ABSTRACT
This chapter presents the concept of dynamic variable-order fractional differential equations. It describes the mathematical formulation of fractional continuously variable-order spring-mass damping systems. The chapter also presents displacement-time graphs for fractional continuously variable-order mass-spring damper systems for free oscillation with viscoelastic damping, viscous-viscoelastic damping and forced oscillation with viscous-viscoelastic damping, and viscoelastic-viscous damping. The fractional differential equations examples include in describing the dynamics of visco-elastic materials, turbulence, phase transition, complex networks, di-electric relaxations, control systems, and several other physical phenomena. The fractional differential equations have been often used to model the behavior of dynamic systems. Several analytical and numerical approaches have been used for analyzing the solutions of fractional damper systems. Such methods like Fourier transform and Laplace transform have been proposed by researchers to find the solution of fractional damper systems. The generalization of linear oscillator to form the fractional oscillator has been studied by Stanislavsky.
