0

Full Content is available to subscribers

Subscribe/Learn More  >

An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory

[+] Author Affiliations
Ashu Sharma, Subhash C. Sinha

Auburn University, Auburn, AL

Paper No. DETC2017-68041, pp. V006T10A065; 15 pages
doi:10.1115/DETC2017-68041
From:
  • ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 6: 13th International Conference on Multibody Systems, Nonlinear Dynamics, and Control
  • Cleveland, Ohio, USA, August 6–9, 2017
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5820-2
  • Copyright © 2017 by ASME

abstract

Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been 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. Although Floquét theory is applicable only to 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 two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the 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. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.

Copyright © 2017 by ASME

Figures

Tables

Interactive Graphics

Video

Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature

Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal

NOTE:
Citing articles are presented as examples only. In non-demo SCM6 implementation, integration with CrossRef’s "Cited By" API will populate this tab (http://www.crossref.org/citedby.html).

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 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