Linear Fractionally Damped Oscillator

Abstract

The linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative. The order of the derivative being considered is $0\le v\le 1$. At the lower end $\left(v=0\right)$ the equation represents an undamped oscillator and at the upper end $\left(v=1\right)$ the ordinary linearly damped oscillator equation is recovered. A solution is found analytically, and a comparison with the ordinary linearly damped oscillator is made. It is found that there are nine distinct cases as opposed to the usual three for the ordinary equation (damped, over-damped, and critically damped). For three of these cases it is shown that the frequency of oscillation actually increases with increasing damping order before eventually falling to the limiting value given by the ordinary damped oscillator equation. For the other six cases the behavior is as expected, the frequency of oscillation decreases with increasing order of the derivative (damping term).