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Research Papers

Computing Numerical Solutions of Delayed Fractional Differential Equations With Time Varying Coefficients

[+] Author and Article Information
Venkatesh Suresh Deshmukh

Department of Mechanical Engineering,
Villanova University,
800 Lancaster Avenue,
Villanova, PA 19085
e-mail: venkatesh.deshmukh@villanova.edu

Manuscript received March 5, 2013; final manuscript received April 21, 2014; published online September 12, 2014. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 10(1), 011004 (Sep 12, 2014) (5 pages) Paper No: CND-13-1056; doi: 10.1115/1.4027489 History: Received March 05, 2013; Revised April 21, 2014

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.

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References

Segalman, D. J., and Butcher, E. A., 2000, “Suppression of Regenerative Chatter Via Impedance Modulation,” J. Vib. Control, 6, pp. 243–256. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Gaul, L., Klein, P., and Kemple, S., 1989, “Damping Description Using Fractional Derivatives,” Mech. Syst. Signal Process., 5(2), pp. 8–88.
Hwang, C., and Cheng, Y.-C., 2006, “A Numerical Algorithm for Stability Testing of Fractional Delay Systems,” Automatica, 42, pp. 825–831. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic Press, New York.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, San Diego, CA.
Kleinz, M., and Osler, T., 2001, “A Child's Garden of Fractional Derivatives,” Coll. Math. J., 31(2), pp. 82–88. [CrossRef]
Chaterjee, A., 2005, “Statistical Origins of Fractional Derivatives in Viscoelasticity,” J. Sound Vib., 283(3–5), pp. 1239–1245. [CrossRef]
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. [CrossRef]
Trefethen, L. N., 2000, Spectral Methods in Matlab, SIAM, Philadelphia.
Shampine, L. F., and Thompson, F., 2001, “Solving DDEs in Matlab,” Appl. Numer. Math., 37, pp. 441–458. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Comparison of exact and approximate solutions of Eq. (22)

Grahic Jump Location
Fig. 2

The trajectories of underlying ordinary differential equation (ODE), delay differential equation (DDE), fractional ordinary differential equation(FODE) of Eq. (23), and fractional delay differential equation (FDDE (23))

Grahic Jump Location
Fig. 3

Displacements of underlying ODE, DDE, FODE of Eq. (24), and the FDDE (24) Eq. (1b)

Grahic Jump Location
Fig. 4

Displacements of underlying ODE, DDE, FODE of Eq. (24), and the FDDE (24)

Grahic Jump Location
Fig. 5

Displacements of underlying ODE, DDE, FODE of Eq. (24), and the FDDE (24) Eq. (1d)

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