Properties of Vibration with Fractional Derivative Damping of Order 1 / 2

Abstract

The general properties of a harmonic oscillation with damping proportional to the fractional time derivative of the displacement of order 1 / 2 is studied. It is shown that vibration decays when the damping coefficient is positive, implying that the model is thermodynamically valid. The center of the oscillation is a curve that approaches the time axis algebraically, so that after the oscillation damps out, there is a long trailing tail. There is no critical value of the damping coefficient that distinguishes the pattern of damping. For a negative damping coefficient, oscillation grows exponentially. An equivalent oscillator model with ordinary damping may be constructed for small damping, which shows that fractional damping acts partly as a supplementary spring, reflecting the recoil effect of a viscoelastic material.