A Study on the Damped Free Vibration with Fractional Calculus
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Abstract
Fractional calculus theory includes the definitions of derivatives and integrals with arbitrary order. This theory is used to solve some classes of differential equations and fractional differential equations. One of these equations is the damped free vibration equation. This equation is a linear homogeneous differential equation with constant coefficients. In this paper, we intend to solve this equation by means of N-fractional calculus method.Downloads
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Podlubny I, ‘‘Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Methods of Their Solution and Some of Their Applications’’, Mathematics in Science and Enginering, USA: Academic Press, 1999.
Yilmazer R, Ozturk O, ‘‘Explicit Solutions of Singular Differential Equation by means of Fractional Calculus Operators’’, Abstract and Applied Analysis, vol. 2013, 6 pages, 2013.
Tu S.T, Chyan D.K, Srivastava H.M, ‘‘Some Families of Ordinary and Partial Fractional Differintegral Equations’’, Integral Transform. Spec. Funct. vol. 11, p.291-302, 2001.
Nishimoto K, ‘‘Kummer’s Twenty-Four Functions and NFractional Calculus’’, NonlinearAnalysis, Theory, Methods & Applications, vol. 30, p.1271-1282, 1997.
Yilmazer R, ‘‘N-Fractional Calculus Operator ܰఓ method to a Modified Hydrogen Atom Equation’’, Math. Commun., vol. 15, p.489-501, 2010.
Whittaker E.T, Watson G.N, ‘‘A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions (Fourth Edition)’’, Cambridge: Cambridge University Press, 1927.
Fukuhara M, ‘‘Ordinary Differential Equations’’, Tokyo: Iwanami Shoten, 1941.
Tricomi F.G, ‘‘Funzioni Ipergeometriche Confluenti’’, Roma: Edizioni Cremonese, 1954.
Watson G.N, ‘‘A Treatise on the Theory of Bessel Functions (Second Edition)’’, Cambridge: Cambridge University Press, 1944.
Feldman M, ‘‘Non-linear System Vibration Analysis Using Hilbert Transform’’, Mechanical Systems and Signal Processing. vol 8 (2), p.119-127, 1994.
Bagley R.L, Torvik J, ‘‘Fractional Calculus - A Different Approach to the Analysis of Viscoelastically Damped Structure’’, AIAA Journal. vol. 21 (5), p.741-748, 1983.
Gough C, ‘‘The nonlinear free vibration of a damped elastic string’’, J. Acoust. Soc. Am. 75, 1984.
Rossikhin Y, Shitikova M, ‘‘Application of Fractional Derivatives to the Analysis of Damped Vibrations of Viscoelastic Single Mass Systems’’, Acta Mechanica vol. 120 (1), p.109-125, 1997.
Rossikhin Y, Shitikova M, ‘‘Application of Fractional Calculus for Analysis of Nonlinear Damped Vibrations of Suspension Bridges’’, J. Eng. Mech. vol. 124 (9), p.1029- 1036, 1998.
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