Analysis of stationary random vibrating systems using smooth decomposition

International audienceA modified Karhunen-Loève Decomposition/Proper Orthogonal Decomposition method, named Smooth Decomposition (SD) (also named smooth Karhunen-Lo'eve decomposition), was recently introduced to analyze stationary random signal. It is based on a generalized eigenproblem defined from the covariance matrix of the random process and the covariance matrix of the associated time-derivative random process. The SD appears to be an interesting tool in terms of modal analysis. In this paper, the SD will be described in case of stationary random processes and extended also to stationary random fields. The main properties will be discussed and illustrated on a randomly excited clamped-free beam

Nonlinear normal modes of a two degree of freedom oscillator with a bilateral elastic stop

A study of the non linear modes of a two degree of freedom mechanical system with bilateral elastic stop is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. In order to obtain the Nonlinear Normal Mode (NNM), the harmonic balance method with a large number of harmonics, combined with the asymptotic numerical method, is used to solve the regularized problem. These methods are present in the software "package" MANLAB. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non regular problem. The two NNMs starting respectively from the two linear normal modes of the associated underlying linear system are discussed. The energy-frequency plot is used to present a global vision of the behavior of the modes. The dynamics of the modes are also analyzed comparing each periodic orbits and modal lines. The first NNM shows an elaborate dynamics with the occurrence of multiple impacts per period. On the other hand, the second NNM presents a more simple dynamics with a localization of the displacement on the first mass

Free vibrations of an uncertain energy pumping system

International audienceThe aim of this paper is to study the energy pumping (the irreversible energy transfer from one structure, linear, to another structure, nonlinear) robustness considering the uncertainties of the parameters of a two DOF mass-spring-damper, composed of two subsystems, coupled by a linear spring: one linear subsystem, the primary structure, and one nonlinear subsystem, the so-called NES (nonlinear energy sink). Three parameters of the system will be considered as uncertain: the nonlinear stiffness and the two dampers. Random variables are associated to the uncertain parameters and probability density functions are constructed for the random variables applying the Maximum Entropy Principle. A sensitivity analysis is then performed, considering different levels of dispersion, and conclusions are obtained about the influence of the uncertain parameters in the robustness of the system

Passive suppression of helicopter ground resonance instability by means of a strongly nonlinear absorber

International audienceIn this paper, we study a problem of passive suppression of helicopter Ground Resonance (GR) using a single degree freedom Nonlinear Energy Sink (NES). GR is a dynamic instability involving the coupling of the blades motion in the rotational plane (i.e. the lag motion) and the helicopter fuselage motion. A reduced linear system reproducing GR instability is used. It is obtained using successively Coleman transformation and binormal transformation. The analysis of the steadystate responses of this model is performed when a NES is attached on the helicopter fuselage. The NES involves an essential cubic restoring force and a linear damping force. The analysis is achieved applying complexification-averaging method. The resulting slow-flow model is finally analyzed using multiple scale approach. Four steady-state responses corresponding to complete suppression, partial suppression through strongly modulated response, partial suppression through periodic response and no suppression of the GR are highlighted. An algorithm based on simple criterions is developed to predict these steady-state response regimes. Numerical simulations of the complete system confirm this analysis of the slow-flow dynamics. A parametric analysis of the influence of the NES damping coefficient and the rotor speed on the response regime is finally proposed

Nonlinear modes of clarinet-like musical instruments

The concept of nonlinear modes is applied in order to analyze the behavior of a model of woodwind reed instruments. Using a modal expansion of the impedance of the instrument, and by projecting the equation for the acoustic pressure on the normal modes of the air column, a system of second order ordinary differential equations is obtained. The equations are coupled through the nonlinear relation describing the volume flow of air through the reed channel in response to the pressure difference across the reed. The system is treated using an amplitude-phase formulation for nonlinear modes, where the frequency and damping functions, as well as the invariant manifolds in the phase space, are unknowns to be determined. The formulation gives, without explicit integration of the underlying ordinary differential equation, access to the transient, the limit cycle, its period and stability. The process is illustrated for a model reduced to three normal modes of the air column

NUMERICAL AND EXPERIMENTAL STUDY OF A HYBRID ELECTRO- ACOUSTIC NONLINEAR SOUND ABSORBER IN A RESONANT ROOM

International audienceWe present a hybrid Electro-Acoustic Nonlinear Energy Sink (EA-NES) coupled to a resonant room. It is made of a baffled nonlinear membrane with its front face exposed to the noise in the room, and the rear face enclosed. The enclosure includes a feedback loop associating a microphone and a loudspeaker that control the acoustic pressure applied to the rear face of the membrane. The EA-NES action is based on the concept of Targeted Energy Transfer resulting from the nonlinear coupling between the absorber and an acoustic mode in the room (primary system). The nonlinear dynamics of the system is studied theoretically. The asymptotic study shows the possible existence of relaxation oscillations, which can be tuned by the control loop settings. The experimental study confirms theoretical predictions and shows that a hybrid EA-NES can reduce the sound level in a concrete building. Harmonic regime and Strongly Modulated Regime (SMR) are observed. The SMR responses are simulated with a good agreement

Modes non-linéaires de structures élastiques dotées de non-linéarités de contact

National audienceCet article propose une méthode de calcul des modes non-linéaires de structures élastiques dotées de non-linéaritées de contact localisées. Les lois de contact sont régularisées au moyen de relations polynomiales comportant un paramètre de régularisation. La procédure de calcul est basée sur le couplage de la méthode de balance harmonique et de la méthode asymptotique numérique

Toward an adjustable nonlinear low frequency acoustic absorber

International audienceThis paper presents experimental results showing that a loudspeaker used as a suspended piston working outside its range of linearity can also be used as a nonlinear acoustic absorber. The main advantage of this technology of absorber is the perspective to adjust independently the device parameters (mass, nonlinear stiffness and damping) according to the operational conditions. To achieve this purpose, quasi-static and dynamic tests have been performed on three types of commercial devices (one with structural modifications), in order to define the constructive characteristics that it should present. An experimental setup has been developed using a one-dimensional acoustic linear system coupled through a box (acting as a weak spring) to a loudspeaker used as a suspended piston acting as an essentially nonlinear oscillator. The tests carried out on the whole vibro-acoustic system have showed the occurrence of the acoustic TET from the acoustic media to the suspended piston and demonstrated the efficiency of this new kind of absorber at low frequencies over a wide frequency range. Moreover, the experimental analyses conducted with different NES masses have confirmed that it is possible to optimize the noise absorption with respect to the excitation level of the acoustic resonator

PWR fuel assembly modal testing and analysis

ABSTRACT Pressurized Water Reactor (PWR) seismic or Lost Of Coolant Accident (LOCA) loads could result in impacts between nuclear fuel assemblies or between fuel assemblies and the core baffles. Forces generated during these shocks are often the basis for the determination of the maximum loads and of the spacer grid and fuel rod design. The knowledge of the fuel assembly kinematics is essential to compute these maximum loads, and this requires experimental tests. Our study aims at characterizing the behavior of a fullscale fuel assembly subjected to various excitations. The effect of the assembly environment (air, still water and water under flow) is studied. The French Nuclear Reactor Directorate experimental facility HERMES T allows hydraulic and mechanical testing of full-scale fuel assemblies. It is designed for flow rate up to 1200 m3/h and temperature up to 170°C. Specific excitation devices allow mechanical tests with amplitudes of motion up to 20 ram. Laser vibrometry, displacement transducers and tracking camera apparatus measure the fuel assembly displacement. To identify this Multi Degree Of Freedom (MDOF) system (assembly or assembly + fluid), two dependent problems have to be addressed: the linear or non-linear model selection, and the estimation of the corresponding parameters. Under different environments and excitation types, it is shown that the mechanical system is strongly non-linear. The damping term, essentially fluid, increases with flow rate and with motion amplitude, while the stiffness decreases with amplitude. The main results, the measuring and identification methods and the extrapolation to the reactor thermohydraulic conditions are presented and discussed

Modélisation d'un réseau de tubes par un milieu poreux: comparaison expérience théorie

Dans cet article nous proposons un modèle numérique du comportement mécanique d'un coeur de réacteur nucléaire incluant la dynamique complète des assemblages combustibles et du fluide. Chaque assemblage combustible est assimilé à un milieu poreux déformable avec un comportement visco-élastique non linéaire, les équations globales du fluide sont obtenues par moyenne spatiale, le champ de vitesse du fluide et le champ de déplacement de la structure sont définis sur tout le domaine spatial. Les équations du mouvement de la structure sont obtenues par une formulation Lagrangienne, tandis que, pour permettre le coulage fluide structure, les équations de mouvement du fluide sont obtenues par une formulation Arbitrary Lagrangian Eulerian. Des résultats numériques sont confrontés aux données expérimentales