Fractional differential equation modeling of viscoelastic fluid in mass-spring-magnetorheological damper mechanical syst
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Fractional differential equation modeling of viscoelastic fluid in mass-spring-magnetorheological damper mechanical system J. E. Escalante-Martínez1,a , L. J. Morales-Mendoza2 , M. I. Cruz-Orduña1, M. Rodriguez-Achach3 , D. Behera4,b , J. R. Laguna-Camacho1, H. D. López-Calderón5 , V. M. López-Cruz1 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
Received: 29 June 2020 / Accepted: 23 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 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 γ changes by varying the viscosity damping coefficient β. 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.
1 Introduction The mass spring damper system is a physical system used to study mechanical vibrations. When the viscous properties of the fluid in the buffer remain constant, an ordinary differential equation of the second order serves as a mathematical model; that is to say, the evolution of said mechanical oscillations can be predicted when studying the analytical solution of said equation. Three typical behaviors such as over-damped, critically damped and under-damped movement are observed from the values of the elastic spring constants.
a e-mail: [email protected] b e-mails: [email protected]; [email protected] (corresponding author)
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The viscous damping coefficient is an artificial parameter, which cannot be accessed by a physical measurement such as the mass or spring elasticity constant. It is an ad hoc parameter, which serves to justify all t
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