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Claude-henri Lamarque - One of the best experts on this subject based on the ideXlab platform.

  • Extended complexification method to study nonlinear passive control
    Nonlinear Dynamics, 2020
    Co-Authors: Diala Bitar, Alireza Ture Savadkoohi, Claude-henri Lamarque, Emmanuel Gourdon, Manuel Collet
    Abstract:

    The present research work aims to design a passive vibration control based on nonlinear energy pumping. An extended asymptotic approach is introduced based on the invariant manifold approach for the case of 1:1 resonance. It consists in introducing an extended form of Manevitch's complex variables, taking into consideration higher harmonics, enabling the detection of the invariant manifold of the system at Fast Time Scale. At the slow Time Scale, equilibrium points and singularities are identified analytically in order to predict periodic regimes and strongly modulated responses. The example of a passive shunt loudspeaker using a nonlinear absorber is studied. Unlike classical investigations, the first and third harmonics are taken into consideration. It is demonstrated that the presence of the third harmonic improves the approximations of the results. Different cases are considered, where the obtained analytical results are in good agreement with those obtained via direct numerical integration of the principal system of equations.

  • Energy exchange between a nonlinear absorber and a pendulum under parametric excitation
    2019
    Co-Authors: Gabriel Hurel, Alireza Ture Savadkoohi, Claude-henri Lamarque
    Abstract:

    The studied system is a planar pendulum coupled with a nonlinear absorber and parametrically excited at itsbasis. The dynamical equations are treated with a multiple Scale method. At Fast Time Scale, a slow invariantmanifold represents the asymptotic behavior. At slow Time Scale, the equilibrium points and their stability areinvestigated. Several phase portraits complete the analysis of the dynamical behavior of the system. Finally, numerical examples are given to confirm analytic predictions.

  • Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber
    Nonlinear Dynamics, 2019
    Co-Authors: Gabriel Hurel, Alireza Ture Savadkoohi, Claude-henri Lamarque
    Abstract:

    A pendulum, which can oscillate in two directions, is subjected to a generalized external force. A non-smooth absorber is coupled to the pendulum with an arbitrary location and orientation. The equations of the system are derived and are treated with a multiple Scale method. At Fast Time Scale, the topology of the slow invariant manifold is described with its stable and unstable zones. The equilibrium and singular points of the system are detected at the first slow Time Scale. The responses of the main system, given as a function of the frequency of the external force, show reductions of the vibration levels. The analytic predictions are compared by direct numerical Time integration of the equations of the system. They illustrate the operationality of the non-smooth absorber in several cases.

  • Nonlinear vibratory energy exchanges between a two degrees-of-freedom pendulum and a nonlinear absorber
    Journal of Engineering Mechanics, 2019
    Co-Authors: Gabriel Hurel, Alireza Ture Savadkoohi, Claude-henri Lamarque
    Abstract:

    The multi-Time-Scale responses of a two-degree-of-freedom pendulum coupled with a nonlinear absorber were studied. The absorber was positioned in an arbitrary direction with respect to those of the pendulum oscillations. The phase-dependent slow invariant manifold of the system and its stable zones were traced at a Fast Time Scale while system responses were studied at a slow Time Scale around the slow, invariant manifold, leading to detection of equilibrium and singular points. Moreover, the amplitude–frequency curves of the system were detected, showing the possibility of the existence of isolated branches, which could correspond to high energy levels for pendulum oscillations. All analytic developments were confronted with numerical results collected from direct numerical integration of system equations. Depending on the characteristics of external excitations, the system can face periodic or modulated regimes.

  • Nonlinear vibratory interactions between a linear and a non-smooth forced oscillator in the gravitational field
    Mechanical Systems and Signal Processing, 2017
    Co-Authors: Claude-henri Lamarque, S. Charlemagne, Alireza Ture Savadkoohi, Pierre Abdoulhadi
    Abstract:

    Nonlinear interactions of two coupled forced oscillators in the gravitational field are studied.The first oscillator that is supposed to be linear is coupled to a system which possesses multi-phase non-smooth non-conservative and restoring forces. The mass ratio of two oscillators is very small. Studying the system at Fast Time Scale reveals an invariant manifold which depends on the amplitude of applied force on the non-smooth oscillator while slow dynamics of the system around its invariant demonstrates its equilibrium and singular points. Detected Time multi-Scale dynamics of the system can be endowed for designing proper controller and/or harvester non-smooth oscillators which presents final desirable periodic and/or strongly modulated responses.

Alireza Ture Savadkoohi - One of the best experts on this subject based on the ideXlab platform.

  • Extended complexification method to study nonlinear passive control
    Nonlinear Dynamics, 2020
    Co-Authors: Diala Bitar, Alireza Ture Savadkoohi, Claude-henri Lamarque, Emmanuel Gourdon, Manuel Collet
    Abstract:

    The present research work aims to design a passive vibration control based on nonlinear energy pumping. An extended asymptotic approach is introduced based on the invariant manifold approach for the case of 1:1 resonance. It consists in introducing an extended form of Manevitch's complex variables, taking into consideration higher harmonics, enabling the detection of the invariant manifold of the system at Fast Time Scale. At the slow Time Scale, equilibrium points and singularities are identified analytically in order to predict periodic regimes and strongly modulated responses. The example of a passive shunt loudspeaker using a nonlinear absorber is studied. Unlike classical investigations, the first and third harmonics are taken into consideration. It is demonstrated that the presence of the third harmonic improves the approximations of the results. Different cases are considered, where the obtained analytical results are in good agreement with those obtained via direct numerical integration of the principal system of equations.

  • Energy exchange between a nonlinear absorber and a pendulum under parametric excitation
    2019
    Co-Authors: Gabriel Hurel, Alireza Ture Savadkoohi, Claude-henri Lamarque
    Abstract:

    The studied system is a planar pendulum coupled with a nonlinear absorber and parametrically excited at itsbasis. The dynamical equations are treated with a multiple Scale method. At Fast Time Scale, a slow invariantmanifold represents the asymptotic behavior. At slow Time Scale, the equilibrium points and their stability areinvestigated. Several phase portraits complete the analysis of the dynamical behavior of the system. Finally, numerical examples are given to confirm analytic predictions.

  • Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber
    Nonlinear Dynamics, 2019
    Co-Authors: Gabriel Hurel, Alireza Ture Savadkoohi, Claude-henri Lamarque
    Abstract:

    A pendulum, which can oscillate in two directions, is subjected to a generalized external force. A non-smooth absorber is coupled to the pendulum with an arbitrary location and orientation. The equations of the system are derived and are treated with a multiple Scale method. At Fast Time Scale, the topology of the slow invariant manifold is described with its stable and unstable zones. The equilibrium and singular points of the system are detected at the first slow Time Scale. The responses of the main system, given as a function of the frequency of the external force, show reductions of the vibration levels. The analytic predictions are compared by direct numerical Time integration of the equations of the system. They illustrate the operationality of the non-smooth absorber in several cases.

  • Nonlinear vibratory energy exchanges between a two degrees-of-freedom pendulum and a nonlinear absorber
    Journal of Engineering Mechanics, 2019
    Co-Authors: Gabriel Hurel, Alireza Ture Savadkoohi, Claude-henri Lamarque
    Abstract:

    The multi-Time-Scale responses of a two-degree-of-freedom pendulum coupled with a nonlinear absorber were studied. The absorber was positioned in an arbitrary direction with respect to those of the pendulum oscillations. The phase-dependent slow invariant manifold of the system and its stable zones were traced at a Fast Time Scale while system responses were studied at a slow Time Scale around the slow, invariant manifold, leading to detection of equilibrium and singular points. Moreover, the amplitude–frequency curves of the system were detected, showing the possibility of the existence of isolated branches, which could correspond to high energy levels for pendulum oscillations. All analytic developments were confronted with numerical results collected from direct numerical integration of system equations. Depending on the characteristics of external excitations, the system can face periodic or modulated regimes.

  • Nonlinear vibratory interactions between a linear and a non-smooth forced oscillator in the gravitational field
    Mechanical Systems and Signal Processing, 2017
    Co-Authors: Claude-henri Lamarque, S. Charlemagne, Alireza Ture Savadkoohi, Pierre Abdoulhadi
    Abstract:

    Nonlinear interactions of two coupled forced oscillators in the gravitational field are studied.The first oscillator that is supposed to be linear is coupled to a system which possesses multi-phase non-smooth non-conservative and restoring forces. The mass ratio of two oscillators is very small. Studying the system at Fast Time Scale reveals an invariant manifold which depends on the amplitude of applied force on the non-smooth oscillator while slow dynamics of the system around its invariant demonstrates its equilibrium and singular points. Detected Time multi-Scale dynamics of the system can be endowed for designing proper controller and/or harvester non-smooth oscillators which presents final desirable periodic and/or strongly modulated responses.

Antoine Girard - One of the best experts on this subject based on the ideXlab platform.

  • Singular perturbation approach for linear coupled ODE-PDE systems
    2019
    Co-Authors: Ying Tang, Christophe Prieur, Antoine Girard
    Abstract:

    This paper focuses on a class of linear coupled ODE-PDE systems whose dynamics evolve in two Time Scales. The Fast Time Scale modeled by a small positive perturbation parameter is introduced to the dynamics either of the ODE or of the PDE. By setting the perturbation parameter to zero, two subsystems, namely the reduced and the boundary-layer subsystems, are formally computed. Firstly, we propose a sufficient stability condition for the full coupled system. This stability condition implies the stability of both subsystems. Then, we state an approximation of the full coupled ODE-PDE systems by the subsystems based on the singular perturbation method. The error between the solution of the full system and that of the subsystems is the order of the perturbation parameter. Finally, numerical simulations on academic examples illustrate the theoretical results.

  • Singular perturbation approximation by means of a H 2 Lyapunov function for linear hyperbolic systems
    Systems & Control Letters, 2016
    Co-Authors: Ying Tang, Christophe Prieur, Antoine Girard
    Abstract:

    Abstract A linear hyperbolic system of two conservation laws with two Time Scales is considered in this paper. The Fast Time Scale is modeled by a small perturbation parameter. By formally setting the perturbation parameter to zero, the full system is decomposed into two subsystems, the reduced subsystem (representing the slow dynamics) and the boundary-layer subsystem (standing for the Fast dynamics). The solution of the full system can be approximated by the solution of the reduced subsystem. This result is obtained by using a H 2 Lyapunov function. The estimate of the errors is the order of the perturbation parameter for all initial conditions belonging to H 2 and satisfying suitable compatibility conditions. Moreover, for a particular subset of initial conditions, more precise estimates are obtained. The main result is illustrated by means of numerical simulations.

  • Singular perturbation approximation by means of a $H^2$ Lyapunov function for linear hyperbolic systems
    Systems and Control Letters, 2016
    Co-Authors: Ying Tang, Christophe Prieur, Antoine Girard
    Abstract:

    A linear hyperbolic system of two conservation laws with two Time Scales is considered in this paper. The Fast Time Scale is modeled by a small perturbation parameter. By formally setting the perturbation parameter to zero, the full system is decomposed into two subsystems, the reduced subsystem (representing the slow dynamics) and the boundary-layer subsystem (standing for the Fast dynamics). The solution of the full system can be approximated by the solution of the reduced subsystem. This result is obtained by using a H 2 Lyapunov function. The estimate of the errors is the order of the perturbation parameter for all initial conditions belonging to H 2 and satisfying suitable compatibility conditions. Moreover, for a particular subset of initial conditions, more precise estimates are obtained. The main result is illustrated by means of numerical simulations.

Ying Tang - One of the best experts on this subject based on the ideXlab platform.

  • Singular perturbation approach for linear coupled ODE-PDE systems
    2019
    Co-Authors: Ying Tang, Christophe Prieur, Antoine Girard
    Abstract:

    This paper focuses on a class of linear coupled ODE-PDE systems whose dynamics evolve in two Time Scales. The Fast Time Scale modeled by a small positive perturbation parameter is introduced to the dynamics either of the ODE or of the PDE. By setting the perturbation parameter to zero, two subsystems, namely the reduced and the boundary-layer subsystems, are formally computed. Firstly, we propose a sufficient stability condition for the full coupled system. This stability condition implies the stability of both subsystems. Then, we state an approximation of the full coupled ODE-PDE systems by the subsystems based on the singular perturbation method. The error between the solution of the full system and that of the subsystems is the order of the perturbation parameter. Finally, numerical simulations on academic examples illustrate the theoretical results.

  • Singular perturbation approximation by means of a H 2 Lyapunov function for linear hyperbolic systems
    Systems & Control Letters, 2016
    Co-Authors: Ying Tang, Christophe Prieur, Antoine Girard
    Abstract:

    Abstract A linear hyperbolic system of two conservation laws with two Time Scales is considered in this paper. The Fast Time Scale is modeled by a small perturbation parameter. By formally setting the perturbation parameter to zero, the full system is decomposed into two subsystems, the reduced subsystem (representing the slow dynamics) and the boundary-layer subsystem (standing for the Fast dynamics). The solution of the full system can be approximated by the solution of the reduced subsystem. This result is obtained by using a H 2 Lyapunov function. The estimate of the errors is the order of the perturbation parameter for all initial conditions belonging to H 2 and satisfying suitable compatibility conditions. Moreover, for a particular subset of initial conditions, more precise estimates are obtained. The main result is illustrated by means of numerical simulations.

  • Singular perturbation approximation by means of a $H^2$ Lyapunov function for linear hyperbolic systems
    Systems and Control Letters, 2016
    Co-Authors: Ying Tang, Christophe Prieur, Antoine Girard
    Abstract:

    A linear hyperbolic system of two conservation laws with two Time Scales is considered in this paper. The Fast Time Scale is modeled by a small perturbation parameter. By formally setting the perturbation parameter to zero, the full system is decomposed into two subsystems, the reduced subsystem (representing the slow dynamics) and the boundary-layer subsystem (standing for the Fast dynamics). The solution of the full system can be approximated by the solution of the reduced subsystem. This result is obtained by using a H 2 Lyapunov function. The estimate of the errors is the order of the perturbation parameter for all initial conditions belonging to H 2 and satisfying suitable compatibility conditions. Moreover, for a particular subset of initial conditions, more precise estimates are obtained. The main result is illustrated by means of numerical simulations.

Luc Deike - One of the best experts on this subject based on the ideXlab platform.

  • Turbulence of capillary waves forced by steep gravity waves
    Journal of Fluid Mechanics, 2018
    Co-Authors: Michael Berhanu, Eric Falcon, Luc Deike
    Abstract:

    We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in the laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness (0.2–0.3), power-law spectra are observed both in space and Time, defining a turbulent regime of capillary waves transferring energy from the large Scale to the small Scale. Analysis of temporal fluctuations of the spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to the small Scale in a Fast Time Scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of the wave spectra are found to be in agreement with those of the weakly nonlinear wave turbulence theory. However, the energy flux is not constant through the Scales and the wave spectrum scaling with this flux is not in good agreement with wave turbulence theory. These results suggest that theoretical developments beyond the classic wave turbulence theory are necessary to describe the dynamics and statistics of capillary waves in a natural environment. In particular, in the presence of broad-Scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions should be reconsidered.

  • Turbulence of capillary waves forced by steep gravity waves
    Journal of Fluid Mechanics, 2018
    Co-Authors: Michael Berhanu, Eric Falcon, Luc Deike
    Abstract:

    We study experimentally the dynamics and statistics of capillary waves forced by random steep gravity waves mechanically generated in laboratory. Capillary waves are produced here by gravity waves from nonlinear wave interactions. Using a spatio-temporal measurement of the free-surface, we characterize statistically the random regimes of capillary waves in the spatial and temporal Fourier spaces. For a significant wave steepness ($0.2-0.3$), power-law spectra are observed both in space and Time, defining a turbulent regime of capillary waves transferring energy from large Scale to small Scale. Analysis of temporal fluctuations of spatial spectrum demonstrates that the capillary power-law spectra result from the temporal averaging over intermittent and strong nonlinear events transferring energy to small Scale in a Fast Time Scale, when capillary wave trains are generated in a way similar to the parasitic capillary wave generation mechanism. The frequency and wavenumber power-law exponents of wave spectrum are found to be in agreement with those of the weakly nonlinear Wave Turbulence Theory. However, the energy flux is not constant through the Scales and the wave spectrum scaling with this flux is not in good agreement with Wave Turbulence Theory. These results suggest that theoretical developments beyond the classic Wave Turbulence Theory are necessary to describe the dynamics and statistics of capillary waves in natural environment. In particular, in presence of broad Scale viscous dissipation and strong nonlinearity, the role of non-local and non-resonant interactions could be reconsidered.