0
research-article

An Approximate Analysis of Quasi-Periodic Systems via Floqu?t Theory

[+] Author and Article Information
Ashu Sharma

Student Member, ASME, Graduate Student, Department of Mechanical Engineering, Auburn University, AL 36849, USA
azs0111@auburn.edu

Subhash C. Sinha

Life Fellow, ASME, Alumni Professor Emeritus, Department of Mechanical Engineering, Auburn University, AL 36849, USA
ssinha@eng.auburn.edu

1Corresponding author.

ASME doi:10.1115/1.4037797 History: Received March 06, 2017; Revised August 17, 2017

Abstract

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

Copyright (c) 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Figures

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In