Fractional differential equation modeling a viscoelastic fluid in
mass-spring-magnetorheological damper mechanical system
Abstract
The mass-spring-damper system is the minimum complexity scenario that
characterizes almost all the mechanical vibration phenomena, it is well
known that a second-order differential equation model its dynamics.
However, if the damper has a magnetorheological fluid in the presence of
a magnetic field then the fluid shows viscoelastic properties. Hence the
mathematical model that best reflects the dynamics of this system is a
fractional order differential equation. Naturally, the Mittag-Leffler
function appears as analytical solution. Accordingly we present here the
mathematical modeling of the mass-spring-magnetorheological damper
system. The main result of our investigation is to show how the
fractional order γ changes when the viscosity damping coefficient β
changes, this was found when varying current intensity in the range of
0.2 to 2 Amperes. A Helmholtz coil is used to produce the magnetic
field. We consider that this document has a high pedagogical value in
connecting the fractional calculation to mechanical vibrations and can
be used as a starting point for a more advanced treatment of
\textit{fractional mechanical oscillations