In this note, a closed-form solution of periodic motions in a periodically forced oscillator with quadratic nonlinearity is presented without any small parameters. The perturbation method is based on one harmonic term plus perturbation modification, and the traditional harmonic balance is to arbitrarily select harmonic terms with constant coefficients. If harmonic terms are not enough included in the approximate solution, such a solution is not an appropriate, analytical solution for periodic motions, and some analytical solutions cannot be caught.

Issue Section:
Technical Brief
Topics:
Bifurcation, Eigenvalues, Excitation, Stability, Fourier series, Steady state

References

1.
Poincare
,
H.
,
1899
,
Methodes Nouvelles de la Mecanique Celeste
, Vol.
3
,
Gauthier-Villars
,
Paris
.
2.
van der Pol
,
B.
,
1920
, “
A Theory of the Amplitude of Free and Forced Triode Vibrations
,”
Radio Rev.
,
1
, pp.
701
–
710
, 754–762.
3.
Hayashi
,
G.
,
1964
,
Nonlinear Oscillations in Physical Systems
,
McGraw-Hill Book Company
,
New York
.
4.
Nayfeh
,
A. H.
,
1973
,
Perturbation Methods
,
John Wiley
,
New York
.
5.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillation
,
John Wiley
,
New York
.
6.
Luo
,
A. C. J.
, and
Han
,
R. P. S.
,
1997
, “
A Quantitative Stability and Bifurcation Analyses of a Generalized Duffing Oscillator With Strong Nonlinearity
,”
J. Franklin Inst.
,
334B
, pp.
447
–
459
.10.1016/S0016-0032(96)00089-0
Google Scholar
Crossref
Search ADS
 
7.
Peng
,
Z. K.
,
Lang
,
Z. Q.
,
Billings
,
S. A.
, and
Tomlinson
,
G. R.
,
2008
, “
Comparisons Between Harmonic Balance and Nonlinear Output Frequency Response Function in Nonlinear System Analysis
,”
J. Sound Vib.
,
311
, pp.
56
–
73
.10.1016/j.jsv.2007.08.035
Google Scholar
Crossref
Search ADS
 
8.
Luo
,
A. C. J.
, and
Huang
,
J. Z.
,
2012
, “
Approximate Solutions of Periodic Motions in Nonlinear Systems via a Generalized Harmonic Balance
,”
J. Vibr. Control
,
18
, pp.
1661
–
1674
.10.1177/1077546311421053
Google Scholar
Crossref
Search ADS
 
9.
Luo
,
A. C. J.
, and
Huang
,
J. Z.
,
2012
, “
Analytical Dynamics of Period-m Flows and Chaos in Nonlinear Systems
,”
Int. J. Bifurcation Chaos
,
22
, p. 1250093.10.1142/S0218127412500939
10.
Luo
,
A. C. J.
,
2012
,
Continuous Dynamical Systems
,
HEP-L&H Scientific
,
Glen Carbon, IL
.
Copyright © 2013 by ASME
You do not currently have access to this content.