Fractional differential equations with time varying coefficients and delay are encountered in the analysis of models of metal cutting processes such as milling and drilling with viscoelastic damping elements. Viscoelastic damping is modeled as a fractional derivative. In the present paper, delayed fractional differential equations with bounded time varying coefficients in four different forms are analyzed using series solution and Chebyshev spectral collocation. A fractional differential equation with a known exact solution is then solved by the methodology presented in the paper. The agreement between the two is found to be excellent in terms of point-wise error in the trajectories. Solutions to the described fractional differential equations are computed next in state space and second order forms.

References

1.
Segalman
,
D. J.
, and
Butcher
,
E. A.
,
2000
, “
Suppression of Regenerative Chatter Via Impedance Modulation
,”
J. Vib. Control
,
6
, pp.
243
256
.10.1177/107754630000600205
2.
Rossikhin
,
Y. A.
, and
Shitikova
,
M. V.
,
1997
, “
Application of Fractional Derivatives to the Analysis of Damped Vibrations of Viscoelastic Single Mass Systems
,”
Acta Mech.
,
120
(
1–4
), pp.
109
125
.10.1007/BF01174319
3.
Yuan
,
L.
, and
Agrawal
,
O. P.
,
2002
, “
A Numerical Scheme for Dynamic Systems Containing Fractional Derivatives
,”
ASME J. Vib. Acoust.
,
124
(
2
), pp.
321
324
.10.1115/1.1448322
4.
Gaul
,
L.
,
Klein
,
P.
, and
Kemple
,
S.
,
1989
, “
Damping Description Using Fractional Derivatives
,”
Mech. Syst. Signal Process.
,
5
(
2
), pp.
8
88
.
5.
Hwang
,
C.
, and
Cheng
,
Y.-C.
,
2006
, “
A Numerical Algorithm for Stability Testing of Fractional Delay Systems
,”
Automatica
,
42
, pp.
825
831
.10.1016/j.automatica.2006.01.008
6.
Fioravanti
,
A. R.
,
Bonnet
,
C.
,
Ozbay
,
H.
, and
Nicolescu
,
S.-I.
,
2012
, “
A Numerical Method for Stability Windows and Unstable Root Locus Calculation for Linear Fractional Time Delay Systems
,”
Automatica
,
48
, pp.
2824
2830
.10.1016/j.automatica.2012.04.009
7.
Chen
,
Y.
, and
Moore
,
K. L.
,
2002
, “
Analytical Stability Bound for a Class of Delayed Fractional Order Dynamic Systems
,”
Nonlinear Dyn.
,
29
, pp.
191
200
.10.1023/A:1016591006562
8.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
9.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
10.
Kleinz
,
M.
, and
Osler
,
T.
,
2001
, “
A Child's Garden of Fractional Derivatives
,”
Coll. Math. J.
,
31
(
2
), pp.
82
88
.10.2307/2687575
11.
Chaterjee
,
A.
,
2005
, “
Statistical Origins of Fractional Derivatives in Viscoelasticity
,”
J. Sound Vib.
,
283
(
3–5
), pp.
1239
1245
.10.1016/j.jsv.2004.09.019
12.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
,
27
(
3
), pp.
201
210
.10.1122/1.549724
13.
Trefethen
,
L. N.
,
2000
,
Spectral Methods in Matlab
,
SIAM
,
Philadelphia
.
14.
Shampine
,
L. F.
, and
Thompson
,
F.
,
2001
, “
Solving DDEs in Matlab
,”
Appl. Numer. Math.
,
37
, pp.
441
458
.10.1016/S0168-9274(00)00055-6
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