zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Linearized Galerkin and artificial parameter techniques for the determination of periodic solutions of nonlinear oscillators. (English) Zbl 1135.65344
Summary: Linearized harmonic balance-Galerkin methods for the determination of the periodic solutions of nonlinear oscillators are presented and the results are compared with those of an artificial parameter method based on the introduction of a linear stiffness term and the Linstedt-Poincaré technique. The linearized harmonic balance methods presented in this paper are not iterative and are based on Taylor series expansions, the neglect of nonlinear terms in the resulting series, a Fourier approximation to the solution, and orthogonality conditions. It is shown that the first approximation provided by the linearized harmonic balance technique is identical to that of the artificial parameter method, whereas the second one depends on how the equation is linearized and written, and the discrepancies have been found to be caused by the neglect of the nonlinear terms in the Taylor series expansion because a Fourier series expansion of these terms introduces harmonics which are not accounted for in the linearization procedure. It is also shown that the first approximation of the artificial parameter method and the linearized harmonic balance procedure are asymptotically equivalent to that derived from a modified Linstedt-Poincaré method, when the nonlinearities are small, and that these techniques may not be applicable to some oscillators.

MSC:
65L05Initial value problems for ODE (numerical methods)
34A34Nonlinear ODE and systems, general
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
WorldCat.org
Full Text: DOI
References:
[1] Nayfeh, A. H.: Perturbation methods. (1973) · Zbl 0265.35002
[2] Nayfeh, A. H.; Mook, D. T.: Nonlinear oscillations. (1979) · Zbl 0418.70001
[3] Kevorkian, J.; Cole, J. D.: Multiple scale and singular perturbation methods. (1996) · Zbl 0846.34001
[4] Mickens, R. E.: An introduction to nonlinear oscillations. (1981) · Zbl 0459.34002
[5] Wu, B. S.; Sun, W. P.; Lim, C. W.: An analytical approximate technique for a class of strongly non-linear oscillators. Int. J. Non-linear mech. 41, 766-774 (2006) · Zbl 1160.70340
[6] Wu, B. S.; Lim, C. W.; Ma, Y. F.: Analytical approximation to large-amplitude oscillation of a non-linear conservative system. Int. J. Non-linear mech. 38, 1037-1043 (2003) · Zbl 05138203
[7] Wu, B. S.; Lim, C. W.: Large amplitude non-linear oscillations of a general conservative system. Int. J. Non-linear mech. 39, 859-870 (2004) · Zbl 05138496
[8] Wu, B.; Li, P.: A method for obtaining approximate analytic periods for a class of nonlinear oscillators. Meccanica 36, 167-176 (2001) · Zbl 1008.70016
[9] Mickens, R. E.: A generalized iteration procedure for calculating approximations to periodic solutions of ”truly nonlinear oscillators”. J. sound vibr. 287, 1045-1051 (2005) · Zbl 1243.65079
[10] Lim, C. W.; Wu, B. S.: A modified Mickens procedure for certain non-linear oscillators. J. sound vibr. 257, 202-206 (2002) · Zbl 1237.70109
[11] Mickens, R. E.: Iteration method solutions for conservative and limit-cycle x1/3 force oscillators. J. sound vibr. 292, 964-968 (2006) · Zbl 1243.34051
[12] Hu, H.: Solutions of nonlinear oscillators with fractional powers by an iterative procedure. J. sound vibr. 294, 608-614 (2006) · Zbl 1243.34005
[13] Marinca, V.; Herinasu, N.: A modified iteration perturbation method for some nonlinear oscillation problems. Acta mech. 184, 142-231 (2006)
[14] He, Ji-H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Modern phys. 20, 1141-1199 (2006) · Zbl 1102.34039
[15] Bender, C. M.; Milton, K. A.; Pinsky, S. S.; Simmons, L. M.: A new perturbative approach to nonlinear problems. J. math. Phys. 30, 1447-1455 (1989) · Zbl 0684.34008
[16] J.I. Ramos, On Linstedt -- Poincaré techniques for the quintic Duffing equation, Appl. Math. Comput., in press, doi:10.1016/j.amc.2007.03.050. · Zbl 1193.65142
[17] Adomian, G.: Stochastic systems. (1983) · Zbl 0523.60056
[18] Adomian, G.: Nonlinear stochastic operator equations. (1986) · Zbl 0609.60072
[19] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122
[20] He, Ji-H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. math. Comput. 135, 73-79 (2003) · Zbl 1030.34013
[21] He, Ji-H.: Addendum: new interpretation of homotopy perturbation method. Int. J. Modern phys. 20, 2561-2568 (2006)
[22] He, Ji-H.: Homotopy perturbation technique. Comput. methods appl. Mech. eng. 178, 257-262 (1999) · Zbl 0956.70017
[23] Liao, S. -J.: An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude. Int. J. Nonlinear mech. 38, 1173-1183 (2003) · Zbl 05138214
[24] Liao, S. J.: Beyond perturbation. (2003)
[25] Xu, L.: He’s parameter-expanding methods for strongly nonlinear oscillators. J. comput. Appl. math. 207, 148-154 (2007) · Zbl 1120.65084
[26] Hu, H.: A classical perturbation technique that works even when the linear part of restoring force is zero. J. sound vibr. 271, 1175-1179 (2004) · Zbl 1236.34082
[27] Pierre, C.; Dowell, E. H.: A study of dynamic instability of plates by an extended incremental harmonic balance method. ASME J. Appl. mech. 52, 693-697 (1985) · Zbl 0568.73053
[28] Pierre, C.; Ferri, A. A.; Dowell, E. H.: Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. ASME J. Appl. mech. 52, 958-964 (1985) · Zbl 0584.73109
[29] Lau, S. L.; Cheung, Y. K.: Amplitude incremental variational principle for nonlinear vibration of elastic systems. ASME J. Appl. mech. 48, 959-964 (1981) · Zbl 0468.73066
[30] Lau, S. L.; Cheung, Y. K.; Wu, S. Y.: A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. ASME J. Appl. mech. 49, 849-853 (1982) · Zbl 0503.73031
[31] Ferri, A. A.: On the equivalence of the incremental harmonic balance method and the harmonic balance-Newton raphson method. ASME J. Appl. mech. 53, 455-457 (1986)