of the solution onto the mode shape basis of the plate without prestresses. It's response is numerically computed by a continuation method, in order to predict the system's non-linear behavior.

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.

International audience; Non-linear normal modes (NNMs) are used in order to derive reduced-order models for large amplitude, geometrically non-linear vibrations of thin shells. The main objective of the paper is to compare the accuracy of different truncations, using linear and non-linear modes, in order to predict the response of shells structures subjected to harmonic excitation. For an exhaustive comparison, three different shell problems have been selected: (i) a doubly curved shallow shell, simply supported on a rectangular base; (ii) a circular cylindrical panel with simply supported, in-plane free edges; and (iii) a simply supported, closed circular cylindrical shell. In each case, the models are derived by using refined shell theories for expressing the strain-displacement relationship. As a consequence, in-plane inertia is retained in the formulation. Reduction to one or two NNMs shows perfect results for vibration amplitude lower or equal to the thickness of the shell in the three cases, and this limitation is extended to two times the thickness for two of the selected models. © 2008 Elsevier B.V. All rights reserved.

International audience; A vibrating plate is set into a chaotic state of wave turbulence by either a periodic or a random local forcing. Correlations between the forcing and the local velocity response of the plate at the forcing point are studied. Statistical models with fairly good agreement with the experiments are proposed for each forcing. Both distributions of injected power have a logarithmic cusp for zero power, while the tails are Gaussian for the periodic driving and exponential for the random one. The distributions of injected work over long time intervals are investigated in the framework of the fluctuation theorem, also known as the Gallavotti-Cohen theorem. It appears that the conclusions of the theorem are verified only for the periodic, deterministic forcing. Using independent estimates of the phase space contraction, this result is discussed in the light of available theoretical framework.

International audience; The aim of the present paper is to compare two different methods available for reducing the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD), and an asymptotic approximation of the nonlinear normal modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the partial differential equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed. The response is investigated also for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation. Bifurcation diagrams of Poincaré maps obtained from direct time integration and calculation of the maximum Lyapunov exponent have been used to characterize the system. © 2007 Elsevier Ltd. All rights reserved.

International audience; This study is devoted to the experimental validation of a theoretical model of large amplitude vibrations of thin spherical shells described in a previous study by the same authors. A modal analysis of the structure is first detailed. Then, a specific mode coupling due to a 1:1:2 internal resonance between an axisymmetric mode and two companion asymmetric modes is especially addressed. The structure is forced with a simple-harmonic signal of frequency close to the natural frequency of the axisymmetric mode. The experimental setup, which allows precise measurements of the vibration amplitudes of the three involved modes, is presented. Experimental frequency response curves showing the amplitude of the modes as functions of the driving frequency are compared to the theoretical ones. A good qualitative agreement is obtained with the predictions given by in the model. Some quantitative discrepancies are observed and discussed, and improvements of the model are proposed.

International audience; In 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.

International audience; The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

International audience; The 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 audience; Non-linear normal modes (NNMs), defined as invariant manifolds, are introduced through Normal Form theory. In a conservative framework, it is shown that all NNMs, as well as the attendant dynamics onto the manifolds, are computed in a single operation. The general third-order approximation of the dynamics onto a single NNM is derived. It is underlined that single linear mode truncation can lead to erroneous results which are corrected when considering NNMs. These results are illustrated by studying the vibrations of a linear beam resting on a non-linear elastic foundation.

International audience; 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 Kàrmàn 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 $\omega_n$ of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to $\omega_n$, 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.

Geometrically nonlinear vibrations of free edge circular plates subjected to a harmonic excitation are discussed. Particularly, transition from periodic to chaotic motion is observed when increasing the amplitude of the forcing. The present work is devoted to reproduce numerically these highly nonlinear behaviours. The temporal integration of such dynamics, including instabilities and chaotic regimes, is not straight forward because a stiff problem with a very large number of dofs is at hand. Consequently, numerical instabilities are observed when typical Runge‐Kutta schemes are applied. To settle the matter, two methods have been tested and compared. They both rely on a modal approach applied to the von Karman's model for large amplitude vibrations of plate. For the first one, the energies of the plates are expressed at the continuous level. The Hamiltonian of the system is then derived and discretized using the eigenmodes. The Hamiltonian formulation ensures the conservation of energy. An implicit time ...

This study is devoted to the steelpan, a harmonically tuned percussion from Trinidad and Tobago. It consists in a steel barrel subjected to several steps of irreversible metal forming to obtain a main spherical bowl within wich convex domes are formed. This work is mainly focused on two topics. Firstly, the study of the making processes allows us to gain a better understanding of the work of the tuner in order to propose a model of the first step of steelpan making. Secondly, a vibration analysis of the steelpan, at the end of this process, is performed to understand its dynamics which is responsible of its particular tone. In the first part of the manuscript we study the evolution of the dynamical parameters during manufacturing. In particular, mode localization after note forming, strong variations of modal dampings, natural frequencies after burning and harmonic relationships after fine tuning are observed. The second part of the manuscript focusses on the first step of manufacturing during which the top of the barrel is hammered to obtain the spherical shell. The shell is analytically modeled as a nonlinear von Karman plate and the plastic strains are introduced as initial inelastic strains. The effect of the geometry and of the residual stress fields are separately quantified on the dynamical parameters of the structure in its final equilibrium state. The last part is concerned with the complex nonlinear dynamics of the steelpan at the end of the forming process. During normal playing, energy exchanges between modes are clearly identifiable: they result in a rich sound due to numerous internal resonances. Modal coupling are measured even for small amplitudes of vibrations. Original models of internal resonances are proposed.; Cette étude porte sur le steeldrum, aussi appelé steelpan, percussion mélodique de Trinidad et Tobago, fabriquée à partir de bidons métalliques dont une des faces subit un ensemble de déformations irréversibles afin d'obtenir une cuve principale à l'intérieur de laquelle sont façonnées les différentes notes de l'instrument. Cette thèse s'organise autour de deux axes. D'une part les processus de fabrication sont étudiés afin de mieux comprendre le travail du facteur et de proposer une modélisation de la première étape de fabrication. D'autre part, des études vibratoires sont menées sur l'instrument fini, pour comprendre la dynamique complexe responsable de son timbre. La première partie rend compte de l'évolution des caractéristiques vibratoires au cours de la fabrication. On remarque une forte localisation vibratoire au moment de la création des notes, des évolutions marquées des amortissements et des fréquences propres lors du chauffage, et des relations harmoniques après l'accordage final. La seconde partie se concentre sur la première étape de fabrication au cours de laquelle le haut du bidon est martelé pour obtenir une calotte sphérique concave. Ce processus de plasticité est modélisé analytiquement sous les hypothèses de von Karman par une structure mince soumise à des contraintes initiales. L'influence du changement géométrique et celle de l'état de contraintes résiduelles sont alors quantifiées sur les paramètres dynamiques de la structure dans son nouvel état d'équilibre. Enfin la dernière partie s'intéresse aux vibrations de l'instrument terminé qui présente un comportement dynamique caractéristique des systèmes non linéaires géométriques. En mode de jeu usuel, des échanges d'énergie entre les modes sont clairement audibles. Cette richesse de timbre provient des résonances internes dues à l'accordage, si bien que des couplages complexes sont mesurés pour des amplitudes vibratoires très faibles. Des modèles originaux de résonances internes sont proposés.

This study is devoted to the steelpan, a harmonically tuned percussion from Trinidad and Tobago. It consists in a steel barrel subjected to several steps of irreversible metal forming to obtain a main spherical bowl within wich convex domes are formed. This work is mainly focused on two topics. Firstly, the study of the making processes allows us to gain a better understanding of the work of the tuner in order to propose a model of the first step of steelpan making. Secondly, a vibration analysis of the steelpan, at the end of this process, is performed to understand its dynamics which is responsible of its particular tone. In the first part of the manuscript we study the evolution of the dynamical parameters during manufacturing. In particular, mode localization after note forming, strong variations of modal dampings, natural frequencies after burning and harmonic relationships after fine tuning are observed. The second part of the manuscript focusses on the first step of manufacturing during which the top of the barrel is hammered to obtain the spherical shell. The shell is analytically modeled as a nonlinear von Karman plate and the plastic strains are introduced as initial inelastic strains. The effect of the geometry and of the residual stress fields are separately quantified on the dynamical parameters of the structure in its final equilibrium state. The last part is concerned with the complex nonlinear dynamics of the steelpan at the end of the forming process. During normal playing, energy exchanges between modes are clearly identifiable: they result in a rich sound due to numerous internal resonances. Modal coupling are measured even for small amplitudes of vibrations. Original models of internal resonances are proposed., Cette étude porte sur le steeldrum, aussi appelé steelpan, percussion mélodique de Trinidad et Tobago, fabriquée à partir de bidons métalliques dont une des faces subit un ensemble de déformations irréversibles afin d'obtenir une cuve principale à l'intérieur de laquelle sont façonnées les différentes notes de l'instrument. Cette thèse s'organise autour de deux axes. D'une part les processus de fabrication sont étudiés afin de mieux comprendre le travail du facteur et de proposer une modélisation de la première étape de fabrication. D'autre part, des études vibratoires sont menées sur l'instrument fini, pour comprendre la dynamique complexe responsable de son timbre. La première partie rend compte de l'évolution des caractéristiques vibratoires au cours de la fabrication. On remarque une forte localisation vibratoire au moment de la création des notes, des évolutions marquées des amortissements et des fréquences propres lors du chauffage, et des relations harmoniques après l'accordage final. La seconde partie se concentre sur la première étape de fabrication au cours de laquelle le haut du bidon est martelé pour obtenir une calotte sphérique concave. Ce processus de plasticité est modélisé analytiquement sous les hypothèses de von Karman par une structure mince soumise à des contraintes initiales. L'influence du changement géométrique et celle de l'état de contraintes résiduelles sont alors quantifiées sur les paramètres dynamiques de la structure dans son nouvel état d'équilibre. Enfin la dernière partie s'intéresse aux vibrations de l'instrument terminé qui présente un comportement dynamique caractéristique des systèmes non linéaires géométriques. En mode de jeu usuel, des échanges d'énergie entre les modes sont clairement audibles. Cette richesse de timbre provient des résonances internes dues à l'accordage, si bien que des couplages complexes sont mesurés pour des amplitudes vibratoires très faibles. Des modèles originaux de résonances internes sont proposés.