Abstract
As a semi-analytical approach, the incremental harmonic balance (IHB) method is widely implemented for solving steady-state (including both periodic and quasi-periodic) responses through an iteration process. The iteration is carried out through a Jacobian matrix (JM) and a residual vector, both updated in each iteration. Though the JM is known to be singular at certain bifurcation points, the singularity is still an open question and could play a pivotal role in real applications. In this study, we define and calculate an expanded JM (EJM) by applying an expanded solution expression in the IHB iteration. The singularity of the EJM at several different bifurcation points is proved in a general manner, according to the bifurcation theory for equilibria in nonlinear dynamical systems. Given the possible bifurcation type, furthermore, the singularity is applied to locate the corresponding bifurcation point directly and precisely. Considered are the cases of the period-doubling, symmetry breaking, and Neimark-Sacker bifurcations of periodic and/or quasi-periodic responses.
Issue Section:
Research Papers
Keywords:
incremental harmonic balance method,
expanded Jacobian matrix,
singularity,
bifurcation point,
steady-state response
Topics:
Bifurcation
References
1.
Lau
,
S. L.
, and
Cheung
,
Y. K.
, 1981
, “
Amplitude Incremental Variational Principle for Nonlinear Vibration of Elastic Systems
,” ASME J. Appl. Mech.
,
48
(4
), pp. 959
–964
.10.1115/1.31577622.
Lau
,
S. L.
,
Cheung
,
Y. K.
, and
Wu
,
S. Y.
, 1983
, “
Incremental Harmonic Balance Method With Multiple Time Scales for Aperiodic Vibration of Nonlinear System
,” ASME J. Appl. Mech.
,
50
(4a
), pp. 871
–876
.10.1115/1.31671603.
Ferri
,
A. A.
, 1986
, “
On the Equivalence of the Incremental Harmonic Balance Method and the Harmonic balance-Newton Raphson Method
,” ASME J. Appl. Mech.
,
53
(2
), pp. 455
–457
.10.1115/1.31717804.
Shen
,
J. H.
,
Lin
,
K. C.
,
Chen
,
S. H.
, and
Sze
,
K. Y.
, 2008
, “
Bifurcation and Route-to-Chaos Analyses for Mathieu–Duffing Oscillator by the Incremental Harmonic Balance Method
,” Nonlinear Dyn.
,
52
(4
), pp. 403
–414
.10.1007/s11071-007-9289-z5.
Zheng
,
Z. C.
,
Lu
,
Z. R.
,
Liu
,
G.
, and
Chen
,
Y. M.
, 2022
, “
Twice Harmonic Balance Method for Stability and Bifurcation Analysis of Quasi-Periodic Responses
,” ASME J. Comput. Nonlinear Dyn.
,
17
(12
), p. 121006
.10.1115/1.40559236.
Raghothama
,
A.
, and
Narayanan
,
S.
, 1999
, “
Non-Linear Dynamics of a Two Dimensional Airfoil by Incremental Harmonic Balance Method
,” J. Sound Vib.
,
226
(3
), pp. 493
–517
.10.1006/jsvi.1999.22607.
Wu
,
G. Y.
, 2013
, “
Non-Linear Vibration of Bimaterial Magneto-Elastic Cantilever Beam With Thermal Loading
,” Int. J. Non-Linear Mech.
,
55
, pp. 10
–18
.10.1016/j.ijnonlinmec.2013.04.0098.
Ju
,
R.
,
Fan
,
W.
,
Zhu
,
W. D.
, and
Huang
,
J. L.
, 2017
, “
A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems
,” ASME J. Comput. Nonlinear Dyn.
,
12
(5
), p. 051007
.10.1115/1.40361189.
Chen
,
Y. M.
,
Liu
,
Q. X.
, and
Liu
,
J. K.
, 2020
, “
A Study on Multi-Frequency Patterns in Nonlinear Network Oscillators Using Incremental Harmonic Balance Method
,” Int. J. Non-Linear Mech.
,
121
, p. 103435
.10.1016/j.ijnonlinmec.2020.10343510.
Ma
,
W. L.
,
Li
,
Y.
, and
Wang
,
M. Q.
, 2023
, “
Study on the Dynamics of Two-Degree-of-Freedom Fractional Differential Piecewise Nonlinear Systems Under Harmonic Excitation
,” Int. J. Non-Linear Mech.
,
149
, p. 104302
.10.1016/j.ijnonlinmec.2022.10430211.
Sun
,
P.
,
Zhao
,
X. L.
,
Yu
,
X. F.
,
Huang
,
Q.
,
Feng
,
Z. P.
, and
Zhou
,
J. X.
, 2023
, “
Incremental Harmonic Balance Method for Multi-Harmonic Solution of High-Dimensional Delay Differential Equations: Application to Cross Flow-Induced Nonlinear Vibration of Steam Generator Tubes
,” Appl. Math. Modell.
,
118
, pp. 818
–831
.10.1016/j.apm.2023.02.01812.
Ju
,
R.
,
Fan
,
W.
, and
Zhu
,
W. D.
, 2021
, “
Comparison Between the Incremental Harmonic Balance Method and Alternating Frequency/Time-Domain Method
,” ASME J. Vib. Acoust.
,
143
(2
), pp. 1
–24
.10.1115/1.404817313.
Wu
,
H.
,
Zeng
,
X.
,
Liu
,
Y.
, and
Lai
,
J.
, 2021
, “
Analysis of Harmonically Forced Duffing Oscillator With Time Delay State Feedback by Incremental Harmonic Balance Method
,” J. Vib. Eng. Tech.
,
9
(6
), pp. 1239
–1251
.10.1007/s42417-021-00293-y14.
Wang
,
Y.
,
Zhang
,
L.
, and
Zhou
,
J.
, 2020
, “
Incremental Harmonic Balance Method for Periodic Forced Oscillation of a Dielectric Elastomer Balloon
,” Appl. Math. Mech.
,
41
(3
), pp. 459
–470
.10.1007/s10483-020-2590-715.
Ri
,
K.
,
Han
,
W. J.
,
Pak
,
C.
,
Kim
,
K.
, and
Yun
,
C.
, 2021
, “
Nonlinear Forced Vibration Analysis of the Composite Shaft-Disk System Combined the Reduced-Order Model With the IHB Method
,” Nonlinear Dyn.
,
104
(4
), pp. 3347
–3364
.10.1007/s11071-021-06510-316.
Li
,
C. F.
,
Li
,
P. Y.
, and
Miao
,
X. Y.
, 2021
, “
Research on Nonlinear Vibration Control of Laminated Cylindrical Shells With Discontinuous Piezoelectric Layer
,” Nonlinear Dyn.
,
104
(4
), pp. 3247
–3267
.10.1007/s11071-021-06497-x17.
Huang
,
J. L.
, and
Zhu
,
W. D.
, 2017
, “
A New Incremental Harmonic Balance Method With Two Time Scales for Quasi-Periodic Motions of an Axially Moving Beam With Internal Resonance Under Single-Tone External Excitation
,” ASME J. Vib. Acoust.
,
139
(2
), p. 021010
.10.1115/1.403513518.
Liu
,
G.
,
Lv
,
Z. R.
,
Liu
,
J. K.
, and
Chen
,
Y. M.
, 2018
, “
Quasi-Periodic Aeroelastic Response Analysis of an Airfoil With External Store by Incremental Harmonic Balance Method
,” Int. J. Non-Linear Mech.
,
100
, pp. 10
–19
.10.1016/j.ijnonlinmec.2018.01.00419.
Fontanela
,
F.
,
Grolet
,
A.
,
Salles
,
L.
, and
Hoffmann
,
N.
, 2019
, “
Computation of Quasi-Periodic Localised Vibrations in Nonlinear Cyclic and Symmetric Structures Using Harmonic Balance Methods
,” J. Sound Vib.
,
438
, pp. 54
–65
.10.1016/j.jsv.2018.09.00220.
Huang
,
J. L.
,
Wang
,
T.
, and
Zhu
,
W. D.
, 2021
, “
An Incremental Harmonic Balance Method With Two Time-Scales for Quasi-Periodic Responses of a Van Der Pol–Mathieu Equation
,” Int. J. Non-Linear Mech.
,
135
, p. 103767
.10.1016/j.ijnonlinmec.2021.10376721.
Ri
,
K.
,
Jang
,
J.
,
Yun
,
C.
,
Pak
,
C.
, and
Kim
,
K.
, 2022
, “
Analysis of Subharmonic and Quasi-Periodic Vibrations of a Jeffcott Rotor Supported on a Squeeze-Film Damper by the IHB Method
,” AIP Adv.
,
12
(5
), p. 055328
.10.1063/5.008833422.
Leung
,
A. Y. T.
, and
Chui
,
S. K.
, 1995
, “
Nonlinear Vibration of Coupled Duffing Oscillators by an Improved Incremental Harmonic Balance Method
,” J. Sound Vib.
,
181
(4
), pp. 619
–633
.10.1006/jsvi.1995.016223.
Lu
,
W.
,
Ge
,
F.
,
Wu
,
X. D.
, and
Hong
,
Y. S.
, 2013
, “
Nonlinear Dynamics of a Submerged Floating Moored Structure by Incremental Harmonic Balance Method With FFT
,” Mar. Struct
,
31
, pp. 63
–81
.10.1016/j.marstruc.2013.01.00224.
Leung
,
A. Y. T.
, and
Ge
,
T.
, 1995
, “
Toeplitz Jacobian Matrix for Nonlinear Periodic Vibration
,” ASME J. Appl. Mech.
,
62
(3
), pp. 709
–717
.10.1115/1.289700425.
Wang
,
X. F.
,
Zhao
,
X.
, and
Zhu
,
W. D.
, 2019
, “
An Incremental Harmonic Balance Method With a General Formula of Jacobian Matrix and a Direct Construction Method in Stability Analysis of Periodic Responses of General Nonlinear Delay Differential Equations
,” ASME J. Appl. Mech.
,
86
(6
), p. 061011
.10.1115/1.404283626.
Wang
,
X. F.
, and
Zhu
,
W. D.
, 2015
, “
A Modified Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Broyden's Method
,” Nonlinear Dyn.
,
81
(1–2
), pp. 981
–989
.10.1007/s11071-015-2045-x27.
Ju
,
R.
,
Fan
,
W.
, and
Zhu
,
W. D.
, 2020
, “
An Efficient Galerkin Averaging-Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Tensor Contraction
,” ASME J. Vib. Acoust.
,
142
(6
), p. 061011
.10.1115/1.404723528.
Dimitriadis
,
G.
, 2008
, “
Continuation of Higher-Order Harmonic Balance Solutions for Nonlinear Aeroelastic Systems
,” J. Aircr.
,
45
(2
), pp. 523
–537
.10.2514/1.3047229.
Yuan
,
T. C.
,
Yang
,
J.
, and
Chen
,
L. Q.
, 2019
, “
Nonlinear Vibration Analysis of a Circular Composite Plate Harvester Via Harmonic Balance
,” Acta Mech. Sin.
,
35
(4
), pp. 912
–925
.10.1007/s10409-019-00863-030.
Dunne
,
J. F.
, 2006
, “
Subharmonic-Response Computation and Stability Analysis for a Nonlinear Oscillator Using a Split-Frequency Harmonic Balance Method
,” ASME J. Comput. Nonlinear Dyn.
,
1
(3
), pp. 221
–229
.10.1115/1.219887531.
Liu
,
L.
,
Wong
,
Y. S.
, and
Lee
,
B. H. K.
, 2000
, “
Application of the Center Manifold Theory in Nonlinear Aeroelasticity
,” J. Sound Vib.
,
234
(4
), pp. 641
–659
.10.1006/jsvi.1999.289532.
Chen
,
Y. M.
, and
Liu
,
J. K.
, 2008
, “
Supercritical as Well as Subcritical Hopf Bifurcation in Nonlinear Flutter Systems
,” Appl. Math. Mech.
,
29
(2
), pp. 199
–206
.10.1007/s10483-008-0207-x33.
Agnihotri
,
K.
, and
Kaur
,
H.
, 2019
, “
The Dynamics of Viral Infection in Toxin Producing Phytoplankton and Zooplankton System With Time Delay
,” Chaos, Solitons Fract.
,
118
, pp. 122
–133
.10.1016/j.chaos.2018.11.01834.
Chen
,
Y. M.
, and
Liu
,
J. K.
, 2016
, “
A Precise Calculation of Bifurcation Points for Periodic Solution in Nonlinear Dynamical Systems
,” Appl. Math. Comput.
,
273
, pp. 1190
–1195
.10.1016/j.amc.2015.08.13035.
Yuan
,
T. C.
,
Yang
,
J.
, and
Chen
,
L. Q.
, 2019
, “
A Harmonic Balance Approach With Alternating Frequency Time Domain Progress for Piezoelectric Mechanical Systems
,” Mech. Syst. Signal Process.
,
120
, pp. 274
–289
.10.1016/j.ymssp.2018.10.02236.
Traversa
,
F. L.
,
Bonani
,
F.
, and
Guerrieri
,
S. D.
, 2008
, “
A Frequency-Domain Approach to the Analysis of Stability and Bifurcations in Nonlinear Systems Described by Differential-Algebraic Equations
,” Int. J. Circuit Theory Appl.
,
36
(4
), pp. 421
–439
.10.1002/cta.44037.
Maggio
,
G. M.
,
Kennedy
,
M. P.
, and
Gilli
,
M.
, 1998
, “
An Approximate Analytical Approach for Predicting Period-Doubling in the Colpitts Oscillator
,” IEEE International Symposium on Circuits and Systems (ISCAS1998),
Monterey, CA, May 31–June 3, Vol.
3
, pp. 671
–674
.38.
Bonani
,
F.
, and
Gilli
,
M.
, 1999
, “
Analysis of Stability and Bifurcations of Limit Cycles in Chua's Circuit Through the Harmonic-Balance Approach
,” IEEE Tran. Circuits Syst.-I: Fundam. Theory Appl.
,
46
(8
), pp. 881
–890
.10.1109/81.78037039.
Anastasio
,
D.
, and
Marchesiello
,
S.
, 2023
, “
Nonlinear Frequency Response Curves Estimation and Stability Analysis of Randomly Excited Systems in the Subspace Framework
,” Nonlinear Dyn.
,
111
(9
), pp. 8115
–8133
.10.1007/s11071-023-08280-640.
Bayer
,
F.
, and
Leine
,
R. I.
, 2023
, “
Sorting-Free Hill-Based Stability Analysis of Periodic Solutions Through Koopman Analysis
,” Nonlinear Dyn.
,
111
(9
), pp. 8439
–8466
.10.1007/s11071-023-08247-741.
Detroux
,
T.
,
Renson
,
L.
,
Masset
,
L.
, and
Kerschen
,
G.
, 2015
, “
The Harmonic Balance Method for Bifurcation Analysis of Large-Scale Nonlinear Mechanical Systems
,” Comput. Methods Appl. Mech. Eng.
,
296
, pp. 18
–38
.10.1016/j.cma.2015.07.01742.
Melot
,
A.
,
Rigaud
,
E.
, and
Liaudet
,
J. P.
, 2023
, “
Robust Design of Vibro-Impacting Geared Systems With Uncertain Tooth Profile Modifications Via Bifurcation Tracking
,” Int. J. Nonlinear Mech.
,
149
, p. 104336
.10.1016/j.ijnonlinmec.2022.10433643.
Melot
,
A.
,
Rigaud
,
E.
, and
Liaudet
,
J. P.
, 2022
, “
Bifurcation Tracking of Geared Systems With Parameter-Dependent Internal Excitation
,” Nonlinear Dyn.
,
107
(1
), pp. 413
–431
.10.1007/s11071-021-07018-644.
Xie
,
L.
,
Baguet
,
S.
,
Prabel
,
B.
, and
Dufour
,
R.
, 2017
, “
Bifurcation Tracking by Harmonic Balance Method for Performance Tuning of Nonlinear Dynamical Systems
,” Mech. Syst. Signal Process.
,
88
, pp. 445
–461
.10.1016/j.ymssp.2016.09.03745.
Alcorta
,
R.
,
Baguet
,
S.
,
Prabel
,
B.
,
Piteau
,
P.
, and
Richardet
,
G. J.
, 2019
, “
Period Doubling Bifurcation Analysis and Isolated Sub-Harmonic Resonances in an Oscillator With Asymmetric Clearances
,” Nonlinear Dyn.
,
98
(4
), pp. 2939
–2960
.10.1007/s11071-019-05245-646.
Tehrani
,
G. G.
,
Gastaldi
,
C.
, and
Berruti
,
T. M.
, 2021
, “
A Forced Response-Based Method to Track Instability of Rotating Systems
,” Eur. J. Mech./A Solids
,
90
, p. 104319
.10.1016/j.euromechsol.2021.10431947.
Liao
,
H. T.
, 2023
, “
A Nonlinear Optimization Bifurcation Tracking Method Forperiodic Solution of Nonlinear Systems
,” Mech. Based Des. Struct. Mach.
,
51
(3
), pp. 1201
–1225
.10.1080/15397734.2020.186323048.
Renson
,
L.
,
Barton
,
D. A. W.
, and
Neild
,
S. A.
, 2017
, “
Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation
,” Int. J. Bifur. Chaos
,
27
(01
), p. 1730002
.10.1142/S021812741730002649.
Basso
,
M.
,
Genesio
,
R.
, and
Tesi
,
A.
, 1997
, “
A Frequency Method for Predicting Limit Cycle Bifurcations
,” Nonlinear Dyn.
,
13
(4
), pp. 339
–360
.10.1023/A:100829820578650.
Lanza
,
V.
,
Bonnin
,
M.
, and
Gilli
,
M.
, 2007
, “
On the Application of the Describing Function Technique to the Bifurcation Analysis of Nonlinear Systems
,” IEEE Tran. Circuits Syst. II: Express Briefs
,
54
(4
), pp. 343
–347
.10.1109/TCSII.2006.89040651.
Innocenti
,
G.
,
Di Marco
,
M.
,
Forti
,
M.
, and
Tesi
,
A.
, 2019
, “
Prediction of Period Doubling Bifurcations in Harmonically Forced Memristor Circuits
,” Nonlinear Dyn.
,
96
(2
), pp. 1169
–1190
.10.1007/s11071-019-04847-452.
Chen
,
Y. M.
, and
Liu
,
J. K.
, 2023
, “
Circularly Distributed Multipliers With Deterministic Modulus Assessing the Stability of Quasiperiodic Responses
,” Phys. Rev. E
,
107
(1
), p. 014218
.10.1103/PhysRevE.107.01421853.
Olson
,
C. L.
, and
Olsson
,
M. G.
, 1991
, “
Dynamical Symmetry Breaking and Chaos in Duffing's Equation
,” Am. J. Phys.
,
59
(10
), pp. 907
–911
.10.1119/1.1666954.
Schilder
,
F.
,
Vogt
,
W.
,
Schreiber
,
S.
, and
Osinga
,
H. M.
, 2006
, “
Fourier Methods for Quasi-Periodic Oscillations
,” Int. J. Numer. Methods Eng
,.
67
(5
), pp. 629
–671
.10.1002/nme.163255.
Guckenheimer
,
J.
, and
Holmes
,
P.
, 2013
, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields
,
Springer Science & Business Media
, New York.Copyright © 2023 by ASME
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