Linear Fractionally Damped Oscillator

TL;DR

This paper explores a generalized damped oscillator with a fractional damping term, revealing nine distinct dynamic cases and unusual frequency behaviors as the fractional order varies.

Contribution

It introduces an analytical solution for the fractional damping oscillator and identifies nine cases, expanding the understanding beyond the traditional three damping regimes.

Findings

Nine distinct damping cases identified

Frequency can increase with fractional damping in some cases

Analytical solution provided for fractional damping oscillator

Abstract

In this paper 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 less than or equal to nu which is less than or equal to 1 . At the lower end, nu = 0, the equation represents an un-damped oscillator and at the upper end, nu = 1, 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).