Mechanical Energy and Equivalent Viscous Damping for Fractional Zener Oscillator

Abstract

This paper presents mechanical energy and equivalent viscous damping for a single-degree-of-freedom fractional Zener oscillator. Differential equation of motion is derived in terms of fractional Zener constitutive equation of viscoelastic materials. A virtual fractional oscillator is generated via a state transformation. Then, based on the diffusive model for fractional integrators, the stored energy in fractional derivatives with orders lying in (0, 1) and (2, 3) is determined. Thus, the total mechanical energy in the virtual oscillator is determined. Finally, fractional derivatives are split into three parts: the equivalent viscous damping, equivalent stiffness, and equivalent mass. In this way, the fractional differential equation is simplified into an integer-order differential equation, which is much more convenient to handle in engineering.