Abstract
Fractional derivative models, which are used to describe the viscoelastic behavior of material, have received considerable attention. Thus it is necessary to put forward the analysis solutions of dynamic systems containing a fractional derivative. Although previously reported such kind of fractional calculus-based constitutive models, it only handles the particularity of rational number in part, has great limitation by reason of only handling with particular rational number field. Simultaneously, the former study has great unreliability by reason of using the complementary error function which can’t ensure uniform real number. In this paper, a new approach is proposed for an analytical scheme for dynamic system of a spring-mass-damper system of single-degree of freedom under general forcing conditions, whose damping is described by a fractional derivative of the order 0< α< 1 which can be both irrational number and rational number. The new approach combines the fractional Green’s function and Laplace transform of fractional derivative. Analytical examples of dynamic system under general forcing conditions obtained by means of this approach verify the feasibility very well with much higher reliability and universality.
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References
Agrawal, O. P., 2001, “Stochastic Analysis of Dynamic System Containing Fractional Derivatives,”Journal of Sound and Vibration, Vol. 247, No. 5, pp. 927–938.
Bagley, R. L. and Torvik, P. J., 1983, “Fractional Calculus-a Different Approach to the Analysis of Viscoelastically Damped Structures,”AIAA Journal, Vol. 21, pp. 741–748.
Elshehawey, E. F., Elbarbary, E. M. E., Afifl, N. A. S. and El-Shahed, M., 2001, “On the Solution of Theendolymph Equation Using Fractional Calculus,”Applied Mathematics and Computation, Vol. 124, pp. 337–341.
Enelund, M. and Josefson, B. L., 1997, “Timedomain Finite Element Analysis of Viscoelastic Structures with Fractional Derivative Constitutive Relations,”American Institute of Aeronautics and Astronautics Journal, Vol. 35, pp. 1630–1637.
Enelund, M., Ahler, L. M., Runesson, K. and Jonsefson, B. L., 1999, “Formulation and Integration of the Standard Linear Viscoelastic Solid with Fractional Order Rate Laws,”International Journal of Solid and Structures, Vol. 36, pp. 2417–2442.
Ingman, D. and Suzdalnitsky, J., 2001, “Iteration Method for Equation of Viscoelastic Motion with Fractional Differential Operator of Damping,”Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 5027–5036.
Miller, K. S., 1993, “The Mittag-Leffier and Related Functions,”Integral Transforms and Special Functions, Vol. 1, pp. 41–49.
Narahari, Achar, B. N., Hanneken, J. W., Enck, T. and Clarke, T., 2001,” Dynamics of the fractional oscillator,” Physica A, Vol. 297, pp. 361–367.
Oldham, K. B. and Spanier, J., 1974, The Fractional Calculus, New York: Academic Press.
Rossikhin, Y. A. and Shitikova, M. V., 1997, “Application of Fractional Operators to the Analysis of Damped Vibrations of Viscoelastic Single-mass Systems,”Journal of Sound and vibration, Vol. 199, No. 4, pp. 567–586.
Samko, S. G., Kilbas, A. A. and Marichev, O. I., 1993,Fractional Integrals and Derivatives, Yverdon, Switzerland: Gordon and Breach.
Samuel W. J. Welch, Ronald A. L. Rorrer and Ronald G. Duren, 1999, “Application of TimeBasedfractional Calculus Methods to Viscoelastic Creep and Stress Relaxation of Materials,”Mechanics of Time-Dependent Materials, Vol. 3, pp. 279–303.
Slater, L. J., 1966,Generalized Hypergeometric Functions, Cambridge, England, Cambridge University Press.
Suarez, L. and Shokooh, A., 1997, “An Eigenvector Expansion Method for the Solution of Motion Containing Derivatives,”ASME Journal of Applied Mechanics, Vol. 64, pp. 629–635.
Sweldens, W. and Piessens, R., 1994, “Quadrature Formulae and Asymptotic and Asymptotic Error Expansions for wavelet Approximations of Smooth Functions,”SIAM Journal on Numerical Analysis, Vol. 31, pp. 1240–1264.
Wang, J. and Zhou, Y. H., 1998, “Error Estimation for the Generalized Gaussian Integral Method Weighted by Scaling Functions of wavelets,”Journal of Lanzhou University, natural science, Vol. 34, pp. 26–30.
Wang Jizeng, 2001, “Generalized Theory and Arithmetic of Orthogonal wavelets and Applications to Researches of Mechanics Including Piezoelectric Smart Structures,”Ph. D. Thesis, Lanzhou University, China.
Wim Sweldens, 1995, “The Construction and Application of wavelets in Numerical Analysis,”Ph. D. Thesis, Columbia University.
Xu Mingyu and Tan Wenchang, 2001, “Theoretical Analysis of the Velocity Field, Stress Field and Vortex Sheet of Generalized Second Order Fluid with Fractional Anomalous Diffusion,”Science in China, Series A, Vol. 44, No. 7, pp. 1387–1499.
Zhou, Y. H., Wang, J. and Zheng, X. J., 1998, “Application of Wavelets Galerkin FEM to bending of Beam and Plate Structures,”Applied Mathematics and Mechanics, Vol. 19, pp. 697–706.
Zhou, Y. H., Wang, J. and Zheng, X. J., 1999, “Applications of wavelet Galerkin FEM to bending of plate structure,”Acta Mechanica Solida Sinica, Vol. 12, pp. 136–143.
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Hong, DP., Kim, YM. & Wang, JZ. A new approach for the analysis solution of dynamic systems containing fractional derivative. J Mech Sci Technol 20, 658–667 (2006). https://doi.org/10.1007/BF02915983
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DOI: https://doi.org/10.1007/BF02915983