Order reduction of nonlinear systems with periodic-quasiperiodic coefficients

Sangram Redkar, S. C. Sinha

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this work, some techniques for order reduction of nonlinear systems involving periodic/quasiperiodic coefficients are presented. The periodicity of the linear terms is assumed non-commensurate with the periodicity of either the nonlinear terms or the forcing vector. The dynamical evolution equations are transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of the resulting equations become time-invariant while the nonlinear parts and forcing take the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-modulated functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states. Three methods are proposed to carry out this model order reduction (MOR). First type of MOR technique is a linear method similar to the 'Guyan reduction', the second technique is a nonlinear projection method based on singular perturbation while the third method utilizes the concept of 'quasiperiodic invariant manifold'. Order reduction approach based on invariant manifold technique yields a unique 'generalized reducibility condition'. If this 'reducibility condition' is satisfied only then an accurate order reduction via invariant manifold is possible. Next, the proposed methodologies are extended to solve the forced problem. All order reduction approaches except the invariant manifold technique can be applied in a straightforward way. The invariant manifold formulation is modified to take into account the effects of forcing and nonlinear coupling. This approach not only yields accurate reduced order models but also explains the consequences of various 'primary' and ''secondary resonances' present in the system. One can also recover all 'resonance conditions' obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handing systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. These methodologies are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems.

Original languageEnglish (US)
Title of host publicationProceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conferences - DETC2005
Subtitle of host publication20th Biennial Conf. on Mech. Vibration and Noise
PublisherAmerican Society of Mechanical Engineers
Pages1933-1942
Number of pages10
ISBN (Print)0791847381, 9780791847381
DOIs
StatePublished - 2005
Externally publishedYes
EventDETC2005: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - Long Beach, CA, United States
Duration: Sep 24 2005Sep 28 2005

Publication series

NameProceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005
Volume1 C

Other

OtherDETC2005: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
CountryUnited States
CityLong Beach, CA
Period9/24/059/28/05

Keywords

  • L-F Transformation
  • Nonlinear Forced Systems
  • Order reduction

ASJC Scopus subject areas

  • Engineering(all)

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  • Cite this

    Redkar, S., & Sinha, S. C. (2005). Order reduction of nonlinear systems with periodic-quasiperiodic coefficients. In Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conferences - DETC2005: 20th Biennial Conf. on Mech. Vibration and Noise (pp. 1933-1942). (Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005; Vol. 1 C). American Society of Mechanical Engineers. https://doi.org/10.1115/detc2005-85306