The periodic response of a fractional oscillator with a spring-pot and an inerter-pot

The fractional oscillation system with two Weyl-type fractional derivative terms -infinity D-t(beta) x (0< beta < 1) and -infinity D(t)(alpha)x (1< alpha < 2), which portray a "spring-pot" and an "inerter-pot" and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders beta (0 < beta < 1) and alpha (1 < alpha < 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude-frequency and phase-frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a "spring-pot", and also describing viscous inertia and serving as an "inerter-pot".