Eur. Phys. J. Plus (2020) 135: 847
https://doi.org/10.1140/epjp/s13360-020-00802-0

Regular Article

Fractional differential equation modeling of viscoelastic fluid in mass-spring-magnetorheological damper mechanical system

¹ Faculty of Mechanical and Electrical Engineering, Universidad Veracruzana, 93390, Poza Rica, Mexico
² Faculty of Electronics and Communications Engineering, Universidad Veracruzana, 93390, Poza Rica, Mexico
³ Marist University of Merida, 97300, Mérida, Yucatán, Mexico
⁴ Department of Mathematics, The University of the West Indies, Mona Campus, Kingston 7, Jamaica
⁵ Institute of Biotechnology, Universidad Autónoma de Nuevo León, San Nicolás, Mexico

^e diptiranjanb@gmail.com, diptiranjan.behera@uwimona.edu.jm

Received: 29 June 2020
Accepted: 23 September 2020
Published online: 19 October 2020

Abstract

The mass-spring-damper system has the minimum complexity scenario which characterizes almost all the mechanical vibration phenomena. Also it is well known that a second-order differential equation can model its dynamics. However, if the damper has a magnetorheological fluid, then it shows viscoelastic properties in the presence of a magnetic field. Hence the mathematical model that best reflects the dynamics of this system is a fractional order differential equation. Naturally, here the Mittag–Leffler function appears in the analytical solution. Mathematical modeling of the mass-spring-magnetorheological damper mechanical system has been presented here. The main focus of the investigation is to show how the fractional order changes by varying the viscosity damping coefficient . These observations have been made by varying current intensity in the range of 0.2–2 A. A Helmholtz coil has been used to produce the magnetic field. It is worth mentioning that, this work has a high pedagogical value in the connection of fractional calculus to mechanical vibrations as well as it can be used as a starting point for a more advanced treatment of fractional mechanical oscillations.