Formulation and solutions of fractional continuously variable order mass spring damper systems controlled by viscoelastic and viscous-viscoelastic dampers

The article presents the formulation and a new approach to find analytic solutions for fractional continuously variable order dynamic models viz. 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 iteration method, here we find the solution of fractional continuously variable order mass-spring damper systems, and then give a close form solution. We then present and discuss the solutions obtained in the cases with continuously variable order of damping for this oscillator with graphical plots.