Order reduction of parametrically excited nonlinear systems

Techniques and applications

S. C. Sinha, Sangram Redkar, Vankatesh Deshmukh, Eric Butcher

Research output: Contribution to journalArticle

19 Citations (Scopus)

Abstract

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique 'reducibility condition' that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L-F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same 'reducibility conditions' obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'true combination resonances' are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

Original languageEnglish (US)
Pages (from-to)237-273
Number of pages37
JournalNonlinear Dynamics
Volume41
Issue number1-3
DOIs
StatePublished - Aug 2005
Externally publishedYes

Fingerprint

Order Reduction
Nonlinear systems
Nonlinear Systems
Invariant Manifolds
State Space
Reducibility
Lyapunov
Internal
Parametric Excitation
Canonical Transformation
Invariant
Reduced Order Model
Methodology
Periodic Coefficients
Time varying systems
Second-order Systems
Singular Perturbation
Reduction Method
Post-processing
Alternate

Keywords

  • Lyapunov-Floquet transformation
  • Nonlinear time periodic systems
  • Order reduction

ASJC Scopus subject areas

  • Mechanical Engineering
  • Mechanics of Materials
  • Computational Mechanics

Cite this

Order reduction of parametrically excited nonlinear systems : Techniques and applications. / Sinha, S. C.; Redkar, Sangram; Deshmukh, Vankatesh; Butcher, Eric.

In: Nonlinear Dynamics, Vol. 41, No. 1-3, 08.2005, p. 237-273.

Research output: Contribution to journalArticle

Sinha, S. C. ; Redkar, Sangram ; Deshmukh, Vankatesh ; Butcher, Eric. / Order reduction of parametrically excited nonlinear systems : Techniques and applications. In: Nonlinear Dynamics. 2005 ; Vol. 41, No. 1-3. pp. 237-273.
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