Claude-Henri Lamarque

Vibratory Energy Localization by Non-smooth Energy Sink with Time-Varying Mass

We study nonlinear interactions of two coupled oscillators at different time scales. The main oscillator which is linear is coupled to a nonlinear energy sink with non-smooth (piecewise linear) potential and time-dependent mass. The overall time is embedded to fast and slow time scales and the behavior of the system at each one of them is revealed.The invariant of the system at fast time scale is detected. Then we try to have further information about the overall system behavior at the first slow time scale. Finally, analytical developments are compared with numerical results and the possibility of the passive control of the main system by means of the time-dependent NES is commented upon.

Multiple resonance capture of a compound system under harmonic excitation

Vibration exchange between a master linear oscillator and a slave Nonlinear Energy Sink (NES) under harmonics excitation by means of several resonance capture is investigated and commented upon. The importance of higher frequencies by introducing the concept of the relative mode is pinpointed. A fast and special iterative technique and then the more general form of it is introduced in order to evaluate the system unknowns. Finally, the effect of the correct resonance paring on detecting the actual system behavior is illustrated by an example.

Suppressing Aeroelastic Instability in a Suspension Bridge Using a Nonlinear Absorber

We study a problem of passive nonlinear targeted energy transfer between a two degree of freedom suspension bridge model and a single degree of freedom nonlinear energy sink (NES). The system is studied under 1:1:1 nonlinear resonance involved in targeted energy transfer mechanisms. Analytical expansions are performed by mean of complexification methods, multiple scales expansions and exploits also the concept of limiting phase trajectories (LPTs). Several control mechanisms for aeroelastic instability are identified, and analytical calculations bring to efficient parameters for the absorber design. Numerical simulations are performed and good agreement with analytical predictions is observed. It results that the concept of Limiting Phase Trajectories (LPT) allows formulating adequately the problem of intensive energy transfer from a bridge to a nonlinear energy sink.

Targeted energy transfer in a 2-DOF mechanical system coupled to a non-linear energy sink with varying stiffness

The dynamical behavior of a non-linear mechanical system with two degrees of freedom (DOFs) during free and forced excitations is studied analytically and numerically. The non-linearity of the system is represented intentionally by a smooth non-linear simple function with periodically varying stiffness around a constant value for the sake of practical investigations. Analysis of the system leads to a method that could be used to design the non-linear energy sink (NES) so that the behavior of the system during relaxation and its strongly modulated response (SMR) could be improved versus the constant stiffness configuration.

Passive Control of Differential Algebraic Inclusions - General Method and a Simple Example

In this chapter, we consider a master system consisting of a nonlinear differential inclusion and an algebraic equation of constraint (resulting in a Differential Algebraic Inclusion (DAI) system). This system is coupled to a nonlinear energy sink (NES) corresponding to a one degree-of-freedom essentially nonlinear differential equation. We examine how a resonance capture can lead to a reduced order dynamical system. To obtain this reduced order model, we describe a multiple time scale analysis governed by the introduction of multi-timescales via a small parameter \(\varepsilon \) that is finite and strictly positive. The mass of the NES is small versus the mass of the master system, and it governs a mass ratio defining the small parameter \(\varepsilon \). The first timescale is the fast scale. Introducing the Manevitch complexification leads to the definition of slow time envelope coordinates. These envelope coordinates either do not directly depend on the fast time scale or do not depend on this fast time scale via introduction of the so-called Slow Invariant Manifold (SIM). The slow time dynamics of the master system components is analyzed through introduction of equilibrium points, corresponding to periodic solutions, or singular points (governing bifurcations around the SIM), corresponding to quasi-periodic behaviors. We present a simple example of semi-implicit Differential Algebraic Equation (DAE), including a friction term coupled to a cubic NES. Analytical developments of a 1:1:1 resonance case permit us to predict passive control of a DAI by a NES.

Experiment-based motion reconstruction and restitution coefficient estimation of a vibro-impact system

The objective of this paper is to demonstrate the motion reconstruction and the parameter estimation of a vibro-impact (VI) system from limited experimental information. Based on the measured displacement and acceleration of its linear main system, the rest motion information such as the displacement and velocity of the attached VI energy sink can be calculated rather than difficult direct measurement, and therefore, different response regimes from the strongly modulated response to the classic regime with two impacts per cycle are reconstructed. Consequently, it provides comprehensive experimental data for the validation of analytical and numerical results and for any experimental bifurcation analysis. Moreover, a procedure to estimate the restitution coefficient from periodic impacts is demonstrated. This new experimental approach to estimate the value of the restitution coefficient is simple and this accurate value could play an important role in analytical and numerical study.

Real time sensing in the non linear regime of nems resonators

This paper reports the proof of concept of a nonlinear detection scheme for Nanoelectromechanical (NEMS) resonators for sensing applications. This set-up increases the dynamic range of a resonant sensor by operating it at amplitudes beyond its limit of linearity. Unlike other works in non-linear sensing, this method allows the tracking of the resonance frequency in real time, while being suitable for multi-mode operation and hence, single-particle detection for example.

LOCALIZATION OF VIBRATORY ENERGY OF A LINEAR SYSTEM IN A CHAIN OF FOUR NONLINEAR OSCILLATORS

Abstract. In this paper, dynamics of a five degree-of-freedom system formed by a main linear oscillator coupled to four light nonlinear systems in series is studied. The aim is to control and/or to harvest the energy of the main structure under harmonic excitations around its resonance. A multiple scales method is used to derive the behavior of the system at different time scales. At fast time scale, detected slow invariant manifold gives an overall comprehension of the possible behaviors that the system can undergo. At slow time scale, the modulated behavior of the system around its invariant is described by traced equilibrium and singular points. The former predict periodic regimes, while the latter hint at strongly modulated responses characterized by persisting bifurcations of the system around its unstable zones. All analytical results are validated by numerical simulations.

Experimental and theoretical investigation on nonlinear behavior of cable-stayed bridges

This paper deals with experimental study and with understanding via a finite number of degrees of freedom model of the vibrations of an inclined cable linked to a continuous beam. This is a simplified version of deck and cable of a bridge. External excitation is exerted on the beam. The cable attached to the end of the beam is submitted to a vertical sinusoidal solicitation due to the response of the finite stiffness beam. The excitation of the cable though it is more complex looks similar to the excitation used in previous works. A guided device located at the end of the beam ensures the excitation with a variation of the horizontal component of the cable tension that introduces a new parametric excitation. Analysis of preliminary experimental results for main and secondary resonances permits us to consider simple modeling with one degree of freedom systems obtained by projection of the continuous three-dimensional model of the cable on adapted Irvine mode. Analytical treatment of these models involving data from the experimental devices shows a correct qualitative agreement between preliminary experiments and theoretical. Continuation techniques are used to highlight the influence of physical parameters.