The Periodic Solution of Fractional Oscillation Equation with Periodic Input

The periodic solution of fractional oscillation equation with periodic input is considered in this work. The fractional derivative operator is taken as -∞Dtα, where the initial time is -∞; hence, initial conditions are not needed in the model of the present fractional oscillation equation. With the input of the harmonic oscillation, the solution is derived to be a periodic function of time t with the same circular frequency as the input, and the frequency of the solution is not affected by the system frequency c as is affected in the integer-order case. These results are similar to the case of a damped oscillation with a periodic input in the integer-order case. Properties of the periodic solution are discussed, and the fractional resonance frequency is introduced.