Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel
José Francisco Gómez-Aguilar,
Huitzilin Yépez-Martínez,
Celia Calderón-Ramón,
Ines Cruz-Orduña,
Ricardo Fabricio Escobar-Jiménez,
Victor Hugo Olivares-Peregrino
Affiliations
José Francisco Gómez-Aguilar
CONACYT, Centro Nacional de Investigación y Desarrollo Tecnológico, TecnológicoNacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca,Morelos, Mexico
Huitzilin Yépez-Martínez
Universidad Autónoma de la Ciudad de México, Prolongación San Isidro 151, Col. San LorenzoTezonco, Del. Iztapalapa, 09790 México D.F., Mexico
Celia Calderón-Ramón
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
Ines Cruz-Orduña
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
Ricardo Fabricio Escobar-Jiménez
Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico
Victor Hugo Olivares-Peregrino
Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico
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.
