Abstract
In this study, vibration characteristics of a single-degree-of-freedom linear oscillator with the fractional order derivative are examined in terms of the relationship between the order of the derivative, the damping coefficient of the equation of motion and a critical damping over the wide range of the order by using numerical analysis. Two types of critical dampings proposed in the previous studies are considered. No critical value of the damping coefficient is observed in some range of the order. We explain analytically the reason why the impulsive response of the system always vibrates when the order is 1/2. It is also shown that the difference of the existence of the critical damping between the order 1/3 and 2/3 is mainly caused by the change of the characteristics of the roots of the characteristic polynomial of the oscillator. The response characteristics are classified into three cases in the view of the existence of the two types of critical dampings. The relationship between the order of the derivative, the coefficient of the motion equation and the resonance frequency is also obtained by using numerical calculation and is compared with the classical mass-spring-damper system. The relationship between the order, the coefficient and the peak value of the frequency response function is also investigated.
