Furthermore, we find that when [alpha] = 1, the critical damping coefficients [c.sub.c] = 2[square root of mk], r = [square root of k/m], and s = -[square root of k/m] = -[[omega].sub.n] are obtained, which are consistent with the critical damping in an integer order system.
Figure 3 shows the curves of decaying free motions of critical damping systems with different orders under the initial state [x.sub.0] = 0.1, [x.sub.0] = 0.
where u is the control force, [[alpha].sub.1] and [[alpha].sub.2] are the orders of fractional derivative, and [c.sub.1] and [c.sub.2] are the corresponding fractional derivative critical damping coefficients, x is the displacement.
where the control force [c.sub.isky][[eta].sup.([alpha])] used to keep the system in the case of critical damping. According to the method in Section 2, the relation between the damping coefficient and the order is obtained
To verify the characteristics of fractional critical damping, a work condition is designed as follows: when the simulation goes to 5s, on the left side of the vehicle, the front and rear wheels have been raised successively by road bump shaped like a sine wave with a height of 0.1m.
Figure 6 shows that the vibration with fractional derivative critical damping has a better performance on amplitude responses than that with integer one.
The strategy with integer order critical damping coefficients has a good effect, and the fractional one is seen as a supplement, which provides more parameter selection and has a better performance on amplitude responses.
Conditions of existing critical damping are given and the relation between the critical damping coefficient and the order fractional derivative is derived.
The fractional order critical damping coefficient is selected as the skyhook damping coefficient to clarify the superiority of proposed fractional order critical damping in practical application.
The results not only confirm the superiority of fractional critical damping, but also validate the effectiveness of this control strategy.
Beskos, "Critical Damping Surfaces of Linear Dynamic Systems," Engineering Mechanics: ASCE, pp.
Bulatovic, "On the critical damping in multi-degree-of-freedom systems," Mechanics Research Communications, vol.