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 (STMs) 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.
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February 2018
Research-Article
An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory
S. C. Sinha
S. C. Sinha
Life Fellow ASME
Alumni Professor Emeritus,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu
Alumni Professor Emeritus,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu
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Ashu Sharma
S. C. Sinha
Life Fellow ASME
Alumni Professor Emeritus,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu
Alumni Professor Emeritus,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu
Contributed by Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 6, 2017; final manuscript received August 17, 2017; published online November 9, 2017. Assoc. Editor: Bogdan I. Epureanu.
J. Comput. Nonlinear Dynam. Feb 2018, 13(2): 021008 (18 pages)
Published Online: November 9, 2017
Article history
Received:
March 6, 2017
Revised:
August 17, 2017
Citation
Sharma, A., and Sinha, S. C. (November 9, 2017). "An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory." ASME. J. Comput. Nonlinear Dynam. February 2018; 13(2): 021008. https://doi.org/10.1115/1.4037797
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