A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses

Abstract

The fractional oscillator equation with the sinusoidal excitation mx″(t)+bDtαx(t)+kx(t)=Fsin(ωt), m,b,k,ω>0 and 0<α<2 is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation xW(t). In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case α∈(0,1)∪(1,2), the response to excitation, S(t), is a superposition of three parts: the steady periodic oscillation xW(t), an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values, S0(t) and S1(t), either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense.