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Fractional differential equation modeling a viscoelastic fluid in mass-spring-magnetorheological damper mechanical system
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  • Jesús Escalante-Martínez,
  • Luis Morales-Mendoza,
  • María Cruz-Orduña,
  • Manuel Rodriguez-Achach,
  • Diptiranjan Behera,
  • Juan Laguna-Camacho,
  • Héctor López-Calderón,
  • Víctor López-Cruz
Jesús Escalante-Martínez
Universidad Veracruzana
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Luis Morales-Mendoza
Universidad Veracruzana
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María Cruz-Orduña
Universidad Veracruzana
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Manuel Rodriguez-Achach
Marist University of Merida
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Diptiranjan Behera
University of the West Indies at Mona Faculty of Science and Technology
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Juan Laguna-Camacho
Universidad Veracruzana
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Héctor López-Calderón
Universidad Autonoma de Nuevo Leon Facultad de Ciencias Biologicas
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Víctor López-Cruz
Universidad Veracruzana
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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