Vibrations of fractional half- and single-degree of freedom systems
Author affiliations
DOI:
https://doi.org/10.15625/0866-7136/9772Keywords:
fractional oscillator, fractional differential equation, numerical solutionAbstract
In this paper we study vibrations of fractional oscillators by two methods: the triangular strip matrix approach, based on the Grunwald-Letnikov discretization of the fractional term, and the state variable analysis, which is suitable for systems with fractional derivatives of rational order. Some examples are solved in order to compare the two approaches and to conduct comparison with benchmark problems.
Downloads
References
A. Schmidt and L. Gaul. On the numerical evaluation of fractional derivatives in multi-degree-of-freedom systems. Signal Processing, 86, (10), (2006), pp. 2592–2601. doi:10.1016/j.sigpro.2006.02.006.
M. Ciesielski and T. Błaszczyk. Numerical solution of non-homogenous fractional oscillator equation in integral form. Journal of Theoretical and Applied Mechanics, 53, (2015), pp. 959–968. doi:10.15632/jtam-pl.53.4.959.
T. Blaszczyk and M. Ciesielski. Fractional oscillator equation–transformation into integral equation and numerical solution. Applied Mathematics and Computation, 257, (2015), pp. 428– 435. doi:10.1016/j.amc.2014.12.122.
L. Yuan and O. P. Agrawal. A numerical scheme for dynamic systems containing fractional derivatives. Journal of Vibration and Acoustics, 124, (2), (2002), pp. 321–324. doi:10.1115/1.1448322.
A. Tofighi. An especial fractional oscillator. International Journal of Statistical Mechanics, 2013, (2013). doi:10.1155/2013/175273.
A. Tofighi. The intrinsic damping of the fractional oscillator. Physica A: Statistical Mechanics and its Applications, 329, (1), (2003), pp. 29–34. doi:10.1016/s0378-4371(03)00598-3.
B. N. N. Achar, J. W. Hanneken, and T. Clarke. Damping characteristics of a fractional oscillator. Physica A: Statistical Mechanics and its Applications, 339, (3), (2004), pp. 311–319. doi:10.1016/j.physa.2004.03.030.
B. N. N. Achar, J. W. Hanneken, and T. Clarke. Response characteristics of a fractional oscillator. Physica A: Statistical Mechanics and its Applications, 309, (3), (2002), pp. 275–288. doi:10.1016/s0378-4371(02)00609-x.
A. A. Stanislavsky. Fractional oscillator. Physical Review E, 70, (5), (2004). doi:10.1103/physreve.70.051103.
I. Podlubny. Matrix approach to discrete fractional calculus. Fractional Calculus and Applied Analysis, 3, (4), (2000), pp. 359–386.
M. Di Paola, F. P. Pinnola, and P. D. Spanos. Analysis of multi-degree-of-freedom systems with fractional derivative elements of rational order. In International Conference on Fractional Differentiation and Its Applications (ICFDA). IEEE, (2014), pp. 1–6. doi:10.1109/icfda.2014.6967364.
K. B. Oldham and J. Spanier. The fractional calculus. Academic Press, New York, (1974).
K. S. Miller and B. Ross. An introduction to the fractional calculus and fractional differential equations. John Wiley and Sons, Inc., New York, (1993).
R. Gorenflo and F. Mainardi. Fractional calculus: Integral and differential equations of fractional order, pp. 223–276. Springer Verlag, (1997).
S. G. Samko, A. A. Kilbas, and O. I. Marichev. Fractional integrals and derivatives–theory and applications. Gordon and Breach Science Publishers, Longhorne, PA, USA, (1993).
V. S. Kiryakova. Generalized fractional calculus and applications. Longman Scientific & Technical, Longman House, Burnt Mill, Harlow, England, (1993).
I. Podlubny. Fractional differential equations. Academic Press, New York, (1999).
A. M. Mathai and H. J. Haubold. Mittag-Leffler functions and fractional calculus, chapter 2, pp. 79–134. Springer, (2008). doi:10.1007/978-0-387-75894-7 2.
O. P. Agrawal. Analytical solution for stochastic response of a fractionally damped beam. Journal of Vibration and Acoustics, 126, (4), (2004), pp. 561–566. doi:10.1115/1.1805003.
J. Freundlich and A. Tylikowski. Transient resonance oscillations of a system with fractional derivative damping of order 1/2. Machine Dynamics Research, 37, (4), (2015), pp. 27–33.
Downloads
Published
Issue
Section
License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
