Fourier series expansion (FSE) plays a pivotal role in frequency domain analysis of a wide variety of nonlinear dynamical systems. To the best of our knowledge, there are two general approaches for FSE, i.e., a collocation method (CM) previously proposed by the authors and the classical discrete FSE. Though there are huge applications of these methods, it still remains much less understood in their relationship and error estimation. In this study, we proved that they are equivalent if time points are uniformly chosen. Based on this property, more importantly, the error was analytically estimated for both discrete Fourier expansion (DFE) and CM. Furthermore, we revealed that the accuracy of frequency domain solutions cannot be improved by increasing the number of time points alone, whereas it absolutely depends upon the truncated number of harmonics. It indicates that an appropriate number of time points should be chosen in FSE if frequency domain solutions are targeted for nonlinear dynamical systems, especially those with complicated functions.

References

References
1.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
Wiley
,
New York
.
2.
Ali Akbar
,
M.
,
Shamsul Alam
,
M.
, and
Sattar
,
M. A.
,
2006
, “
KBM Unified Method for Solving an nth Order Non-Linear Differential Equation Under Some Special Conditions Including the Case of Internal Resonance
,”
Int. J. Non-Linear Mech.
,
41
(
1
), pp.
26
42
.
3.
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
.
4.
Akgün
,
D.
, and
Çankaya
,
İ.
,
2010
, “
Frequency Response Investigations of Multi-Input Multi-Output Nonlinear Systems Using Automated Symbolic Harmonic Balance Method
,”
Nonlinear Dyn.
,
61
(
4
), pp.
803
818
.
5.
Liao
,
S. J.
,
2004
, “
An Analytic Approximate Approach for Free Oscillations of Self-Excited Systems
,”
Int. J. Non-Linear Mech.
,
39
(
2
), pp.
271
280
.
6.
Zhen
,
Y. X.
, and
Fang
,
B.
,
2015
, “
Nonlinear Vibration of Fluid-Conveying Single-Walled Carbon Nanotubes Under Harmonic Excitation
,”
Int. J. Non-Linear Mech.
,
76
, pp.
48
55
.
7.
Lim
,
C. W.
, and
Wu
,
B. S.
,
2005
, “
Accurate Higher-Order Approximations to Frequencies of Nonlinear Oscillators With Fractional Powers
,”
J. Sound Vib.
,
281
(3–5), pp.
1157
1162
.
8.
Chung
,
K. W.
,
He
,
Y. B.
, and
Lee
,
B. H. K.
,
2009
, “
Bifurcation Analysis of a Two-Degree-of-Freedom Aeroelastic System With Hysteresis Structural Nonlinearity by a Perturbation-Incremental Method
,”
J. Sound Vib.
,
320
(1–2), pp.
163
183
.
9.
Howcroft
,
C.
,
Lowenberg
,
M.
,
Neild
,
S.
,
Krauskopf
,
B.
, and
Coetzee
,
E.
,
2015
, “
Shimmy of an Aircraft Main Landing Gear With Geometric Coupling and Mechanical Freeplay
,”
ASME J. Comput. Nonlinear Dyn.
,
10
(
5
), p.
051011
.
10.
Yang
,
J. Y.
,
Peng
,
T.
, and
Lim
,
T. C.
,
2014
, “
An Enhanced Multi-Term Harmonic Balance Solution for Nonlinear Period Dynamic Motions in Right-Angle Gear Pairs
,”
Nonlinear Dyn.
,
76
(
2
), pp.
1237
1252
.
11.
Kim
,
Y. B.
, and
Noah
,
S. T.
,
1991
, “
Stability and Bifurcation Analysis of Oscillators With Piecewise-Linear Characteristics: A General Approach
,”
ASME J. Appl. Mech.
,
58
(
2
), pp.
545
553
.
12.
Villa
,
C.
,
Sinou
,
J. J.
, and
Thouverez
,
F.
,
2008
, “
Stability and Vibration Analysis of a Complex Flexible Rotor Bearing System
,”
Commun. Nonlinear Sci. Numer. Simul.
,
13
(
3
), pp.
804
821
.
13.
Zhang
,
Z. Y.
, and
Chen
,
Y. S.
,
2014
, “
Harmonic Balance Method With Alternating Frequency/Time Domain Technique for Nonlinear Dynamical System With Fractional Exponential
,”
Appl. Math. Mech.
,
35
(
4
), pp.
423
436
.
14.
Zhang
,
Z. Y.
,
Chen
,
Y. S.
, and
Cao
,
Q. J.
,
2015
, “
Bifurcations and Hysteresis of Varying Compliance Vibrations in the Primary Parametric Resonance for a Ball Bearing
,”
J. Sound Vib.
,
350
(
18
), pp.
171
184
.
15.
Chen
,
Y. M.
,
Meng
,
G.
, and
Liu
,
J. K.
,
2011
, “
A New Method for Fourier Series Expansions: Applications in Rotor-Seal Systems
,”
Mech. Res. Commun.
,
38
(
5
), pp.
399
403
.
16.
Liu
,
J. K.
,
Chen
,
F. X.
, and
Chen
,
Y. M.
,
2012
, “
Bifurcation Analysis of Aeroelastic Systems With Hysteresis by Incremental Harmonic Balance Method
,”
Appl. Math. Comput.
,
219
, pp.
2398
2411
.
17.
Davis
,
P. J.
, and
Rabinowitz
,
P.
,
2007
,
Methods of Numerical Integration
, 2nd ed., Dover Publications, Mineola, NY.
18.
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
.
19.
Lu
,
C. J.
, and
Lin
,
Y. M.
,
2011
, “
A Modified Incremental Harmonic Balance Method for Rotary Periodic Motions
,”
Nonlinear Dyn.
,
66
(
4
), pp.
781
788
.
20.
Chen
,
Y. M.
,
Liu
,
J. K.
, and
Meng
,
G.
,
2010
, “
Relationship Between the Homotopy Analysis Method and Harmonic Balance Method
,”
Commun. Nonlinear Sci. Numer. Simul.
,
15
(
8
), pp.
2017
2025
.
21.
Gottlieb
,
D.
, and
Shu
,
C. W.
,
1997
, “
On the Gibbs Phenomenon and Its Resolution
,”
SIAM Rev.
,
39
(
4
), pp.
644
668
.
22.
Eckhoff
,
K. S.
,
1995
, “
Accurate Reconstruction of Functions of Finite Regularity From Truncated Fourier Series Expansions
,”
Math. Comput.
,
64
(
210
), pp.
671
690
.
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