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Entropy 2015, 17(9), 6289-6303; doi:10.3390/e17096289

Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

1
CONACYT, Centro Nacional de Investigación y Desarrollo Tecnológico, TecnológicoNacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca,Morelos, Mexico
2
Universidad Autónoma de la Ciudad de México, Prolongación San Isidro 151, Col. San LorenzoTezonco, Del. Iztapalapa, 09790 México D.F., Mexico
3
Facultad de Ingeniería Mecánica y Eléctrica (FIME), Facultad de Ingeniería en Electrónica y Comunicaciones (FIEC), Universidad Veracruzana, Venustiano Carranza S/N., C.P. 93396 Poza RicaVeracruz, Mexico
4
Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editors: J. A. Tenreiro Machado and António M. Lopes
Received: 3 August 2015 / Revised: 2 September 2015 / Accepted: 7 September 2015 / Published: 10 September 2015
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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Abstract

In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo–Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter σ. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag–Leffler function; for the Caputo–Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one. View Full-Text
Keywords: Caputo fractional derivative; Caputo–Fabrizio fractional derivative; Mittag–Leffler function; fractional-order dynamics; oscillations; mechanical oscillators Caputo fractional derivative; Caputo–Fabrizio fractional derivative; Mittag–Leffler function; fractional-order dynamics; oscillations; mechanical oscillators
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Gómez-Aguilar, J.F.; Yépez-Martínez, H.; Calderón-Ramón, C.; Cruz-Orduña, I.; Escobar-Jiménez, R.F.; Olivares-Peregrino, V.H. Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel. Entropy 2015, 17, 6289-6303.

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