The harmonic balance method for bifurcation analysis of nonlinear mechanical systems

Detroux, Thibaut; Renson, Ludovic; Masset, Luc; Kerschen, Gaëtan

doi:10.1007/978-3-319-15221-9_5

Download

Paper published in a book (Scientific congresses and symposiums)

The harmonic balance method for bifurcation analysis of nonlinear mechanical systems

Detroux, Thibaut; Renson, Ludovic; Masset, Luc et al.

2015 • In Proceedings of the International Modal Analysis Conference (IMAC) XXXIII

Permalink
https://hdl.handle.net/2268/173556

DOI
10.1007/978-3-319-15221-9_5

Files (1)Send to Details Statistics Bibliography Similar publications

Files

Full Text

IMAC2015_TDetroux_V2.pdf

Author preprint (1.62 MB)

Download

All documents in ORBi are protected by a user license.

Send to

RIS BibTex APA Chicago Permalink X Linkedin

Details

Keywords :

Continuation of periodic solutions; Bifurcation tracking; Harmonic balance method; Quasiperiodic oscillations; Detached resonance curves

Abstract :

[en] Because nowadays structural engineers are willing to use or at least understand nonlinearities instead of simply avoiding them, there is a need for numerical tools performing analysis of nonlinear large-scale structures. Among these techniques, the harmonic balance (HB) method is certainly one of the most commonly used to study finite element models with reasonably complex nonlinearities. However, in its classical formulation the HB method is limited to the approximation of periodic solutions. For this reason, the present paper proposes to extend the method to the detection and tracking of codimension-1 bifurcations in the system parameters space. As an application, the frequency response of a spacecraft is studied, together with two nonlinear phenomena, namely quasiperiodic oscillations and detached resonance curves. This example illustrates how bifurcation tracking using the HB method can be employed as a promising design tool for detecting and eliminating such undesired behaviors.

Disciplines :

Aerospace & aeronautics engineering

Author, co-author :

Detroux, Thibaut ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux

Renson, Ludovic ; Université de Liège - ULiège > R&D Direction : Chercheurs ULiège en mobilité

Masset, Luc ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux

Kerschen, Gaëtan ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux

Language :

English

Title :

The harmonic balance method for bifurcation analysis of nonlinear mechanical systems

Publication date :

2015

Event name :

SEM IMAC XXXIII A Conference and Exposition on Structural Dynamics

Event organizer :

Society for Experimental Mechanics

Event place :

Orlando, Florida, United States

Event date :

2 - 5 February 2015

Audience :

International

Main work title :

Proceedings of the International Modal Analysis Conference (IMAC) XXXIII

European Projects :

FP7 - 307265 - NOVIB - The Nonlinear Tuned Vibration Absorber

Funders :

EU - European Union

Available on ORBi :

since 04 November 2014

Statistics

Number of views

332 (40 by ULiège)

Number of downloads

1149 (11 by ULiège)

More statistics

Scopus citations^®

Scopus citations^®
without self-citations

OpenCitations

OpenAlex citations

Bibliography

Nayfeh AH, Mook DT (2008) Nonlinear oscillations. Wiley, New York
Noël J, Renson L, Kerschen G (2014) Complex dynamics of a nonlinear aerospace structure: experimental identification and modal interactions. J Sound Vib 333(12):2588–2607
Renson L, Noël J, Kerschen G (2015) Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes. Nonlinear Dyn 79(2):1293–1309
Denegri CM (2000) Limit cycle oscillation flight test results of a fighter with external stores. J Aircraft 37(5):761–769
Ahlquist JR, Carreño JM, Climent H, de Diego R, de Alba J (2011) Assessment of nonlinear structural response in A400M GVT. In: Proulx T (ed) Structural dynamics, vol 3. Springer, New York, pp 1147–1155
Noël J-P, Renson L, Kerschen G, Peeters B, Manzato S, Debille J (2013) Nonlinear dynamic analysis of an F-16 aircraft using GVT data. In: Proceedings of the international forum on aeroelasticity and structural dynamics
Padmanabhan C, Singh R (1995) Analysis of periodically excited non-linear systems by a parametric continuation technique. J Sound Vib 184(1):5–58
Peeters M, Viguié R, Sérandour G, Kerschen G, Golinval J-C (2009) Nonlinear normal modes, part II: toward a practical computation using numerical continuation techniques. Mech Syst Signal Process 23(1):195–216
Doedel EJ, Champneys AR, Fairgrieve TF, Kuznetsov YA, Sandstede B, Wang X (1997) AUTO97: continuation and bifurcation software for ordinary differential equations (with HomCont), User’s guide. Concordia University, Montreal. http://www.indy.cs.concordia.ca
Ascher U, Christiansen J, Russell RD (1979) A collocation solver for mixed order systems of boundary value problems. Math Comput 33: 659–679
Kuznetsov YA, Levitin VV (1995/1997) CONTENT: a multiplatform environment for analyzing dynamical systems, User’s guide. Dynamical Systems Laboratory, CWI, Amsterdam. http://www.ftp.cwi.nl/pub/CONTENT
Dhooge A, Govaerts W, Kuznetsov YA (2003) MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans Math Softw (TOMS) 29(2):141–164
Dankowicz H, Schilder F (2011) An extended continuation problem for bifurcation analysis in the presence of constraints. J Comput Nonlinear Dyn 6(3):031003
Kundert KS, Sangiovanni-Vincentelli A (1986) Simulation of nonlinear circuits in the frequency domain. IEEE Trans Comput Aided Des Integr Circ Syst 5(4):521–535
Cardona A, Coune T, Lerusse A, Geradin M (1994) A multiharmonic method for non-linear vibration analysis. Int J Numer Methods Eng 37(9):1593–1608
von Groll G, Ewins DJ (2001) The harmonic balance method with arc-length continuation in rotor/stator contact problems. J Sound Vib 241(2):223–233
Laxalde D, Thouverez F (2009) Complex non-linear modal analysis for mechanical systems: application to turbomachinery bladings with friction interfaces. J Sound Vib 322(4):1009–1025
Grolet A, Thouverez F (2012) On a new harmonic selection technique for harmonic balance method. Mech Syst Signal Process 30:43–60
Grolet A, Thouverez F (2013) Vibration of mechanical systems with geometric nonlinearities: solving harmonic balance equations with Groebner basis and continuations methods. In: Proceedings of the colloquium calcul des structures et Modélisation CSMA, Giens
Arquier R (2007) Une méthode de calcul des modes de vibrations non-linéaires de structures. Ph.D. thesis, Université de la méditerranée (Aix-Marseille II), Marseille
Cochelin B, Vergez C (2009) A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions. J Sound Vib 324(1):243–262
Karkar S, Cochelin B, Vergez C (2014) A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems. J Sound Vib 333(12):2554–2567
Detroux T, Renson L, Kerschen G (2014) The harmonic balance method for advanced analysis and design of nonlinear mechanical systems. In: Nonlinear dynamics, vol 2. Springer, Cham, pp 19–34
Petrov E, Ewins D (2003) Analytical formulation of friction interface elements for analysis of nonlinear multi-harmonic vibrations of bladed disks. J Turbomach 125(2):364–371
Lau S, Zhang W-S (1992) Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method. J Appl Mech 59:153
Pierre C, Ferri A, Dowell E (1985) Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. J Appl Mech 52(4):958–964
Cameron T, Griffin J (1989) An alternating frequency/time domain method for calculating the steady-state response of nonlinear dynamic systems. J Appl Mech 56(1):149–154
Narayanan S, Sekar P (1998) A frequency domain based numeric–analytical method for non-linear dynamical systems. J Sound Vib 211(3):409–424
Bonani F, Gilli M (1999) Analysis of stability and bifurcations of limit cycles in Chua’s circuit through the harmonic-balance approach. IEEE Trans Circ Syst I Fundam Theory Appl 46(8):881–890
Duan C, Singh R (2005) Super-harmonics in a torsional system with dry friction path subject to harmonic excitation under a mean torque. J Sound Vib 285(4):803–834
Kim T, Rook T, Singh R (2005) Super-and sub-harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method. J Sound Vib 281(3):965–993
Lazarus A, Thomas O (2010) A harmonic-based method for computing the stability of periodic solutions of dynamical systems. C R Mec 338(9):510–517
Moore G (2005) Floquet theory as a computational tool. SIAM J Numer Anal 42(6):2522–2568
Seydel R (2010) Practical bifurcation and stability analysis. Springer, New York
Govaerts W, Kuznetsov YA, De Witte V, Dhooge A, Meijer H, Mestrom W, Riet A, Sautois B (2011) MATCONT and CL MATCONT: continuation toolboxes in matlab. Technical report, Gent University and Utrech University
Guckenheimer J, Myers M, Sturmfels B (1997) Computing hopf bifurcations I. SIAM J Numer Anal 34(1):1–21
Govaerts WJ (2000) Numerical methods for bifurcations of dynamical equilibria, vol 66. SIAM, Philadelphia
Beyn W-J, Champneys A, Doedel E, Govaerts W, Kuznetsov YA, Sandstede B (2002) Numerical continuation, and computation of normal forms. In: Fiedler B (ed) Handbook of dynamical systems, vol. 2. Elsevier Science, North-Holland, pp 149–219
Doedel EJ, Govaerts W, Kuznetsov YA (2003) Computation of periodic solution bifurcations in ODEs using bordered systems. SIAM J Numer Anal 41(2):401–435
Van der Aa N, Ter Morsche H, Matthei J R (2007) Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem. Electron J Linear Algebra 16(1):300–314
Masri S, Caughey T (1979) A nonparametric identification technique for nonlinear dynamic problems. J Appl Mech 46(2):433–447