Vibration Equation of Fractional Order Describing Viscoelasticity and Viscous Inertia

Abstract

Abstract The steady state response of a fractional order vibration system subject to harmonic excitation was studied by using the fractional derivative operator − ∞ D t β , ${}_{-\infty} D_t^\beta,$ where the order β is a real number satisfying 0 ≤ β ≤ 2. We derived that the fractional derivative contributes to the viscoelasticity if 0 < β < 1, while it contributes to the viscous inertia if 1 < β < 2. Thus the fractional derivative can represent the “spring-pot” element and also the “inerterpot” element proposed in the present article. The viscosity contribution coefficient, elasticity contribution coefficient, inertia contribution coefficient, amplitude-frequency relation, phase-frequency relation, and influence of the order are discussed in detail. The results show that fractional derivatives are applicable for characterizing the viscoelasticity and viscous inertia of materials.