Dynamical analysis of linear SDOF oscillator with fractional-order derivative (Ⅱ)

Abstract

A linear single degree-of-freedom (SDOF) oscillator with two kinds of fractional-order derivatives is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical properties, including the fractional coefficients and the fractional orders in the two kinds of fractional-order derivatives, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and the results is entirely different from the results given in the existing literature. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The following analysis of the effects of the fractional parameters on the amplitude-frequency is presented, and it is found that the fractional coefficients and the fractional orders can affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness. Finally, the effects of the fractional coefficient in the second fractional-order derivative on resonance frequency are analyzed, and the design rule for the fractional coefficient in the second fractional-order derivative to meet the satisfactory vibration control performance is pointed out.