1. Introduction
The vehicle suspension system plays a crucial role in supporting the static weight of a vehicle while determining its overall performance regarding the ride-comfort and road-holding. Nowadays, vehicle suspension systems can be categorized into three types: passive suspension, semi-active suspension, and active suspension. Active suspension provides optimal performance but also requires a lot of energy to drive [
1]. Passive suspension is less expensive and does not require additional energy input to operate compared to active suspension. However, passive suspension has limitations when a vehicle requires greater performance [
2]. Comparatively, a semi-active suspension can provide better performance than passive suspension while incurring less energy consumption compared to an active suspension [
3,
4,
5]. To achieve the required performance, semi-active suspensions mainly utilize adjustable mechanical elements, such as semi-active damping [
6,
7,
8], a semi-active spring [
9], or a semi-active inerter [
10,
11].
In recent years, network synthesis has regained its position as a research hotspot with the proposal of the inerter [
12]. The emergence of inerters has completed the mechanical–electrical analogy, making it possible to apply electrical network synthesis to mechanical networks [
13,
14]. Papageorgiou and Smith [
15] were the first to propose a procedure for the synthesis of positive real controllers based on matrix inequalities, in which the
and
problems were considered. Passive elements were used to realize the admittance function. The feasibility of this procedure was verified in the design of quarter vehicle passive suspension. Chen et al. [
16] investigated the effect of admittance functions of different orders on the performance of the suspension system, finding that the performance improves with an increase in the order of the admittance functions, but the structure also becomes more complex. Wang et al. [
17], using up to four elements, investigated the network synthesis problem of biquadratic impedances and derived the sufficient conditions for the implementation of arbitrary biquadratic impedances. Chen et al. [
18] designed the structure of the ISD suspension system using Linear Matrix Inequality (LMI) and optimized the ISD suspension using the quantum genetic algorithm. Through simulations, they demonstrated that the suspension performance can effectively reduce the spring mass acceleration in the low-frequency band. Based on the inerter, Jason et al. [
19] analyzed the positive-real biquadratic functions that can be implemented by five components in mechanical networks, and provided their implementable conditions. Based on the research above, the method of combining the inerter, spring, and damping to construct various structural forms has shifted from the “structural method” to the “impedance method”, also known as the “black box method” [
20,
21,
22].
For traditional semi-active suspensions, the usual focus to improve their performance is on the semi-active control algorithm and optimization [
23]. However, with the advent of the inerter, a semi-active inerter can be used instead of a semi-active spring or semi-active damping. Hu et al. [
24] proposed a ball-screw semi-active inerter that can continuously adjust its inertance by adjusting the radius of the flywheel. Li et al. [
25] investigated semi-active suspension with a controlled inerter and designed an
state feedback controller. This resulted in a substantial reduction in the sprung mass acceleration on the spring at the intrinsic body frequency. Overall, it is clear that suspension systems with semi-active inerter can perform better than traditional suspension systems. To further explore this technique, in this paper, we design a semi-active ISD suspension using network synthesis, where we introduce a semi-active inerter under Sky-hook control.
This article is organized as follows. In
Section 2, we model and analyze the proposed semi-active ISD suspension based on the quarter vehicle model. We will also obtain the positive real controller’s BMI corresponding to it. In
Section 3, we solve the LMI and implement the solution to obtain the passive configuration of the considered suspension. In
Section 4, the parameters of the mechanical network from
Section 3 are optimized using PSO. In
Section 5, we introduce the semi-active Sky-hook control law and simulate the suspension’s overall performance for analysis.
Section 6 concludes the paper.
5. Performance Analysis and Discussion of Semi-Active ISD Suspensions
In this section, the simulation and performance analysis of the semi-active ISD suspension will be performed. The random road in
Figure 4 is selected as the input of the suspension system, and the semi-active ISD suspension proposed is simulated and compared with the traditional passive suspension and the traditional semi-active suspension. First of all, the ideal Sky-hook inerter suspension will be realized by the semi-active method. The maximum and minimum damping coefficient of traditional semi-active suspension are 300 and 2500, respectively. The damping coefficient of traditional passive suspension is 2200. The other parameters are the same as those of the suspension studied in this paper.
In this figure, the of the passive part will be replaced by the and mechanical networks, respectively, for the purpose of obtaining the first-order and second-order suspensions.
5.1. Semi-Active Realization of the Ideal Sky-Hook Inerter
The suspension is equipped with a semi-active inerter as shown in
Figure 5, whose first-order and second-order suspension dynamics equations are given in the following equation:
where
is the inertance of the semi-active inerter, and
represents the forces transmitted by the first-order and second-order mechanical networks to the suspension system,
.
The inerter is in proportion to the relative acceleration of the two ends. In this case, the force of the Sky-hook inertance is then in proportion to the acceleration of the sprung mass, as the acceleration of the sky is zero. Therefore, the Sky-hook inerter can improve the ride comfort of the vehicle by the force that the Sky-hook inerter produces. The force conducted by the ideal Sky-hook inerter in
Figure 1 is
. The force conducted by the semi-active inerter in
Figure 5 is
.
In order to make the semi-active suspension has the similar performance as the ideal Sky-hook inerter suspension, set
, where:
Similar to the Sky-hook damping control algorithm, for the Sky-hook inerter control algorithm, ON–OFF control and continuous control can be introduced according to the adjustable inertance of the semi-active inerter.
In order to make the force produced by the semi-active inerter equal to the one produced by the ideal Sky-hook inerter, take ; in ideal conditions, .
5.2. Suspension Performance Analysis
With two road classes and two velocities, the ISD suspension system is simulated using both ON–OFF control and continuous control, and the simulation time is chosen to be 20 s.
The RMS values of the performance are shown in
Table 5, and the performance improvement rate of semi-active ISD suspension compared to traditional suspension is shown in
Table 6. Taking the condition of class B road at 30 m/s for example, the system performance of different suspensions with time is shown in
Figure 6.
Figure 7 shows the performance curves in the frequency domain.
As can be seen from
Table 5 and
Table 6, the RMS value of sprung acceleration in
Table 5 is the main performance index to the ride comfort as shown in
Table 6. The tire dynamic load reflects the road-holding in
Table 6. Compared to the traditional passive suspension, the ride comfort from the first-order ON–OFF and Continuous ISD suspensions improves around 19% on the Class A road and 13% on the Class B road, while there is also a slight advantage compared to the traditional semi-active suspension. On the other hand, the ride comfort from the second-order ON-OFF and Continuous ISD suspension, compared to the traditional passive suspension, reduced about 45% on the Class A road and 33% on the Class B road, respectively. The performance is also much better than the traditional semi-active suspension. In addition, the semi-active ISD suspension proposed in this paper benefits the road-holding as well; the tire dynamic loads are smaller than the traditional passive suspension; and the improvements are over 10% with first-order suspensions and 20% for second-order suspensions on the Class A road. On the Class B road, improvement depends on different orders and control methods, which are about 1%, 4%, 7%, and 11%. The road-holding is about the same level as the traditional semi-active suspension. In both classes of roads, the velocities contribute little influence to the ride comfort and road-holding performance. Furthermore, for the suspension deflection, the first-order ISD suspension is the same as the traditional one, while the second-order ISD suspension needs a little more space, which is in an acceptable range.
The simulated performances for the traditional and semi-active ISD suspensions on the class B road at 30 m/s are shown in
Figure 6.
It can be seen in
Figure 6a that the sprung acceleration curve of the traditional passive suspension has the largest maximum peak to peak value, followed by the traditional semi-active suspension. When the first-order semi-active ISD suspension system proposed in this paper chooses ON–OFF control, its dynamic response of the sprung acceleration will have a jitter problem during the direction reversing, which is similar to the traditional ON–OFF type of semi-active suspension. It is caused by the fact that the ON–OFF control can only switch the inertance. Moreover, ON–OFF control of the inertance may also destroy control elements such as the control valve of the semi-active inerter. The curves of the second-order semi-active ISD suspension have a lower maximum peak to peak value than the traditional suspensions and first-order ones, in which the jitter problem still goes with the ON–OFF control. Thus, the second-order semi-active ISD suspension with continuous control performs the best. In
Figure 6b,c, the dynamic tire load and suspension deflection and the response curves of first-order and second-order semi-active ISD suspensions are consistent with the traditional ones. However, in the suspension deflection curve, the second-order semi-active ISD suspensions show the largest maximum peak to peak value. It can also be seen by the RMS value in the
Table 5, which means that the second-order semi-active ISD suspensions need more space in the design.
From
Figure 7a, it can be seen that the system responses excited by the road input are mainly in the frequency band of 1–5 Hz; the semi-active ISD suspension systems proposed in this paper have much lower PSD amplitude of the sprung acceleration compared to the traditional ones (passive, semi-active), and in the frequency band higher than 5 Hz, PSDs of the first-order, second-order semi-active ISD and traditional suspension are roughly the same. For the PSD of tire dynamic load in
Figure 7b, it can be seen that the first-order and second-order semi-active ISD suspensions can significantly reduce the tire dynamic load in the resonance frequency band compared with traditional suspension, which enhances the safety of car driving. The second-order semi-active ISD suspension produces a little deterioration in the frequency band of 0–1 Hz. In addition, the first-order semi-active ISD suspension with ON–OFF control has a partial deterioration in the middle frequency band of 6–10 Hz. In
Figure 7c, the body resonance frequency of suspension deflection in the second-order semi-active ISD suspension system is mainly concentrated between the 0 Hz and 1 Hz band compared with the traditional and first-order semi-active ISD suspensions, while the PSD of the latter two in the full frequency band have the same curve trend. It means that the second-order semi-active ISD suspension system has the lowerest resonance frequency and broadest frequency band for vibration attenuation. It can be shown by our simulation results that the semi-active ISD suspension proposed in this paper can effectively improve the ride comfort of the vehicle without reducing the road-holding of the vehicle. Notably, the second-order semi-active ISD suspension system shows a better performance than the first-order one.