Forced Oscillations of a Single Degree of Freedom System withFractional Dissipation
Authors: T. M. Atanacković, D. Ć . Dolićanin, S. Pilipović
Keywords: Fractional derivative, distributed-order fractional derivative.
Abstract:
We study motion of a single degree of freedom mechanical system consisting of a visco-elastic rod of finite length with concentrated mass at the free end. If the deformation of the rod is approximated so that rod is considered in a state of quasi-static deformation or, equivalently, if it is assumed that the rod is light (density is equal to zero) we show that the several known oscillation equations can be derived.
References:
[1] B. S. Baclić, T. M. Atanacković, Stability and creep of a Fractional order viscoelastic rod,
Bull. de l’Académie Serbe des Sciences et des Arts. Classe des Sciences mathématiques et
naturelles 25, 115-131 (2000).
[2] T. M. Atanacković, A generalized model for the uniaxial isothermal deformation of a viscoelastic
body, Acta Mech. 159, 77–86 (2002).
[3] T. M.Atanacković, On a distributed derivative model of a viscoelastic body, Cr. Acad. Sci. II
B-Mec. 331, 687–692 (2003).
[4] T. M. Atanacković, M. Budinčević, S. Pilipović, On a fractional distributed-order oscillator,
J. Phys. A: Math. Gener. 38, 6703–6713 (2005).
[5] T. M. Atanacković, S. Pilipović, D. Zorica, Time distributed order diffusion-wave equation, I.
Voltera type equation. Proc. R. Soc. A 465, 1869–1891 (2009).
[6] T.M. Atanacković, Lj. Oparnica, S. Pilipović, Distributional framework for solving fractional
differential equations, Integral Transforms and Special Functions 20, 215-222 (2009).
[7] T. M. Atanacković, S. Pilipović, D. Zorica, Time distributed order diffusion-wave equation,
II. Applications of the Laplace and Fourier transformations. Proc. R. Soc. A 465, 1893–1917
(2009).
[8] A.Hanyga, Fractional-order relaxation laws in non-linear viscoelasticity, Continuum Mech.
Termodyn. 19: 25-36 (2007).
[9] A. Hanyga, M. Seredynska, Hamiltonian and Lagrangian theory of viscoelasticity, Continuum
Mech. Thermodyn. 19: 475-492 (2008).
[10] T. T. Hartley, C. F. Lorenzo, Fractional-order system identification based on continuous orderdistributions, Signal Process. 83, 2287–2300 (2003).
[11] A. Lion,On the thermodynamics of fractional damping elements, Continuum Mech. Thermodyn.
9, 83-96 (1997).
[12] I. Podlubny,Fractional Differential Equations, Academic Press, San Diego (1999).
[13] Yu. A. Rossikhin, M. V. Shitikova, Analysis of dynamic behaviour of viscoelastic rods whose
rheological models contain fractional derivatives of two different orders, Z. Angew. Math.
Mech. 81, 363–376 (2001).
[14] Yu. A. Rossikhin, M. V. Shitikova, A new method for solving dynamic problems of fractional
derivative viscoelasticity, Int. J. Eng. Sci. 39, 149–176 (2001).
[15] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and
Breach, Amsterdam (1993).
[16] V. S. Vladimirov, Equations of Mathematical Physics, Mir Publishers, Moscow (1984).
[17] T.M. Atanacković, S. Pilipović, D. Zorica, Distributed-order fractional wave equation on a
finite domain, Creep and forced oscillations of a rod. Continuum Mech. and Thermodynamics.
DOI 10.1007/s00161-010-0177-2 (2011).
[18] T. M. Atanacković, S. Pilipović, D. Zorica, Distributed-order fractional wave equation on a
finite domain, Stress relaxation in a rod. International Journal of Engineering Science, (2010).
[19] W. Arendt, C. J. K. Batty, M. Hieber, N. Neubrander,Vector-valued Laplace Transforms and
Cauchy problems, Monographs in Mathematics, Vol 96, Birkh¨auser, Basel, Boston, Berlin (2001).
[20] D.W. Dreisigmeyer, P.M. Young, Extending Bauer’s corollary to fractional derivatives, J.
Phys. A: Math. Gen. 37 117–21 (2003).
[21] J. T. Katsikadelis, Numerical solution of multi-term fractional differential equations, Z.
Angew. Math. Mech. (ZAMM) 89, No. 7, 593 – 608 (2009).
[22] B. Stanković, T. M. Atanacković, On an inequality arising in Fractional oscillator theory,
Fractional Calculus and Applied Analysis, 7, 11–19 (2004).
[23] T. M. Atanacković, D. Dolićanin, S. Konjik, S. Pilipović, Dissipativity and stability for a nonlinear
differential equation with distributed order symmetrized fractional derivative, Applied
Mathematics Letters 24, 1020–1025 (2011).
[24] D. Ingman, J. Suzdalnitsky, Iteration method for equation of viscoleastic motion with fractional
differential operator of damping, Comput. Methods Appl. Mech. Engrg. 190. 5027-5036 (2001).