# 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 language English (US) 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 American Society of Mechanical Engineers 1933-1942 10 0791847381, 9780791847381 https://doi.org/10.1115/detc2005-85306 Published - 2005 Yes DETC2005: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - Long Beach, CA, United StatesDuration: Sep 24 2005 → Sep 28 2005

### Publication series

Name Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference - DETC2005 1 C

### Other

Other DETC2005: ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference United States Long Beach, CA 9/24/05 → 9/28/05

## Keywords

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

## ASJC Scopus subject areas

• Engineering(all)

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