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Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems

[+] Author Affiliations
W. Grant Kirkland, S. C. Sinha

Auburn University, Auburn, AL

Paper No. DETC2015-47486, pp. V006T10A065; 12 pages
doi:10.1115/DETC2015-47486
From:
  • ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control
  • Boston, Massachusetts, USA, August 2–5, 2015
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5716-8
  • Copyright © 2015 by ASME

abstract

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix Φ(t,α) associated with the linear part of the equation can be expressed in terms of the periodic Lyapunov-Floquét transformation matrix Q(t,α) and a time-invariant matrix R(α). Computation of Q(t,α) and R(α) in a symbolic form as a function of system parameters α is of paramount importance in stability, bifurcation analysis, and control system design. In the past, a methodology has been presented for computing Φ(t,α) in a symbolic form, however Q(t,α) and R(α) have never been calculated in a symbolic form. Since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In this work a technique for symbolic computation of Q(t,α), and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then R(α) is computed using the Gaussian quadrature integral formula. Finally Q(t,α) is computed using the matrix exponential summation method. Using Mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

Copyright © 2015 by ASME

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