Shock and Vibration

Volume 2017, Article ID 2738976, 8 pages

https://doi.org/10.1155/2017/2738976

## Fractional Critical Damping Theory and Its Application in Active Suspension Control

^{1}College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China^{2}College of Automobile and Traffic Engineering, Nanjing Forestry University, Nanjing 210037, China

Correspondence should be addressed to Ning Chen; nc.moc.ufjn@gninnehc

Received 9 June 2017; Revised 8 October 2017; Accepted 23 October 2017; Published 19 November 2017

Academic Editor: Evgeny Petrov

Copyright © 2017 Peng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1]. Considering the particularity of critical damping, it is frequently studied in other systems. The criterion for critical damping of viscously damped multi-degree-of-freedom systems is provided by Bulatovic [2]. The existence conditions for the critical damping in second-order pendulum-like systems are established by Li et al. [3]. A general method that determines the “critical damping surfaces” of a certain linear continuous dynamic system is proposed by Beskos and Boley [4]. However, so far, there are only a few researches on the critical damping in fractionally damped systems. In 1984, Torvik and Bagley [5] proposed a mechanical model with fractional derivatives in the study of the motion of a rigid plate immersed in a Newton fluid, and the study results in [6, 7] make the fractional calculus attractive for many engineers and technicians [8].

Vehicle suspension is an important component for improving the driving comfort and the handling performance [9], the research on its control strategy is a hot spot. In these control approaches, the skyhook control strategy proposed by Karnopp et al. [10, 11] is widely applied because of its simple algorithm and good control performance. The classical skyhook control principle is based on a SDOF vibration system, which is suitable for the vertical vibration control of two DOFs quarter-car models. In recent years, many scholars have studied the application of skyhook algorithm in full-car suspension model. The mainstream skyhook control strategies for full-car suspension systems are based on physical thinking; these strategies are the application extensions of the classical skyhook method that is widely used to control the quarter-vehicle suspension systems. The full-car suspension model is regarded as a simple combination of four 1/4 subsuspension models, and it is assumed that there is a “skyhook” connected with each 1/4 car body by four skyhook dampers to control the vibration of the car body, whereas the full-vehicle suspension has requirements of multiobjective suspension performances involving the vertical, pitch, and roll motions [12]. Therefore, the problem of how to coordinate the forces of the four independent controllers to keep a good body posture [13] needs to be solved and the typical solution is adding a decision-making system.

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 2, the conditions of fractional damped systems being in critical damping case are given first. Then the properties of the vibration with critical damping are studied. In Section 3, a new fractional skyhook control algorithm for full-car suspension systems is proposed. In Section 4, the simulation results are discussed. Conclusions are given in Section 5.

#### 2. Critical Damping of the System with Fractional Derivative Damping

##### 2.1. Formula Derivation

The free vibration differential equation of a SDOF system with fractional derivative damping has the form where is the displacement, is the fractional time derivative of , and , , and are the mass, damping, and stiffness coefficient, respectively.

There are many definitions for fractional derivatives [14], among which Riemann-Liouville definition and Caputo definitions are most widely used [15]. The former is frequently used for problem description because of its demand moderately for the continuity of the function. The latter has the same Laplace transform as the integer order one, so it is widely used in control theory. In this paper, the fractional derivative damping force is regarded as a control force to study the properties of free damped vibration of the system, so the Caputo definition is used here.

By the Laplace transform method, the characteristic equation of the system takes the formwhere is the complex variable. By substituting its polar form into (2), we have

Considering the Euler formula , (3) takes the form

The establishment-condition of (4) is that both the real and imaginary parts are equal to zero, so we obtain

It is known that when the imaginary part of the roots of (2) is non-zero, the damped free motion of the system is always oscillating. To avoid the oscillation, the characteristic roots must lie in the negative real axis. Assume that where is an integer, so and are obtained, then (5) can be simplified as

The establishing condition for (7) is , which means that . Therefore, it can be obtained thatwhere is an integer. As a result, we have

We find that the set of is dense, but the probability density of any locating in this domain is small, so the existence condition of critical damping is strict.

From (6), a negative damping coefficient is obtained when , which represents an energy input to the system. In this case, the system oscillation is strengthened, and there is no critical damping, while it is the opposite when ; that is, is odd, so substituting into (6) and then (10) is obtained. In summary, in (9) is an integer, is odd, and . The existence conditions of critical damping in the vibration systems with fractional derivative damping and its calculation formula are presented.

For linear 1 DOF fractionally damped systems, only when (9) is satisfied by the order of fractional operator, there is a critical value of damping coefficients. To make the solutions of (1) be without oscillation, the relation between the damping coefficient and the order iswhere . In (10), when , i.e. , we have the minimum value of damping coefficient which represents the critical damping coefficient* c*_{c}.

The curves that represent the relation between the variables in (10) are plotted in Figure 1. Take , for example, the lowest point of the curve represents the critical damping point and its corresponding damping coefficient is the critical value of damping coefficient. It is worth noting that many previous researches on 1 DOF fractionally damped systems focus on the solutions of the characteristic equations. From this perspective, we find when , the characteristic equations only have complex or conjugate roots and they have negative real roots when . Therefore, when , it represents the overdamping coefficient, and when , it is the underdamping coefficient. In the case of critical damping, the characteristic equation has the root , which represents the convergence rate. When increases from 0 to 2, the critical damping point is shifted to the lower right in the figure, which indicates that with larger , it turns out a smaller and larger ; that is, with a smaller eigenvalue, the system is a faster convergent.