Fractional mechanical oscillators
J.F. Gómez-Aguilar a, *, J.J. Rosales-Garcíab, J.J. Bernal-Alvaradoa, T. Córdova-Fragaa and R. Guzmán-Cabreraa
a Departamento de Física, División de Ciencias e Ingenierías Campus León, Universidad de Guanajuato, Lomas del Bosque s/n, Lomas del Campestre, León Guanajuato, México,e-mail: jfga@fisica.ugto.mx, bernal@fisica.ugto.mx, teo@fisica.ugto.mx
b Departamento de Ingeniería Eléctrica, División de Ingenierías Campus Irapuato-Salamanca, Universidad de Guanajuato, Carretera Salamanca-Valle de Santiago, km. 3.5 + 1.8 km, Comunidad de Palo Blanco, Salamanca Guanajuato México, e-mail: rosales@ugto.mx, guzmanc@ugto.mx *Tel: +52 (477) 788-5100 ext. 8449.
]]> Recibido el 15 de febrero de 2012;
Abstract
In this contribution we propose a new fractional differential equation to describe the mechanical oscillations of a simple system. In particular, we analyze the systems mass-spring and spring-damper. The order of the derivatives is 0 < γ ≤ 1. In order to be consistent with the physical equation a new parameter σ is introduced. This parameter characterizes the existence of fractional structures in the system. A relation between the fractional order time derivative γ and the new parameter a is found. Due to this relation the solutions of the corresponding fractional differential equations are given in terms of the Mittag-Leffler function depending only on the parameter γ. The classical cases are recovered by taking the limit when γ = 1.
Keywords: Fractional calculus; mechanical oscillators; caputo derivative; fractional structures.
Resumen
En esta contribución se propone una nueva ecuación diferencial fraccionaria que describe las oscilaciones mecánicas de un sistema simple. En particular, se analizan los sistemas masa-resorte y resorte-amortiguador. El orden de las derivadas es 0 < γ≤ 1. Para mantener la consistencia con la ecuación física se introduce un nuevo parámetro σ. Este parámetro caracteriza la existencia de estructuras fraccionarias en el sistema. Se muestra que existe una relación entre el orden de la derivada fraccionaria γ y el nuevo parámetro a. Debido a esta relación las soluciones de las correspondientes ecuaciones diferenciales fraccionarias estan dadas en terminos de la función de Mittag-Leffler, cuyas soluciones dependen solo del orden fraccionario γ. Los casos clásicos son recuperados en el límite cuando γ = 1.
Descriptores: Cálculo fraccionario; oscilaciones mecanicas; derivada de caputo; estructuras fraccionarias.
]]>PACS: 45.10.Hj; 46.40.Ff; 45.20.D-
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Acknowledgments
This research was supported by CONACYT and PROMEP under the Grant: Fortalecimiento de CAs., 2011, UGTO-CA-27.
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