Research on Energy Regeneration Characteristics of Multi-Link Energy-Fed Suspension

Abstract

Inspired by the single-blade hyperboloid, a new type of multi-bar shock absorber was designed, which can recover vibration energy. Its principle is to convert the droop reciprocating vibration of the vehicle in the spatial domain into the reciprocating rotational motion in the plane through the trajectory and force characteristics of the single-blade hyperboloid moving along the space. To improve the efficiency of energy regeneration, a mechanical motion filtering mechanism was designed. Through theoretical derivation, the energy regeneration formula of a new type of multi-rod shock absorber was obtained. After simulation analysis and experimental verification, under the input excitation of 1.82 Hz, the maximum instantaneous output voltage can reach 29 V, the maximum excitation current is 0.58 A, and the maximum power is 16.84 W. The efficient recovery and utilization of energy have been achieved, and the ride comfort of the vehicle has been improved.

1. Introduction

Automobile suspension refers to the mechanical component connecting the body and the wheel, which plays the role of supporting the weight load and alleviating the random road vibration impact in order to ensure ride comfort and operation stability. The traditional suspension designed based on experience cannot adjust the stiffness and damping coefficient independently during driving, so it cannot balance the contradiction [1,2,3]. Nowadays, the vehicle is moving towards the direction of intelligence, electrification, and the Internet of Things integration, and thus an active suspension that can adjust the stiffness and damping coefficient independently has become a main research goal. The implementation of an active suspension system improves cornering stability and tire adhesion, thereby reducing body tilt. However, in order to achieve fast and precise control, external energy input needs to be adjusted and regulated, resulting in an increase in overall vehicle energy consumption. At present, some manufacturers have begun to develop vehicles equipped with active suspension systems and are actively exploring the combination of active suspension and intelligent control and other technologies, hoping to achieve the energy efficiency and intelligence of the suspension system. If the combination of an energy-fed suspension and an active suspension can be realized, then, the goal of active control in suspension operation and the offset of external energy input can be achieved, reducing the research cost of active suspension. The study of the energy conversion mechanism of suspension systems and the exploration of their regenerative potential in harnessing vibration energy are therefore highly significant [4,5].

According to the form of energy storage, the energy recovery methods proposed in the current research can be roughly divided into three forms: air pressure, hydraulic pressure, and electromagnetic. Li Zhongxing of Jiangsu University simulated the ride comfort of large trucks by Matlab and deeply studied the dynamic characteristics [6]. The active hydraulic interconnected suspension studied by Guo Yaohua et al. proposed a new type of active hydraulically interconnected suspension (HIS) system actuator; the stabilization speed is 20% faster than the passive HIS system, and the side inclination is 55% [7]. The electromagnetic induction force of the DC motor is proportional to its speed, which makes it possible to provide damping force while regenerating energy in generator mode. As shown in Figure 1, the linear motor acts as an actuator to ensure stability and comfort when encountering bumpy road conditions and can also act as a generator to recover energy [8]. The simplified single-phase equivalent model proposed by Chen Shian et al. reduces the “zero zone” of the linear motor as the actuator by using the variable voltage charging control and improves the control accuracy [9]. The energy management unit designed by Pei Jinshun et al. of Chongqing University not only improves the energy recovery ratio of the linear motor type of active suspension system but also expands the operable frequency range of the system’s energy recovery and improves the energy recovery performance of the system [10].

The rotary electromagnetic collecting mechanism is more compact and has higher energy density than the linear collecting mechanism. Gupta et al. have experimentally confirmed that the feed power of the rotary motor is much greater than that of the linear motor [11]. Common linear–rotary motions include mechanical and hydraulic transmission. Xie et al. [12] introduced an energy collection absorber featuring ball screw transmission and multiple controllable generators, as shown in Figure 2. The dissipated kinetic energy in the damping process is recovered, and the damping coefficient is adjusted independently. When the displacement input frequency is 3 Hz and the amplitude is 20 mm, the average power reaches 32 W [12]. Both can enhance the driving performance effectively; control performance of the ball screw mechanism is not ideal under high-frequency excitation when the nuts of the DC motor are rotating at high speed, and thus the efficiency is significantly reduced [13]. Li et al. [14] carried out laboratory test analysis and actual road field tests on a newly designed improved rack-and-pinion collecting mechanism, as shown in Figure 3. Under the excitation of 30 mm amplitude and 0.5 Hz vibration frequency, the total power conversion efficiency is approximately 56% [14]. Although the rack-and-pinion structure is relatively simple and reliable, its friction problems and maintenance reduce the efficiency and increase the cost of the vehicle.

The hydraulically based shock absorber primarily uses hydraulic oil to transfer the vertical displacement of the hydraulic cylinder to the hydraulic pump or motor in the generating circuit, converting translational motion into power. Achieving stability in hydraulic regeneration-based damper performance requires the use of necessary components such as gas accumulators and hydraulic check valves as motion rectifiers [15]. Fang et al. built a prototype of a hydraulic electromagnetic damper compacted by a hydraulic rectifier and an internal accumulator, a system with a regeneration efficiency of 16 [16]. Li et al. first developed a hydraulic-based regenerative damper, as shown in Figure 4, and they then went on to develop a hydraulic motion rectifier (HMR) that relies on four sets of check valves to correct rotation direction [17]. Therefore, the motion rectifier is an indispensable part of the rotary collection mechanism, which plays a role in improving the energy recovery efficiency and maintaining the stability of the system.

In various vibration energy collection structures, electromagnetic vibration energy collectors have been widely used in the field of automotive regenerative suspension for their high energy conversion efficiency, quick response, adjustable control, and robust energy recovery capability [18]. Therefore, this paper establishes a two-degree-of-freedom (2-DOF) 1/4 vehicle model for random excitation of complex road surfaces according to vehicle dynamics analysis and analyzes the transmission characteristics of a multi-link active suspension system. Through the establishment of a top lifting coordinate system and a rotating coordinate system, the relationship between the falling height h and the rotation angle $\alpha$ can be obtained. Further, a mathematical model of the multi-link system is established based on material mechanics, and the relationship between the vibration amplitude and the rotation angle is analyzed and obtained. According to the working mode of the permanent magnet brushless three-phase motor and the typical star-shaped connection structure, the mathematical model of the permanent magnet brushless three-phase motor and its electromagnetic torque characteristics are established. At the same time, the relationship between active control force F and current I can be obtained by combining the mechanical characteristics of the brushless three-phase motor. The feasibility of energy recovery of the multi-link intelligent suspension system is verified by experiments, and the supercapacitor is considered as the electric energy storage unit.

2. System Modeling

2.1. Two-Degree-of-Freedom 1/4 Vehicle Modeling

The introduction of wheel dynamics resulted in a 2-DOF model and a corresponding reduction in the ride or handling benefits obtained by the active suspension. Methods based on vehicle dynamics analysis generally use 1/4 or 1/2 simplified vehicle models; this type of analysis is very successful in reducing the computational effort. The simplified 1/4 vehicle model can fully reflect the vertical vibration caused by external disturbance, and the corresponding parametric response can be obtained in time. Therefore, to effectively obtain the system dynamic response caused by road excitation, the 2-DOF model of a 1/4 vehicle equipped with a multi-link energy-fed active suspension system was created, as illustrated in Figure 5a. The multi-link energy-fed suspension system consists of a multi-link conversion mechanism, a transmission system, and a permanent magnet three-phase brushless motor. Figure 5b displays the schematic of the multi-link structure, and its mechanical model can be regarded as a 2-DOF parallel mechanism. At the same time, the following assumptions are made:

(1)

The interior and various parts of the vehicle are rigid bodies, without deformation;

(2)

The vehicle’s mass is uniformly distributed between the two centers of mass, located in the body and the chassis;

(3)

The vehicle only moves in the vertical direction, without considering the movement of the vehicle in the horizontal direction;

(4)

The suspension system is a linear elastic element, without considering nonlinear effects;

(5)

The wheels remain vertical in the vertical direction, ignoring the rolling and pitching of the wheels;

(6)

The vehicle maintains the same speed when turning, without considering the change in the acceleration of the vehicle in actual driving.

Based on automobile dynamics theory, the dynamic differential equation for a two-degree-of-freedom quarter vehicle model is derived as follows:

$$\left\{\begin{array}{c}{m}_b{\ddot{x}}_b=-k_s\left(x_b-x_w\right)-c_s\left({\dot{x}}_b-{\dot{x}}_w\right)+F\\ m_w{\ddot{x}}_w=k_s\left(x_b-x_w\right)+c_s\left({\dot{x}}_b-{\dot{x}}_w\right)-k_t\left(x_w-x_g\right)-F\end{array}\right.$$

(1)

where $m_b$ represents the body mass; $m_w$ is the wheel quality; $x_b$ stands for body displacement; $x_w$ indicates wheel displacement; $x_g$ refers to road surface input; $k_s$ is the suspension stiffness; $k_t$ is the tire stiffness; $c_s$ stands for the suspension damping; and F stands for active control.

2.2. Modeling of Random Pavement Input

According to the statistical analysis, the random unevenness of continuous pavement can be regarded as random event excitation; the potholes and bumps on the pavement are discrete event excitations. This paper mainly analyzes the random event excitation of the road surface, and the type of road surface can be divided into eight different grade classifications according to the road roughness coefficient. Generally, the road power spectral density $G_q\left(n\right)$ can be used to represent the roughness characteristics of the driving road surface. The greater the power spectral magnitude, the greater the recoverable power; therefore, the potential recoverable power of active suspension can be determined by power spectral density analysis. Based on the measured random road unevenness data, the range and average value of the vertical displacement power spectral density $G_q\left(n\right)$ for various road surface levels are determined through algorithmic processing.

Table 1. Eight grade classification standards of road roughness.

Grade of Pavement	Lower Limit	Geometric Mean	Upper Limit
A	8	16	32
B	32	64	128
C	128	256	512
D	512	1024	2048
E	2048	4096	8192
F	8192	16,384	32,768
G	32,768	65,536	131,072
H	131,072	262,144	524,288

defines the upper and lower bounds, as well as the geometric mean values, of road roughness coefficients for all pavement levels when W = 2 [19,20].

Road roughness is the vibration input in the process of vehicle driving, usually using the road vertical displacement power spectrum density $G_q\left(n\right)$. In addition, the statistical characteristics of road roughness can also be supplemented by the speed power spectral density $G_{\dot{q}}\left(n\right)$ and acceleration power spectral density $G_{\ddot{q}}\left(n\right)$:

$$G_q\left(n\right)=G_q\left(n_0\right){\left({\displaystyle \frac{n}{n_0}}\right)}^{-W}$$

(2)

$$G_{\dot{q}}\left(n\right)={(2\pi n_0)}^2{G}_q\left(n_0\right)$$

(3)

$$G_{\ddot{q}}\left(n\right)=16{\pi }^4{n}_0^2{n}^2{G}_q\left(n_0\right)$$

(4)

where n represents the spatial frequency (${\mathrm{m}}^{-1}$); $n_0$ is the reference spatial frequency (${\mathrm{m}}^{-1}$); $G_q\left(n_0\right)$ stands for the road roughness coefficient (${\mathrm{m}}^3$); and W stands for the frequency index.

When the vehicle is in the actual movement process of driving, the speed $v_0$ of the vehicle’s vibration input factor should also be considered. By applying Equations (5) and (6), the spatial frequency power spectral density $G_q\left(n\right)$ is converted into time-frequency power spectral density $G_q\left(f\right)$, which can be obtained when the frequency index W = 2:

$$f=v_{0}n$$

(5)

$$G_q\left(f\right)={\displaystyle \frac{G_q\left(n\right)}{v_0}}=G_q\left(n_0\right)n_0^2{\displaystyle \frac{v_0}{f^2}}$$

(6)

Then, the velocity power spectral density of the time-frequency domain $G_{\dot{q}}\left(f\right)$ is:

$$G_{\dot{q}}\left(f\right)={(2\pi n_0)}^2{v}_0{G}_q\left(n_0\right)$$

(7)

The power spectral density of velocity at the acceleration frequency $G_{\ddot{q}}\left(f\right)$ is:

$$G_{\ddot{q}}\left(f\right)=16{\pi }^4{n}_0^2{f}^2{v}_0{G}_q\left(n_0\right)$$

(8)

It is evident that the velocity power spectral density in the time-frequency domain $G_{\dot{q}}\left(f\right)$ is proportional to the road roughness coefficient $G_q\left(n_0\right)$ and the speed $v_0$ of the vehicle and does not change with the frequency; therefore, the velocity power spectral density is “white noise”. The random pavement profile in this paper is generated by filtering unit white noise, with the filter expression for white noise given as follows:

$${\dot{x}}_g\left(t\right)=-2\pi f_0{x}_g\left(t\right)+2\pi n_0\sqrt{G_q\left(n_0\right)v_0}w\left(t\right)$$

(9)

where $w\left(t\right)$ is the time domain signal of unit white noise; $f_0$ stands for the lower cutoff time-frequency domain; and $x_g\left(t\right)$ stands for the time domain signal of the road surface spectrum.

In this paper, the random pavement input is based on filtered white noise, and the equation can be expressed as an equation of state:

$$\left\{\begin{array}{c}{\dot{X}}_r=A_r{X}_r+B_{r}w\\ Y_r=C_r{X}_r+D_{r}w\end{array}\right.$$

(10)

where $X_r=x_g$; $A_r=-2\pi f_0$; $B_r=2\pi n_0\sqrt{G_q\left(n_0\right)v_0}$; $C_r=1$; and $D_r=0$.

For the system motion and road input equations, the fixed parameters are constrained by the matrix form, making it challenging to convert the fixed parameters into variable ones for input when using the 1/4 suspension system state-space equation to analyze vehicle suspension performance evaluation metrics. That is, $G_q\left(n\right)=G_q\left(n_0\right){\left({\displaystyle \frac{n}{n_0}}\right)}^{-W}$, in the chosen road, is the vertical displacement power spectral density, and the road roughness coefficient $G_q\left(n_0\right)$ is a fixed parameter during input. However, for the variable road surface model $G_q\left(n_0\right)$, it must change over time during input and must be variable.

In the chosen state vector of the system, where $\mathit{X}={\left[{\dot{x}}_b,{\dot{x}}_w,x_b,x_w,x_g\right]}^{\mathrm{T}}$, the system motion equation and road surface input equation are expressed in matrix form based on vehicle system dynamics, from which the system state-space equation is derived as follows:

$$\dot{\mathit{X}}=\mathit{A}\mathit{X}+\mathit{B}\mathit{U}+\mathit{H}\mathit{W}$$

(11)

where $\mathit{W}=\left[w\left(t\right)\right]$ is the Gaussian white noise input matrix; $\mathit{U}=\left[F\left(t\right)\right]$ is the control input matrix of active suspension; A is the system state matrix; B is the control input matrix; and H is the pavement input matrix.

The system motion equation can be written in matrix form:

$$\mathit{A}=\left[\begin{array}{ccccc}-c_s/m_b& c_s/m_b& -K_s/m_b& K_s/m_b& 0\\ c_s/m_w& -c_s/m_w& K_s/m_w& \left(-K_t-K_s\right)/m_w& K_t/m_w\\ 1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& -2\pi f_0\end{array}\right]$$

(12)

$$\mathit{B}=\left[\begin{array}{c}1/m_b\\ -1/m_w\\ 0\\ 0\\ 0\end{array}\right]$$

(13)

$$\mathit{H}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 2\pi n_0\sqrt{G_q\left(n_0\right)v_0}\end{array}\right]$$

(14)

In a similar manner, the system’s output equation can be derived as follows:

$$\mathit{Y}=\mathit{C}\mathit{X}+\mathit{D}\mathit{U}$$

(15)

Typically, BA (body vertical acceleration), SWS (suspension dynamic stroke), and DTD (wheel dynamic displacement) are chosen as key performance evaluation metrics in vehicle suspension design. Therefore, where $\mathit{Y}={\left[{\ddot{x}}_b,x_b-x_w,x_w-x_g\right]}^{\mathrm{T}}$, Equation (15) is written in matrix form to obtain:

$$\mathit{C}=\left[\begin{array}{ccccc}-c_s/m_b& c_s/m_b& -K_s/m_b& K_s/m_b& 0\\ 0& 0& 1& -1& 0\\ 0& 0& 0& 1& -1\end{array}\right]$$

(16)

$$\mathit{D}=\left[\begin{array}{c}1/m_b\\ 0\\ 0\end{array}\right]$$

(17)

X and Y are expressed in the form of a state-space equation:

$$\left\{\begin{array}{c}\dot{X}=AX+Bu+Hw\\ Y=CX+Du\end{array}\right.$$

(18)

3. Analysis of Motion Characteristics

3.1. Multi-Link Parallel Mechanism

Multi-link suspension is a kind of complex suspension system composed of multiple rods and rotating connections. Compared with the traditional single-link suspension system, multi-link suspension can provide more freedom and more accurate dynamic control of vehicles. The conversion device proposed in this paper is simple and reliable in structure, small in space, and light in weight. It can adjust the stiffness by actively adjusting the elastic rod and has strong self-adaptability. In comparison with the series mechanism, the parallel mechanism has a higher positioning accuracy, dynamic performance, strength-to-weight ratio, and lower manufacturing cost in theory, and its positioning accuracy will directly affect the actual performance.

According to reference [21], the 6-DOF structure can be used to realize complex control of robot joints, while the 2-DOF structure is sufficient for small vehicles. Therefore, a simplified new 2-DOF multi-link parallel mechanism is proposed. Figure 6 is an assembly drawing of the parallel mechanism; the multi-link mechanism is separated from the permanent magnet motor by a rotating disk in order to ensure that the motion mode conversion of suspension and the operation energy storage of electromagnetic motor are independent of each other. The multi-link mechanism is composed of an upper disk platform, a torsion disk, a universal joint, a spring bar, and a torsion bar. The spring bar is composed of a rigid rod with an embedded spring to provide resilience and play a role in supporting rebound, ensuring that the torsion bar can be quickly restored to its original state after torsion. The upper end of the rigid rod is threaded to realize active adjustment of length to cope with different vehicle conditions. The lower-end rod head is slightly larger than the 5 mm optical hole to prevent falling off. The torsion rod is composed of a universal joint and a rigid rod, which is connected between the upper plate and the torsion plate by screws. There are unidirectional bearings and thrust bearings between the torsion disk and the bottom disk. Unidirectional bearings are used to transfer the torque of unidirectional rotation to ensure maximum power, and unidirectional bearings are connected with the input end of the electromagnetic motor for energy recovery and storage. Thrust bearings bear axial forces to ensure the transmission of torque and the reliability of the structure. Figure 7 is the overall framework flow chart of this study.

3.2. Modeling of Algebraic Helical Mechanisms

In this paper, in the conversion mechanism, the bar connecting the upper and lower platforms is modeled as a rigid two-force bar, installed between the body chassis and the wheel to reduce vibration and transform the body’s motion into a reciprocating rotational motion, which transfers the motion energy to the permanent magnet motor for storage and release for active control. The unidirectional bearing between the torsion disc and the thrust bearing will amplify and transmit the torque of the torsion disc, which greatly improves the efficiency of energy feed.

Figure 8 is the motion diagram of the three-bar mechanism. Combined with the motion characteristics of the three-bar mechanism, the rotating coordinate system $O_A(x,y,z)$ and the top lifting coordinate system $O_B(x^{\prime },y^{\prime },z^{\prime })$ are established. The rotating coordinate system $O_A(x,y,z)$ is responsible for rotation, that is, all the coordinate z values in the coordinate system are kept from becoming zero. The top lifting coordinate system $O_B(x^{\prime },y^{\prime },z^{\prime })$ is responsible for the lifting, that is, $z^{\prime }$, the median value of all coordinate values is constantly changing, and $x^{\prime },y^{\prime }$ does not change with movement. The orientation of the X-axis $O_A{A}_1$ is parallel; the Z-axis is aligned with the central axis and points upward; and the Y-axis direction is defined by the right-hand rule. The $x^{\prime }$ axis, $y^{\prime }$ axis, and $z^{\prime }$ axis directions in the top lifting coordinate system $O_B(x^{\prime },y^{\prime },z^{\prime })$ are parallel to the directions of the X-axis, Y-axis, and Z-axis in the rotating coordinate system $O_A(x,y,z)$, respectively [22].

In the top lifting coordinate system ${O^{\prime }}_B(x^{\prime },y^{\prime },z^{\prime })$, the initial installation position of three universal joints can be expressed as:

$$B_1^{\prime }=M\times \left[\begin{array}{c}{x}_{B_1^{\prime }}\\ y_{B_1^{\prime }}\\ 0\end{array}\right],B_2^{\prime }=M\times \left[\begin{array}{c}{x}_{B_2^{\prime }}\\ y_{B_2^{\prime }}\\ 0\end{array}\right],B_3^{\prime }=M\times \left[\begin{array}{c}{x}_{B_3^{\prime }}\\ y_{B_3^{\prime }}\\ 0\end{array}\right]$$

(19)

where $M=\left[\begin{array}{c}i\\ j\\ k\end{array}\right]$. By the same token, the coordinates of the initial installation position in the rotating coordinate system are:

$$A_1=M\times \left[\begin{array}{c}{x}_{A_1}\\ y_{A_1}\\ 0\end{array}\right],A_2=M\times \left[\begin{array}{c}{x}_{A_2}\\ y_{A_2}\\ 0\end{array}\right],A_3=M\times \left[\begin{array}{c}{x}_{A_3}\\ y_{A_3}\\ 0\end{array}\right]$$

(20)

By setting the height of suspension drop with the vehicle moving process to h, the new coordinate in the top lifting coordinate system can be obtained:

$$B_{1\prime }^{\prime }=M\times \left[\begin{array}{c}{x}_{B_1^{\prime }}\\ y_{B_1^{\prime }}\\ -h\end{array}\right],B_{2\prime }^{\prime }=M\times \left[\begin{array}{c}{x}_{B_2^{\prime }}\\ y_{B_2^{\prime }}\\ -h\end{array}\right],B_{3\prime }^{\prime }=M\times \left[\begin{array}{c}{x}_{B_3^{\prime }}\\ y_{B_3^{\prime }}\\ -h\end{array}\right]$$

(21)

The new coordinates in the rotating coordinate system are:

$$A_1^{\prime }=M\times \left[\begin{array}{c}{x}_{A_1^{\prime }}\\ y_{{A_1^{\prime }}_1}\\ 0\end{array}\right],A_2^{\prime }=M\times \left[\begin{array}{c}{x}_{A_2^{\prime }}\\ y_{A_2^{\prime }}\\ 0\end{array}\right],A_3^{\prime }=M\times \left[\begin{array}{c}{x}_{A_3^{\prime }}\\ y_{A_3^{\prime }}\\ 0\end{array}\right]$$

(22)

In the process of vehicle movement, the top lifting platform of the suspension is always parallel to the rotating platform, and the spring rod and torsion rod connecting the two platforms are rigid rods. Therefore, in the process of converting the upper and lower vibrations of the suspension into rotating motion, the coordinate points in the rotating coordinate system $O_A(x,y,z)$ can be considered to be obtained by the point $B_i$ rotating around the origin of a certain rotation angle; this rotation angle is the angle of rotation. In the rotating coordinate system $O_A(x,y,z)$, the coordinate transformation of each point $B_i{B}_i$ has the same characteristics of the curve trajectory of any point on the upper and lower platform. This paper only focuses on the anchor points $A_1$ and $B_1^{\prime }$ and simplifies the three-dimensional coordinate system into a two-dimensional coordinate system for research.

Let the angle between $A_1$ and the X-axis be $\theta$ and the rotation angle of the rotating disc be $\alpha$ when the suspension moves with the vehicle. That is, the rotation angle of each coordinate point in the rotating coordinate system around the coordinate origin is $\alpha$, and the length of the $A_1$ distance from the coordinate origin is r.

$$r=\sqrt{{x_{A_1^{\prime }}}^2+{y_{{A_1^{\prime }}_1}}^2}$$

(23)

$$r={\displaystyle \frac{x_{A1}}{\cos \theta }}={\displaystyle \frac{y_{A1}}{\sin \theta }}$$

(24)

$$r={\displaystyle \frac{x_{A_1^{\prime }}}{\cos (\theta +\alpha )}}={\displaystyle \frac{y_{A_1^{\prime }}}{\sin (\theta +\alpha )}}$$

(25)

Expand the equation according to the trigonometric Equation (25) to obtain:

$$\left\{\begin{array}{c}{x}_{A_1^{\prime }}=r(\cos \theta \cos \alpha -\sin \theta \sin \alpha )\\ y_{A_1^{\prime }}=r(\sin \theta \cos \alpha +\cos \theta \sin \alpha )\end{array}\right.$$

(26)

You can get:

$$\left\{\begin{array}{c}{x}_{A_1^{\prime }}=x_{A_1}\cos \alpha -y_{A_1}\sin \alpha \\ y_{A_1^{\prime }}=y_{A_1}\cos \alpha +x_{A_1}\sin \alpha \end{array}\right.$$

(27)

It is not difficult to see that the coordinate point after rotation is only related to the original coordinate point and the rotation angle, and then new coordinate in the rotating coordinate system can be represented as:

$$\begin{array}{l}{A}_1^{\prime }=M\times \left[\begin{array}{c}{x}_{A_1}\cos \alpha -y_{A_1}\sin \alpha \\ y_{A_1}\cos \alpha +x_{A_1}\sin \alpha \\ 0\end{array}\right],\\ A_2^{\prime }=M\times \left[\begin{array}{c}{x}_{A_2}\cos \alpha -y_{A_2}\sin \alpha \\ y_{A_2}\cos \alpha +x_{A_2}\sin \alpha \\ 0\end{array}\right],\\ A_3^{\prime }=M\times \left[\begin{array}{c}{x}_{A_3}\cos \alpha -y_{A_3}\sin \alpha \\ y_{A_3}\cos \alpha +x_{A_3}\sin \alpha \\ 0\end{array}\right]\end{array}$$

(28)

Let Q represent the vector connecting the two points $A_1^{\prime }$ and $B_{1\prime }^{\prime }$ in the coordinate system—the vector where the rigid rod is located:

$$Q=A_1^{\prime }{B}_{1\prime }^{\prime }=M\times \left[\begin{array}{c}{x}_{A_1}\cos \alpha -y_{A_1}\sin \alpha -x_{B_1^{\prime }}\\ y_{A_1}\cos \alpha +x_{A_1}\sin \alpha -y_{B_1^{\prime }}\\ h\end{array}\right]$$

(29)

Combined with the structural characteristics of the rigid rod, the length of the rigid rod can be set as l, and then l can be represented as:

$$l=\sqrt{\begin{array}{l}{(x_{A_1}\cos \alpha -y_{A_1}\sin \alpha -x_{B_1^{\prime }})}^2+{(y_{A_1}\cos \alpha +x_{A_1}\sin \alpha -y_{B_1^{\prime }})}^2\\ +h^2\end{array}}$$

(30)

Therefore, the relationship between the falling height h and the rotation angle $\alpha$ can be obtained as follows:

$$h=\sqrt{l^2-a^2-b^2}$$

(31)

where $a=x_{A_1}\cos \alpha -y_{A_1}\sin \alpha -x_{B_1^{\prime }}$ and $b=y_{A_1}\cos \alpha +x_{A_1}\sin \alpha -y_{B_1^{\prime }}$.

3.3. Simulation of Motion Transformation Relationship

Considering the energy dissipation and wear of the motion conversion device in the actual rotating operation process, $r_A=r_B=r_C$ is used to simplify the motion transformation model during the simulation. The simulation software is used to analyze the rotation angle of the rotating coordinate system $O_A(x,y,z)$ of the three-bar mechanism, the top lifting coordinate system $O_B(x^{\prime },y^{\prime },z^{\prime })$ in the XOY plane, and the relationship between the top lifting coordinate system $O_B(x^{\prime },y^{\prime },z^{\prime })$ and the falling height $\Delta h$ of the rotating coordinate system $O_A(x,y,z)$. Figure 9 illustrates the connection between rod length, radius, rotation angle, and height change.

After analyzing the relation surface diagram, it can be obtained that the value range of the rotation angle α is [0,4π] and that there are two periods in the value range. In other words, the smallest positive period is 2π. As the rotation angle α increases, the changing height $\Delta h$ initially rises and then falls. That is, the value range conforms to the movement trend of the top lifting plane falling first and then rising. At the same time, the value range reflects that a different rotating rod radius r or length l have an impact on the rotating angle α and the changing height $\Delta h$. The changing height $\Delta h$ is inversely proportional to the length l, and the changing height $\Delta h$ is proportional to the rotating rod radius r.

The data of the multi-link type energy-fed suspension prototype are as follows: D = 5 mm, $l_i=140$ mm, $r_A=r_B=r_C=35$ mm, and n = 3. According to the relation conversion equation obtained in the previous section, the curve relationship diagram between the rotation angle α and the changing height $\Delta h$ can be obtained by substituting the data.

The blue solid line represents the relationship between the rotation angle and the variation height considering the diameter of the member, while the red dashed line represents the situation without considering the diameter of the member. Ignoring the diameter interference of the rotating member, the rotation angle α ranges from [−π,π] and the changing height $\Delta h$ is 18.76 mm. Observing the multi-link motion conversion characteristic figure in Figure 10, it can be observed that this is a sectional view of the aforementioned diagram, which confirms the accuracy of the relationship between the rotation angle α and the changing height $\Delta h$ of the multi-link conversion mechanism. Usually, a suspension dynamic stroke value within 10 mm can ensure the comfort of the ride and the stability of control. Through the calculation of the design of the multi-link mechanism, the changing height $\Delta h$ is greater than 10 mm; therefore, the design of this multi-link mechanism meets the needs of driving and has certain structural advantages of motion conversion.

4. Analysis of Energy Feed Characteristics

4.1. Torsion Model of Imaginary Thin-Walled Cylinder

If the initial state of a multi-link mechanism is regarded as a thin-walled cylinder torsion, because the middle part of the deformation is too large, it does not conform to the rigid body hypothesis. Therefore, this paper seeks a hypothetical rigid body to replace the initial state of torsion. Through the simulation experiment, we obtain the angle and falling height data; through numerical analysis software, we are able to fit the relationship between the two; and then, finally, through the torsion part of the torque and torsion angle relationship, we obtain torque. The multi-link mechanism has two “dead points” when it is torsional, that is, two special positions that cannot be torsional or interfere with the torsional process. To prevent contact collisions between the three rotating rods from affecting motion conversion, the minimum torsion circle at the center of the active energy-fed automobile suspension in the two-degree-of-freedom multi-link parallel mechanism is designed as the circle tangent to the three rotating rods, as shown in Figure 11. This is the minimum space formed by the three rotating rods. The small circle tangential to the three rotating rods in the middle is the smallest circle, which is also the limit position before the interference of the three-bar mechanism.

In the process of motion transformation, the shortest distance between the two rods creates a plane of rotational change, referred to as the minimum rotation plane or the unnecessary collision surface. Thus, the radius of this plane is defined as $r_m$. Assume the rod’s cross-section is circular, with the radius of the large circle denoted as R; it can be seen by geometric relations that: the center of the three big circles can form a regular triangle; the center of the small circle can be connected to the center of the big circle; and the center of the small circle is the inner (outer center, vertical center) of the triangle. If the side length of the triangle is 2R and the distance between the center of the small circle and the center of the big circle is r + R, then the geometric relation can be obtained as: $R+r={\displaystyle \frac{2R}{\sqrt{3}}}$, and r = 0.4 can be obtained by solving. Through simulation analysis, it can be seen that no matter how the three-bar mechanism is transformed, the shape and size of its smallest circle will not change; therefore, the plane where the smallest circle is located is supposed to have a thin-walled cylinder and the three-bar mechanism is supposed to twist together, that is, to replace the initial state for torsion transformation. The material of the imaginary thin-walled cylinder is the same as that of the rotating rod, and its radius is r + D. With the three-bar structure, the rotation angle of the upper and lower rotating disk is the imaginary rigid body torsion angle.

The equation of material mechanics can be obtained:

$$\phi ={\displaystyle \frac{Ml}{G{I}_P}}$$

(32)

where $\phi$ is the torsion angle; $M$ is the torque; $l$ is the length of the imaginary thin-walled cylinder, i.e., the initial rod length; $G$ is the shear modulus; and $I_P$ is the polar moment of inertia.

Then, the torque is:

$$M={\displaystyle \frac{G{I}_P\phi }{l}}$$

(33)

The shear modulus can be obtained by the relationship between the three elastic constants, i.e.,

$$G={\displaystyle \frac{E_t}{2\left(1+\mu \right)}}$$

(34)

where the elastic modulus $E_t=210$ and Poisson’s ratio $\mu =0.3$.

The polar moment of inertia for a hollow circular section is:

$$I_P={\displaystyle \frac{\pi D^4}{32}}(1-{\alpha }^4)$$

(35)

where D is the outside diameter of the imaginary thin-walled cylinder and $\alpha ={\displaystyle \frac{d}{D}}$ is the ratio of the inside diameter to the outside diameter.

The torque obtained by substituting the initial equation is:

$$M={\displaystyle \frac{E_t\pi D^4\phi (1-{\alpha }^4)}{64l(1+\mu )}}$$

(36)

The miniature three-phase brushless generator is used in the model machine, the internal structure of which is shown in Figure 12. The gear mechanism is added to the interior of the permanent magnet generator to increase the speed and thus improve the power. The number of teeth of the main shaft of the motor L = 10; the module of the gear m = 0.5; the pressure angle $\alpha ={20}^{\circ }$; i represents the gear mechanism’s transmission ratio; and the transmission efficiency of the multi-link mechanism is $\eta$. The torque transmitted to the rotor is:

$$T\prime =\mathrm{M}\cdot \eta \cdot i={\displaystyle \frac{E_t\pi D^4\phi (1-{\alpha }^4)\eta i}{64l(1+\mu )}}$$

(37)

The relationship between torque, speed, and linear speed can be obtained:

$$T\prime ={\displaystyle \frac{2\pi }{\omega }}\hspace{1em}\hspace{1em}\upsilon =\omega r$$

(38)

We can obtain:

$$\upsilon ={\displaystyle \frac{2\pi }{T\prime }}r={\displaystyle \frac{128\pi rl(1+\mu )}{E_t\pi D^4\phi (1-{\alpha }^4)\eta i}}$$

(39)

It can be known from the internal structure of a permanent magnet brushless three-phase motor that its rotor belongs to a ring magnetic field, and its magnetic induction intensity can be obtained by the magnetic charge method. Figure 13 shows a model of the magnetic charge method:

Surface magnetic charge magnetic field strength:

$$H=-\nabla \phi =-\nabla {\displaystyle \underset{s}{\oint }\frac{{\sigma }_{m}dS}{4\pi \mu r}}=-{\displaystyle \underset{s}{\oint }\frac{{\sigma }_{m}dS}{4\pi \mu }}\nabla {\displaystyle \frac{1}{r}}={\displaystyle \underset{s}{\oint }\frac{{\sigma }_m}{4\pi \mu }}{\displaystyle \frac{1}{r^3}}dS$$

(40)

Magnetic induction intensity:

$$B=\mu H=\mu H_{+}+\mu H_{-}={\displaystyle \underset{S_{+}}{\iint }\frac{{\sigma }_m}{4\pi }}{\displaystyle \frac{r_{+}}{r_{+}^3}}dS-{\displaystyle \underset{S_{-}}{\iint }\frac{{\sigma }_m}{4\pi }}{\displaystyle \frac{r_{-}}{r_{-}^3}}dS$$

(41)

B expanded in the rectangular coordinate system:

$$\begin{array}{l}{\displaystyle \underset{S_{+}}{\iint }\frac{{\sigma }_m}{4\pi }}{\displaystyle \frac{r_{+}}{r_{+}^3}}dS=\\ {\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z-h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z-h/2)}^2]}^{3/2}}rdrd\theta }}\end{array}$$

(42)

$$\begin{array}{l}{\displaystyle \underset{S_{-}}{\iint }\frac{{\sigma }_m}{4\pi }}{\displaystyle \frac{r_{-}}{r_{-}^3}}dS=\\ {\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z+h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z+h/2)}^2]}^{3/2}}rdrd\theta }}\end{array}$$

(43)

We can obtain:

$$\begin{array}{l}B={\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z-h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z-h/2)}^2]}^{3/2}}rdrd\theta }}\\ -{\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z+h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z+h/2)}^2]}^{3/2}}rdrd\theta }}\end{array}$$

(44)

The induced electromotive force obtained from the law of electromagnetic induction is:

$$E=NBlv\sin \theta$$

(45)

where N represents the number of coil turns; B represents the magnetic induction intensity of the uniform magnetic field, unit (T, t); l stands for conductor length, unit (m, m); $\theta$ represents the angle between the magnetic field direction and the conductor direction of movement, unit (°, degree); and E represents the induced electromotive force at both ends of the conductor, unit (V, volt).

Substituting the induced electromotive force equation gives:

$$\begin{array}{l}E=({\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z-h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z-h/2)}^2]}^{3/2}}rdrd\theta }}\\ -{\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z+h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z+h/2)}^2]}^{3/2}}rdrd\theta }})\cdot \\ {\displaystyle \frac{N128\pi r{l}^2(1+\mu )}{E_t\pi D^4\phi (1-{\alpha }^4)\eta i}}\end{array}$$

(46)

Since the rotating disk in the suspension system will dynamically adjust in real time with the motion state of the vehicle, it is necessary to use the three-phase full-bridge rectifier circuit, as illustrated in Figure 14 for rectification. The electrical angle during one cycle of the AC power supply, where the thyristor does not conduct due to the positive anode voltage, is referred to as the control angle or phase shift angle, which is expressed by α.

(1) $\alpha \le {60}^{\circ }$

$$U_d={\displaystyle \frac{1}{\pi /3}}{\displaystyle {\int }_{\frac{\pi }{3}+\alpha }^{\frac{2\pi }{3}+\alpha }\sqrt{2}}\sqrt{3}{U}_2\sin \omega td(\omega t)=2.34U_2\cos \alpha$$

(47)

(2) $\alpha >{60}^{\circ }$

$$U_d={\displaystyle \frac{1}{\pi /3}}{\displaystyle {\int }_{\frac{\pi }{3}+\alpha }^{\pi }\sqrt{3}}\sqrt{2}{U}_2\sin \omega td(\omega t)=2.34U_2[1+\cos (\pi /3+\alpha )]$$

(48)

where $U_2$ is the input AC phase voltage and the relationship between the induced electromotive force and voltage is equal to large reverse, obtaining:

(1) $\alpha \le {60}^{\circ }$

$$\begin{array}{l}{U}_d={\displaystyle \frac{299.52N\pi r{l}^2(1+\mu )\cos \alpha }{E_t\pi D^4\phi (1-{\alpha }^4)\eta i}}\cdot \\ \hspace{1em} ({\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z-h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z-h/2)}^2]}^{3/2}}rdrd\theta }}\\ \hspace{1em} -{\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z+h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z+h/2)}^2]}^{3/2}}rdrd\theta }})\end{array}$$

(49)

(2) $\alpha >{60}^{\circ }$

$$\begin{array}{l}{U}_d=[1+\cos (\pi /3+\alpha )]{\displaystyle \frac{299.52N\pi r{l}^2(1+\mu )}{E_t\pi D^4\phi (1-{\alpha }^4)\eta i}}\cdot \\ \hspace{1em} ({\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z-h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z-h/2)}^2]}^{3/2}}rdrd\theta }}\\ \hspace{1em} -{\displaystyle \frac{{\sigma }_m}{4\pi }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{R_1}{\overset{R_2}{\int }}\frac{(x-r\cos \theta )i+(y-r\sin \theta )j+(z+h/2)k}{{[{(x-r\cos \theta )}^2+{(y-r\sin \theta )}^2+{(z+h/2)}^2]}^{3/2}}rdrd\theta }})\end{array}$$

(50)

Average output current:

$$I_d={\displaystyle \frac{U_d}{R}}$$

(51)

4.2. Modeling of Permanent Magnet Brushless Three-Phase Motor

The energy recovered by the energy-fed suspension system during the dynamic adjustment of the vehicle will be reused for active suspension control, aiming to recycle energy and reduce the need for additional energy input in the active suspension. Therefore, according to the working mode of the three-phase six states of the permanent magnet brushless three-phase motor and the typical star-shaped connection structure characteristics, this paper establishes the mathematical model of the permanent magnet brushless three-phase motor and its electromagnetic torque characteristics. To facilitate the analysis and calculation, it is especially assumed that:

(1)

The electrical conductivity of permanent magnet material is zero;

(2)

The armature winding is distributed on the inner side of the stator;

(3)

There is no saturation in the magnetic circuit of the motor;

(4)

The armature magnetic motive force generated when the motor winding through the current is not considered;

(5)

The parameters of the motor are not changed by temperature fluctuations;

(6)

The motor parameters do not change with frequency changes;

(7)

The stator current and rotor magnetic field are arranged in square wave symmetry in the motor;

(8)

The three-phase stator is wound to form a center symmetrical arrangement.

Permanent magnet brushless three-phase motor stator voltage balance equation:

$$\left[\begin{array}{l}{u}_a\\ u_b\\ u_c\end{array}\right]=\left[\begin{array}{l}\begin{array}{ccc}{R}_a& 0& 0\end{array}\\ \begin{array}{ccc}0& R_b& 0\end{array}\\ \begin{array}{ccc}0& 0& R_c\end{array}\end{array}\right]\left[\begin{array}{l}{i}_a\\ i_b\\ i_c\end{array}\right]+\left[\begin{array}{l}{\begin{array}{ccc}{L}_a& L_{ab}& L\end{array}}_{ac}\\ \begin{array}{ccc}{L}_{ba}& L_b& L_{bc}\end{array}\\ {\begin{array}{ccc}{L}_{ca}& L_{cb}& L\end{array}}_c\end{array}\right]P\left[\begin{array}{l}{i}_a\\ i_b\\ i_c\end{array}\right]+\left[\begin{array}{l}{e}_a\\ e_b\\ e_c\end{array}\right]$$

(52)

In the equation: $u_a、u_b、u_c$ represents the stator terminal voltage of each phase (unit is V); $e_a,e_b,e_c$ is the stator induced electromotive force of each phase (unit is V); $i_a,i_b,i_c$ represents the stator phase current of each phase (unit is A); $L_a,L_b,L_c$ represents the stator self-inductance of each phase (unit is H); $L_{ab},L_{ba},L_{ca},L_{ac},L_{bc},L_{cb}$ is the mutual inductance between each phase stator winding (unit is H); $R_a,R_b,R_c$ represents the phase resistance of each phase stator winding (unit is $\Omega$); and P represents the differential operator (unit is $d/dt$).

In a working cycle, the position of the rotor does not change with the change of the working state, ensuring that the three phases are wound to form a symmetrical installation of the center. Then, $L=L_a=L_b=L_c$, $M=L_{ab}=L_{ba}=L_{ca}=L_{ac}=L_{bc}=L_{cb}$, $R=R_a=R_b=R_c$, $i_a+i_b+i_c=0$, and $M_{ia}+M_{ib}+M_{ic}=0$ can be substituted into the above equation, obtaining:

$$\left[\begin{array}{l}{u}_a\\ u_b\\ u_c\end{array}\right]=\left[\begin{array}{l}\begin{array}{ccc}R& 0& 0\end{array}\\ \begin{array}{ccc}0& R& 0\end{array}\\ \begin{array}{ccc}0& 0& R\end{array}\end{array}\right]\left[\begin{array}{l}{i}_a\\ i_b\\ i_c\end{array}\right]+\left[\begin{array}{l}\begin{array}{ccc}L-M& 0& 0\end{array}\\ \begin{array}{ccc}0& L-M& 0\end{array}\\ \begin{array}{ccc}0& 0& L-M\end{array}\end{array}\right]P\left[\begin{array}{l}{i}_a\\ i_b\\ i_c\end{array}\right]+\left[\begin{array}{l}{e}_a\\ e_b\\ e_c\end{array}\right]$$

(53)

The equivalent circuit of the stator of the permanent magnet brushless three-phase motor can be drawn, as shown in Figure 15, according to the stator voltage equation of the motor.

Electromagnetic power equation:

$$P_e=e_a{i}_a+e_b{i}_b+e_c{i}_c$$

(54)

Electromagnetic torque equation:

$$T_e={\displaystyle \frac{9550P_e}{n}}={\displaystyle \frac{9550(e_a{i}_a+e_b{i}_b+e_c{i}_c)}{n}}$$

(55)

Thus, the active control of the output can be obtained as follows:

$$F_e={\displaystyle \frac{T_e}{r}}={\displaystyle \frac{9550(e_a{i}_a+e_b{i}_b+e_c{i}_c)}{nr}}$$

(56)

4.3. Modeling of Control Circuit

The active suspension actuator discussed in this paper consists of a multi-link energy-fed suspension and a permanent magnet brushless DC mechanism, as shown Figure 16a,b. These components form the control circuit model when used as a generator and a motor, where M represents a permanent magnet brushless DC motor; L denotes inductance; U is the supply voltage; I is the armature circuit current and R refers to the internal resistance of the armature circuit.

An ideal actuator possesses the following characteristics:

$$\begin{array}{l}{U}_M=K_M{\dot{x}}_M\\ T_e=K_{M}I\end{array}$$

(57)

where $U_M$ represents the electromotive force of the motor; ${\dot{x}}_M$ stands for the axial speed of the multi-link mechanism. The multi-link mechanism is converted to the transmission motor constant, when $K_M$ is unchanged, the electromagnetic torque $T_e$ is proportional to the current I. Therefore, the size of the active control force can be achieved by controlling the motor armature current.

Combined with the mechanical characteristics of the brushless motor, the relationship between the active control force F and the current I can be obtained as follows:

$$F_a={\displaystyle \frac{2\pi {\eta }_t{\eta }_b{K}_T}{P_h}}{I}_a$$

(58)

$$K_T={\displaystyle \frac{T_{\max }}{I_{\max }}}$$

(59)

The maximum no-load speed is:

$$n_{\max }={\displaystyle \frac{v_{\max }}{P_h}}$$

(60)

The maximum output torque is:

$$T_{\max }={\displaystyle \frac{k_s{l}_f{P}_h}{4\pi {\eta }_t{\eta }_b}}$$

(61)

The maximum output power is:

$$P_{\max }=2\pi T_{\max }{n}_{\max }$$

(62)

The winding wire resistance is:

$$R_1={\displaystyle \frac{U_s}{I_{pv}}}$$

(63)

The maximum input current is:

$$I_{\max }={\displaystyle \frac{P_{\max }}{U_s}}$$

(64)

where ${\eta }_t$ is the transmission efficiency of the multi-link mechanism; ${\eta }_b$ is the comprehensive transmission efficiency of rolling bearings and one-way bearings; $K_T$ is the equivalent torque constant; $P_h$ is the up and down vibration stroke of the multi-link mechanism; $I_a$ is active control armature current; $v_{\max }$ is the maximum line speed of the suspension vibration up and down under bad working conditions; $k_s$ is the equivalent static stiffness of the multi-link suspension; $l_f$ is the free travel of the suspension; and $I_{pv}$ is the peak locked rotor current. Other motor parameters are electromagnetic load, stator core diameter, rotor magnetic steel length, etc. The mechanical characteristic curve of the brushless motor is illustrated in Figure 17.

This paper explores the use of supercapacitors as energy storage units. While they have higher self-discharge rates and lower energy density compared with rechargeable batteries, supercapacitors can handle very high charge and discharge rates, and their charging process is relatively straightforward. Supercapacitors are not affected by memory effects, so their actual service life is very long. The charging requirements and methods of rechargeable batteries are usually stricter, and if the charging method is not proper, it will lead to overcharging, which may damage the battery. Combined with the circuit schematic according to Faraday’s induction law, we can conclude that the back electromotive force (e) is inversely proportional to the speed of the moving part of the electromagnetic damper, although with an opposite sign:

$$U_0=-e=K_e\dot{x}$$

(65)

where K_e is the back electromotive force constant (V·s/m), depending on the geometry and magnetic properties of the electromagnetic damper. Once the circuit is closed, the electromotive force (i.e., the open circuit voltage) will drive the current I to flow through the circuit, and according to Kirchhoff’s voltage law (KVL), we can obtain:

$$U_0=L\dot{I}+RI+U_1$$

(66)

Considering that the vibration frequency of multi-link structures is generally low (e.g., 0.1–10 Hz) and that the L value of the dampers being measured is relatively small, the effect of coil inductance will be ignored unless otherwise stated. Based on Lorentz’s law, an electromagnetic damping force, which is proportional to the coil current, will act on the moving part:

$$F_{em}=K_{f}I$$

(67)

where I is the current in the coil. The proportionality factor K_f = K_e is the force constant (N/A) of the damper; therefore, it is also called the motor constant.

The control circuit includes a basic energy harvesting setup, featuring a full-wave rectifier, a supercapacitor, and a resistor. In this configuration, the current generated by the electromagnetic damper is stored in the supercapacitor for later use in active control. The full-wave rectifier, composed of six diodes, converts the AC power from the damper into DC power, while the resistor in the schematic represents the power consumption of any electrical component. Supercapacitors, similar to rechargeable batteries, serve as common energy storage devices.

$$E_c={\displaystyle \frac{1}{2}}C{U}_c^2$$

(68)

where C stands for capacitor. U_C is the voltage of the supercapacitor, and the voltage U_C increases with the increase of energy storage.

Diodes always cause a forward voltage drop V_F (ranging from 0.2 to 0.7 V), leading to power loss in the system. This voltage drop should be considered, unless the electromotive force is much higher than V_F. To simplify the calculations, only copper losses are considered in the analysis, as they typically contribute the most to heat generation. For three-phase brushless motors, the resistance losses are:

$$P_R=3R_P{I}_P^2$$

(69)

where R_p is the winding phase resistance and I_p is the phase current. Power electronics are also the main heat source of electromechanical servo systems in motor drive electronics, and only considering the conduction loss related to the research is approximate:

$$P_{Pe}=3R_{on}{I}_P^2$$

(70)

5. Prototype Test and Analysis

Figure 18 shows the structure of the test equipment and prototype. In this paper, the reciprocating vibration device is used to build a test platform for a multi-link intelligent suspension energy recovery test. The test is carried out under sine excitation and records the obtained three-phase brushless motor power generation; therefore, the current and voltage can be solved according to the resistance value.

$$P={\displaystyle \frac{U^2}{R}}$$

(71)

To study the energy conversion efficiency of multi-link active suspension under different excitations, the power values under different excitations were recorded by a bench test. Through processing the power data, the corresponding current and time curves under different excitations were obtained, as shown in Figure 19. These curves helped us analyze the electric energy conversion of multi-link active suspension under different excitation and further evaluate the performance and reliability.

As shown in the I–t curve under different excitations, the horizontal axis represents the sampling time of the test sensor, and the vertical axis represents the current value obtained during the test. As shown in the table, according to the requirements of the test equipment, the excitation frequencies were selected as 0.83 Hz, 1.16 Hz, 1.49 Hz, and 1.82 Hz, respectively, and the resistance specification was 50 Ω. Under different excitation frequencies, the test results show that with the increase of the excitation frequency, the maximum excitation current, the maximum excitation power, and the maximum excitation voltage all show an upward trend, indicating that the multi-link energy-fed suspension can produce a good response effect under different excitation frequencies. This phenomenon indicates that as the excitation frequency of the road surface increases, the response effect of the suspension system becomes more significant, demonstrating the excellent performance of this suspension system when adapting to the excitation of different road conditions. Different excitation frequencies correspond to the excitation of uneven road surfaces. Therefore, when excited at 0.83 Hz on a Grade B road surface, the maximum excitation current generated is 0.43 A; the maximum excitation current per second is 0.0172 A; the maximum power is 9.39 W; and the maximum excitation voltage is 21.66 V. When excited at 1.16 Hz under Class B pavement conditions, the maximum excitation current generated is 0.50 A; the maximum excitation current per second is 0.0192 A; the maximum power is 12.44 W; and the maximum excitation voltage is 24.94 V. When excited at 1.49 Hz under Class B pavement conditions, the maximum excitation current generated is 0.54 A; the maximum excitation current per second is 0.0216 A; the maximum power is 14.73 W; and the maximum excitation voltage is 27.14 V. When excited at 1.82 Hz under Class B pavement conditions, the maximum excitation current generated is 0.58 A; the maximum excitation current per second is 0.0276 A; the maximum power is 16.84 W; and the maximum excitation voltage is 29.01 V. Through the comprehensive analysis of these data, it can be clearly seen that as the excitation frequency increases, the response capability of the system gradually enhances. Moreover, with the increase of road surface unevenness, the excitation frequency has a more significant impact on the excitation effect of the suspension system, further verifying the efficiency and stability of the multi-link energy-fed suspension when adapting to different frequencies and different road conditions. Therefore, the multi-link active suspension has a good energy conversion efficiency. Especially after selecting the appropriate energy storage mode, it can realize the efficient recovery and utilization of energy.

After the real car test and data collection and analysis, Figure 20 shows the body acceleration of the passive suspension driving at different speeds under the B-class road surface conditions. Figure 21 shows the relationship between the body acceleration and chassis acceleration and time when the vehicle is traveling at 10 Km/h, 30 Km/h, and 60 Km/h, respectively, under the B-class road surface conditions. The horizontal axis indicates the time of the test, while the vertical axis shows the recorded acceleration values during the test. It can be analyzed from the figure that when driving at different speeds on the same road surface, the fluctuation of body acceleration and chassis acceleration becomes larger and larger with the increase in speed, among which the fluctuation is the largest when driving at 60 Km/h, which is also in line with the overall motion characteristics of the car when driving at high speed. When driving at 10 Km/h, the maximum fluctuation of chassis acceleration is ±4 m/s², and the maximum fluctuation of body acceleration is ±1 m/s². When driving at 30 Km/h, the maximum fluctuation of chassis acceleration is ±5 m/s², and that of body acceleration is ±1 m/s². When driving at 40 Km/h, the maximum fluctuation of chassis acceleration is ±6 m/s², and that of body acceleration is ±3 m/s². At 60 Km/h, the maximum fluctuation of chassis acceleration is ±10 m/s², and the maximum fluctuation of body acceleration is ±2 m/s².

Table 2. Summary of test parameters and results.

Excitation Frequency (Hz)	Maximum Excitation Current (A)	Maximum Excitation Current Per Second (A)	Maximum Excitation Power (W)	Maximum Excitation Voltage (V)	Drive Motor Speed (r/min)	Resistance (Ω)
0.83	0.43	0.0172	9.39	21.66	50	50
1.16	0.50	0.0192	12.44	24.94	70	50
1.49	0.54	0.0216	14.73	27.14	90	50
1.82	0.58	0.0276	16.84	29.01	110	50

and

Table 3. Root-mean-square value of performance index of passive suspension and energy regenerative suspension.

Performance Indicator (Unit)	RMS		Inhibition Ratio
BA (m/s2)-10 km/h	0.9789	0.5441	44.42%
BA (m/s2)-30 km/h	1.5970	0.8311	47.96%
BA (m/s2)-40 km/h	2.0134	0.9690	51.87%
BA (m/s2)-60 km/h	2.4834	1.8628	24.99%

shows the root-mean-square values of performance indicators of passive suspension and the energy-fed suspension designed in this paper at different speeds. In the root-mean-square value column, the left side is passive suspension and the right side is energy-fed suspension. It can be seen from the data comparison in the table that when driving at 10 km/h, the inhibition ratio of energy-fed suspension to body acceleration reaches 44.42%; when driving at 30 km/h, the inhibition ratio of energy-fed suspension to body acceleration reaches 47.96%; and when driving at 40 km/h, the inhibition ratio of energy-fed suspension to body acceleration reaches 51.87%. When driving at is 60 km/h, the inhibition ratio of energy-fed suspension to vehicle acceleration reaches 24.99%.

6. Conclusions

By establishing the three-dimensional spatial structure model of the three-bar mechanism, the complexity of the structure is greatly reduced, and the motion conversion process of the suspension is completed with a simple three-bar mechanical linkage structure, which provides a low-cost and simple design idea for the deeper study of energy feed characteristics. The intelligent automobile active suspension based on the multi-link mechanism, as designed in this paper, fully considers several factors, including road conditions, kinematic characteristics of a multi-link mechanism, the mathematical model and electromagnetic characteristics of a permanent magnet brushed three-phase motor, energy recovery, application of supercapacitors, system parameter identification, and so on. The innovative 2-DOF parallel mechanism system uses unidirectional bearings and thrust bearings to transmit the road excitation to the permanent magnet three-phase brushless motor and to recover that energy, which is then used as the output active control force. Hence, this mechanism solves the problem of the high cost and low energy recovery of the linear motor. In this paper, the kinematic characteristics of the system and its mathematical model are analyzed, and the feasibility of energy recovery and the use of the supercapacitor as an energy storage unit are validated through a bench test. With the increase of excitation frequency, the current fed by the multi-link active suspension also increases, and the maximum growth rate can reach 65.71%. Therefore, the multi-link active suspension has a good energy conversion efficiency. Especially after selecting a suitable energy storage mode, it can realize the efficient recovery and utilization of energy. At the same time, the acceleration of the passive suspension and multi-link active suspension at different speeds is compared, which shows that the multi-link active suspension has good energy recovery characteristics and better vehicle riding comfort.