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In this paper, the existence condition of critical damping in 1 DOF systems with fractional damping is presented, and the relationship between critical damping coefficient and the order of the fractional derivative is derived. It shows only when the order of fractional damping and its coefficient meet certain conditions, the system is in the critical damping case. Then the vibration characteristics of the systems with different orders located in the critical damping set are discussed. Based on the results, the classical skyhook damping control strategy is extended to the fractional one, where a switching control law is designed to obtain a more ideal control effect. Based on the principle of modal coordinate transformation, a new design method of fractional skyhook damping control for full-car suspension is given. The simulation results show that the proposed control method has a good control effect, even in some special cases, such as roads bumps.

The vibrations of linear 1 DOF systems with ordinary damping can be classified as underdamped, critically damped, and overdamped according to the magnitude of the damping coefficient. Critical damping is defined as the threshold between overdamping and underdamping. In the case of critical damping, the oscillator returns to the equilibrium position as quickly as possible, without oscillating, and passes it once at most [

Vehicle suspension is an important component for improving the driving comfort and the handling performance [

Although mainstream algorithms can achieve a good control performance, it is inconsistent with the original skyhook control principle. From the perspective of mathematical principles, the classic skyhook control principle is used to control a SDOF system with one skyhook damper. Whereas the vehicle suspension is a system with multi-DOFs (the existing models have seven or more DOFs), thus the same number of controllers is required. However, in reality, there are only four controllers. How to tackle this problem?

This work is divided into two parts. In the first part, the critical damping in fractional order system is studied. The existence conditions of the critical damping are given, and the relationship order is derived. Then the vibration attenuation characteristics of fractional critical damping systems with different order are discussed. In the second part, the fractional critical damping is applied to the control strategy of the vehicle suspension system. The method of modal decoupling is used to solve the problem that the number of required controllers is not consistent with that of the actual ones. In the modal space, the classical skyhook control strategy is used for depressing the decoupled single mode vibration. Here, the fractional critical damping coefficients are chosen as the skyhook damping coefficients. In this way, the number of designed controllers is consistent with that of DOFs of the system, then these modes are recoupled and the actual controllers are used to control the suspension. A four-wheel-correlated random road time domain model is built to test the effect of fractional derivative skyhook control strategy; a road bump is especially designed to demonstrate the advantages of the fractional derivative critical damping.

The organization of the paper is as follow. In Section

The free vibration differential equation of a SDOF system with fractional derivative damping has the form

There are many definitions for fractional derivatives [

By the Laplace transform method, the characteristic equation of the system takes the form

Considering the Euler formula

The establishment-condition of (

It is known that when the imaginary part of the roots of (

The establishing condition for (

We find that the set of

From (

For linear 1 DOF fractionally damped systems, only when (_{c}.

The curves that represent the relation between the variables in (

The relation between the three parameters in (

It should be noted also that Sakakibara [

When

When

Figure

The curves of free damped motion of critical damping system with different orders.

Step response curves of critical damping systems with different orders.

It is expected that under the premise of nonoscillatory, the system is not easy to be aroused by external excitation and can return back to the equilibrium position as quickly as possible when there is no external force. A switch control law is designed to make the displacement as small as possible when the system is away from the equilibrium position and to limit the time it takes to reach the asymptotically stable position when there is no external force. The designed control law is

Impulse response curves of integer order and fractional orders switched systems.

According to vehicle dynamics theory, the dynamic model of the vehicle with seven DOFs is established. The seven DOFs

According to the linear vibration theory, the decoupled suspension system turns into isolated linear subsystems that can be controlled independently [

The free vibration equations of the modal systems are considered, namely,

When

In practice, with a larger or smaller

A four-wheels-correlated random road time domain model [

Figures

Body motion amplitude response of three kinds of suspension.

Heave displacement

Pitch displacement

Roll displacement

Body motion acceleration response of three kinds of suspension.

Heave acceleration

Pitch acceleration

Roll acceleration

Compared with many other full-car suspension control strategies, there are two main advantages for the method in this paper. Firstly, the proposed method is much more simple than most of the control methods. For example, these methods presented in [

In a word, the proposed skyhook control has a simple algorithm and is consistent with the original skyhook damping scheme in principle. 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.

(1) The free damped motion of SDOF systems with fractional derivative damping is firstly studied. Conditions of existing critical damping are given and the relation between the critical damping coefficient and the order fractional derivative is derived. It is also found that when the order increases from 0 to 2, the critical damping coefficient is getting small, but it is faster to return back to equilibrium position.

(2) Based on the mathematical thinking, a new full-car skyhook damping control strategy is proposed, which is different from the logical thinking of most scholars. The mainstream algorithm can also achieve a good performance; here, it is not the purpose to deny its effectiveness, but to give a new perspective for scholars to re-examine the intrinsic mathematical logic of classic skyhook damping principle. 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.

(3) Simulation results show that compared with the passive suspension, the skyhook controlled active suspension has a better performance on vibration suppression. Furthermore, the fractional skyhook controlled suspension has better responses of the body vibrating, especially when the vehicle passes the road bump. The results not only confirm the superiority of fractional critical damping, but also validate the effectiveness of this control strategy.

Sprung mass, 810 kg

Inertia moment of vehicle pitch, 300 kg·m^{2}

Inertia moment of vehicle roll, 1058 kg·m^{2}

Distance from axle to 1.14 m

Center of gravity, 1.22 m

Front suspension stiffness, 20600 N/m

Rear suspension stiffness, 15200 N/m

Front suspension damping, 1570 N/m

Rear suspension damping, 1760 N/m

Tire stiffness, 138000 N/m

Front tire mass, 26.5 kg

Rear tire mass, 24.4 kg

Distance between two tires, 1.3 m

Vehicle speed, 50 km/h.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 11272159) and (Grant no. 51605228).