Research Papers

Transition Curve Analysis of Linear Fractional Periodic Time-Delayed Systems Via Explicit Harmonic Balance Method

[+] Author and Article Information
Eric A. Butcher

Associate Professor
Department of Aerospace
and Mechanical Engineering,
University of Arizona,
Tucson, AZ 85721
e-mail: ebutcher@email.arizona.edu

Arman Dabiri

Department of Aerospace
and Mechanical Engineering,
University of Arizona,
Tucson, AZ 85721
e-mail: armandabiri@arizona.email.edu

Morad Nazari

Adjunct Faculty
Department of Mechanical,
Industrial, and Manufacturing Engineering,
University of Toledo,
Toledo, OH 43606
e-mail: morad.nazari@utoledo.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 28, 2015; final manuscript received August 28, 2015; published online November 13, 2015. Assoc. Editor: Gabor Stepan.

J. Comput. Nonlinear Dynam 11(4), 041005 (Nov 13, 2015) (6 pages) Paper No: CND-15-1146; doi: 10.1115/1.4031840 History: Received May 28, 2015; Revised August 28, 2015

This paper presents a technique to obtain the transition curves of fractional periodic time-delayed (FPTD) systems based on a proposed explicit harmonic balance (EHB) method. This method gives the analytical Hill matrix of FPTD systems explicitly with a symbolic computation-free algorithm. Furthermore, all linear operations on Fourier basis vectors including fractional order derivative operators and time-delayed operators for a linear FPTD system are obtained. This technique is illustrated with parametrically excited simple and double pendulum systems, with both time-delayed states and fractional damping.

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Grahic Jump Location
Fig. 1

(a) A parametrically excited pendulum with integer and fractional damping and delayed proportional feedback. (b) Convergence of the solution by increasing the number of Fourier terms. The shaded regions are the stability boundaries for N = 17. (c) Effect of integer damping coefficient on the transition curves. The shaded regions are the stability boundaries for cb=0. (d) Effect of fractional derivative coefficient on the transition curves. The shaded regions are the stability boundaries for cf=0. (e) Effect of order of the fractional damping on transition curves. The shaded regions are the stability boundaries for α = 0. (f) Effect of the coefficient of the delayed term on the transition curves. The shaded regions are the stability boundaries for cd=0. (g) Effect of time delay on the transition curves. The shaded regions are the stability boundaries for τ = 0.

Grahic Jump Location
Fig. 2

(a) A double pendulum with a retarded follower force equivalent and fractional damping. (b) Effect of damping coefficient on the flip and fold transition curves of the double pendulum. (c) Effect of fractional damping coefficient on the transition curves of the double pendulum. (d) Coincidence of two flip and fold curves in the circle at cf=0.5 generates another tongue. (e) Effect of fractional order on the transition curves of the double pendulum. (f) Transition curves for the double pendulum by using the EHB method. The secondary Hopf curves are shown explicitly.



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