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

Entropy

José Francisco Gómez-Aguilar⁵  Huitzilin Yépez-Martínez⁴  Celia Calderón-Ramón³  Ines Cruz-Ordu༚³  Ricardo Fabricio Escobar-Jiménez²  Victor Hugo Olivares-Peregrino²  J. A. Tenreiro Machado¹ 

[1]id="af1-entropy-17-06289">1Cátedras CONACYT, Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexi

[2]Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico

[3] E-Mails:

[4]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 Rica Veracruz, Mexico

[5] E-Mails:

[6]Universidad Autónoma de la Ciudad de México, Prolongación San Isidro 151, Col. San Lorenzo Tezonco, Del. Iztapalapa, 09790 México D.F., Mexico

[7] E-Mail:

[8]Cátedras CONACYT, Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico

关键词: Caputo fractional derivative;    Caputo–Fabrizio fractional derivative;    Mittag–Leffler function;    fractional-order dynamics;    oscillations;    mechanical oscillators;   

DOI  :  10.3390/e17096289

来源: mdpi

【 摘 要 】

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 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.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.