Abstract
This article presents the formulation and a new approach to find analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously variable order mass–spring–damper systems. Here, we use the viscoelastic and viscous–viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of fractional derivative varies continuously. Here, we handle the continuous changing nature of fractional order derivative for dynamic systems, which has not been studied yet. By successive recursive method, here we find the solution of fractional continuously variable order mass–spring–damper systems and then obtain closed-form solutions. We then present and discuss the solutions obtained in the cases with continuously variable order of damping for oscillator through graphical plots.
Highlights
In the past few years, the fractional order physical models[1,2,3,4] have seen much attention by researchers due to dynamic behaviour and the viscoelastic behaviour of material.[5]
We point out here the definitions regarding natural frequency, damped frequency, under-damped oscillation, critically damped oscillation and over-damped oscillations in the continuously variable fractional order
We modelled the fractional continuously variable order mass–spring–damper systems for free oscillation with viscoelastic damping and forced oscillation with viscous–viscoelastic damping
Summary
In the past few years, the fractional order physical models[1,2,3,4] have seen much attention by researchers due to dynamic behaviour and the viscoelastic behaviour of material.[5]. The exact solution of fractional order of 1/2 was obtained by Elshehawey et al.[12] The Green function approach for finding solution of dynamic system was studied by Agrawal,[13] which was followed by the Mittag-Leffler function proposed by Miller.[14] By using fractional Green function and Laplace transform, Hong et al.[15] have obtained the solution of single-degree-of-freedom mass–spring system of order 0\a\1. The analytical solution of fractional systems mass–spring and spring–damper system formed by using Mittag-Leffler function was analysed by Gomez-Aguilar et al.[16] The fractional Maxwell model for viscous-damper model and its analytical solution was proposed by Makris and Constantinou[17] and Choudhury et al.[18]. The successful implementations of proposed successive recursive method for finding the analytical solutions of fractional dynamic systems have been discussed in section ‘Application of proposed successive recursive method for solution of fractional continuously variable order mass–spring–damper system’. Ð1Þ and the fractional derivative, namely, Riemann– Liouville derivative of order a, is defined as
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