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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

In this paper the linearly damped oscillator equation is considered with the damping term replaced by a fractional derivative [

In this paper the Caputo formulation of the fractional derivative will be used. The Caputo derivative is preferred over the Riemann-Liouville derivative for physical reasons. Consider the Laplace transform of the two formulations of the fractional derivative for

If the order of the fractional damping term is allowed to become

The analytic solution to the fractionally damped equation is found by means of Laplace transform. For the sake of clarity, and for pointing out some unique difficulties with the factional equation, a comparison with the Laplace transform method as applied to the nonfractional case is made. It is found that there are nine distinct cases for the fractionally damped equation as opposed to the usual three cases for the Nonfractional equation. In six of the nine cases the results are as expected; increasing the order of the factional derivative increases the effects of the damping (i.e., the frequency of the damping slows as the order of the derivative increases). However, in three cases, the frequency of the damping actually increases as the order of the fractional derivative increases until a peak value is reached after which the frequency falls to its Nonfractional limit. The physical reason for this increase in the oscillation frequency is not yet clear.

Before a solution to the linear fractionally damped oscillator equation is constructed it will be useful to review the Laplace transform method of solution for the linearly damped oscillator equation

Note that in cases one and three the poles will be of order one and in case two the pole will be of order two. Note also that if there were no damping the poles would be on the imaginary axis at

Computing the residue for case one gives

Case three is computed the same way as case one. Now the poles are complex so the exponential function can be expressed using sine and cosine with an over-all exponential damping factor

For case two the pole is of order 2, and the residue is given by

Now consider (

Now the question is, where are the poles? This is a somewhat more involved question than in the standard linearly damped model. To find the poles the following equation needs to be solved:

Given values for

Now it needs to be seen if there is a

Now the question of the poles that has been settled the solution to (

For the contour integral the only contributions come from the paths along the negative real axis

Consider the frequency of the oscillation component of the solution,

In the Nonfractional case increasing

Now the question arises, does

This is not entirely what might have been expected. In the first case the oscillation frequency actually increases before falling. Hence there will be some values of

There are some graphs of solutions to the imaginary part of (

Figure

Figure

Figure

In this paper the linear fractionally damped oscillator equation was solved analytically. It was found that the solution is very similar to the Nonfractional case (decayed oscillations but with the inclusion of an additional decay function). It was found that there are nine distinct cases, as opposed to the usual three for the ordinary damped oscillator. An unexpected result was that for three of the cases the oscillation frequency actually increases with increasing order of derivative of the damping term (till a peak value is reached, then the frequency decreases as expected). The physical reason for this increase in oscillation frequency is not yet clear.