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.
Issue Section:
Research Papers
References
1.
Mesbahi
, A.
, Haeri
, M.
, Nazari
, M.
, and Butcher
, E. A.
, 2015
, “Fractional Delayed Damped Mathieu Equation
,” Int. J. Control
, 88
(3
), pp. 622
–630
.2.
Rand
, R. H.
, Sah
, S. M.
, and Suchorsky
, M. K.
, 2010
, “Fractional Mathieu Equation
,” Commun. Nonlinear Sci. Numer. Simul.
, 15
(11
), pp. 3254
–3262
.3.
Khalil
, H. K.
, and Grizzle
, J.
, 2002
, Nonlinear Systems
, Vol. 3
, Prentice Hall
, Upper Saddle River, NJ
.4.
Nayfeh
, A. H.
, and Balachandran
, B.
, 2008
, Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods
, Wiley
, Hoboken, NJ
.5.
Montagnier
, P.
, Spiteri
, R. J.
, and Angeles
, J.
, 2004
, “The Control of Linear Time-Periodic Systems Using Floquet–Lyapunov Theory
,” Int. J. Control
, 77
(5
), pp. 472
–490
.6.
Genesio
, R.
, and Tesi
, A.
, 1992
, “Harmonic Balance Methods for the Analysis of Chaotic Dynamics in Nonlinear Systems
,” Automatica
, 28
(3
), pp. 531
–548
.7.
Insperger
, T.
, and Stépán
, G.
, 2002
, “Stability Chart for the Delayed Mathieu Equation
,” Proc. R. Soc. London, Ser. A
, 458
(2024
), pp. 1989
–1998
.8.
Liu
, L.
, Thomas
, J. P.
, Dowell
, E. H.
, Attar
, P.
, and Hall
, K. C.
, 2006
, “A Comparison of Classical and High Dimensional Harmonic Balance Approaches for a Duffing Oscillator
,” J. Comput. Phys.
, 215
(1
), pp. 298
–320
.9.
Butcher
, E. A.
, Bobrenkov
, O. A.
, Bueler
, E.
, and Nindujarla
, P.
, 2009
, “Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels
,” ASME J. Comput. Nonlinear Dyn.
, 4
(3
), p. 031003
.10.
Nazari
, M.
, Butcher
, E. A.
, and Schaub
, H.
, 2013
, “Spacecraft Attitude Stabilization Using Nonlinear Delayed Multiactuator Control and Inverse Dynamics
,” J. Guid., Control, Dyn.
, 36
(5
), pp. 1440
–1452
.11.
Samiei
, E.
, Nazari
, M.
, Butcher
, E.
, and Schaub
, H.
, 2012
, “Delayed Feedback Control of Rigid Body Attitude Using Neural Networks and Lyapunov-Krasovskii Functionals
,” AAS/AIAA Spaceflight Mechanics Meeting
, Charleston, SC, pp. 12
–168
.12.
Samko
, S. G.
, Kilbas
, A. A.
, and Marichev
, O. I.
, 1993
, Fractional Integrals and Derivatives: Theory and Applications
, Gordon & Breach Science Publishers
, Yverdon
.13.
Das
, S.
, 2011
, Functional Fractional Calculus
, Springer Science & Business Media
, Berlin, Heidelberg
.14.
Tavazoei
, M. S.
, and Haeri
, M.
, 2009
, “A Proof for Non Existence of Periodic Solutions in Time Invariant Fractional Order Systems
,” Automatica
, 45
(8
), pp. 1886
–1890
.15.
Hilfer
, R.
, 2008
, “Threefold Introduction to Fractional Derivatives
,” Anomalous Transport: Foundations and Applications
, pp. 17
–73
.16.
Mees
, A. I.
, 1981
, Dynamics of Feedback Systems
, Wiley
, Hoboken, NJ
.17.
Moosavian
, S. A. A.
, and Dabiri
, A.
, 2010
, “Dynamics and Planning for Stable Motion of a Hexapod Robot
,” 2010 IEEE/ASME International Conference on Advanced Intelligent Mechatronics (AIM)
, IEEE
, Montreal, Canada
, July 6–9, pp. 818
–823
.Copyright © 2016 by ASME
You do not currently have access to this content.