# Cyril TouzéENSTA Paris · IMSIA - Institute of Mechanical Sciences and Industrial Applications

Cyril Touzé

HDR

## About

146

Publications

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Introduction

I am professor at ENSTA Paris, IMSIA (Institute of Mechanical Sciences and Industrial Applications). My work is mainly concerned with nonlinear dynamics with application to structural vibrations and model order reduction.

Additional affiliations

February 2001 - present

## Publications

Publications (146)

This article considers the nonlinear dynamics of coupled oscillators featuring strong coupling in 1:2 internal resonance. In forced oscillations, this particular interaction is the source of energy exchange, leading to a particular shape of the response curves, as well as quasi-periodic responses and a saturation phenomenon. These main features are...

This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to mechanical problems discretised by the finite eleme...

In this contribution we present a method to directly compute asymptotic expansion of invariant manifolds of large finite element models from physical coordinates and their reduced-order dynamics on the manifold. We show the accuracy of the reduction on selected models, exhibiting large rotations and internal resonances. The results obtained with th...

The derivation of periodic orbits together with stability analysis of large dimensional finite element models of continuum structures represents and important goal during the design of structures operating at resonance as for instance MEMS resonators. However, the huge computational requirements involved for full-order simulations make model order...

La méthode de paramétrisation des variétés invariantes permet de proposer des modèles d'ordre réduit efficaces, avec en particulier un calcul direct permettant de passer sans pré-calcul de la discrétisation éléments finis à la dynamique réduite le long des sous-espaces invariants. La méthode générale, permettant le calcul à un ordre arbitraire, est...

A high order direct parametrisation of invariant manifolds is exploited to operate dimen-sionality reduction for vibrating structures subjected to geometric nonlinearities. The method defines a nonlinear coordinate change between the nodal degrees-of-freedom and the normal coordinates, hence expressing the dynamics in an invariant-based span of the...

The direct parametrisation method for invariant manifolds is used for model order reduction of forced-damped mechanical structures subjected to geometric nonlinearities. Nonlinear mappings are introduced, allowing one to pass from the degrees of freedom of the finite-element model to the normal coordinates. Arbitrary order of expansions are conside...

A unimorph piezoelectric cantilever equipped with an Acoustic Black Hole (ABH) termination is designed for broadband energy harvesting. The ABH termination, with its tapered region, induces a focusing of the flexural vibrations which can be used to increase the efficiency of an energy harvesting device. A modal-based analytical model is presented,...

The direct computation of the third-order normal form for a geometrically nonlinear structure discretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate reduced-order models (ROM) relying on invariant manifold theory. The proposed reduction strategy is direct and simu...

This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems expressed in the physical basis, so that the technique is directly applicable to problems discretised by the finite element method. Two nonlinear mappin...

Micro-Electro-Mechanical Systems revolutionized the consumer market for their small dimensions, high performances and low costs. In recent years, the evolution of the Internet of Things is posing new challenges to MEMS designers that have to deal with complex multiphysics systems experiencing highly nonlinear dynamic responses. To be able to simula...

This paper aims at reviewing nonlinear methods for model order reduction of 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. Inv...

Dimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in ope...

This paper aims at reviewing nonlinear methods for model order reduction of 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. Inv...

Nonlinear vibrations of free-edge shallow spherical shells with large amplitudes are investigated, with the aim of predicting the type of nonlinearity (hardening/softening behaviour) for each mode of the shell, as a function of the radius R of curvature of the shell, from the plate case (R →∞) to the limit of non-shallow shell. Two different models...

Quasi-periodic solutions can arise in assemblies of nonlinear oscillators as a consequence of Neimark-Sacker bifurcations. In this work, the appearance of Neimark-Sacker bifurcations is investigated analytically and numerically in the specific case of a system of two coupled oscillators featuring a 1:2 internal resonance. More specifically, the loc...

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 dur...

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 additio...

Dimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in ope...

The aim of this contribution is to present numerical comparisons of model-order reduction
methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the...

The objective of this contribution is to compare two methods proposed recently in order to build efficient reduced-order models for geometrically nonlinear structures. The first method relies on the normal form theory that allows one to obtain a nonlinear change of coordinates for expressing the reduced-order dynamics in an invariant-based span of...

Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of th...

A comparison between two methods to derive reduced-order models (ROM) for geometrically nonlinear structures is proposed. The implicit condensation and expansion (ICE) method relies on a series of applied static loadings. From this set, a stress manifold is constructed for building the ROM. On the other hand, nonlinear normal modes rely on invarian...

The direct computation of the third-order normal form for a geometrically nonlinear structure dis-cretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate reduced-order models (ROM) relying on invariant manifold theory. The proposed reduction strategy is direct and sim...

Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of th...

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 a...

The objective of this contribution is to compare two methods proposed recently in order to build efficient reduced-order models for geometrically nonlinear structures. The first method relies on the normal form theory that allows one to obtain a nonlinear change of coordinates for expressing the reduced-order dynamics in an invariant-based span of...

Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure(STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of the...

A system composed of two cubic nonlinear oscillators with close natural frequencies, and thus displaying a 1:1 internal resonance, is studied both theoretically and experimentally, with a special emphasis on the free oscillations and the backbone curves. The instability regions of uncoupled solutions are derived and the bifurcation scenario as a fu...

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 blac...

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 gener...

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 p...

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 occurring 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 c...

For particular playing techniques such as “pop” or “slap” in the electric bass guitar, the string collides with frets, producing a percussive sound used in different music styles. The string/frets contacts introduce a nonlinearity which is investigated both numerically and experimentally in this paper. A physical model, based on a modal description...

Cette étude porte sur la caractérisation expérimentale et numérique d’un amortisseur de vibrations non linéaire magnétique avec raideurs linéaires et non linéaires ajustables. Le système primaire dont on souhaite réduire les oscillations est une plaque mince vibrant en grande amplitude. Le réglage des caractéristiques de l’amortisseur magnétique so...

The influence of a nonlinear tuned vibration absorber (NLTVA) on the airfoil flutter is investigated. In particular, its effect on the instability threshold and the potential subcriticality of the bifurcation is analyzed. For that purpose, the airfoil is modeled using the classical pitch and plunge aeroelastic model together with a linear approach...

The influence of a hysteretic damper on the airfoil flutter instability is investigated. In particular, its effect on the post-critical limit cycle oscillations (LCOs) is emphasized. For that purpose, an aeroelastic model including large amplitude motions and dynamic stall phenomenon, is considered for a rigid flat plate having two degrees of freed...

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...

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...

A complete vibroacoustic model is presented in order to compute numerically the sound pressure generated by a thin circular plate vibrating with large amplitude motions. The vibratory part relies on a modal approach for the von Kármán thin plate equations. A special emphasis is put in this paper on the inclusion of a geometrical imperfection descri...

The design and characterisation of a magnetic vibration absorber (MVA), completely relying on magnetic forces, is addressed. A distinctive feature of the absorber is the ability of tuning the linear stiffness together with the nonlinear cubic and quintic stiffnesses by means of repulsive magnets located in the axis of the main vibrating magnetic ma...

In the previous chapters, it was assumed that the amplitude of both air and structural oscillations in musical instruments were sufficiently small so that the assumption of linearity for their underlying models was fulfilled. However, this assumption is no longer valid in a number of situations encountered in musical acoustics, and a nonlinear appr...

Turbulence is a general term used for describing the erratic motions displayed by nonlinearsystems that are driven far from their equilibrium position and thus display complicatedmotions involving different time and length scales. Wave turbulence (WT) share many common ideas with turbulence, in particular asbeing a statistical theory for out-of-equ...

This paper presents simulations of nonlinear plate vibrations in relation to sound synthesis of gongs and cymbals. The von Kármán equations are shown and then solved in terms of the modes of the associated linear system. The modal equations obtained constitute a system of nonlinearly coupled Ordinary Differential Equations which are completely gene...

This paper presents simulations of nonlinear plate vibrations in relation to sound synthesis of gongs and cymbals. The von Kármán equations are shown and then solved in terms of the modes of the associated linear system. The modal equations obtained constitute a system of nonlinearly coupled Ordinary Differential Equations which are completely gene...

Time-domain simulation of musical instruments has shown promising results in recent years. Particularly attractive from a sound synthesis perspective is the resolution of system displaying some degree of nonlinearity, because of the richness of the perceptual information that nonlinearities produce. In this work, the focus is on one such system, na...

Les contacts entre une corde vibrante et un obstacle rigide sont fréquemment rencontrés dans divers instruments de musique (basse électrique, contrebasse, sitar, tampoura...), ce qui donne lieu à des so-norités riches et variées. Alors qu'un certain nombre d'études analytiques et numériques ont été menées pour modéliser ces contacts, on ne trouve q...

Cette étude porte sur la réalisation d'un amortisseur de vibrations constitué d'une masse oscillante magnétique placée dans un champ créé par des aimants statiques. En ajustant les positions de ces derniers, il est possible de contrôler les valeurs des raideurs linéaires et non linéaires afin de couvrir les cas d'un amortisseur à masse accordée, d'...

Nous nous intéressons ici à l'instabilité de flottement sur une aile à deux degrés de liberté, cette instabilité est à l'origine de cycles limites dus aux non linéarités provenant des efforts aérodynamiques et/ou de la structure. Nous proposons une méthode de con-trôle passif non linéaire utilisant le caractère pseudoélastique des alliages à mémoir...

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 nonlinea...

This article is concerned with the numerical solution of the full dynamical von K{\'a}rm{\'a}n plate equations for geometrically nonlinear (large-amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement f...