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.

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
.
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