A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure, and complex two-dimensional (2D) integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude–frequency response surfaces of quasi-periodic responses and avoid nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge–Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

References

References
1.
Fan
,
Q.
,
Leung
,
A. Y. T.
, and
Lee
,
Y. Y.
,
2016
, “
Periodic and Quasi-Periodic Responses of Van der Pol–Mathieu System Subject to Various Excitations
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
17
(
1
), pp.
29
40
.
2.
Wang
,
X.
,
2015
, “
Quasi-Periodic Solutions for Second Order Differential Equation With Superlinear Asymmetric Nonlinearities and Nonlinear Damping Term
,”
Boundary Value Probl.
,
2015
(
1
), pp.
1
12
.
3.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
2008
,
Nonlinear Oscillations
,
Wiley
, New York.
4.
Mamandi
,
A.
,
Kargarnovin
,
M. H.
, and
Farsi
,
S.
,
2012
, “
Dynamic Analysis of a Simply Supported Beam Resting on a Nonlinear Elastic Foundation Under Compressive Axial Load Using Nonlinear Normal Modes Techniques Under Three-to-One Internal Resonance Condition
,”
Nonlinear Dyn.
,
70
(
2
), pp.
1147
1172
.
5.
Guennoun
,
K.
,
Houssni
,
M.
, and
Belhaq
,
M.
,
2002
, “
Quasi-Periodic Solutions and Stability for a Weakly Damped Nonlinear Quasi-Periodic Mathieu Equation
,”
Nonlinear Dyn.
,
27
(
3
), pp.
211
236
.
6.
Shen
,
Y.-J.
,
Yang
,
S.-P.
,
Xing
,
H.-J.
, and
Pan
,
C.-Z.
,
2013
, “
Analytical Research on a Single Degree-of-Freedom Semi-Active Oscillator With Time Delay
,”
J. Vib. Control
,
19
(
12
), pp.
1895
1905
.
7.
Xu
,
P.
, and
Jing
,
Z.
,
1999
, “
Quasi-Periodic Solutions and Sub-Harmonic Bifurcation of Duffing's Equations With Quasi-Periodic Perturbation
,”
Acta Math. Appl. Sin.
,
15
(
4
), pp.
374
384
.
8.
Belhaq
,
M.
, and
Houssni
,
M.
,
1999
, “
Quasi-Periodic Oscillations, Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations
,”
Nonlinear Dyn.
,
18
(
1
), pp.
1
24
.
9.
Cheung
,
Y. K.
, and
Lau
,
S. L.
,
1982
, “
Incremental Time—Space Finite Strip Method for Non-Linear Structural Vibrations
,”
Earthquake Eng. Struct. Dyn.
,
10
(
2
), pp.
239
253
.
10.
Lau
,
S. L.
,
Cheung
,
Y. K.
, and
Wu
,
S. Y.
,
1983
, “
Incremental Harmonic Balance Method With Multiple Time Scales for Aperiodic Vibration of Nonlinear Systems
,”
ASME J. Appl. Mech.
,
50
(
4a
), pp.
871
876
.
11.
Gadella
,
M.
,
Giacomini
,
H.
, and
Lara
,
L. P.
,
2015
, “
Periodic Analytic Approximate Solutions for the Mathieu Equation
,”
Appl. Math. Comput.
,
271
, pp.
436
445
.
12.
Huang
,
J. L.
, and
Zhu
,
W. D.
,
2014
, “
Nonlinear Dynamics of a High-Dimensional Model of a Rotating Euler–Bernoulli Beam Under the Gravity Load
,”
ASME J. Appl. Mech.
,
81
(
10
), p.
101007
.
13.
Dai
,
H.
,
Yue
,
X.
,
Yuan
,
J.
, and
Xie
,
D.
,
2014
, “
A Fast Harmonic Balance Technique for Periodic Oscillations of an Aeroelastic Airfoil
,”
J. Fluids Struct.
,
50
, pp.
231
252
.
14.
Mitra
,
R. K.
,
Banik
,
A. K.
, and
Chatterjee
,
S.
,
2013
, “
Dynamic Stability of Time-Delayed Feedback Control System by FFT Based IHB Method
,”
WSEAS Trans. Appl. Theor. Mech.
,
4
(
8
), pp.
292
303
.
15.
Huang
,
J. L.
,
Su
,
R. K. L.
,
Li
,
W. H.
, and
Chen
,
S. H.
,
2011
, “
Stability and Bifurcation of an Axially Moving Beam Tuned to Three-to-One Internal Resonances
,”
J. Sound Vib.
,
330
(
3
), pp.
471
485
.
16.
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
.
17.
Lau
,
S. L.
,
Cheung
,
Y. K.
, and
Wu
,
S. Y.
,
1982
, “
A Variable Parameter Incrementation Method for Dynamic Instability of Linear and Nonlinear Elastic Systems
,”
ASME J. Appl. Mech.
,
49
(
4
), pp.
849
853
.
18.
Pierre
,
C.
,
Ferri
,
A. A.
, and
Dowell
,
E. H.
,
1985
, “
Multi-Harmonic Analysis of Dry Friction Damped Systems Using an Incremental Harmonic Balance Method
,”
ASME J. Appl. Mech.
,
52
(
4
), pp.
958
964
.
19.
Raghothama
,
A.
, and
Narayanan
,
S.
,
1999
, “
Bifurcation and Chaos in Geared Rotor Bearing System by Incremental Harmonic Balance Method
,”
J. Sound Vib.
,
226
(
3
), pp.
469
492
.
20.
Xu
,
L.
,
Lu
,
M. W.
, and
Cao
,
Q.
,
2003
, “
Bifurcation and Chaos of a Harmonically Excited Oscillator With Both Stiffness and Viscous Damping Piecewise Linearities by Incremental Harmonic Balance Method
,”
J. Sound Vib.
,
264
(
4
), pp.
873
882
.
21.
Azizi
,
Y.
,
Bajaj
,
A. K.
,
Davies
,
P.
, and
Sundaram
,
V.
,
2015
, “
Prediction and Verification of the Periodic Response of a Single-Degree-of-Freedom Foam-Mass System by Using Incremental Harmonic Balance
,”
Nonlinear Dyn.
,
82
(
4
), pp.
1933
1951
.
22.
Luo
,
A. C. J.
, and
Huang
,
J.
,
2013
, “
Analytical Solutions for Asymmetric Periodic Motions to Chaos in a Hardening Duffing Oscillator
,”
Nonlinear Dyn.
,
72
(
1
), pp.
417
438
.
23.
Luo
,
A. C. J.
, and
Huang
,
J.
,
2011
, “
Approximate Solutions of Periodic Motions in Nonlinear Systems Via a Generalized Harmonic Balance
,”
J. Vib. Control
,
18
(
11
), pp.
1661
1674
.
24.
Szemplińska-Stupnicka
,
W.
,
1978
, “
The Generalized Harmonic Balance Method for Determining the Combination Resonance in the Parametric Dynamic Systems
,”
J. Sound Vib.
,
58
(
3
), pp.
347
361
.
25.
Lu
,
W.
,
Ge
,
F.
,
Wu
,
X.
, and
Hong
,
Y.
,
2013
, “
Nonlinear Dynamics of a Submerged Floating Moored Structure by Incremental Harmonic Balance Method With FFT
,”
Mar. Struct.
,
31
, pp.
63
81
.
26.
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
), pp.
981
989
.
27.
Wang
,
X. F.
, and
Zhu
,
W. D.
,
2017
, “
Dynamic Analysis of an Automotive Belt-Drive System With a Noncircular Sprocket by a Modified Incremental Harmonic Balance Method
,”
ASME J. Vib. Acoust.
,
139
(
1
), p.
011009
.
28.
Rizzoli
,
V.
,
Cecchetti
,
C.
, and
Lipparini
,
A.
,
1987
, “
A General-Purpose Program for the Analysis of Nonlinear Microwave Circuits Under Multitone Excitation by Multidimensional Fourier Transform
,”
17th European Microwave Conference
(
EUMA
), Rome, Italy, Sept. 7–11, pp.
635
640
.
29.
Sorkin
,
G. B.
,
Kundert
,
K. S.
, and
Sangiovanni-Vincentelli
,
A.
,
1987
, “
An Almost-Periodic Fourier Transform for Use With Harmonic Balance
,” IEEE MTT-S International Microwave Symposium Digest (
MWSYM
), Las Vegas, NV, June 9–11, pp.
717
720
.
30.
Guskov
,
M.
, and
Thouverez
,
F.
,
2012
, “
Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis
,”
ASME J. Vib. Acoust.
,
134
(
3
), p.
031003
.
31.
Hente
,
D.
, and
Jansen
,
R. H.
,
1986
, “
Frequency Domain Continuation Method for the Analysis and Stability Investigation of Nonlinear Microwave Circuits
,”
IEE Proceedings H-Microwaves, Antennas and Propagation
,
133
(5), pp.
351
362
.
32.
Borich
,
V.
,
East
,
J.
, and
Haddad
,
G.
,
1999
, “
An Efficient Fourier Transform Algorithm for Multitone Harmonic Balance
,”
IEEE Trans. Microwave Theory Tech.
,
47
(
2
), pp.
182
188
.
33.
Cheung
,
Y. K.
, and
Chen
,
S. H.
,
1990
, “
Application of the Incremental Harmonic Balance Method to Cubic Non-Linearity Systems
,”
J. Sound Vib.
,
140
(
2
), pp.
273
286
.
34.
Kundert
,
K. S.
,
Sorkin
,
G. B.
, and
Sangiovanni-Vincentelli
,
A.
,
1988
, “
Applying Harmonic Balance to Almost-Periodic Circuits
,”
IEEE Trans. Microwave Theory Tech.
,
36
(
2
), pp.
366
378
.
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