Comptes Rendus
no. 9
A harmonic-based method for computing the stability of periodic solutions of dynamical systems
[Une méthode fréquentielle pour le calcul de stabilité des solutions périodiques des systèmes dynamiques]
Arnaud Lazarus1  ; Olivier Thomas1
1 Structural Mechanics and Coupled Systems Laboratory, Cnam, 2, rue Conté, 75003 Paris, France
Comptes Rendus. Mécanique, Volume 338 (2010) no. 9, pp. 510-517.

Dans cette Note, nous présentons une méthode numérique fréquentielle pour déterminer la stabilité des solutions périodiques d'un système dynamique. La méthode, basée sur la théorie de Floquet et le développement en série de Fourier (méthode de Hill), consiste à extraire les valeurs propres physiques de l'ensemble des valeurs propres numériques du système perturbé étendu dans le domaine fréquentiel. En combinant alors la méthode de l'équilibrage harmonique et la méthode asymptotique numérique avec la précédente méthode de Hill, on obtient un outils de continuation purement fréquentiel où le calcul de la stabilité des solutions suivies est quasiment immédiat. Afin de valider la méthode, nous appliquons la méthode de continuation à un oscillateur de Duffing forcé.

In this Note, we present a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on the Floquet theory and the Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the harmonic-balance method and asymptotic numerical method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. To validate the general methodology, we investigate the dynamical behavior of the forced Duffing oscillator with the developed continuation technique.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2010.07.020
Keywords: Dynamical systems, Stability, Hill's method, Continuation procedure, Harmonic-balance method
Mots-clés : Systèmes dynamiques, Stabilité, Méthode de Hill, Méthode de continuation, Méthode de l'équilibrage harmonique

Arnaud Lazarus 1 ; Olivier Thomas 1

1 Structural Mechanics and Coupled Systems Laboratory, Cnam, 2, rue Conté, 75003 Paris, France 
@article{CRMECA_2010__338_9_510_0,
     author = {Arnaud Lazarus and Olivier Thomas},
     title = {A harmonic-based method for computing the stability of periodic solutions of dynamical systems},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {510--517},
     publisher = {Elsevier},
     volume = {338},
     number = {9},
     year = {2010},
     doi = {10.1016/j.crme.2010.07.020},
     language = {en},
}
TY  - JOUR
AU  - Arnaud Lazarus
AU  - Olivier Thomas
TI  - A harmonic-based method for computing the stability of periodic solutions of dynamical systems
JO  - Comptes Rendus. Mécanique
PY  - 2010
SP  - 510
EP  - 517
VL  - 338
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crme.2010.07.020
LA  - en
ID  - CRMECA_2010__338_9_510_0
ER  - 
%0 Journal Article
%A Arnaud Lazarus
%A Olivier Thomas
%T A harmonic-based method for computing the stability of periodic solutions of dynamical systems
%J Comptes Rendus. Mécanique
%D 2010
%P 510-517
%V 338
%N 9
%I Elsevier
%R 10.1016/j.crme.2010.07.020
%G en
%F CRMECA_2010__338_9_510_0
Arnaud Lazarus; Olivier Thomas. A harmonic-based method for computing the stability of periodic solutions of dynamical systems. Comptes Rendus. Mécanique, Volume 338 (2010) no. 9, pp. 510-517. doi : 10.1016/j.crme.2010.07.020. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2010.07.020/

[1] A.H. Nayfeh; B. Balachandran Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, John Wiley & Sons, Inc., 1995

[2] M. Peeters; R. Viguié; G. Sérandour; G. Kerschen; J.-C. Golinval Nonlinear normal modes. Part II: Toward a practical computation using numerical continuation techniques, Mechanical Systems and Signal Processing, Volume 23 (2009), pp. 195-216

[3] G.W. Hill On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Mathematica, Volume 8 (1886) no. 1, pp. 1-36

[4] J. Killingbeck On the Hill determinant method, Journal of Physics A: Mathematical and General, Volume 19 (1986), pp. 2903-2906

[5] B. Deconinck; J.N. Kutz Computing spectra of linear operators using the Floquet–Fourier–Hill method, Journal of Computational Physics, Volume 219 (2006), pp. 296-321

[6] F. Bonani; M. Gilli Analysis of stability and bifurcations of limit cycles in Chua's circuit through the harmonic-balance approach, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume 46 (1999), pp. 881-890

[7] C. Villa; J.-J. Sinou; F. Thouverez Stability and vibration analysis of a complex flexible rotor bearing system, Communications in Nonlinear Science and Numerical Simulation, Volume 13 (2008), pp. 804-821

[8] A. Lazarus, D. Combescure, B. Prabel, A 3D finite element model for the vibration analysis of asymmetric rotating machines, Journal of Sound and Vibration, . | DOI

[9] B. Cochelin; C. Vergez A high order purely-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, Volume 324 (2009), pp. 243-262

[10] G. Floquet Sur la théorie des équations différentielles, Annales Scientifiques de l'École Normale Supérieure, Volume 8 (1879), pp. 3-132

[11] P. Hartman Ordinary Differential Equations, Society for Industrial and Applied Mathematics, 2002

[12] J. Argyris; G. Faust; M. Haase An Exploration of Chaos, Texts on Computational Mechanics, vol. VII, North-Holland, 1994

[13] A.H. Nayfeh; D.T. Mook Nonlinear Oscillations, Wiley, 1979

[14] H. Poincaré Sur les déterminants d'ordre infini, Bulletin de la Société Mathématique de France, Volume 14 (1886), pp. 77-90

[15] J. Zhou; T. Hagiwara; M. Araki Spectral analysis and eigenvalues computation of the harmonic state operators in continuous-time periodic systems, Systems & Control Letters, Volume 53 (2004), pp. 141-155

[16] C.W. Curtis; B. Deconinck On the convergence of Hill's method, Mathematics of Computation, Volume 79 (2010), pp. 169-187

[17] W. Szemplinska-Stupnicka The Behavior of Nonlinear Vibrating Systems, vol. 1: Fundamental Concepts and Methods: Applications to Single-Degree-of-Freedom Systems, Kluwer Academic Publishers, 1990

[18] E. de Langre Frequency lock-in is caused by coupled-mode flutter, Journal of Fluids and Structures, Volume 22 (2006), pp. 783-791

  • Yi Yang; Hongjun Xiang A New Analysis Framework for Solving Multiple Frequencies and Solutions of Nonlinear Piezoelectric Energy Harvesters, Communications in Nonlinear Science and Numerical Simulation, Volume 140 (2025), p. 108433 | DOI:10.1016/j.cnsns.2024.108433
  • Olivier Thomas Understanding, Computing and Identifying the Nonlinear Dynamics of Elastic and Piezoelectric Structures Thanks to Nonlinear Modes, Model Order Reduction for Design, Analysis and Control of Nonlinear Vibratory Systems, Volume 614 (2025), p. 151 | DOI:10.1007/978-3-031-67499-0_4
  • Michael McGurk; Jie Yuan Prediction and validation of aeroelastic limit cycle oscillations using harmonic balance methods and Koopman operator, Nonlinear Dynamics (2025) | DOI:10.1007/s11071-025-11065-8
  • Yi Yang; Hongjun Xiang A New Magnetic Model and a New Analysis Framework for Solving Magnetically Nonlinear Piezoelectric Energy Harvesting Problems, The 5th International Conference on Vibration and Energy Harvesting Applications (VEH 2024) (2025), p. 59 | DOI:10.1007/978-981-96-1191-1_6
  • Martin Pégeot; Tom Colinot; Jean-Baptiste Doc; Vincent Fréour; Christophe Vergez Playability of self-sustained musical instrument models: statistical approaches, Acta Acustica, Volume 8 (2024), p. 74 | DOI:10.1051/aacus/2024075
  • Fabia Bayer; Remco I. Leine Optimal Projection in a Koopman-Based Sorting-Free Hill Method, Advances in Nonlinear Dynamics, Volume I (2024), p. 409 | DOI:10.1007/978-3-031-50631-4_35
  • S. Westerbeek; S. Hulshoff; H. Schuttelaars; M. Kotsonis DeHNSSo: The Delft Harmonic Navier-Stokes Solver for nonlinear stability problems with complex geometric features, Computer Physics Communications, Volume 302 (2024), p. 109250 | DOI:10.1016/j.cpc.2024.109250
  • Adrien Mélot; Enora Denimal Goy; Ludovic Renson Control of isolated response curves through optimization of codimension-1 singularities, Computers Structures, Volume 299 (2024), p. 107394 | DOI:10.1016/j.compstruc.2024.107394
  • Hanwen Gu; Yueming Ji; Hao Qi; Xintong Liu; Zaibin Jiao; Yinan Xiang; Jun Liu Linear Time-Periodic theory-based novel stability analysis method for voltage-source converter under unbalanced grid conditions, International Journal of Electrical Power Energy Systems, Volume 161 (2024), p. 110197 | DOI:10.1016/j.ijepes.2024.110197
  • B.E. Saunders; R.J. Kuether; R. Vasconcellos; A. Abdelkefi Nonlinear analysis and vibro-impact characteristics of a shaft-bearing assembly, International Journal of Non-Linear Mechanics, Volume 159 (2024), p. 104618 | DOI:10.1016/j.ijnonlinmec.2023.104618
  • Fabia Bayer; Remco I. Leine; Olivier Thomas; Aurélien Grolet Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation, International Journal of Non-Linear Mechanics, Volume 167 (2024), p. 104894 | DOI:10.1016/j.ijnonlinmec.2024.104894
  • Shivangi Mittal; Yogesh M. Joshi; Sachin Shanbhag Harmonic balance for differential constitutive models under oscillatory shear, Physics of Fluids, Volume 36 (2024) no. 5 | DOI:10.1063/5.0207942
  • Michel Potier-Ferry Asymptotic numerical method for hyperelasticity and elastoplasticity: a review, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 480 (2024) no. 2285 | DOI:10.1098/rspa.2023.0714
  • Jie Yuan; Chiara Gastaldi; Enora Denimal Goy; Benjamin Chouvion Friction damping for turbomachinery: A comprehensive review of modelling, design strategies, and testing capabilities, Progress in Aerospace Sciences, Volume 147 (2024), p. 101018 | DOI:10.1016/j.paerosci.2024.101018
  • Yu Zhou; Jianliang Huang; Li Wang Event-driven Gauss quadrature and stability analysis for fast alternating frequency–time harmonic balance of non-smooth systems, Communications in Nonlinear Science and Numerical Simulation, Volume 120 (2023), p. 107189 | DOI:10.1016/j.cnsns.2023.107189
  • G. Ghannad Tehrani; C. Gastaldi; T. M. Berruti Numerical and Experimental Stability Investigation of a Parametrically Excited Cantilever Beam at Combination Parametric Resonance, Experimental Mechanics, Volume 63 (2023) no. 1, p. 177 | DOI:10.1007/s11340-022-00903-0
  • Adrien Mélot; Emmanuel Rigaud; Joël Perret-Liaudet Robust design of vibro-impacting geared systems with uncertain tooth profile modifications via bifurcation tracking, International Journal of Non-Linear Mechanics, Volume 149 (2023), p. 104336 | DOI:10.1016/j.ijnonlinmec.2022.104336
  • B.E. Saunders; R.J. Kuether; R. Vasconcellos; A. Abdelkefi Evaluating route to impact convergence of the harmonic balance method for piecewise-smooth systems, International Journal of Non-Linear Mechanics, Volume 152 (2023), p. 104398 | DOI:10.1016/j.ijnonlinmec.2023.104398
  • Kyriakos D. Kantarakias; George Papadakis Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach, Journal of Computational Physics, Volume 474 (2023), p. 111757 | DOI:10.1016/j.jcp.2022.111757
  • Tiago S. Martins; Francesco Trainotti; Andreas Zwölfer; Frederico Afonso A Python Implementation of a Robust Multi-Harmonic Balance With Numerical Continuation and Automatic Differentiation for Structural Dynamics, Journal of Computational and Nonlinear Dynamics, Volume 18 (2023) no. 7 | DOI:10.1115/1.4062424
  • Cui Chao; Baiyang Shi; Wei Dai; Jian Yang Performance Analysis of Frictional Inerter-Based Vibration Isolator, Journal of Vibration Engineering Technologies, Volume 11 (2023) no. 6, p. 2793 | DOI:10.1007/s42417-023-01051-y
  • Manuel Gadella; Luis Pedro Lara A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents, Mathematical Methods in the Applied Sciences, Volume 46 (2023) no. 8, p. 8956 | DOI:10.1002/mma.9029
  • Lukas Woiwode; Malte Krack Are Chebyshev-based stability analysis and Urabe’s error bound useful features for Harmonic Balance?, Mechanical Systems and Signal Processing, Volume 194 (2023), p. 110265 | DOI:10.1016/j.ymssp.2023.110265
  • Yu. D. Selyutsky Dynamics of a Wind Turbine with Two Moving Masses Using the Galloping Effect, Mechanics of Solids, Volume 58 (2023) no. 2, p. 426 | DOI:10.3103/s0025654422600507
  • V. Eclerová; L. Přibylová; A. E. Botha Embedding nonlinear systems with two or more harmonic phase terms near the Hopf–Hopf bifurcation, Nonlinear Dynamics, Volume 111 (2023) no. 2, p. 1537 | DOI:10.1007/s11071-022-07906-5
  • B. E. Saunders; R. Vasconcellos; R. J. Kuether; A. Abdelkefi Nonlinear dynamics, bifurcations, and multi-stability in a vibro-impact system with geometric and multi-segmented freeplay nonlinearities, Nonlinear Dynamics, Volume 111 (2023) no. 20, p. 18655 | DOI:10.1007/s11071-023-08823-x
  • Fabia Bayer; Remco I. Leine Sorting-free Hill-based stability analysis of periodic solutions through Koopman analysis, Nonlinear Dynamics, Volume 111 (2023) no. 9, p. 8439 | DOI:10.1007/s11071-023-08247-7
  • Brian Evan Saunders; Rui M. G. Vasconcellos; Robert J. Kuether; Abdessattar Abdelkefi Stability and Convergence Analysis of the Harmonic Balance Method for a Duffing Oscillator with Free Play Nonlinearity, Nonlinear Structures Systems, Volume 1 (2023), p. 249 | DOI:10.1007/978-3-031-04086-3_36
  • Yu. D. Selyutsky Dynamics of a Wind Turbine with Two Moving Masses Using the Galloping Effect, Известия Российской академии наук. Механика твердого тела (2023) no. 2, p. 55 | DOI:10.31857/s0572329922100117
  • Vincent Fréour; Louis Guillot; Hideyuki Masuda; Christophe Vergez; Bruno Cochelin Parameter identification of a physical model of brass instruments by constrained continuation, Acta Acustica, Volume 6 (2022), p. 9 | DOI:10.1051/aacus/2022004
  • Suguang Dou An Improved Tensorial Implementation of the Incremental Harmonic Balance Method for Frequency-Domain Stability Analysis, Advances in Nonlinear Dynamics (2022), p. 443 | DOI:10.1007/978-3-030-81162-4_39
  • Martin Volvert; Gaëtan Kerschen Resonant phase lags of a Duffing oscillator, International Journal of Non-Linear Mechanics, Volume 146 (2022), p. 104150 | DOI:10.1016/j.ijnonlinmec.2022.104150
  • Lucas Franceschini; Denis Sipp; Olivier Marquet; Johann Moulin; Julien Dandois Identification and reconstruction of high-frequency fluctuations evolving on a low-frequency periodic limit cycle: application to turbulent cylinder flow, Journal of Fluid Mechanics, Volume 942 (2022) | DOI:10.1017/jfm.2022.376
  • Adrien Mélot; Youness Benaïcha; Emmanuel Rigaud; Joël Perret-Liaudet; Fabrice Thouverez Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train, Journal of Sound and Vibration, Volume 516 (2022), p. 116495 | DOI:10.1016/j.jsv.2021.116495
  • Y. Colaïtis; A. Batailly Stability analysis of periodic solutions computed for blade-tip/casing contact problems, Journal of Sound and Vibration, Volume 538 (2022), p. 117219 | DOI:10.1016/j.jsv.2022.117219
  • Junqing Wu; Ling Hong; Jun Jiang A robust and efficient stability analysis of periodic solutions based on harmonic balance method and Floquet-Hill formulation, Mechanical Systems and Signal Processing, Volume 173 (2022), p. 109057 | DOI:10.1016/j.ymssp.2022.109057
  • Di Zhou; D. Zeb Rocklin; Michael Leamy; Yugui Yao Topological invariant and anomalous edge modes of strongly nonlinear systems, Nature Communications, Volume 13 (2022) no. 1 | DOI:10.1038/s41467-022-31084-y
  • Adrien Mélot; Emmanuel Rigaud; Joël Perret-Liaudet Bifurcation tracking of geared systems with parameter-dependent internal excitation, Nonlinear Dynamics, Volume 107 (2022) no. 1, p. 413 | DOI:10.1007/s11071-021-07018-6
  • Thomas Breunung Asymptotic stability of quasi-periodic orbits, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 478 (2022) no. 2259 | DOI:10.1098/rspa.2021.0787
  • Dongxiao Hong; Thomas L. Hill; Simon A. Neild Existence and location of internal resonance of two-mode nonlinear conservative oscillators, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Volume 478 (2022) no. 2260 | DOI:10.1098/rspa.2021.0659
  • Jinhong Noh; Minh Sang Nguyen; Pilkee Kim; Yong-Jin Yoon Harmonic balance analysis of magnetically coupled two-degree-of-freedom bistable energy harvesters, Scientific Reports, Volume 12 (2022) no. 1 | DOI:10.1038/s41598-022-10061-x
  • Tom Colinot; Christophe Vergez; Philippe Guillemain; Jean-Baptiste Doc Multistability of saxophone oscillation regimes and its influence on sound production, Acta Acustica, Volume 5 (2021), p. 33 | DOI:10.1051/aacus/2021026
  • Jianwang Shao; Qimeng Luo; Guoming Deng; Tao Zeng; Jinmeng Yang; Xian Wu; Chang Jin Experimental study on influence of wall acoustic materials of 3D cavity for targeted energy transfer of a nonlinear membrane absorber, Applied Acoustics, Volume 184 (2021), p. 108342 | DOI:10.1016/j.apacoust.2021.108342
  • Jie Xu; Xuegang Yuan; Jia Jiao; Hongwu Zhang Computation of axisymmetric nonlinear low-frequency resonances of hyperelastic thin-walled cylindrical shells, Applied Mathematical Modelling, Volume 94 (2021), p. 332 | DOI:10.1016/j.apm.2021.01.027
  • Arthur Givois; Jean-François Deü; Olivier Thomas Dynamics of piezoelectric structures with geometric nonlinearities: A non-intrusive reduced order modelling strategy, Computers Structures, Volume 253 (2021), p. 106575 | DOI:10.1016/j.compstruc.2021.106575
  • Yichang Shen; Natacha Béreux; Attilio Frangi; Cyril Touzé Reduced order models for geometrically nonlinear structures: Assessment of implicit condensation in comparison with invariant manifold approach, European Journal of Mechanics - A/Solids, Volume 86 (2021), p. 104165 | DOI:10.1016/j.euromechsol.2020.104165
  • Ghasem Ghannad Tehrani; Chiara Gastaldi; Teresa M. Berruti A forced response-based method to track instability of rotating systems, European Journal of Mechanics - A/Solids, Volume 90 (2021), p. 104319 | DOI:10.1016/j.euromechsol.2021.104319
  • Huoming Yang; Hendrik Just; Malte Eggers; Sibylle Dieckerhoff Linear Time-Periodic Theory-Based Modeling and Stability Analysis of Voltage-Source Converters, IEEE Journal of Emerging and Selected Topics in Power Electronics, Volume 9 (2021) no. 3, p. 3517 | DOI:10.1109/jestpe.2020.3003379
  • Chen Zhang; Sjur Foyen; Jon Are Suul; Marta Molinas Modeling and Analysis of SOGI-PLL/FLL-Based Synchronization Units: Stability Impacts of Different Frequency-Feedback Paths, IEEE Transactions on Energy Conversion, Volume 36 (2021) no. 3, p. 2047 | DOI:10.1109/tec.2020.3041797
  • J. Sierra; P. Jolivet; F. Giannetti; V. Citro Adjoint-based sensitivity analysis of periodic orbits by the Fourier–Galerkin method, Journal of Computational Physics, Volume 440 (2021), p. 110403 | DOI:10.1016/j.jcp.2021.110403
  • Ghasem Ghannad Tehrani; Chiara Gastaldi; Teresa M. Berruti Trained Harmonic Balance Method for Parametrically Excited Jeffcott Rotor Analysis, Journal of Computational and Nonlinear Dynamics, Volume 16 (2021) no. 1 | DOI:10.1115/1.4048578
  • Georgios Rigas; Denis Sipp; Tim Colonius Nonlinear input/output analysis: application to boundary layer transition, Journal of Fluid Mechanics, Volume 911 (2021) | DOI:10.1017/jfm.2020.982
  • Allen Anilkumar; V. Kartik Eigen and temporal characteristics of multi-frequency parametrically-excited rotor-oscillator systems, Journal of Sound and Vibration, Volume 493 (2021), p. 115824 | DOI:10.1016/j.jsv.2020.115824
  • Alessandra Vizzaccaro; Loïc Salles; Cyril Touzé Comparison of nonlinear mappings for reduced-order modelling of vibrating structures: normal form theory and quadratic manifold method with modal derivatives, Nonlinear Dynamics, Volume 103 (2021) no. 4, p. 3335 | DOI:10.1007/s11071-020-05813-1
  • Mattia Cenedese; George Haller Stability of forced–damped response in mechanical systems from a Melnikov analysis, Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 30 (2020) no. 8 | DOI:10.1063/5.0012480
  • Martin Jahn; Merten Stender; Sebastian Tatzko; Norbert Hoffmann; Aurélien Grolet; Jörg Wallaschek The extended periodic motion concept for fast limit cycle detection of self-excited systems, Computers Structures, Volume 227 (2020), p. 106139 | DOI:10.1016/j.compstruc.2019.106139
  • Ahmed Ghorbel; Becem Zghal; Moez Abdennadher; Lassâad Walha; Mohamed Haddar Analysis of Strongly Nonlinear Systems by Using HBM-AFT Method and Its Comparison with the Five-Order Runge–Kutta Method: Application to Duffing Oscillator and Disc Brake Model, International Journal of Applied and Computational Mathematics, Volume 6 (2020) no. 2 | DOI:10.1007/s40819-020-0803-z
  • Louis Guillot; Arnaud Lazarus; Olivier Thomas; Christophe Vergez; Bruno Cochelin A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems, Journal of Computational Physics, Volume 416 (2020), p. 109477 | DOI:10.1016/j.jcp.2020.109477
  • Attilio Frangi; Andrea Opreni; Nicolo Boni; Patrick Fedeli; Roberto Carminati; Massimiliano Merli; Gianluca Mendicino Nonlinear Response of PZT-Actuated Resonant Micromirrors, Journal of Microelectromechanical Systems, Volume 29 (2020) no. 6, p. 1421 | DOI:10.1109/jmems.2020.3022557
  • Arthur Givois; Jin-Jack Tan; Cyril Touzé; Olivier Thomas Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identification, Meccanica, Volume 55 (2020) no. 3, p. 481 | DOI:10.1007/s11012-020-01132-2
  • Torsten Heinze; Lars Panning-von Scheidt; Jörg Wallaschek Global detection of detached periodic solution branches of friction-damped mechanical systems, Nonlinear Dynamics, Volume 99 (2020) no. 3, p. 1841 | DOI:10.1007/s11071-019-05425-4
  • Tom Colinot; Philippe Guillemain; Christophe Vergez; Jean-Baptiste Doc; Patrick Sanchez Multiple two-step oscillation regimes produced by the alto saxophone, The Journal of the Acoustical Society of America, Volume 147 (2020) no. 4, p. 2406 | DOI:10.1121/10.0001109
  • Vincent Fréour; Louis Guillot; Hideyuki Masuda; Satoshi Usa; Eiji Tominaga; Yutaka Tohgi; Christophe Vergez; Bruno Cochelin Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons, The Journal of the Acoustical Society of America, Volume 148 (2020) no. 2, p. 748 | DOI:10.1121/10.0001603
  • Malte Krack; Johann Gross Application to Mechanical Systems, Harmonic Balance for Nonlinear Vibration Problems (2019), p. 47 | DOI:10.1007/978-3-030-14023-6_3
  • Zhihua Wu; Yimin Zhang; Guo Yao; Zhou Yang Nonlinear primary and super-harmonic resonances of functionally graded carbon nanotube reinforced composite beams, International Journal of Mechanical Sciences, Volume 153-154 (2019), p. 321 | DOI:10.1016/j.ijmecsci.2019.02.015
  • P. Vincent; A. Descombin; S. Dagher; T. Seoudi; A. Lazarus; O. Thomas; A. Ayari; S. T. Purcell; S. Perisanu Nonlinear polarization coupling in freestanding nanowire/nanotube resonators, Journal of Applied Physics, Volume 125 (2019) no. 4 | DOI:10.1063/1.5053955
  • E. P. Petrov A Method for Parametric Analysis of Stability Boundaries for Nonlinear Periodic Vibrations of Structures With Contact Interfaces, Journal of Engineering for Gas Turbines and Power, Volume 141 (2019) no. 3 | DOI:10.1115/1.4040850
  • Shuai Wang; Chuanxing Bi; Jiazhu Li; Changjun Zheng Parametric instability of anisotropic rotor-bearing systems with a transverse crack, Journal of Sound and Vibration, Volume 443 (2019), p. 253 | DOI:10.1016/j.jsv.2018.11.052
  • A. Renault; O. Thomas; H. Mahé Numerical antiresonance continuation of structural systems, Mechanical Systems and Signal Processing, Volume 116 (2019), p. 963 | DOI:10.1016/j.ymssp.2018.07.005
  • Pierre Vigué; Christophe Vergez; Bruno Lombard; Bruno Cochelin Continuation of periodic solutions for systems with fractional derivatives, Nonlinear Dynamics, Volume 95 (2019) no. 1, p. 479 | DOI:10.1007/s11071-018-4577-3
  • Louis Guillot; Bruno Cochelin; Christophe Vergez A Taylor series-based continuation method for solutions of dynamical systems, Nonlinear Dynamics, Volume 98 (2019) no. 4, p. 2827 | DOI:10.1007/s11071-019-04989-5
  • Haitao Liao; Wenwang Wu The Reduced Space Shooting Method for Calculating the Peak Periodic Solutions of Nonlinear Systems, Journal of Computational and Nonlinear Dynamics, Volume 13 (2018) no. 6 | DOI:10.1115/1.4039682
  • E. P. Petrov Frequency-Domain Sensitivity Analysis of Stability of Nonlinear Vibrations for High-Fidelity Models of Jointed Structures, Journal of Engineering for Gas Turbines and Power, Volume 140 (2018) no. 1 | DOI:10.1115/1.4037708
  • Barend Bentvelsen; Arnaud Lazarus Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column, Nonlinear Dynamics, Volume 91 (2018) no. 2, p. 1349 | DOI:10.1007/s11071-017-3949-4
  • Malte Krack; Loic Salles; Fabrice Thouverez Vibration Prediction of Bladed Disks Coupled by Friction Joints, Archives of Computational Methods in Engineering, Volume 24 (2017) no. 3, p. 589 | DOI:10.1007/s11831-016-9183-2
  • E. P. Petrov Stability Analysis of Multiharmonic Nonlinear Vibrations for Large Models of Gas Turbine Engine Structures With Friction and Gaps, Journal of Engineering for Gas Turbines and Power, Volume 139 (2017) no. 2 | DOI:10.1115/1.4034353
  • A.M. Panunzio; Loic Salles; C.W. Schwingshackl Uncertainty propagation for nonlinear vibrations: A non-intrusive approach, Journal of Sound and Vibration, Volume 389 (2017), p. 309 | DOI:10.1016/j.jsv.2016.09.020
  • Pierre Vigué; Christophe Vergez; Sami Karkar; Bruno Cochelin Regularized friction and continuation: Comparison with Coulomb's law, Journal of Sound and Vibration, Volume 389 (2017), p. 350 | DOI:10.1016/j.jsv.2016.11.002
  • Louis Guillot; Pierre Vigué; Christophe Vergez; Bruno Cochelin Continuation of quasi-periodic solutions with two-frequency Harmonic Balance Method, Journal of Sound and Vibration, Volume 394 (2017), p. 434 | DOI:10.1016/j.jsv.2016.12.013
  • A.B. Sabater; J.F. Rhoads Parametric system identification of resonant micro/nanosystems operating in a nonlinear response regime, Mechanical Systems and Signal Processing, Volume 84 (2017), p. 241 | DOI:10.1016/j.ymssp.2016.06.003
  • Suguang Dou; Jakob Søndergaard Jensen Optimization of hardening/softening behavior of plane frame structures using nonlinear normal modes, Computers Structures, Volume 164 (2016), p. 63 | DOI:10.1016/j.compstruc.2015.11.001
  • L. Renson; G. Kerschen; B. Cochelin Numerical computation of nonlinear normal modes in mechanical engineering, Journal of Sound and Vibration, Volume 364 (2016), p. 177 | DOI:10.1016/j.jsv.2015.09.033
  • Loïc Salles; Bernard Staples; Norbert Hoffmann; Christoph Schwingshackl Continuation techniques for analysis of whole aeroengine dynamics with imperfect bifurcations and isolated solutions, Nonlinear Dynamics, Volume 86 (2016) no. 3, p. 1897 | DOI:10.1007/s11071-016-3003-y
  • Simon Peter; Robin Riethmüller; Remco I. Leine Tracking of Backbone Curves of Nonlinear Systems Using Phase-Locked-Loops, Nonlinear Dynamics, Volume 1 (2016), p. 107 | DOI:10.1007/978-3-319-29739-2_11
  • T. Detroux; L. Renson; L. Masset; G. Kerschen The Harmonic Balance Method for Bifurcation Analysis of Nonlinear Mechanical Systems, Nonlinear Dynamics, Volume 1 (2016), p. 65 | DOI:10.1007/978-3-319-15221-9_5
  • Morten Hartvig Hansen Modal dynamics of structures with bladed isotropic rotors and its complexity for two-bladed rotors, Wind Energy Science, Volume 1 (2016) no. 2, p. 271 | DOI:10.5194/wes-1-271-2016
  • T. Detroux; L. Renson; L. Masset; G. Kerschen The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems, Computer Methods in Applied Mechanics and Engineering, Volume 296 (2015), p. 18 | DOI:10.1016/j.cma.2015.07.017
  • Malte Krack Nonlinear modal analysis of nonconservative systems: Extension of the periodic motion concept, Computers Structures, Volume 154 (2015), p. 59 | DOI:10.1016/j.compstruc.2015.03.008
  • Shuai Wang; Yu Wang; Yanyang Zi; Zhengjia He A 3D finite element-based model order reduction method for parametric resonance and whirling analysis of anisotropic rotor-bearing systems, Journal of Sound and Vibration, Volume 359 (2015), p. 116 | DOI:10.1016/j.jsv.2015.08.027
  • Haitao Liao Optimization analysis of Duffing oscillator with fractional derivatives, Nonlinear Dynamics, Volume 79 (2015) no. 2, p. 1311 | DOI:10.1007/s11071-014-1744-z
  • Jan Dupal; Martin Zajíc̆ek Analytical periodic solution and stability assessment of 1 DOF parametric systems with time varying stiffness, Applied Mathematics and Computation, Volume 243 (2014), p. 138 | DOI:10.1016/j.amc.2014.05.089
  • Jianwang Shao; Bruno Cochelin Theoretical and numerical study of targeted energy transfer inside an acoustic cavity by a non-linear membrane absorber, International Journal of Non-Linear Mechanics, Volume 64 (2014), p. 85 | DOI:10.1016/j.ijnonlinmec.2014.04.008
  • Orhan Ozcelik; Peter J. Attar Effect of non-linear damping on the structural dynamics of flapping beams, International Journal of Non-Linear Mechanics, Volume 65 (2014), p. 148 | DOI:10.1016/j.ijnonlinmec.2014.05.005
  • Malte Krack; Lars Panning-von Scheidt; Jörg Wallaschek On the computation of the slow dynamics of nonlinear modes of mechanical systems, Mechanical Systems and Signal Processing, Volume 42 (2014) no. 1-2, p. 71 | DOI:10.1016/j.ymssp.2013.08.031
  • Mélodie Monteil; Cyril Touzé; Olivier Thomas; Simon Benacchio Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances, Nonlinear Dynamics, Volume 75 (2014) no. 1-2, p. 175 | DOI:10.1007/s11071-013-1057-7
  • T. Detroux; L. Renson; G. Kerschen The Harmonic Balance Method for Advanced Analysis and Design of Nonlinear Mechanical Systems, Nonlinear Dynamics, Volume 2 (2014), p. 19 | DOI:10.1007/978-3-319-04522-1_3
  • Malte Krack; Lars Panning-von Scheidt; Jörg Wallaschek A Framework for the Computational Dynamic Analysis of Coupled Structures Using Nonlinear Modes, Nonlinear Dynamics, Volume 2 (2014), p. 45 | DOI:10.1007/978-3-319-04522-1_5
  • O. Thomas; F. Mathieu; W. Mansfield; C. Huang; S. Trolier-McKinstry; L. Nicu Efficient parametric amplification in micro-resonators with integrated piezoelectric actuation and sensing capabilities, Applied Physics Letters, Volume 102 (2013) no. 16 | DOI:10.1063/1.4802786
  • S. Stoykov; P. Ribeiro Non-linear vibrations of beams with non-symmetrical cross sections, International Journal of Non-Linear Mechanics, Volume 55 (2013), p. 153 | DOI:10.1016/j.ijnonlinmec.2013.04.015
  • Haitao Liao Constrained Optimization Shooting Method for Predicting the Periodic Solutions of Nonlinear System, Journal of Computational and Nonlinear Dynamics, Volume 8 (2013) no. 4 | DOI:10.1115/1.4023916
  • Haitao Liao; Jianjun Wang Maximization of the vibration amplitude and bifurcation analysis of nonlinear systems using the constrained optimization shooting method, Journal of Sound and Vibration, Volume 332 (2013) no. 16, p. 3781 | DOI:10.1016/j.jsv.2013.02.034
  • Sami Karkar; Bruno Cochelin; Christophe Vergez A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities, Journal of Sound and Vibration, Volume 332 (2013) no. 4, p. 968 | DOI:10.1016/j.jsv.2012.09.033
  • A. Lazarus; J.T. Miller; P.M. Reis Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method, Journal of the Mechanics and Physics of Solids, Volume 61 (2013) no. 8, p. 1712 | DOI:10.1016/j.jmps.2013.04.002
  • Haitao Liao; Wei Sun A new method for predicting the maximum vibration amplitude of periodic solution of non-linear system, Nonlinear Dynamics, Volume 71 (2013) no. 3, p. 569 | DOI:10.1007/s11071-012-0682-x
  • Loïc Peletan; Sébastien Baguet; Mohamed Torkhani; Georges Jacquet-Richardet A comparison of stability computational methods for periodic solution of nonlinear problems with application to rotordynamics, Nonlinear Dynamics, Volume 72 (2013) no. 3, p. 671 | DOI:10.1007/s11071-012-0744-0
  • Ferhat Bekhoucha; Said Rechak; Laëtitia Duigou; Jean-Marc Cadou Nonlinear forced vibrations of rotating anisotropic beams, Nonlinear Dynamics, Volume 74 (2013) no. 4, p. 1281 | DOI:10.1007/s11071-013-1040-3
  • A. Lazarus; O. Thomas; J.-F. Deü Finite element reduced order models for nonlinear vibrations of piezoelectric layered beams with applications to NEMS, Finite Elements in Analysis and Design, Volume 49 (2012) no. 1, p. 35 | DOI:10.1016/j.finel.2011.08.019
  • J.W. Shao; B. Cochelin Passive control of resonances by nonlinear absorbers, MATEC Web of Conferences, Volume 1 (2012), p. 05006 | DOI:10.1051/matecconf/20120105006
  • Claude-Henri Lamarque; Cyril Touzé; Olivier Thomas An upper bound for validity limits of asymptotic analytical approaches based on normal form theory, Nonlinear Dynamics, Volume 70 (2012) no. 3, p. 1931 | DOI:10.1007/s11071-012-0584-y
  • E. Sarrouy; A. Grolet; F. Thouverez Global and bifurcation analysis of a structure with cyclic symmetry, International Journal of Non-Linear Mechanics, Volume 46 (2011) no. 5, p. 727 | DOI:10.1016/j.ijnonlinmec.2011.02.005
  • S. Stoykov; P. Ribeiro Stability of nonlinear periodic vibrations of 3D beams, Nonlinear Dynamics, Volume 66 (2011) no. 3, p. 335 | DOI:10.1007/s11071-011-0150-z

Cité par 111 documents. Sources : Crossref

Commentaires - Politique

Ces articles pourraient vous intéresser

Many-body quantum electrodynamics networks: Non-equilibrium condensed matter physics with light

Karyn Le Hur; Loïc Henriet; Alexandru Petrescu; ...

C. R. Phys (2016)