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Research Papers

# Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems1

[+] Author and Article Information
W. Grant Kirkland

Nonlinear Systems Research Laboratory,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: gkirkland@auburn.edu

S. C. Sinha

Life Fellow ASME
Professor
Nonlinear Systems Research Laboratory,
Department of Mechanical Engineering,
Auburn University,
Auburn, AL 36849
e-mail: ssinha@eng.auburn.edu

2Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 1, 2015; final manuscript received March 30, 2016; published online May 13, 2016. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 11(4), 041022 (May 13, 2016) (10 pages) Paper No: CND-15-1197; doi: 10.1115/1.4033382 History: Received July 01, 2015; Revised March 30, 2016

## 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 (STM) $Φ(t,α)$, associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix $Q(t,α)$ and a time-invariant matrix $R(α)$ containing a set of symbolic system parameters $α.$ Computation of $Q(t,α)$ and $R(α)$ in symbolic form as a function of $α$ is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, 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 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the $Q(t,α)$ matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient 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 a 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.

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## Figures

Fig. 7

2T-periodic Q̂(t) matrix of the DME (unstable)

Fig. 8

Inverted double pendulum (IDP) [8]

Fig. 9

CPU time to compute Q̂(t,α) versus number of summation terms for the IDP

Fig. 10

Relative error versus number of Gaussian nodes of the R̂ matrix for the IDP

Fig. 11

R̂ matrix over two periods of the IDP

Fig. 12

Relative error versus number of expansion terms of the Q̂ matrix for the IDP

Fig. 13

T-periodic Q̂11 and Q̂12 of the IDP

Fig. 1

CPU time to compute R̂(α) versus number of Gaussian nodes for the DME

Fig. 2

CPU time to compute Q̂(t,α) versus number of summation terms for the DME

Fig. 4

R̂ matrix over two periods of the DME (stable)

Fig. 5

Relative error versus number of summation terms of the Q̂ matrix (stable DME)

Fig. 6

T-periodic Q̂(t) matrix of the DME (stable)

Fig. 3

Relative error versus number of Gaussian nodes of the R̂ matrix (stable DME)

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