Abstract
Ordinary differential equations (ODEs) with fractional order derivatives are infinite dimensional systems and nonlocal in time: the history of the state variable is needed to calculate the instantaneous rate of change. This nonlocal nature leads to expensive long-time computations (O(t 2) computations for solution up to time t). A finite dimensional approximation of the fractional order derivative can alleviate this problem. We present one such approximation using a Galerkin projection. The original infinite dimensional system is replaced with an equivalent infinite dimensional system involving a partial differential equation (PDE). The Galerkin projection reduces the PDE to a finite system of ODEs. These ODEs can be solved cheaply (O(t) computations). The shape functions used for the Galerkin projection are important, and given attention. The approximation obtained is specific to the fractional order of the derivative; but can be used in any system with a derivative of that order. Calculations with both global shape functions as well as finite elements are presented. The discretization strategy is improved in a few steps until, finally, very good performance is obtained over a user-specifiable frequency range (not including zero). In particular, numerical examples are presented showing good performance for frequencies varying over more than 7 orders of magnitude. For any discretization held fixed, however, errors will be significant at sufficiently low or high frequencies. We discuss why such asymptotics may not significantly impact the engineering utility of the method.
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References
Bagley, R. L. and Torvik, P. J., ‘Fractional calculus: A different approach to the analysis of viscoelastically damped structures’, AIAA Journal 21(5), 1983, 741–748.
Bagley, R. L. and Torvik, P. J.,` Fractional calculus in the transient analysis of viscoelastically damped structures’, AIAA Journal 23(6), 1985, 918–925.
Gaul, L., Klein, P., and Kempfle, S., ‘Impulse response function of an oscillator with fractional derivative in damping description’, Mechanics Research Communications 16(5), 1989, 297–305.
Makris, N., ‘Fractional derivative model for viscous damper’, ASCE Journal of Structural Engineering 117, 1991, 2708–2724.
Chen, Y. and Moore, K. L., ‘On D α-type iterative learning control’, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2001, pp. 4451–4456.
Debnath, L., ‘Recent applications of fractional calculus to science and engineering’, International Journal of Mathematics and Mathematical Sciences 2003(54), 2003, 3413–3442.
Moreau, X., Ramus-Serment, C., and Oustaloup, A., ‘Fractional differentiation in passive vibration control’, Nonlinear Dynamics 29(1–4), 2002, 343–362.
Podlubny, I., Petras, I., Vinagre, B. M., Chen, Y., O'Leary, P., and Dorcak, L., ‘Realization of fractional order controllers’, Acta Montanistica Slovaca 8(4), 2003, 233–235.
Suarez, L. E. and Shokooh, A., ‘An eigenvector expansion method for the solution of motion containing fractional derivatives’, ASME Journal of Applied Mechanics 64(3), 1997, 629–635.
Chen, Y., Vinagre, B. M., and Podlubny, I., ‘Continued fraction expansion approaches to discretizing fractional order derivatives–An expository review’, Nonlinear Dynamics 38(1/2), 2004, 155–170.
Oustaloup, A., Levron, F., Mathieu, B., and Nanot, F. M., ‘Frequency-band complex noninteger differentiator: Characterization and synthesis’, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47(1), 2000, 25–39.
Oldham, K. B., The Fractional Calculus, Academic Press, New York, 1974.
Koh, C. G. and Kelly, J. M., ‘Application of fractional derivative to seismic analysis of base-isolated models’, Earthquake Engineering and Structural Dynamics 19, 1990, 229-241.
Wahi, P. and Chatterjee, A., ‘Averaging oscillations with small fractional damping and delay terms’, Nonlinear Dynamics 38(1/2), 2004, 3–22.
Shinozuka, M., ‘Monte carlo solution of structural dynamics’, Computers and Structures 2(5/6), 1972, 855–874.
Yuan, L. and Agrawal, O. P., ‘Numerical scheme for dynamic systems containing fractional derivatives’, Journal of Vibration and Acoustics 124, 2002, 321–324.
Ford, N. J. and Simpson,A. C., ‘The numerical solution of fractional differential equations: Speed versus accuracy’, Numerical Algorithms 26, 2001, 333–346.
Schmidt, A., and Gaul, L.,On a critique of a numerical scheme for the calculation of fractionally damped dynamical systems’, Mechanics Research Communications, 33, 2006, 99-107.
Ogata, K., System Dynamics, Prentice Hall, New Jersey, 1998.
Chatterjee, A., ‘Statistical origins of fractional derivatives in viscoelasticity’, Journal of Sound and Vibration 284, 2005, 1239–1245.
Singh, S. J. and Chatterjee, A., ‘Fractional damping: Statistical origins and galerkin projections’, in Proceedings of the ENOC-2005, the Fifth Euromech Nonlinear Dynamics Conference, Eindhoven, The Netherlands, August 7–12, 2005.
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Singh, S.J., Chatterjee, A. Galerkin Projections and Finite Elements for Fractional Order Derivatives. Nonlinear Dyn 45, 183–206 (2006). https://doi.org/10.1007/s11071-005-9002-z
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DOI: https://doi.org/10.1007/s11071-005-9002-z