Eur. Phys. J. Plus (2018) 133: 78
https://doi.org/10.1140/epjp/i2018-11905-4

Regular Article

A novel fractional derivative with variable- and constant-order applied to a mass-spring-damper system

¹ Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo, Guerrero, Mexico
² CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico
³ Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico

^* e-mail: jgomez@cenidet.edu.mx

Received: 7 December 2017
Accepted: 27 January 2018
Published online: 27 February 2018

Abstract

This paper deals with the application of a novel variable- and constant-order fractional derivative with no singular kernel in the modeling of a mass-spring-damper system. The variable-order fractional derivative can be set as a smooth function, bounded on (0;1] , while the constant-order fractional derivative can be set as a fractional equation, bounded on (0;1] . Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales. In the variable-order model, in contrast to the constant-order fractional mass-spring-damper system, the displacement changes with time. This means that the memory rate of the system changes with time and is determined by the current time instant. For different time periods we have different memory abilities. The integer-order classical model is recovered when the order of the fractional derivative is equal to 1.