A New Vibration-Absorbing Wheel Structure with Time-Delay Feedback Control for Reducing Vehicle Vibration

Abstract

To address the contradiction between ride comfort and handling stability, a new vibration-absorbing wheel structure with time-delay feedback control was proposed by applying a time-delay feedback-controlled dynamic vibration absorber to the wheel structure. Using H₂ optimization and the particle swarm optimization algorithm (PSO), the design parameters of the time-delay vibration-absorbing wheel structure were obtained with sprung mass acceleration (SMA), suspension deflection (SD) and dynamic tire deflection (DTD) as the optimization objective functions. The stability interval of the system was analyzed using the Routh–Hurwitz criterion and the Sturm criterion. The vehicle vibration reduction performance of the new vibration-absorbing wheel structure was simulated and verified in the time domain under harmonic excitation and random excitation. The simulation results show this new structure can effectively solve the contradiction between the ride comfort and handling stability. It provides a new technical idea for reducing vehicle vibration.

1. Introduction

With the development of vehicle technology, people have higher requirements for the ride comfort and handling stability of the vehicle. The suspension has an essential effect on the ride comfort and handling stability of the vehicle [1]. After the conventional passive suspension design is completed, the parameters are determined and cannot be well adapted to different road conditions. In the suspension process design, it is found that there is a contradiction between the ride comfort and handling stability of the vehicle [2]. Conventional suspension has difficulty in solving this problem, and people have had to compromise between ride comfort and handling stability [3]. To solve this problem, much research has been conducted in the past decades and many active suspension control methods have been proposed, such as adaptive control [4,5,6], sliding mode control [7,8], and H_∞ control [9,10]. The active control of the suspension inevitably suffers from time delay. People always ignore the time delay for the convenience of control design [11]. However, even a small time delay can cause the actuator to energy input to the system when the system does not need it, which may reduce the control efficiency or even lead to instability of the controlled system. People have different ways of dealing with time delay.

The conventional view is that time delay is an unfavorable factor that can lead to system instability [12,13]. The inherent time delay of the system is often offset by time-delay compensation [14,15]. However, Olgac et al. ingeniously found that using the time delay as an active control parameter of the system can effectively suppress the vibration of the system [16,17]. The magnitude of time delay in a control system directly affects the stability of the system, and using time delay as an active control parameter requires us to analyze the time-delay interval corresponding to the entire control system under the premise of stability. Therefore, to take advantage of the positive effect of time delay on system vibration reduction, people must explore effective time-delay stability analysis methods.

For stability analysis, time-domain analysis and frequency-domain analysis methods are generally available. The time-domain analysis method is based primarily on Lyapunov functions or functional analysis, and there is a certain conservatism in these methods. Improvements have been made to its methodology [18,19], but there is still some conservativeness. Wang et al. explored the time-delay-independent stability region and the stability switching region of systems [20]. In the time-delay-dependent stability region, the system switches between stability and instability as the time delay increases from zero to infinity, and the correctness is verified numerically on a two-degree-of-freedom model. However, with the increase in system degrees of freedom, the difficulty of stability analysis increases sharply. Li proposed a frequency domain scanning method [21], which can greatly simplify the stability analysis of multi-freedom systems, but it can only assess the stability of a certain control parameter and cannot draw the stability region of the whole system. People’s research on the stability analysis of systems with time delay has developed a lot [22,23,24,25], but it is still being studied in depth.

The use of time delay as an active control parameter to suppress system vibrations has become an important approach. Jalili et al. presented an optimal time delay feedback damper that provides a minimum peak frequency response over a given frequency range [26,27]. Udwadia et al. [28] explored a time delay positive proportional feedback control and verified the effectiveness of a single-degree-of-freedom model. Zhao et al. investigated the vibration reduction mechanism of a time-delay nonlinear dynamic vibration absorber and achieved better results under simple harmonic excitation [29]. Yan et al. explored the suppression of the main system by nonlinear time delay feedback dynamic vibration absorber under superimposed excitation [30]. Huan et al. studied the multi-objective optimal design of an active vibration absorber with delayed feedback [31]. Wu et al. applied time delay feedback to the active control of vehicle suspension [32,33]. Time delay feedback control has been in greater development.

Considering the above problems, the new wheel structure proposed in this paper is based on a conventional wheel structure equipped with a time-delay feedback-controlled dynamic vibration absorber. This new structure works by transferring the vibration energy generated by the road excitation to the tire to the mass block, reducing the vibration of the tire. The reduction in the amount of tire vibration will reduce the transfer of vibration energy to the vehicle’s body through the suspension. This new wheel structure solves the contradiction between vehicle ride comfort and handling stability. In addition, based on Wang’s method for analyzing the stability of the vehicle quarter model [20], it is generalized to the vehicle quarter model with the vibration-absorbing mass block, which provides a method of stability analysis for three-degree-of-freedom time-delay dynamical systems.

The remainder of the paper is organized as follows: Section 2 introduces the mathematical model of the new vibration-absorbing wheel structure and designs some parameters of the new structure; in Section 3, the stability of the system is analyzed; afterwards, the objective function of the system is designed and optimized; Section 5 presents the simulation results to verify the effectiveness of the proposed new vibration-absorbing wheel structure; conclusions are drawn in Section 6.

2. System Model of the New Wheel Structure

2.1. Mathematical Model

The schematic diagram of a new vibration-absorbing wheel structure proposed in this paper is shown in Figure 1a. To facilitate the establishment of a mathematical model, this new structure can be simplified to the model diagram shown in Figure 1b. The actuator provides the time-delay feedback control force. The excitation of the road surface causes the vertical vibration of the vehicle body, mass block and the wheel, and the sensors detect the signals of them. The signal conditioner transmits the displacement signal of the vehicle body, wheel and mass block to the controller. According to the signal from the sensor, the controller uses time-delay feedback control to make comprehensive judgments and input the best feedback control parameters g and τ. The actuator controls the applied force by changing the voltage value.

As shown in Figure 1, $m_s$ is the body mass, $m_t$ is the wheel mass, and $m_a$ is the mass block. $k_t$ is the stiffness of the tire. $k_s$ and $c_s$ are the stiffness and damping coefficients of the suspension; $k_a$ and $c_a$ are the stiffness and damping coefficients of this new structure; $u$ is the time-delay feedback control force; and $x_s$, $x_t$, $x_a$ and $x_r$ are the body displacement, tire displacement, mass block displacement and external excitation displacement, respectively. The model is used to investigate the vibration reduction performance of the vehicle with the new vibration-absorbing wheel structure, and the vehicle system model parameters are given in

Table 1. Vehicle system model parameters.

Parameters	Value	Parameters	Value
$m_s$ (kg)	345	cs (N·s/m)	1500
$m_t$ (kg)	40.5	g (N·s/m)	−30,000~30,000
$k_s$ (N/m)	17,000	τ (s)	0~1
$k_t$ (N/m)	192,000

.

According to D’Alembert’s principle, the dynamic equation of the model can be expressed as

$$\{\begin{array}{l}{m}_s{\ddot{x}}_s+c_s({\dot{x}}_s-{\dot{x}}_t)+k_s(x_s-x_t)=0\\ m_t{\ddot{x}}_t+c_s({\dot{x}}_t-{\dot{x}}_s)+c_a({\dot{x}}_t-{\dot{x}}_a)+k_t(x_t-x_r)+k_s(x_t-x_s)+k_a(x_t-x_a)+g{x}_a(t-\tau )=0\\ m_a{\ddot{x}}_a+c_a({\dot{x}}_a-{\dot{x}}_t)+k_a(x_a-x_t)-g{x}_a(t-\tau )=0\end{array},$$

(1)

where $g$ is the time-delay feedback gain coefficients of the active control force; $\tau$ is the time-delay; and $t$ is the time.

2.2. Preliminary Design of Parameters

The design of the structural parameters of the new vibration-absorbing wheel includes $m_a$, $k_a$, $c_a$, $g$ and $\tau$. These five parameters have a direct impact on the performance of the vehicle vibration reduction. Firstly, the design process of the absorber is borrowed for the design of $m_a$, $k_a$ and $c_a$. The parameters of $g$ and $\tau$ are designed in Section 4.2 using PSO for them. The purpose of the new vibration-absorbing wheel is to allow the mass block to absorb the vibration energy generated by the road excitation. Therefore, the H₂ parameter optimization method is used for parameter design in this chapter. According to the literature [34], the optimal damping and optimal tuning are obtained as follows:

$$\{\begin{array}{l}{\xi }_a=\sqrt{\frac{\mu (4+3\mu )}{8(1+\mu )(2+\mu )}}\\ f_a=\frac{1}{1+\mu }\sqrt{\frac{2+\mu }{2}}\end{array},$$

(2)

where ${\xi }_a$ is the optimal damping and $f_a$ is the optimal tuning. $u$ is the mass ratio. After deduction, the following can be obtained:

$$\{\begin{array}{l}{m}_a=\mu m_t\\ k_a=f_a{}^2\mu (k_t+k_s)\\ c_a=2{\xi }_a\sqrt{k_a{m}_a}\end{array}.$$

(3)

In the design of vibration absorbers, the mass ratio is generally less than 0.4. By selecting different mass ratios, multiple sets of H₂ optimization parameters with different mass ratios can be obtained according to Equation (3), as shown in

Table 2. Optimization parameters of H₂ with different mass ratios.

$\mathbf{Mass} \mathbf{Ratio} \mathit{\mu }$	${\mathit{m}}_{\mathit{a}} \left(\mathbf{kg}\right)$	${\mathit{k}}_{\mathit{s}} (\mathbf{N}/\mathbf{m})$	${\mathit{c}}_{\mathit{a}} (\mathbf{N}·\mathbf{s}/\mathbf{m})$
0.1	4.05	18,136.36	82.62
0.2	8.10	31,930.56	212.28
0.3	12.15	42,665.68	356.97
0.4	16.20	51,183.67	506.61

.

To investigate the effect of different design parameters on the frequency response of the vehicle body in Table 2, a Laplace transform of Equation (1) gives the following:

$$\left[\begin{array}{ccc}{A}_{11}& A_{12}& 0\\ A_{21}& A_{22}& A_{23}\\ 0& A_{31}& A_{32}\end{array}\right]\left[\begin{array}{c}{X}_s(s)\\ X_t(s)\\ X_a(s)\end{array}\right]=\left[\begin{array}{c}0\\ k_t{X}_r\\ 0\end{array}\right],$$

(4)

where $s=\omega \mathrm{i}$, and each coefficient matrix item is

$$\{\begin{array}{l}{A}_{11}=m_s{s}^2+c_{s}s+k_s\\ A_{12}=-c_{s}s-k_s\\ A_{21}=-c_{s}s-k_s\\ A_{22}=m_t{s}^2+(c_a+c_s)s-(k_s+k_t+k_a)\\ A_{23}=-c_{a}s-k_a+g{e}^{-\tau s}\\ A_{31}=m_a{s}^2+c_{a}s+k_a-g{e}^{-\tau s}\\ A_{32}=-c_{a}s-k_a\end{array}.$$

(5)

From Equation (4), the frequency-response function of the SMA of the vehicle is

$$\left|H_1(\omega )\right|=\left|\frac{{\ddot{X}}_s(s)}{X_r(s)}\right|=\left|\frac{{\omega }^2{k}_t{A}_{12}{A}_{32}}{A_{12}{A}_{21}{A}_{32}-A_{11}{A}_{22}{A}_{32}+A_{11}{A}_{23}{A}_{32}}\right|.$$

(6)

The frequency response function curves of different parameters on sprung mass acceleration can be drawn by Equation (6), as shown in Figure 2.

It is concluded that the larger the mass ratio, the better the vibration reduction effect on the main system, but the mass ratio should not be too large. In the design of dynamic vibration absorbers, the mass of the absorber is generally selected below 0.4 [35]. It can be seen from Figure 2 that the larger the mass block, the smaller the value of the frequency response function to sprung mass acceleration, i.e., the more obvious the vibration reduction effect on the vehicle body. However, the mass block is constrained by the structure space. When $\mu =0.1$, the response suppression effect on the vehicle body is not obvious. A better way to improve the vehicle body vibration reduction effect is to increase the mass ratio. When $\mu =0.2$, the response suppression effect on the vehicle body increases significantly. However, the vibration reduction effect does not increase proportionally with the increase in the mass ratio. In this design, the mass block $m_a$ is chosen to be 0.2 times the wheel mass $m_t$ taking into account the structural space of the wheel and the vibration reduction efficiency.

2.3. Dimensionless System Model

To facilitate the stability analysis of the system, the system parameters are dimensionless. The influence of damping $c_a=0~220\hspace{0.17em}\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$ of the new vibration-absorbing wheel on system stability is analyzed, and the mathematical model of system control is

$$\{\begin{array}{l}{\ddot{x}}_s{}^{*}+{\zeta }_1({\dot{x}}_s{}^{*}-{\dot{x}}_t{}^{*})+(x_s{}^{*}-x_t{}^{*})=0\\ {\ddot{x}}_t{}^{*}+{\zeta }_1\gamma ({\dot{x}}_t{}^{*}-{\dot{x}}_s{}^{*})+{\zeta }_2\gamma ({\dot{x}}_t{}^{*}-{\dot{x}}_a{}^{*})+\gamma (x_t{}^{*}-x_s{}^{*})+k_2\gamma (x_t{}^{*}-x_a{}^{*})+k_1\gamma (x_t{}^{*}-x_r{}^{*})+g^{*}\gamma x_a{}^{*}(t^{*}-{\tau }^{*})=0\\ {\ddot{x}}_a{}^{*}+\beta {\zeta }_2({\dot{x}}_a{}^{*}-{\dot{x}}_t{}^{*})+k_2\beta (x_a{}^{*}-x_t{}^{*})-g^{*}\beta x_a{}^{*}(t^{*}-{\tau }^{*})=0\end{array},$$

(7)

where ${\zeta }_1=\frac{c_s}{\sqrt{k_s{m}_s}}$, ${\zeta }_2=\frac{c_a}{\sqrt{k_s{m}_s}}$, $k_1=\frac{k_t}{k_s}$, $k_2=\frac{k_a}{k_s}$, $\gamma =\frac{m_s}{m_t}$, $\beta =\frac{m_s}{m_a}$, $g^{*}=\frac{g}{k_s}$; $x_s{}^{*}$, $x_t{}^{*}$, $x_a{}^{*}$ and $x_r{}^{*}$ are dimensionless body displacement, tire displacement, mass displacement, and external excitation displacement; $t^{*}$ is the dimensionless time; and ${\tau }^{*}$ is the dimensionless time-delay. The body displacement $x_s{}^{*}$, body velocity ${\dot{x}}_s{}^{*}$, tire displacement $x_t{}^{*}$, tire velocity ${\dot{x}}_t{}^{*}$, mass block displacement $x_a{}^{*}$, and mass block velocity ${\dot{x}}_a{}^{*}$ are selected as state variables, and the state space equation of the system can be obtained as

$$\{\begin{array}{l}\dot{\mathit{X}}=\mathit{A}\mathit{X}+\mathit{B}\mathit{U}+\mathit{E}\mathit{w}\\ \mathit{Y}=\mathit{C}\mathit{X}+\mathit{D}\mathit{w}\end{array},$$

(8)

where $\mathit{X}={[x_s{}^{*},{\dot{x}}_s{}^{*},x_t{}^{*},{\dot{x}}_t{}^{*},x_a{}^{*},{\dot{x}}_a{}^{*}]}^{\mathrm{T}}$ is the state vector; $\mathit{Y}=[{\ddot{x}}_s{}^{*}, x_s{}^{*}-x_t{}^{*}, x_t{}^{*}-x_r{}^{*}{]}^{\mathrm{T}}$ is the output vector; $\mathit{w}=[x_r{}^{*}, {\dot{x}}_r{}^{*}{]}^{\mathrm{T}}$ is the excitation vector of the road surface; $\mathit{U}=[x_s{}^{*}(t^{*}-{\tau }^{*}),$${\dot{x}}_s{}^{*}(t^{*}-{\tau }^{*}), x_t{}^{*}(t^{*}-{\tau }^{*}), {\dot{x}}_t{}^{*}(t^{*}-{\tau }^{*}), x_a{}^{*}(t^{*}-{\tau }^{*}), {\dot{x}}_a{}^{*}(t^{*}-{\tau }^{*}{\left)\right]}^{\mathrm{T}}$ is the state time-delay vector. The matrices in the above equation are

$$\begin{gathered} \mathit{A}=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ -1& -{\zeta }_1& 1& {\zeta }_1& 0& 0\\ 0& 0& 0& 1& 0& 0\\ \gamma & \gamma {\zeta }_1& -\gamma (1+k_1+k_2)& -\gamma ({\zeta }_1+{\zeta }_2)& \gamma k_2& \gamma {\zeta }_2\\ 0& 0& 0& 0& 0& 1\\ 0& 0& \beta k_2& \beta {\zeta }_2& -\beta k_2& -\beta {\zeta }_2\end{array}\right], \mathit{B}=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -g^{*}\gamma & 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& g^{*}\beta & 0\end{array}\right], \\ \mathit{E}=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ k_1\gamma & 0\\ 0& 0\\ 0& 0\end{array}\right],\mathit{C}=\left[\begin{array}{cccccc}-1& -{\zeta }_1& 1& {\zeta }_1& 0& 0\\ 1& 0& -1& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right],\mathit{D}=\left[\begin{array}{cc}0& 0\\ 0& 0\\ -1& 0\end{array}\right], \mathrm{respectively}. \end{gathered}$$

The parameters of the vehicle system model in Table 1 can be transformed into dimensionless parameters using Equation (7), as shown in

Table 3. Dimensionless model parameters.

Parameters	Value	Parameters	Value
${\zeta }_1$	0.6194	$g^{*}$	−1.7647~1.7647
${\zeta }_2$	0~0.0909	$k_1$	11.2941
$\beta$	42.5926	$k_2$	1.8783
$\gamma$	8.5185	$\tau$	0~7.0196

.

3. Stability Analysis

3.1. Time-Delay Independent Stability

A time-delay dynamical system is said to be time-delay independent stable if it is asymptotically stable for any given ${\tau }^{*}\ge 0$. According to the theory of time-delay differential equations, we obtain the following characteristic equation [36]:

$$\det (s\mathit{I}-\mathit{A}-\mathit{B}{\mathrm{e}}^{-s{\tau }^{*}})=0,$$

(9)

where $s$ is the characteristic root of the characteristic Equation (9); and $\mathit{I}$ is the identity matrix. The characteristic Equation (9) is expanded into a polynomial form, which can be expressed as

$$D(s,{\tau }^{*})=P(s)+Q(s){\mathrm{e}}^{-s{\tau }^{*}},$$

(10)

$P(s)$ and $Q(s)$ are real coefficient polynomials of the sixth and fourth order. These can be expressed as

$$\begin{array}{l}P(s)=s^6+p_5{s}^5+p_4{s}^4+p_3{s}^3+p_2{s}^2+p_{1}s+p_0\\ Q(s)=q_4{s}^4+q_3{s}^3+q_2{s}^2+q_{1}s+q_0\end{array}$$

(11)

where

$$\begin{array}{l}{p}_5={\zeta }_1+\beta {\zeta }_2+\gamma {\zeta }_1+\gamma {\zeta }_2\\ p_4=1+\gamma +\beta k_2+\gamma k_1+\gamma k_2+\beta {\zeta }_1{\zeta }_2+\gamma {\zeta }_1{\zeta }_2+\beta \gamma {\zeta }_1{\zeta }_2\\ p_3=\beta {\zeta }_2+\gamma {\zeta }_2+\beta \gamma {\zeta }_2+\beta {\zeta }_1{k}_2+\gamma {\zeta }_1{k}_1+\gamma {\zeta }_1{k}_2+\beta \gamma {\zeta }_1{k}_2+\beta \gamma {\zeta }_2{k}_1\\ p_2=\beta k_2+\gamma k_1+\gamma k_2+\beta \gamma k_2+\beta \gamma k_1{k}_2+\beta \gamma {\zeta }_1{\zeta }_2{k}_1\\ p_1=\beta \gamma {\zeta }_2{k}_1+\beta \gamma {\zeta }_1{k}_1{k}_2\\ p_0=\beta \gamma k_1{k}_2\\ q_4=-\beta g^{*}\\ q_3=-(\beta {\zeta }_1{g}^{*}+\beta \gamma {\zeta }_1{g}^{*})\\ q_2=-(\beta \gamma g^{*}+\beta g^{*}+\beta \gamma g^{*}{k}_1)\\ q_1=-\beta {\zeta }_1\gamma k_1{g}^{*}\\ q_0=-\beta \gamma k_1{g}^{*}\end{array}.$$

According to the stability theory, the critical time-delay stability is the characteristic equation with pure imaginary roots $s=i{\omega }_0$. For the time-delay independent stability of Equation (10), the following two conditions need to be satisfied simultaneously:

Condition 1.

When${\tau }^{*}=0$, the characteristic polynomial$D(s,0)=P(s)+Q(s)$is Routh-Hurwitz stable.

Condition 2.

When${\tau }^{*}>0$, the polynomial$D(s,{\tau }^{*})=0$has no real root${\omega }_0$.

For Condition 1, when ${\tau }^{*}=0$, Equation (10) can be expressed as

$$D(s,0)=P(s)+Q(s)=a_6{s}^6+a_5{s}^5+a_4{s}^4+a_3{s}^3+a_2{s}^2+a_{1}s+a_0,$$

(12)

where $a_6=1$, $a_5=p_5$, $a_4=p_4+q_4$, $a_3=p_3+q_3$, $a_2=p_2+q_2$, $a_1=p_1+q_1$, and $a_0=p_0+q_0$. According to the Routh–Hurwitz stability criterion, to ensure the asymptotic stability of the system, it is necessary to satisfy that the principal minor of each order of the determinant is greater than zero. The parameters of Table 3 are substituted into Equation (12), and the Routh–Hurwitz stability dividing line is judged to be obtained, as shown in line R₁ in Figure 3. Figure 3 is divided into two parts by lines R₁. The orange and green areas on the upper are the Routh–Hurwitz stability area, the blue area below does not meet the vehicle selection parameters.

For Condition 2, judge whether $D(i{\omega }_0,{\tau }^{*})=0$ has real roots. To facilitate the calculation, the real and imaginary parts of the polynomial can be separated by bringing Euler’s formula ${\mathrm{e}}^{-i{\omega }_0{\tau }^{*}}=\cos ({\omega }_0{\tau }^{*})-i\sin ({\omega }_0{\tau }^{*})$ into Equation (10). We can obtain

$$D(i{\omega }_0,\tau )=P_R({\omega }_0)+i{P}_I({\omega }_0)+[Q_R({\omega }_0)+\mathrm{i}{Q}_I({\omega }_0)][\cos ({\omega }_0\tau )-i\sin ({\omega }_0\tau )]=0,$$

(13)

where $P_R({\omega }_0)=-{\omega }_0{}^6+p_4{\omega }_0{}^4-p_2{\omega }_0{}^2+p_0$, $Q_R({\omega }_0)=q_4{\omega }_0{}^4-q_2{\omega }_0{}^2+q_0$, $Q_I({\omega }_0)=-q_3{\omega }_0{}^3+q_1{\omega }_0$, $P_I({\omega }_0)=p_5{\omega }_0{}^5-p_3{\omega }_0{}^3+p_1{\omega }_0$.

If Equation (13) is equal to zero, then the real and imaginary parts of the polynomial are zero, respectively. That is,

$$\{\begin{array}{c}{P}_R({\omega }_0)+Q_R({\omega }_0)\cos ({\omega }_0{\tau }^{*})+Q_I({\omega }_0)\sin ({\omega }_0{\tau }^{*})=0\\ P_I({\omega }_0)-Q_R({\omega }_0)\sin ({\omega }_0{\tau }^{*})+Q_I({\omega }_0)\cos ({\omega }_0{\tau }^{*})=0\end{array}.$$

(14)

By solving Equation (14) and eliminating the trigonometric function terms, a polynomial without time-delay ${\tau }^{*}$ can be obtained:

$$P_R{({\omega }_0)}^2+P_I{({\omega }_0)}^2-[Q_R{({\omega }_0)}^2+Q_I{({\omega }_0)}^2]=0.$$

(15)

Let $F({\omega }_0)=P_R{({\omega }_0)}^2+P_I{({\omega }_0)}^2-[Q_R{({\omega }_0)}^2+Q_I{({\omega }_0)}^2]$. If Equation (10) has a pair of purely imaginary roots $\pm \mathrm{i}{\omega }_0$ under the condition of ${\tau }^{*}\ge 0$, it must be satisfied that $F({\omega }_0)=0$ has a positive real root. The polynomial $F({\omega }_0)=0$ does not include the time-delay ${\tau }^{*}$. Through the above description, we can simplify the solution of whether the polynomial $D(i{\omega }_0,{\tau }^{*})=0$ has real roots or not to the solution of whether the polynomial $F({\omega }_0)=0$ has real roots or not. By this method, the judgment of condition 2 can be completed.

Equation (10) can be expressed as

$$F({\omega }_0)={\omega }_0{}^{12}+b_1{\omega }_0{}^{10}+b_2{\omega }_0{}^8+b_3{\omega }_0{}^6+b_4{\omega }_0{}^4+b_5{\omega }_0{}^2+b_6,$$

(16)

where

$$\begin{array}{l}{b}_1=p_5{}^2-2p_4\\ b_2=2p_2-2p_5{p}_3+p_4{}^2-q_4{}^2\\ b_3=p_3{}^2-q_3{}^2+2p_1{p}_5-2p_2{p}_4+2q_4{q}_2-2p_0\\ b_4=p_2{}^2-2p_1{p}_3-q_2{}^2-2q_0{q}_4+2p_0{p}_4+2q_1{q}_3\\ b_5=p_1{}^2+2q_0{q}_2-q_1{}^2-2p_0{p}_2\\ b_6=-q_0{}^2+p_0{}^2\end{array}.$$

Equation (16) is a 12th order polynomial, and the number of real roots of the polynomial equation $F({\omega }_0)=0$ can be determined according to the generalized Sturm criterion. Based on the theory of time-delay independent stability of two-degree-of-freedom vibration system, this paper extends it to the three-degree-of-freedom vibration system. We define the discriminant sequence $F({\omega }_0)$ as $[D_1(F),D_2(F),\dots ,D_{12}(F)]$, i.e., $[1,d_0,d_0{d}_1,d_1{d}_2,\cdots ,d_8{d}_9,d_9{}^2{d}_{10}]$. The formula $d_j(j=1, 2, \dots , 10)$ is shown in Appendix A.

To draw the time-delay independent stable region, The parameters of Table 3 are substituted into Equation (16); then, the coefficients $b_1$, $b_2$, $b_3$, $b_4$, $b_5$, and $b_6$ in Equation (16) are substituted into the Sturm criterion. We can draw the curve of the discriminant sequence coefficient $d_j=0\hspace{0.17em}(j=0, 1, \dots , 10)$. Taking the intersection with the curve satisfying condition 1, we can see that the selected parameter range is divided into 15 regions, as shown in Figure 3.

Sign tables of the discrimination sequence of the polynomial $F({\omega }_0)$ are shown in

Table 4. Sign tables of the discrimination sequence of the polynomial.

Region	${\mathit{d}}_0$${\mathit{d}}_1$${\mathit{d}}_2$$\dots {\mathit{d}}_9$${\mathit{d}}_{10}$	${\mathit{D}}_1$${\mathit{D}}_2$${\mathit{D}}_3$$\dots {\mathit{D}}_{11}$${\mathit{D}}_{12}$	$\mathit{l}-2\mathit{k}$
s1	+ − − − − − + + + − +	1, 1, −1, 1, 1, 1, 1, −1, 1, 1, −1, 1	0
s2	+ + − − − − + + + − +	1, 1, 1, −1, 1, 1, 1, −1, 1, 1, −1, 1	0
s3	+ + − − − − − + + − +	1, 1, 1, −1, 1, 1, 1, 1, −1, 1, −1, 1	0
s4, s5	+ + − − − − − − − + +	1, 1, 1, −1, 1, 1, 1, 1, 1, 1, −1, 1	4
s6, s7	+ + − + − − − − − + +	1, 1, 1, −1, −1, −1, 1, 1, 1, 1, −1, 1	4
s8, s9	+ + + + − − − − − + +	1, 1, 1, 1, 1, −1, 1, 1, 1, 1, −1, 1	4
s10, s11	+ + + + + − − − − + +	1, 1, 1, 1, 1, 1, −1, 1, 1, 1, −1, 1	4
s12, s13	+ + + − + − − − − + +	1, 1, 1, 1, −1, −1, −1, 1, 1, 1, −1, 1	4
s14, s15	+ + + − − − − − − + +	1, 1, 1, 1, −1, 1, 1, 1, 1, 1, −1, 1	4

. $l$ is the number of non-zero terms of the discriminant sequence and $k$ denotes the number of discriminant sequence number changes, then $F({\omega }_0)$ has $l-2k$ distinct real roots. If $l-2k=0$, the polynomial $F({\omega }_0)$ has no real roots, which means that the system increases with time delay, and the characteristic roots do not cross the imaginary axis. The stability of the system does not change, so the stability of the system depends on the stability of the system when ${\tau }^{*}=0$. If $l-2k>0$, it means that the polynomial $F({\omega }_0)$ has real roots, i.e., the stability of the system falls outside the time-delay independent stability region, and the stability of the whole system switches a finite number of times with the increase in time delay.

In the regions s₁, s₂ and s₃, the number of non-zero terms in the discrimination sequence is 12 and the number of sign changes is 6, so the polynomial has no real roots. In the regions of s₄~ s₁₅, the number of non-zero terms in the discrimination sequence is 12 and the number of sign changes is 4, so there are two positive real roots of the polynomial. From the above analysis, it can be concluded that the blue region part of Figure 3 does not meet condition 1, so it is not Routh–Hurwitz stable at ${\tau }^{*}=0$, and this region is unstable. There are real roots in the orange region, so it is necessary to determine the stability interval according to the stability switches. The green region is the time-delay independent stability. It can be found from Figure 3 that the time-delay independent stability region is V-shaped. We find that that the time-delay independent stability region expands with the increase in ${\zeta }_2$, which indicates a close relationship between damping and time-delay independent stability. The larger the damping, the larger the time-delay independent stability region.

3.2. Time-Delay Dependent Stability

From the damping determined in Section 2.2, we obtain ${\zeta }_2=0.0876$. If the feedback gain $g^{*}$ falls within the orange region of Figure 3, the system switches in stability as the time-delay changes. Assuming that the polynomial $F({\omega }_0)$ has positive real roots [37], a set of trigonometric equations can be derived as

$$\begin{array}{l}\cos ({\omega }_0{\tau }^{*})=-\frac{P_R({\omega }_0)Q_R({\omega }_0)+P_I({\omega }_0)Q_I({\omega }_0)}{Q_R{({\omega }_0)}^2+Q_I{({\omega }_0)}^2}\\ \sin ({\omega }_0{\tau }^{*})=\frac{P_I({\omega }_0)Q_R({\omega }_0)-P_R({\omega }_0)Q_I({\omega }_0)}{Q_R{({\omega }_0)}^2+Q_I{({\omega }_0)}^2}\end{array}$$

(17)

where ${\omega }_0{\tau }^{*}\in [0,2\pi )$.

When ${\omega }_0{\tau }^{*}\in [0,\hspace{0.17em}\pi ]$, we can solve for the relationship between ${\omega }_0$ and ${\tau }^{*}$ as follows:

$${\tau }_0{}^{*}=\frac{1}{{\omega }_0}\arccos (-\frac{P_R({\omega }_0)Q_R({\omega }_0)+P_I({\omega }_0)Q_I({\omega }_0)}{Q_R{({\omega }_0)}^2+Q_I{({\omega }_0)}^2}).$$

(18)

When ${\omega }_0{\tau }^{*}\in (\pi , 2\pi )$,

$${\tau }_0{}^{*}=\frac{1}{{\omega }_0}[2\pi -\arccos (-\frac{P_R({\omega }_0)Q_R({\omega }_0)+P_I({\omega }_0)Q_I({\omega }_0)}{Q_R{({\omega }_0)}^2+Q_I{({\omega }_0)}^2})].$$

(19)

The critical time-delay can be obtained:

$${\tau }_{\mathrm{i},}{{}_{\mathrm{j}}}^{*}={\tau }_0{}^{*}+\frac{2\mathrm{i}\pi }{{\omega }_0},\{\begin{array}{c}\mathrm{i}=1,2,\cdots \\ \mathrm{j}=0,1,\cdots \end{array},$$

(20)

where i denotes the i-th positive real root of the polynomial, and j denotes the j-th critical time delay corresponding to the i-th positive real root ${\omega }_0$. In this way, the relationship between ${\omega }_0$ and ${\tau }_{\mathrm{i},\mathrm{j}}{}^{*}$ can be obtained as ${\tau }_{\mathrm{i},\mathrm{j}}{}^{*}=T({\omega }_0)$. Then, the stability interval is determined according to the stability switches theory. The relation between the gain $g^{*}$ and ${\omega }_0$ can be obtained from Equation (16), ${\omega }_0=R(g^{*})$. According to the above, the relationship between $g^{*}$ and ${\tau }_{\mathrm{i},\mathrm{j}}{}^{*}$ can be obtained as ${\tau }_{\mathrm{i},\mathrm{j}}{}^{*}=T(R(g^{*}))$. Therefore, the time-delay dependent stability region can be drawn as shown in Figure 4.

3.3. Numerical Simulation Verification

To verify the time-delay stability analysis method with three degrees of freedom derived in this paper, eight points are randomly selected in Figure 4 for verification, where, ${\mathrm{P}}_1({\mathrm{g}}^{*}=-1.4,{\mathsf{\tau }}^{*}=0.74)$, ${\mathrm{P}}_2({\mathrm{g}}^{*}=-0.85,{\mathsf{\tau }}^{*}=1.988)$, ${\mathrm{P}}_3({\mathrm{g}}^{*}=-0.86,{\mathsf{\tau }}^{*}=4.181)$, ${\mathrm{P}}_4({\mathrm{g}}^{*}=-0.9,{\mathsf{\tau }}^{*}=5.903)$, ${\mathrm{P}}_5({\mathrm{g}}^{*}=0.91,{\mathsf{\tau }}^{*}=0.71)$, ${\mathrm{P}}_6({\mathrm{g}}^{*}=1,{\mathsf{\tau }}^{*}=1.25)$, ${\mathrm{P}}_7({\mathrm{g}}^{*}=0.85,{\mathsf{\tau }}^{*}=4.62)$, and ${\mathrm{P}}_8({\mathrm{g}}^{*}=0.85,{\mathsf{\tau }}^{*}=6.73)$. Applying a small external displacement excitation $\mathit{X}={[0,0,0.01,0,0,0]}^T$ to the system, the body displacement response is shown in Figure 5. From the verification results, we can obtain that the boundary of the critical time delay is accurate.

4. Objective Function and Optimization Design

4.1. Objective Function

Treating both the time-delay term and the external excitation in Equation (8) as excitation terms [30], the solution of the equation can be written as

$${\mathit{x}}^{*}(t^{*})=\exp (\mathit{A}{t}^{*}){\mathit{x}}^{*}(0)+{\displaystyle \underset{0}{\overset{t^{*}}{\int }}\exp [\mathit{A}(t^{*}-s)]\mathit{B}{x}_a{}^{*}(s-{\tau }^{*})}ds+{\displaystyle \underset{0}{\overset{t^{*}}{\int }}\exp [\mathit{A}(t^{*}-s)]\mathit{w}(s)}ds.$$

(21)

Letting the time step $\Delta t=t_{k+1}{}^{*}-t_k{}^{*}$, if the simulation duration is $T$, then the total number of steps is $n=T/\Delta t$. Equation (21) can be discretized into the following stepwise integral function of the recursive form:

$${\mathit{x}}_{k+1}{}^{*}=\exp (\mathit{A}\Delta t){\mathit{x}}_k{}^{*}+{\displaystyle \underset{t_k{}^{*}}{\overset{t_{k+1}{}^{*}}{\int }}\exp [\mathit{A}(t_{k+1}{}^{*}-s)]\mathit{B}{x}_a{}^{*}(s-{\tau }^{*})}ds+{\displaystyle \underset{t_k{}^{*}}{\overset{t_{k+1}{}^{*}}{\int }}\exp [\mathit{A}(t_{k+1}{}^{*}-s)]\mathit{w}(s)}ds.$$

(22)

Then, Equation (22) can be written as

$${\mathit{x}}_{k+1}{}^{*}=\exp (\mathit{A}\Delta t){\mathit{x}}_k{}^{*}+{\displaystyle \underset{k\Delta t}{\overset{(k+1)\Delta t}{\int }}\exp \{\mathit{A}[(k+1)\Delta t-s]\}\mathit{w}(s)}ds+{\displaystyle \underset{(k-n)\Delta t}{\overset{[(k+1)-n]\Delta t}{\int }}\exp \{\mathit{A}[(k+1)\Delta t-s-n\Delta t]\}\mathit{B}x(s)}ds.$$

(23)

By solving the equation, the vibration response of the system at each time point can be derived as

$$\{\begin{array}{c}{x}_s{}^{*}=[x_{s1}{}^{*}, x_{s2}{}^{*}, \dots , x_{sk}{}^{*}, \dots , x_{sn}{}^{*}]\\ {\dot{x}}_s{}^{*}=[{\dot{x}}_{s1}{}^{*}, {\dot{x}}_{s2}{}^{*}, \dots , {\dot{x}}_{sk}{}^{*}, \dots , {\dot{x}}_{sn}{}^{*}]\\ x_t{}^{*}=[x_{t1}{}^{*}, x_{t2}{}^{*}, \dots , x_{tk}{}^{*}, \dots , x_{tn}{}^{*}]\\ {\dot{x}}_t{}^{*}=[{\dot{x}}_{t1}{}^{*}, {\dot{x}}_{t2}{}^{*}, \dots , {\dot{x}}_{tk}{}^{*}, \dots , {\dot{x}}_{tn}{}^{*}]\end{array}k\in [1,n].$$

(24)

Bringing the values obtained from Equation (24) into Equation (7), the sprung mass acceleration can be calculated for each time point as

$${\ddot{\mathit{x}}}_s{}^{*}=[{\ddot{x}}_{s1}{}^{*}, {\ddot{x}}_{s2}{}^{*}, \dots , {\ddot{x}}_{sk}{}^{*}, \dots , {\ddot{x}}_{sn}{}^{*}].$$

(25)

The goal of the new vibration-absorbing wheel structure is to improve the ride comfort and handling stability of the vehicle to the maximum. Thus, SMA, SD, and DTD are selected as the optimization objective function, which can be expressed as

$$J(g, \tau )=\frac{\mathrm{RMS}({\ddot{x}}_s{}^{*})}{\mathrm{RMS}{({\ddot{x}}_s{}^{*})}_{\mathrm{p}}}+\frac{\mathrm{RMS}(x_s{}^{*}-x_t{}^{*})}{\mathrm{RMS}{(x_s{}^{*}-x_t{}^{*})}_{\mathrm{p}}}+\frac{\mathrm{RMS}(x_t{}^{*}-x_r{}^{*})}{\mathrm{RMS}{(x_t{}^{*}-x_r{}^{*})}_{\mathrm{p}}},$$

(26)

where the subscript p is the conventional structure performance index.

4.2. Optimization Procedure

To improve the search capability of the particle swarm algorithm, we use the variable inertia weight particle swarm optimization algorithm [37]. An optimization design method is described in Figure 6. The steps of the optimization procedure are presented as follows:

• Step 1: Initialization parameters, including the time-delay ${\tau }^{*}\in [0, 7.0196]$, feedback gain $g^{*}\in [-1.7647, 1.7647]$, population size (pop_size = 50), number of iterations (N = 200), maximum velocity ($V_{max}$), initial position($X_{id}$), initial velocity ($V_{id}$), inertia weight ($w^k$), and learning factors ($c_1=c_2=1.2$). The position of the particles in each dimension is restricted to the selected area and the maximum velocity is a certain fraction of the search space range in each dimension.
• Step 2: Randomly generate the initial population, and temporarily record it as the population optimum, and then find the optimal particle in the population and record it as the global optimum.
• Step 3: Record the number of iterations of the system and compare the current number of iterations with the maximum number of iterations.
• Step 4: Based on the original population, the population position is updated by the PSO for the iteration, as shown in Equation (27).

  $$\{\begin{array}{c}{V}_{id}{}^{k+1}=w^k{V}_{id}{}^k+c_1{r}_1(P_{id}{}^k-X_{id}{}^k)+c_2{r}_2(P_{gd}{}^k-X_{id}{}^k)\\ X_{id}{}^{k+1}=X_{id}{}^k+V_{id}{}^{k+1}\end{array}$$

  (27)

  where $k$ is the current number of iterations; $r_1$ and $r_2$ are random parameters between $[0, 1]$; $P_{id}$ is the individual optimal particle position; $P_{gd}$ is the global optimal particle position; $V_{id}$ is the particle velocity; and $X_{id}$ is the current position of the particle. The start inertia weight is $w_{\mathit{start}}=0.9$, end inertia weight is $w_{\mathit{end}}=0.4$, and the current inertia weight is

  $$w^k=w_{\mathit{start}}-k(w_{\mathit{start}}-w_{\mathit{end}})/N.$$

  (28)
• Step 5: In the optimization process, the objective function is also called the fitness function. The performance of the new structure is determined by the time delay (${\tau }^{*}$) and the feedback gain coefficient ($g^{*}$). The minimum value of the fitness function is obtained by optimizing these two parameters.
• Step 6: The particle positions updated in Step 4 are brought into the function in Step 5, and the new fitness function values are calculated. Each particle adaptation value is compared in an iterative search process, and the particle that produces a smaller value of fitness function is retained.
• Step 7: When the number of cycles reaches the set number of iterations $N$, the system outputs the optimal parameters.

By the above steps, the global optimal control parameters under harmonic excitation and random excitation are obtained, respectively: $g_h{}^{*}=-1.0316$ and ${\tau }_h{}^{*}=0.1079$ represent the time-delay feedback control parameters under harmonic excitation. $g_r{}^{*}=0.7962$ and ${\tau }_r{}^{*}=3.0446\times {10}^{-4}$ represent the time-delay feedback control parameters under random excitation. Then, transforming the obtained dimensionless parameters into dimensional parameters, we can obtain $g_h=-17537 \mathrm{N}\cdot \mathrm{s}/\mathrm{m}$, ${\tau }_h=1.5373\times {10}^{-2} \mathrm{s}$; $g_r=13537\mathrm{N}\cdot \mathrm{s}/\mathrm{m}$, ${\tau }_r=4.3373\times {10}^{-5} \mathrm{s}$.

5. Results Analysis

5.1. Simulation Analysis under Harmonic Excitation

The wheel without a mass block is called conventional suspension; the vibration-absorbing wheel structure without active control is called passive vibration-absorbing wheel structure (PVAWS); the vibration-absorbing wheel structure with time-delay feedback control is called the time-delay vibration-absorbing wheel structure (TDVAWS). Simulation verification of the vibration reduction efficiency of the absorbing wheel structure is completed near the resonance frequency of the conventional suspension. Take the optimized time-delay control parameters into Equation (1). The SMA, SD, and DTD of the vehicle are simulated under 11.25Hz, as shown in Figure 7. The performance indexes of the vehicle are given in

Table 5. RMS of vehicle performance indexes under harmonic excitation.

Performance Indexes	Conventional Suspension	PVAWS	TDVAWS
$\mathrm{RMS} \mathrm{of} \mathrm{SMA} (\mathrm{m}/{\mathrm{s}}^2)$	19.7211	11.1859	1.4964
$\mathrm{RMS} \mathrm{of} \mathrm{SD} (\mathrm{mm})$	63.3691	35.9532	5.1521
$\mathrm{RMS} \mathrm{of} \mathrm{DTD} (\mathrm{mm})$	72.0473	49.3270	33.0481

.

As illustrated by the simulation results in Figure 7, it can be seen that SMA, SD, and DTD of the vehicle equipped with the new vibration-absorbing wheel structure under harmonic excitation are better than those of the conventional suspension. From Table 5, it can be obtained that the RMS of SMA, SD and DTD of PVAWS are improved by 43.28%, 43.26%, and 31.54%, respectively. However, the RMS values of SMA, SD and DTD of TDVAWS are improved by 92.41%, 91.87% and 54.13%, respectively. Thus, the vibration reduction effect of TDVAWS is significantly better than that of PVAWS. It can be concluded that TDVAWS has an excellent effect on resonance. From the above data analysis, we can see that the new vibration-absorbing wheel structure plays an effective role in both the vehicle ride comfort and handling stability under resonant frequency band.

5.2. Simulation Analysis under Random Excitation

To further analyze the vibration reduction performance of the new vibration-absorbing wheel structure on the vehicle, a class C road surface model was established [38], as shown in Figure 8.

The optimized parameters are substituted into Equation (1) and simulated under the road surface excitation. The simulation pairs under random excitation are shown in Figure 9, and the performance indexes are shown in

Table 6. RMS of vehicle performance indexes under random excitation.

Performance Indexes	Conventional Suspension	PVAWS	TDVAWS
$\mathrm{RMS} \mathrm{of} \mathrm{SMA} (\mathrm{m}/{\mathrm{s}}^2)$	1.2342	1.1158	1.0465
$\mathrm{RMS} \mathrm{of} \mathrm{SD} (\mathrm{mm})$	11.0682	11.0286	11.0089
$\mathrm{RMS} \mathrm{of} \mathrm{DTD} (\mathrm{mm})$	4.0993	3.9222	3.5861

.

It can be seen from Figure 9 that the new wheel absorption structure has a better damping effect on both the sprung mass acceleration and tire dynamic displacement under random excitation. Compared with the conventional suspension, the RMS value of SMA of PVAWS is reduced by 9.59%, the RMS value of SD is reduced by 0.36%, and DTD of tires is reduced by 4.32%; the RMS value of SMA of TDVAWS is reduced by 15.21%, the RMS value of SD is reduced by 0.54%, and the RMS value of DTD is reduced by 12.52%. It can be concluded that both PVAWS and TDVAWS can improve vehicle ride comfort and handling stability, but TDVAWS has a better effect. Ref. [39] is the application of time-delay feedback control to the suspension. Compared with Ref. [39], TDVAWS has a better vibration reduction effect on sprung mass. In conclusion, this structure proposed in this paper can effectively solve the contradiction between vehicle ride comfort and handling stability.

6. Conclusions

Based on the conventional wheel structure, a new vibration-absorbing wheel structure with time-delay feedback control was invented. The time-delay independent stability and time-delay dependent stability of the system are investigated, and the time-delay feedback control parameters are optimized using the PSO algorithm. The proposed new vibration-absorbing wheel structure with time-delay feedback control is proved effective, according to the comprehensive analysis. The following conclusions can be obtained:

(1) Based on the theory of time-delay independent stability of two-degree-of-freedom vibration system, this paper extends it to the three-degree-of-freedom vibration system. In the range of time-delay independent stability, the system is stable at any value of the time delay. The time-delay independent stability of the system is related to the damping, and the range of time-delay independent stability will increase with the increase in damping. However, in the time-delay dependent region, the stability of the system switches with the increase in time delay and eventually becomes unstable.

(2) The size of the time delay affects the stability of a system. Time delay can either make a system unstable or make an unstable system stable. Choosing the proper size of time delay can improve the vehicle vibration reduction performance.

(3) The performance of the new vibration-absorbing wheel structure on the vehicle vibration reduction was analyzed in different external excitation. Compared with the passive suspension, TDVAWS improved the RMS values of SMA and DTD by 92.41% and 54.13% at 11.25 Hz; TDVAWS improved the RMS values of SMA and DTD by 15.21% and 12.52% under random excitation. It can be concluded that the new vibration-absorbing wheel structure with the time-delay feedback control can effectively solve the contradiction between SMA and the DTD, and better improve the ride comfort and handling stability. This can provide a useful reference for further experimental investigation.

TDVAWS has improved ride comfort and handling stability, but its performance needs further improvement. This needs to be considered, as its time delay varies with time. It means that new optimization methods and solution algorithms need to be created.