Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds

S. C. Sinha, Sangram Redkar, Eric A. Butcher

Research output: Contribution to journalArticle

39 Citations (Scopus)

Abstract

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via 'Time Periodic Center Manifold Theory'. A 'reducibility condition' is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show potential applications to real problems. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'combination' resonances are discussed.

Original languageEnglish (US)
Pages (from-to)985-1002
Number of pages18
JournalJournal of Sound and Vibration
Volume284
Issue number3-5
DOIs
StatePublished - Jun 21 2005
Externally publishedYes

Fingerprint

nonlinear systems
Nonlinear systems
coefficients
pendulums
Pendulums
Equations of motion
equations of motion
methodology
perturbation
excitation

ASJC Scopus subject areas

  • Engineering(all)
  • Mechanical Engineering

Cite this

Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds. / Sinha, S. C.; Redkar, Sangram; Butcher, Eric A.

In: Journal of Sound and Vibration, Vol. 284, No. 3-5, 21.06.2005, p. 985-1002.

Research output: Contribution to journalArticle

@article{cf77e907e53e4b529aa95f2656c928a0,
title = "Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds",
abstract = "The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via 'Time Periodic Center Manifold Theory'. A 'reducibility condition' is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show potential applications to real problems. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'combination' resonances are discussed.",
author = "Sinha, {S. C.} and Sangram Redkar and Butcher, {Eric A.}",
year = "2005",
month = "6",
day = "21",
doi = "10.1016/j.jsv.2004.07.027",
language = "English (US)",
volume = "284",
pages = "985--1002",
journal = "Journal of Sound and Vibration",
issn = "0022-460X",
publisher = "Academic Press Inc.",
number = "3-5",

}

TY - JOUR

T1 - Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds

AU - Sinha, S. C.

AU - Redkar, Sangram

AU - Butcher, Eric A.

PY - 2005/6/21

Y1 - 2005/6/21

N2 - The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via 'Time Periodic Center Manifold Theory'. A 'reducibility condition' is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show potential applications to real problems. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'combination' resonances are discussed.

AB - The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via 'Time Periodic Center Manifold Theory'. A 'reducibility condition' is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show potential applications to real problems. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'combination' resonances are discussed.

UR - http://www.scopus.com/inward/record.url?scp=18444364818&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=18444364818&partnerID=8YFLogxK

U2 - 10.1016/j.jsv.2004.07.027

DO - 10.1016/j.jsv.2004.07.027

M3 - Article

AN - SCOPUS:18444364818

VL - 284

SP - 985

EP - 1002

JO - Journal of Sound and Vibration

JF - Journal of Sound and Vibration

SN - 0022-460X

IS - 3-5

ER -