Analysis of free vibrations of a viscoelastic oscillator via the models involving several fractional parameters and relaxation/retardation times

Free damped vibrations of a linear viscoelastic oscillator based on the fractional derivative model involving more than one different fractional parameters and several relaxation (retardation) times are investigated. The analytical solution is obtained in the form of two terms, one of which governs the drift of the systemʹs equilibrium position and is defined by the dynamic relaxation retardation processes occurring in the system, and the other term describes damped vibrations around the equilibrium position and is determined by the systemʹs inertia and energy dissipation. The drift is governed by an improper integral taken along two sides of the cut of the complex plane, while damped vibrations are determined according to the two complex conjugate roots of the characteristic equation, which are located on the left half of the complex plane. The behaviour of the characteristic equation roots as functions of the systemʹs rheological parameters, which enable the control of the dynamic response of the oscillator, is shown in the complex plane.