@article{BERGEOT2024134063,
abstract = {This paper investigates the dynamic behavior of a Van der Pol oscillator (used as an archetypal self-sustained oscillator) coupled to a bistable nonlinear energy sink (BNES). We first show using numerical simulations that this system can undergo a multitude of motions including different types of periodic regimes and so-called strongly modulated responses (SMR) as well as chaotic regimes. We also show that a BNES can be much more efficient than a classical cubic NES but this is not robust since a little perturbation can switch the system from harmless to harmful situations. However, even in the most unfavorable cases, it is possible to find a set of parameters for which the BNES performs better than the NES. A multiple time scales approach is then addressed to analyze the system. In this context, we show that the so-called Multiple Scale/Harmonic Balance Method (MSHBM) must be modified (compared to its usual use) to consider the specific feature of the BNES, i.e., that it can have a nonzero-mean oscillating motion. This allows us to derive a so-called amplitude-phase modulation dynamics (APMD) which can reproduce the complex behavior of the initial system. Because of the presence of a small perturbation parameter (i.e., the mass ratio between the BNES and the VdP oscillator), the APMD is governed by two different time scales. More precisely, it appears as a (3,1)-fast--slow system whose motion is constituted in a succession of slow and fast epochs. Founding a (3,1)-fast--slow APMD is interesting since that implies a more complex dynamics than in the case of a classic NES whose APMD is only (2,1)-fast--slow. A fast--slow analysis is finally conducted within the framework of the geometric singular perturbation theory. From the computation of the so-called critical manifold and the analytical expressions of the APMD fixed points, a global stability analysis is performed. This enables us to interpret a certain number of regimes observed on numerical simulations of the initial system.},
author = {Baptiste Bergeot and S{\'e}bastien Berger},
date-added = {2024-02-08 09:32:54 +0100},
date-modified = {2024-02-08 09:34:33 +0100},
doi = {10.1016/j.physd.2024.134063},
hal = {https://hal.science/hal-04280444},
journal = {Physica D: Nonlinear Phenomena},
pages = {134063},
title = {Fast--slow analysis of passive mitigation of self-sustained oscillations by means of a bistable nonlinear energy sink},
volume = {460},
year = {2024},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0167278924000149},
bdsk-url-2 = {https://doi.org/10.1016/j.physd.2024.134063}
}
@article{BERGEOT2023104351,
abstract = {In this paper the influence of stochasticity (i.e. a Gaussian white noise forcing) on the dynamic behavior of a self-sustained oscillator coupled to a non-linear energy sink is investigated. To this end, the standard stochastic averaging is used to compute the slow flow dynamics of the system. Preliminary results show that the reasoning which allows to predict the system behavior in the deterministic case can be contradicted in presence of stochasticity. Then, by means of the Monte Carlo method, the stochastic averaging procedure is validated. Finally, two quantities are introduced to highlight more precisely the special features of the stochastic system behavior compared to that of the deterministic system. These are the probability of being in a harmless regime and the First-Passage Time to reach a harmful regime which are computed and investigated combining again the Monte Carlo approach with numerical integrations of the slow flow dynamics. The results obtained show afresh that the stochastic forcing can modify significantly the dynamic behavior of the corresponding deterministic system. Indeed, when they are computed on the latter, the two quantities aforementioned have a discontinuity at the mitigation limit (i.e. the value of the bifurcation parameter under consideration below which the NES acts and above which it no longer acts) revealing an abrupt change of behavior of the coupled system. The paper shows that this typical characteristic of the deterministic system is lost in the presence of stochasticity, the stochastic system becoming smooth at the mitigation limit.},
author = {Baptiste Bergeot},
date-added = {2023-02-01 17:06:01 +0100},
date-modified = {2023-02-01 17:20:54 +0100},
doi = {10.1016/j.ijnonlinmec.2023.104351},
hal = {https://hal.science/hal-03937086},
journal = {International Journal of Non-Linear Mechanics},
keywords = {Passive vibration control, Non-linear energy sink, Self-sustained oscillations, Stochastic averaging, Stochastic forcing},
pages = {104351},
title = {Effect of stochastic forcing on the dynamic behavior of a self-sustained oscillator coupled to a non-linear energy sink},
volume = {150},
year = {2023},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0020746223000033},
bdsk-url-2 = {https://doi.org/10.1016/j.ijnonlinmec.2023.104351}
}
@article{BergeotVergezNODY2021,
author = {Bergeot, Baptiste and Vergez, Christophe},
date-added = {2022-01-05 12:00:52 +0100},
date-modified = {2023-02-01 17:19:21 +0100},
doi = {10.1007/s11071-021-07104-9},
hal = {https://hal.archives-ouvertes.fr/hal-03215274v3},
journal = {Accepted in Nonlinear Dynamics},
pages = {3291-3312},
title = {Analytical prediction of delayed Hopf bifurcations in a simplified stochastic model of reed musical instruments},
volume = {107},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1007/s11071-021-07104-9}
}
@article{SnounBergeotBergerEJMA2022,
author = {Snoun, Cherif and Bergeot, Baptiste and Berger, S\'ebastien},
date-added = {2022-01-05 12:00:52 +0100},
date-modified = {2022-02-15 10:34:12 +0100},
doi = {10.1016/j.euromechsol.2022.104529},
hal = {https://hal.archives-ouvertes.fr/hal-03537619},
journal = {Accepted with minor revisions in European Journal of Mechanics - A/Solids},
pages = {104529},
title = {Robust optimization of nonlinear energy sinks used for mitigation of friction-induced limit cycle oscillations},
volume = {93},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1016/j.euromechsol.2022.104529}
}
@article{BERGEOT2021116109,
abstract = {In this paper one first shows that the slow flow of a mechanical system with one unstable mode coupled to a Nonlinear Energy Sink (NES) can be reduced, in the neighborhood of a fold point of its critical manifold, to a normal form of the dynamic saddle-node bifurcation. This allows us to then obtain a scaling law for the slow flow dynamics and to improve the accuracy of the theoretical prediction of the mitigation limit of the NES previously obtained as part of a zeroth-order approximation. For that purpose, the governing equations of the coupled system are first simplified using a reduced-order model for the primary structure by keeping only its unstable modal coordinates. The slow flow is then derived by means of the complexification-averaging method and, by the presence of a small perturbation parameter related to the mass ratio between the NES and the primary structure, it appears as a fast-slow system. The center manifold theorem is finally used to obtain the reduced form of the slow flow which is solved analytically leading to the scaling law. The latter reveals a nontrivial dependence with respect to the small perturbation parameter of the slow flow dynamics near the fold point, involving the fractional exponents 1/3 and 2/3. Finally, a new theoretical prediction of the mitigation limit is deduced from the scaling law. In the end, the proposed methodology is exemplified and validated numerically using an aeroelastic aircraft wing model coupled to one NES.},
author = {Bergeot, Baptiste},
date-added = {2021-04-02 11:20:27 +0200},
date-modified = {2021-04-26 14:24:59 +0200},
doi = {10.1016/j.jsv.2021.116109},
hal = {https://hal.archives-ouvertes.fr/hal-03190849v2},
issn = {0022-460X},
journal = {Journal of Sound and Vibration},
keywords = {Passive vibration control, Nonlinear energy sink, Center manifold reduction, Scaling law, Aeroelastic instability},
pages = {116109},
title = {Scaling law for the slow flow of an unstable mechanical system coupled to a nonlinear energy sink},
volume = {503},
year = {2021},
bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0022460X21001814},
bdsk-url-2 = {https://doi.org/10.1016/j.jsv.2021.116109}
}
@article{BERGEOT2021105623,
abstract = {This paper investigates a problem of passive mitigation of vibratory instabilities caused by two unstable modes by means of a single nonlinear energy sink (NES). For this purpose, a linear four-degree-of-freedom (DOF) primary structure having two unstable modes (reproducing the typical dynamic behavior of a friction system) and undergoing, as it is linear, unbounded motions when it is unstable, is coupled to a NES. In this work, the NES involves an essentially cubic restoring force and a linear damping force. We are interested in characterizing analytically the response regimes resulting from the coupling of the two unstable linear modes of the primary structure and the nonlinear mode of the NES. To this end, from a suitable rescaling of the governing equations of the coupled system in which the dynamics of the primary structure is reduced to its unstable modal coordinates, the complexification-averaging method is applied. The resulting averaged system appears to be a fast-slow system with four fast variables and two slow ones related to the two unstable modes of the primary structure. The critical manifold of the averaged dynamics is obtained through the geometric singular perturbation theory and appears as a two-dimensional parametric surface (with respect to two of the four fast variables) which evolves in the whole six-dimensional variable space. The asymptotic analysis reveals that the NES attachment can produce some bounded responses and suggests that the system may have simultaneous stable attractors. Numerical simulations complement the study, highlighting a possible competition between stable attractors and allowing us to investigate their basins of attraction. In each considered situation, a good agreement has been observed between theoretical results and numerical simulations, which validates the proposed asymptotic analysis.},
author = {Bergeot, Baptiste and Bellizzi, Sergio and Berger, S{\'e}bastien},
date-added = {2021-01-16 11:21:42 +0100},
date-modified = {2021-04-26 14:30:24 +0200},
doi = {10.1016/j.cnsns.2020.105623},
hal = {https://hal.archives-ouvertes.fr/hal-03025795},
issn = {1007-5704},
journal = {Communications in Nonlinear Science and Numerical Simulation},
keywords = {Nonlinear energy sink, Multi-instabilities, Relaxation oscillations, Multiple-scale analysis},
pages = {105623},
title = {Dynamic behavior analysis of a mechanical system with two unstable modes coupled to a single nonlinear energy sink},
volume = {95},
year = {2021},
bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/S1007570420304536},
bdsk-url-2 = {https://doi.org/10.1016/j.cnsns.2020.105623}
}
@article{SNOUN2020103917,
abstract = {In this paper, a friction system with uncertain parameters and coupled to two Nonlinear Energy Sinks (NESs) is studied. The dispersion of some physical parameters due to their uncertain nature may generate a dynamic instability which leads to a Limit Cycle Oscillations (LCO) causing a propensity of squeal. The concept of Targeted Energy Transfer (TET) by means of NESs to mitigate this squealing noise is proposed. In this kind of unstable dynamical system coupled to NES, the transition from harmless regimes (i.e. the LCO is mitigated) to harmful regimes (i.e. the LCO is not mitigated) as a function of the uncertain parameters implies a discontinuity in the steady-state amplitude profiles. In this context, a Multi-Element generalized Polynomial Chaos (ME-gPC) based method is proposed to locate this discontinuity (called mitigation limit) and therefore to predict the Propensity of the system to undergo an Harmless Steady-State Regime (PHSSR). The results obtained with this original method lead to a good compromise between computational cost and accuracy in comparison with a reference method.},
author = {Cherif Snoun and Baptiste Bergeot and S{\'e}bastien Berger},
date-added = {2019-12-03 21:02:31 +0100},
date-modified = {2021-04-26 14:33:50 +0200},
doi = {10.1016/j.euromechsol.2019.103917},
hal = {https://hal.archives-ouvertes.fr/hal-02396312},
issn = {0997-7538},
journal = {European Journal of Mechanics - A/Solids},
keywords = {Friction-induced vibration, Nonlinear Energy Sink, Uncertainty, Robust modeling, Multi-Element generalized polynomial chaos},
pages = {103917},
title = {Prediction of the dynamic behavior of an uncertain friction system coupled to nonlinear energy sinks using a multi-element generalized polynomial chaos approach},
volume = {80},
year = {2020},
bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/S0997753819307132},
bdsk-url-2 = {https://doi.org/10.1016/j.euromechsol.2019.103917}
}
@article{BERGEOT2019728,
abstract = {A general method to predict the steady-state regimes of a multi-degree-of-freedom unstable vibrating system (the primary system) coupled to several nonlinear energy sinks (NESs) is proposed. The method has three main steps. The first step consists in the diagonalization of the primary underline linear system using the so-called biorthogonal transformation. Within the assumption of a primary system with only one unstable mode the dynamics of the diagonalized system is reduced ignoring the stable modes and keeping only the unstable mode. The complexification method is applied in the second step with the aim of obtaining the slow-flow of the reduced system. Then, the third step is an asymptotic analysis of the slow-flow based geometric singular perturbation theory. The analysis shows that the critical manifold of the system can be reduced to a one dimensional parametric curve evolving in a multidimensional space. The shape and the stability properties of the critical manifold and the stability properties of the fixed points of the slow-flow provide an analytical tool to predict the nature of the possible steady-state regimes of the system. Finally, two examples are considered to evaluate the effectiveness and advancement of the proposed method. The method is first applied to the prediction of the mitigation limit of a breaking system subject to friction-induced vibrations coupled to two NESs, and next an airfoil model undergoing an aeroelastic instability coupled to a NESs setup (from one to four) is discussed. Theoretical results are compared, for validation purposes, to direct numerical integration of the system. The comparisons show good agreement.},
author = {Bergeot, Baptiste and Bellizzi, Sergio},
date-added = {2019-07-10 17:46:56 +0200},
date-modified = {2021-04-26 14:38:18 +0200},
doi = {doi.org/10.1016/j.ymssp.2019.05.045},
hal = {https://hal.archives-ouvertes.fr/hal-02177489},
issn = {0888-3270},
journal = {Mechanical Systems and Signal Processing},
keywords = {Multi-degree-of-freedom unstable system, Set of nonlinear energy sinks, Passive mitigation, Relaxation oscillations, Mitigation limit, Asymptotic analysis},
pages = {728--750},
title = {Steady-state regimes prediction of a multi-degree-of-freedom unstable dynamical system coupled to a set of nonlinear energy sinks},
volume = {131},
year = {2019},
bdsk-url-1 = {http://www.sciencedirect.com/science/article/pii/S0888327019303541},
bdsk-url-2 = {https://doi.org/10.1016/j.ymssp.2019.05.045}
}
@article{refId0,
author = {Kieu, Duc Thinh and Bergeot, Baptiste and Gobert, Marie-Laure and Berger, S\'ebastien},
date-added = {2019-03-28 17:41:33 +0100},
date-modified = {2021-04-26 14:37:26 +0200},
doi = {10.1051/meca/2019003},
hal = {https://hal.archives-ouvertes.fr/hal-02089643},
journal = {Mechanics \& Industry},
number = 1,
pages = {104},
title = {Stability analysis of a clutch system with uncertain parameters using sparse polynomial chaos expansions},
volume = 20,
year = 2019,
bdsk-url-1 = {https://doi.org/10.1051/meca/2019003}
}
@article{Almeida2015,
author = {Almeida, Andr{\'e} and Bergeot, Baptiste and Vergez, Christophe and Gazengel, Bruno},
date-added = {2019-02-06 11:59:34 +0100},
date-modified = {2021-04-26 14:28:11 +0200},
doi = {10.3813/AAA.918897},
hal = {https://hal.archives-ouvertes.fr/hal-01216282},
issn = {16101928},
journal = {Acta Acustica united with Acustica},
number = {5},
pages = {1026--1038},
publisher = {S. Hirzel Verlag},
title = {{Analytical Determination of the Attack Transient in a Clarinet With Time-Varying Blowing Pressure}},
volume = {101},
year = {2015},
bdsk-url-1 = {http://www.ingentaconnect.com/search/article?option1=tka%7B%5C&%7Dvalue1=Analytical+Determination+of+the+Attack+Transient+in+a+Clarinet+With+Time-Varying+Blowing+Pressure%7B%5C&%7DpageSize=10%7B%5C&%7Dindex=1},
bdsk-url-2 = {https://doi.org/10.3813/AAA.918897}
}
@article{Bergeot2014,
author = {Bergeot, Baptiste and Almeida, Andr{\'e} and Gazengel, Bruno and Vergez, Christophe and Ferrand, Didier},
date-added = {2019-02-06 11:59:34 +0100},
date-modified = {2021-04-26 14:36:41 +0200},
doi = {10.1121/1.4835755},
hal = {https://hal.archives-ouvertes.fr/hal-00769274v3},
issn = {00014966},
journal = {The Journal of the Acoustical Society of America},
number = {1},
pages = {479},
title = {{Response of an artificially blown clarinet to different blowing pressure profiles}},
volume = {135},
year = {2014},
bdsk-url-1 = {http://link.aip.org/link/JASMAN/v135/i1/p479/s1%7B%5C&%7DAgg=doi},
bdsk-url-2 = {https://doi.org/10.1121/1.4835755}
}
@article{Bergeot2013d,
author = {Bergeot, Baptiste and Almeida, Andr{\'e} and Vergez, Christophe and Gazengel, Bruno},
date-added = {2019-02-06 11:59:34 +0100},
date-modified = {2021-04-26 14:35:52 +0200},
doi = {10.1007/s11071-013-0991-8},
hal = {https://hal.archives-ouvertes.fr/hal-00809293v4},
issn = {0924-090X},
journal = {Nonlinear Dynamics},
keywords = {bifurcation delay,clarinet-like,dynamic bifurcation,instruments,iterated maps,musical acoustics,noise,transient processes},
number = {3},
pages = {591--605},
title = {{Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure: influence of noise}},
volume = {74},
year = {2013},
bdsk-url-1 = {http://link.springer.com/10.1007/s11071-013-0991-8},
bdsk-url-2 = {https://doi.org/10.1007/s11071-013-0991-8}
}
@article{Bergeot2013,
author = {Bergeot, Baptiste and Almeida, Andr{\'e} and Vergez, Christophe and Gazengel, Bruno},
date-added = {2019-02-06 11:59:34 +0100},
date-modified = {2021-04-26 14:35:13 +0200},
doi = {10.1007/s11071-013-0806-y},
hal = {https://hal.archives-ouvertes.fr/hal-00719228v4},
issn = {0924-090X},
journal = {Nonlinear Dynamics},
keywords = {bifurcation delay,clarinet-like,dynamic bifurcation,instruments,iterated maps,musical acoustics,noise,transient processes},
number = {1-2},
pages = {521--534},
title = {{Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure}},
volume = {73},
year = {2013},
bdsk-url-1 = {http://link.springer.com/10.1007/s11071-013-0991-8%20http://link.springer.com/10.1007/s11071-013-0806-y},
bdsk-url-2 = {https://doi.org/10.1007/s11071-013-0806-y}
}
@article{Bergeot2016a,
author = {Bergeot, Baptiste and Bellizzi, Sergio and Cochelin, Bruno},
date-added = {2019-02-06 11:58:42 +0100},
date-modified = {2021-04-26 14:39:48 +0200},
doi = {10.12989/aas.2016.3.3.271},
hal = {https://hal.archives-ouvertes.fr/hal-01343438},
journal = {Advances in Aircraft and Spacecraft Science},
keywords = {helicopter ground resonance,nonlinear energy sink,passive control,relaxation oscillations,strongly modulated response},
number = {3},
pages = {271--298},
title = {{Passive suppression of helicopter ground resonance instability by means of a strongly nonlinear absorber}},
volume = {3},
year = {2016},
bdsk-url-1 = {https://doi.org/10.12989/aas.2016.3.3.271}
}
@article{Bergeot2016,
author = {Bergeot, Baptiste and Bellizzi, Sergio and Cochelin, Bruno},
date-added = {2019-02-06 11:55:28 +0100},
date-modified = {2021-04-26 14:25:49 +0200},
doi = {10.1016/j.ijnonlinmec.2015.10.006},
hal = {https://hal.archives-ouvertes.fr/hal-01091763v5},
issn = {00207462},
journal = {International Journal of Non-Linear Mechanics},
pages = {72--89},
publisher = {Elsevier},
title = {{Analysis of steady-state response regimes of a helicopter ground resonance model including a non-linear energy sink attachment}},
volume = {78},
year = {2016},
bdsk-url-1 = {http://linkinghub.elsevier.com/retrieve/pii/S0020746215002000},
bdsk-url-2 = {https://doi.org/10.1016/j.ijnonlinmec.2015.10.006}
}
@article{Bergeot2017,
abstract = {This paper investigates the passive control of a rotor instability named helicopter Ground Resonance (GR). The passive device consists of a set of essential cubic nonlinear absorbers named Nonlinear Energy Sinks (NES) each of them positioned on a blade. A dynamic model reproducing helicopter GR instability is presented and transformed to a time-in-variant nonlinear system using a multi-blade coordinate transformation based on Fourier transform mapping the dynamic state variables into a non-rotating reference frame. Combining complexification, slow/fast partition of the dynamics and averaging procedure, a reduced model is obtained which allowed us to use the so-called geometric singular perturbation analysis to characterize the steady state response regimes. As in the case of a NES attached to the fuselage, it is shown that under suitable conditions, GR instability can be completely suppressed, partially suppressed through periodic response or strongly modulated response. Relevant analytical results are compared, for validation purposes, to direct integration of the reference and reduced models.},
author = {Bergeot, Baptiste and Bellizzi, Sergio and Cochelin, Bruno},
date-added = {2019-02-06 11:55:07 +0100},
date-modified = {2021-04-26 14:33:13 +0200},
doi = {10.1016/j.jsv.2016.12.039},
hal = {https://hal.archives-ouvertes.fr/hal-01432078},
journal = {Journal of Sound and Vibration},
keywords = {Complex multi-blade coordinate transformation,Helicopter ground resonance,Nonlinear Energy Sink,Passive control,Relaxation oscillations,Strongly modulated response},
mendeley-groups = {Absorbeurs Nonlin{\'{e}}aires},
pages = {41--55},
title = {{Passive suppression of helicopter ground resonance using nonlinear energy sinks attached on the helicopter blades}},
volume = {392},
year = {2017},
bdsk-url-1 = {https://doi.org/10.1016/j.jsv.2016.12.039}
}
@article{Bergeot2017a,
author = {Bergeot, Baptiste and Berger, S{\'e}bastien and Bellizzi, Sergio},
date-added = {2019-02-06 11:16:01 +0100},
date-modified = {2021-04-26 14:31:20 +0200},
doi = {10.1177/1077546317707101},
file = {:Users/baptistebergeot/Documents/Mendeley Desktop/10.1177{\_}1077546317707101.pdf:pdf},
hal = {https://hal.archives-ouvertes.fr/hal-01299991v3},
issn = {1077-5463},
journal = {Journal of Vibration and Control},
keywords = {friction-induced vibration,non linear energy sink,passive control,relaxation oscillations,strongly},
number = {15},
pages = {3487--3511},
title = {{Mode coupling instability mitigation in friction systems by means of nonlinear energy sinks : numerical highlighting and local stability analysis}},
volume = {24},
year = {2017},
bdsk-url-1 = {https://doi.org/10.1177/1077546317707101}
}
@article{Bergeot2018,
author = {Bergeot, Baptiste and Bellizzi, Sergio},
date-added = {2019-02-06 11:06:01 +0100},
date-modified = {2021-04-26 14:29:20 +0200},
doi = {10.1007/s11071-018-4438-0},
hal = {https://hal.archives-ouvertes.fr/hal-01829447},
issn = {0924-090X},
journal = {Nonlinear Dynamics},
keywords = {Nonlinear energy sink network,Passive mitigation,R,mitigation,mitigation limit,nonlinear energy sink network,passive,relaxation oscillations},
number = {2},
pages = {1501--1522},
publisher = {Springer Netherlands},
title = {{Asymptotic analysis of passive mitigation of dynamic instability using a nonlinear energy sink network}},
volume = {94},
year = {2018},
bdsk-url-1 = {http://link.springer.com/10.1007/s11071-018-4438-0},
bdsk-url-2 = {https://doi.org/10.1007/s11071-018-4438-0}
}
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