Abstract

By introducing linear and constant terms with an undetermined parameter and subsequently using certain rules to determine the optimal value of the parameter, we establish analytical approximate frequencies and the corresponding periodic solutions for strongly mixed-parity nonlinear oscillators. A quadratic–cubic nonlinear oscillator is used to verify and illustrate the usefulness and effectiveness of the proposed method.

Issue Section:
Research Papers
Keywords:
nonlinear dynamical systems, nonlinear differential equations, oscillators, perturbation theory, modified Lindstedt–Poincaré method, strong nonlinearity, mixed-parity
Topics:
Parity (Physics), Oscillations
1.
Cole
,
J. D.
, 1968,
Perturbation Methods in Applied Mathematics
,
Waltham
, Blaudell.
2.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
, 1979,
Nonlinear Oscillations
,
Wiley
, New York.
Crossref
Search ADS
 
3.
Hagedorn
,
P.
, 1988,
Nonlinear Oscillations
,
Clarendon
, Oxford.
4.
Mickens
,
R. E.
, 1996,
Oscillations in Planar Dynamic Systems
,
World Scientific
, Singapore.
Crossref
Search ADS
 
5.
Cheung
,
Y. K.
,
Chen
,
S. H.
, and
Lau
,
S. L.
, 1991, “
A Modified Lindstedt-Poincaré Method for Certain Strongly Non-linear Oscillators
,”
Int. J. Non-Linear Mech.
0020-7462,
26
, pp.
367
–
378
.
Crossref
Search ADS
 
6.
Senator
,
M.
, and
Bapat
,
C. N.
, 1993, “
A Perturbation Technique that Works Even when the Non-linearity is Not Small
,”
J. Sound Vib.
0022-460X,
164
, pp.
1
–
27
.
Crossref
Search ADS
 
7.
Amore
,
P.
, and
Aranda
,
A.
, 2003, “
Presenting a New Method for the Solution of Nonlinear Problems
,”
Phys. Lett. A
0375-9601,
316
, pp.
218
–
225
.
Crossref
Search ADS
 
8.
Wu
,
B. S.
,
Lim
,
C. W.
, and
Li
,
P. S.
, 2005, “
A Generalization of the Senator-Bapat Method for Certain Strongly Nonlinear Oscillators
,”
Phys. Lett. A
0375-9601,
341
, pp.
164
–
169
.
Crossref
Search ADS
 
9.
Mickens
,
R. E.
, 1981, “
A uniformly valid perturbation solution of ü+u=A+εu2
,”
J. Sound Vib.
0022-460X,
76
, pp.
150
–
153
.
Crossref
Search ADS
 
10.
Mickens
,
R. E.
, 1993, “
Construction of a Perturbation Solution to a Mixed Parity System that Satisfies the Correct Initial Conditions
,”
J. Sound Vib.
0022-460X,
167
, pp.
564
–
567
.
Crossref
Search ADS
 
11.
Stevenson
,
P. M.
, 1981, “
Optimized Perturbation Theory
,”
Phys. Rev. D
0556-2821,
23
, pp.
2916
–
2944
.
Crossref
Search ADS
 
12.
Amore
,
P.
, and
Saenz
,
R. A.
, 2005, “
The Period of a Classical Oscillator
,”
Europhys. Lett.
0295-5075,
70
, pp.
425
–
431
.
Crossref
Search ADS
 
Copyright © 2007
 by American Society of Mechanical Engineers
You do not currently have access to this content.