Order Reduction of Nonlinear Quasi-periodic Systems Subjected to External Excitations

TL;DR

This paper introduces order reduction methods for nonlinear quasi-periodic systems using Lyapunov-Perrone transformation, enabling analysis via linear time-invariant systems and proposing techniques based on Guyan reduction and invariant manifolds.

Contribution

The paper presents novel order reduction techniques for nonlinear quasi-periodic systems utilizing Lyapunov-Perrone transformation and invariant manifold concepts, advancing systematic model reduction methods.

Findings

L-P transformation converts quasi-periodic systems into LTI systems.

Guyan-like reduction simplifies nonlinear systems effectively.

Invariant manifold approach reveals resonant interaction conditions.

Abstract

In his paper, we present order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov-Perrone (L-P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L-P transformation. This approach is similar to using the Lyapunov-Floquet (L-F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control. Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems' essential dynamics. In this work, reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations are presented. The methods proposed here use the L-P transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems. The 'quasi-periodic invariant manifold' based technique yields' reducibility conditions.' These conditions help us to understand the various types of resonant interactions in the system. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation.