Improvement of Commercial Vehicle Seat Suspension Employing a Mechatronic Inerter Element

Abstract

To further improve the ride comfort of commercial vehicles, a seat ISD (Inerter–Spring–Damper) suspension utilizing a mechatronic inerter is proposed in this paper. Firstly, a five-DOF (degree-of-freedom) commercial vehicle seat ISD model was built. Then, the positive real network constraint conditions of a biquadratic impedance transfer function were determined, and the meta-heuristic intelligent optimization algorithm was used to solve the parameters. According to the solution, the impedance transfer function was obtained and the specific network structure was realized by network synthesis. Lastly, this study compares the vibration isolation performance of the mechatronic ISD suspension of the vehicle seat with that of a passive suspension. In comparison to passive seat suspension, the seat mechatronic ISD suspension reduces seat vibration transmissibility by 16.33% and vertical acceleration by 16.78%. Results indicate that the new suspension system can be an effective improvement in ride comfort.

1. Introduction

In some commercial vehicles, the driving conditions are poor, and long driving hours can seriously harm the driver’s health. Whole body vibration can cause various diseases, such as waist, back, and neck pain, cardiovascular disease, indigestion, and even cancer [1,2]. Therefore, many scholars and experts have studied and designed the vehicle seat suspension to improve the ride comfort and reduce the physical harm caused by vibration [3,4]. Scissor seat suspension offers the advantages of being a cost-efficient, simple structure. The researchers optimized the structural parameters of the scissor seat suspension to solve the problem of the seat acceleration and the dynamic stroke balance of the seat suspension [5,6,7]. In order to improve the ride comfort of the vehicle, some researchers also designed a new structural seat suspension based on the negative stiffness structure [8,9,10,11,12]. Some scholars have carried out research on nonlinear seat suspension [13,14]. For passive vibration isolation devices, Antoniadis et al. [15] proposed a new passive vibration isolation concept called “KDamper”. KDamper is based on an optimal combination of appropriate stiffness elements, including negative stiffness elements. KDamper has received extensive research since it was proposed [16,17]. The performance of passive scissor seat suspension and negative stiffness seat suspension in vibration isolation is limited. Furthermore, the mechanical structure of both suspensions is complex, which hinders engineering implementation.

To further improve vehicle ride comfort, Stefan et al. [18] studied the influence of a sky-hook damping control strategy on vehicle ride comfort, and realized semi-active control through an MR (magnetorheological) damper. Karnopp et al. [19] first proposed sky-hook damping control, and the effectiveness of sky-hook damping control is verified. In the automotive field, scholars have introduced MR dampers into the seat suspension system to further improve the ride comfort of vehicles [20,21,22,23]. However, implementing semi-active control is significantly challenging due to the hysteretic behavior of MR dampers. This hysteresis makes it challenging to attain the desired level of control effectiveness, ultimately impeding the realization of ideal control outcomes. Therefore, Wang et al. [24] developed a five-DOF (degree-of-freedom) model that couples the driver and seat, and developed an optimized PID (proportional-integral-derivative) controller. Dong et al. [25] proposed a human-like intelligent controller based on a seat suspension with a four-DOF human body model, aiming at the vibration of a seat suspension with variable stiffness and variable damping. Li et al. [26] proposed an output feedback controller based on a human seat suspension system and proposed an event triggering scheme to reduce data transmission. A number of studies have shown that low-frequency vibration of commercial vehicle seats is the main factor affecting ride comfort, and the control of low frequency vibration in commercial vehicle seat suspension systems has become a popular research topic [27,28]. In the last few years, the research on suspension has shown that the application of an inerter can enhance the comfort of vehicle rides, especially when the low frequency vibration suppression has a good performance [29,30,31]. The seat vibration is primarily concentrated in low frequencies; therefore, the inerter is incorporated into the vibration isolation system of the commercial vehicle seat suspension to increase the ride comfort of commercial vehicles.

Smith first described the concept of an inerter [32]. The potential vibration isolation performance advantage of an inerter was preliminarily studied in a quarter of a vehicle model and a whole vehicle model. The findings indicate that the suspension vibration isolation performance of an inerter was improved compared with the traditional passive suspension [33]. The appearance of an inerter solved the problem of incomplete matching in the mechanical–electrical analogy theory and formed ISD (inerter–spring–damper) suspension [34,35]. In addition, some scholars combined the semi-active suspension with an inerter to design and study. The research shows that the ride comfort of the vehicle can be effectively improved by the semi-active ISD suspension [36,37,38,39]. In 2011, Wang et al. [40] proposed a mechatronic inerter composed of a mechanical inerter and a rotary motor, and applied the device to the automobile suspension system. The device can integrate mechanical and electrical networks, thereby enabling the complex impedance realization of the suspension system. This integration effectively simplifies the complexity of the suspension’s mechanical structure. Jiang et al. [41] proposed the concept of a regular positive real function to classify the five-element network synthesis of biquadratic impedance transfer function. The results indicate that if the positive real biquadratic impedance transfer function satisfies the regular positive real condition, then the impedance transfer function can be realized by a five-element network comprising two reactive elements. Scholars and experts have carried out extensive research based on ISD suspension network synthesis theory. Yang et al. [42] combined the ADD (acceleration-driven-damping) control strategy and the network synthesis theory of ISD suspension, and proposed an optimal design methodology of ISD vehicle suspension based on the ADD positive real network. In order to further study the impact of mechatronic inerter on the vibration isolation performance of ISD suspension, Shen et al. [43] proposed a mechatronic ISD suspension system optimization design method based on fractional-order electrical ne twork. Zhang et al. [44] used the bridge network structure as the outside electrical network of the mechatronic inerter to effectively improve the ride comfort of the vehicle. All the above studies have shown that mechatronic ISD suspension can greatly enhance the vibration isolation performance of the suspension system. The innovations and main contributions of this paper are as follows:

(1) A five-degree-of-freedom coupling model of human seat ISD suspension with positive real second-order impedance transfer function is built, and its mathematical model is derived.

(2) In the proposed seat suspension design scheme, fitness functions based on seat acceleration, suspension working space, and vibration transmissibility are constructed in the process of particle swarm optimization.

(3) The design scheme of the seat suspension proposed in this paper is based on optimization results and network synthesis theory. The outer electric network is designed to suppress low-frequency vibrations of the seat suspension through electromagnetic force generated by the closed loop formed by the outer electric network and the motor.

The article is structured as follows: In Section 2, a five-DOF human seat ISD model based on the positive real impedance transfer function is built. In Section 3, the particle swarm optimization is used to solve the model key parameters and the seat suspension network structure is designed, and achieved through the mechatronic inerter. In Section 4, the vibration isolation performance of the seat mechatronic ISD suspension is analyzed. Finally, the thesis is summarized in Section 5.

2. Modeling of Vehicle Seat ISD Suspension

In the study of this paper, a seat ISD suspension model is proposed, as illustrated in Figure 1. This model combines the dynamical model of the human sitting position with the seat ISD suspension model, where Figure 1a is a wheel–body–seat suspension model. The function of the model in Figure 1a is to obtain the excitation signal of the seat ISD suspension. Figure 1b represents a human–seat vibration model, where the human body is divided into four parts (head, viscera, upper trunk, and pelvis). In the paper, only the vertical vibration of the human seat model is considered. Consequently, the mass of the seat and all parts of the human body constitute a five-DOF human seat model, as illustrated in Figure 1b. This paper conducts research based on the seat mannequin model in Figure 1b.

In Figure 1, the dashed blue box indicates the mechatronic inerter and the external electrical network. The red dashed line box indicates that the simplified human body model is composed of m₂, m₃, m₄, m₅ in Figure 1b. m_s is vehicle body mass, m_u is unsprung mass, and m₁, m₂, m₃, m₄, and m₅ are the equivalent mass of seat, pelvis, upper trunk, viscera, and head. k and c are the stiffness coefficient and the damping coefficient of vehicle body suspension, respectively. k_t is tire equivalent stiffness and k₁ is the stiffness coefficient of seat suspension. k₂, k₃, k₄, and k₅ and c₂, c₃, c₄, and c₅ are the stiffness and the damping coefficient of each part of human body, respectively. b₀ is the inertance of the inerter. T(s) is the unknown biquadratic impedance transfer function. In this paper, the parameters of the biquadratic impedance transfer function are obtained by particle swarm optimization. In the optimization process, constraint conditions are used to ensure that the T(s) obtained is a positive real function. When the impedance transfer function is positive real, T(s) has a physical meaning and can be realized through the combination of “inerter–spring–damper”. The solution process will be developed in the following contents.

Table 1. Model parameters of seat suspension.

Parameters of Damping Coefficient, Stiffness Coefficient and Equivalent Mass
k1 (N/m)	k2 (N/m)	k3 (N/m)	k4 (N/m)	k5 (N/m)	k (N/m)	kt (N/m)
15,000	49,340	192,000	10,000	134,400	326,000	1,300,000
c1 (Ns/m)	c2 (Ns/m)	c3 (Ns/m)	c4 (Ns/m)	c5 (Ns/m)	c (Ns/m)
1206	2475	200	125	250	14000
m1 (kg)	m2 (kg)	m3 (kg)	m4 (kg)	m5 (kg)	ms (kg)	mu (kg)
22	27	20	9	5.5	1396	150

displays the model parameters [24].

According to the kinematic analysis of the human seat model, the dynamics equation of human seat ISD suspension model is shown in Formulae (1)–(5).

The dynamic equation of the seat is shown in Formula (1):

$$m_1{\ddot{x}}_1+T(s)\left({\dot{x}}_1-{\dot{x}}_s\right)+b_0\left({\ddot{x}}_1-{\ddot{x}}_s\right)+k_1\left(x_1-x_s\right)+c_2\left({\dot{x}}_1-{\dot{x}}_2\right)+k_2\left(x_1-x_2\right)=0$$

(1)

The dynamic equation of the equivalent mass of a human pelvis is shown in Formula (2):

$$m_2{\ddot{x}}_2+c_3\left({\dot{x}}_2-{\dot{x}}_3\right)+k_3\left(x_2-x_3\right)+c_2\left({\dot{x}}_2-{\dot{x}}_1\right)+k_2\left(x_2-x_1\right)=0$$

(2)

The dynamic equation of the equivalent mass of a human upper trunk is shown in Formula (3):

$$\begin{array}{c}{m}_3{\ddot{x}}_3+c_5\left({\dot{x}}_3-{\dot{x}}_5\right)+k_5\left(x_3-x_5\right)+c_3\left({\dot{x}}_3-{\dot{x}}_2\right)\\ +k_3\left(x_3-x_2\right)+c_4\left({\dot{x}}_3-{\dot{x}}_4\right)+k_4\left(x_3-x_4\right)=0\end{array}$$

(3)

The dynamic equation of the equivalent mass of human viscera is shown in Formula (4):

$$m_4{\ddot{x}}_4+c_4\left({\dot{x}}_4-{\dot{x}}_3\right)+k_4\left(x_4-x_3\right)=0$$

(4)

The dynamic equation of the equivalent mass of a human head is shown in Formula (5):

$$m_5{\ddot{x}}_5+c_5\left({\dot{x}}_5-{\dot{x}}_3\right)+k_5\left(x_5-x_3\right)=0$$

(5)

In this paper, the research of seat ISD suspension is carried out by constructing a dynamic equation. The optimal design of seat ISD suspension is achieved by solving the model parameters. The selection and solution of the impedance transfer function T(s) in the model determines the performance of the suspension. The existing research results show that the higher the order of the impedance transfer function, the better the vibration damping performance [44]. However, considering the complexity of the optimization solution, only T(s) is considered as the biquadratic impedance transfer function, where the expression of biquadratic impedance transfer function is shown in Formula (6):

$$T(s)=\frac{A_2{s}^2+A_{1}s+A_0}{B_2{s}^2+B_{1}s+B_0}$$

(6)

where A₀, A₁, A₂, B₀, B₁, and B₂ are the parameters to be solved. The most important point is that T(s) can be realized physically only if it satisfies the positive real condition.

3. Results Optimization of Seat Mechatronic ISD Suspension Model Parameter

3.1. Evaluation Index of Seat Comfort

In this paper, the seat suspension vibration transmissibility, RMS (root-mean-square) of seat acceleration, and RMS of other body parts are the assessment indices of seat ISD suspension performance. Among them, the seat vibration transmissibility reflects the damping ability of the seat ISD suspension, where the seat vibration transmissibility is calculated as follows:

$$T=\frac{B{A}_1}{B{A}_2}\times 100\%$$

(7)

where BA₁ and BA₂ are acceleration RMS of seat and vehicle body acceleration of vehicle, respectively.

3.2. Parameter Solving of Seat Mechatronic ISD Suspension Based on PSO

Particle swarm optimization (PSO) [45] is a meta-heuristic optimization algorithm which simulates the optimal decision process. Birds’ foraging process is similar to human decision-making. The trajectory of birds in the early foraging period is chaotic. Over time, birds in different positions learn from each other and share information about searching for food within the group. Each bird evaluates the worth of its current location for food acquisition based on personal experience. Additionally, it considers the information transmitted by its associates during each foraging excursion. This search method inspired the development of PSO.

The updating rules of particle position and velocity are shown in Formula (8):

$$\{\begin{array}{c}{V}^{k+1}=\omega V^k+d_1{r}_1\left(P_{id}^k-X^k\right)+d_2{r}_2\left(P_{gd}^k-X^k\right)\\ X^{k+1}=X^k+V^{k+1}\end{array}$$

(8)

where ω represents the inertia weight; d₁ and d₂ represent learning factors or acceleration constants; the variables r₁ and r₂ are random numbers between 0 and 1, while k represents the iteration number; $P_{id}^k$ represents the individual best; $P_{gd}^k$ represents the global best; V^k is the velocity of the particle at iteration k; V^k+1 is the velocity of the particle at k + 1; X^k represents the position of the particle at iteration k; and X^k+1 represents the position of the particle at iteration k + 1.

The paper converts the multi-objective function into a single objective function to attain the optimal solution. The parameters for the biquadratic impedance transfer function and the inertance are designated as the solution parameters. This paper presents the establishment of the fitness function of the particle swarm optimization (PSO) algorithm. Based on the weight function, the algorithm is designed to optimize three performance evaluation indexes: seat acceleration, suspension working space, and vibration transmissibility.

The objective function is expressed mathematically as follows:

$$\min J=w_1\frac{BA}{B{A}_p}+w_2\frac{SWS}{SW{S}_p}+w_3\frac{T}{T_p}$$

(9)

where BA_p and SWS_p are RMS values of traditional passive seat suspension acceleration and suspension work space, respectively, and BA and SWS are RMS values of seat mechatronic ISD suspension acceleration and suspension work space, respectively. T_p and T are the seat vibration transmissibility of passive seat suspension and seat mechatronic ISD suspension, respectively.

In the process of particle swarm optimization, it is necessary to ensure both the performance constraint of the seat suspension and the positive real constraint of the transfer function. The suspension performance constraints of PSO are as follows:

$$\mathrm{s}.\mathrm{t}.\{\begin{array}{l}BA<B{A}_p\\ SWS<SW{S}_p\\ T<T_p\end{array}$$

(10)

The positive real constraint of the transfer function of particle swarm optimization is as follows:

$$\{\begin{array}{l}{A}_0,A_1,A_2,B_0,B_1,B_2\ge 0\\ {\left(\sqrt{A_2{B}_0}-\sqrt{A_0{B}_2}\right)}^2\le A_1{B}_1\end{array}$$

(11)

The workflow of the particle swarm optimization algorithm is shown in Figure 2.

Figure 2 shows the flow chart of PSO. This article describes the particle swarm optimization process in detail as follows.

Step 1: Initialize the parameters: inertia coefficient (w), learning factor (d₁, d₂), population quantity (n), iterations (N), initial velocity, and position of the particle (X, V);

Step 2: Judge the number of iterations. If the number of iterations is reached, select “Yes” to output the result; if “No” is selected, the particle position and velocity will continue to be updated;

Step 3: Update particle velocity and position according to Formula (8);

Step 4: The fitness value of individual particle is calculated by Formula (9);

Step 5: Compare and update the optimal positions of individual particles;

Step 6: Compare and update the optimal positions of population particles;

If the number of iterations is not reached, the loop starts again with the Step 3. The results of the optimization are presented in

Table 2. Parameter optimization result of PSO.

A2	A1	A0	B2	B1	B0	b0
1.28 × 107	4.86 × 109	4.66 × 108	2.03 × 104	7.15 × 106	7.35 × 105	87.93 (kg)

.

A₀, A₁, A₂, B₀, B₁, and B₂ are the transfer function parameter values of the seat suspension, and b₀ is the inertance of the mechatronic inerter. The impedance transfer function obtained is shown as follows.

$$T(s)=\frac{1.28\times {10}^7{s}^2+ 4.86\times {10}^{9}s+4.66\times {10}^8}{2.03\times {10}^4{s}^2+7.15\times 10^{6}s+7.35\times {10}^5}$$

(12)

In addition, the biquadratic impedance transfer function satisfies the regular positive real condition, which means that T(s) can be synthesized by a five-element network consisting of two reactive elements [41]. Therefore, the structure shown in Figure 3 is selected as the network structure of T(s) in the paper. Figure 3 depicts a network structure comprising five mechanical components. Additionally, the component parameters of the corresponding mechanical network structure are calculated. The mechanical network structure of biquadratic impedance transfer function is shown in Figure 3.

Formula (13) shows the expression of the velocity impedance transfer function of a mechanical network structure.

$$T(s)=\frac{b_1{c}_{11}(c_{22}+c_{33})s^2+[c_{11}{c}_{22}{c}_{33}+(c_{11}+c_{22}+c_{33})k_{11}{b}_1]+k_{11}{c}_{22}(c_{11}+c_{33})}{b_1(c_{22}+c_{33})s^2+(c_{22}{c}_{33}+k_{11}{b}_1)+k_{11}{c}_{22}}$$

(13)

Formula (12) corresponds exactly to Formula (13). Thus, according to Formulae (12) and (13), the specific parameters of mechanical network structure can be obtained, and component parameters in the structure are displayed in

Table 3. Parameter of mechanical network structure element.

	Parameter Symbol and Units	Value
Damping coefficient	c11 (Ns/m)	633.29
	c22 (Ns/m)	45.09
	c33 (Ns/m)	1.1
Spring stiffness	k11 (N/m)	16291
Inertance	b1 (kg)	439

.

Considering the integrated design of suspension, pure mechanical structure cannot meet the design requirements, so this paper uses equivalent electrical network impedance for simulation. According to the relevant theories in reference [42], the mechatronic inerter enables the mechanical network in Figure 3 to simulate the equivalent impedance through the electrical network in Figure 4.

According to Figure 3 and Figure 4, the damper, spring and inerter in the mechanical network respectively correspond to the resistance, inductance and capacitance components in the electrical network. The conversion relationship between the parameters of mechanical network elements and electrical network elements is shown below.

$$C=\frac{b}{K_m},R=\frac{K_m}{c},L=\frac{K_m}{k}$$

(14)

The electrical network parameters R₁, R₂, R₃, L and C can be solved by Formula (14) respectively. When the motor coefficient of the mechatronic inerter is set as 4055 HN/m. The load circuit component parameter values of the mechatronic inerter are shown in

Table 4. Parameter of external electrical network structure element.

	Parameter Symbol and Units	Value
Resistance	R1 (Ω)	6.4
	R2 (Ω)	89.92
	R3 (Ω)	3686.4
Inductance	L (H)	0.25
Capacitance	C (F)	0.11

.

A mechatronic inerter can simulate an equivalent mechanical network with an electric network, and it has been applied [40,43]. This scheme is used to realize the biquadratic impedance transfer function in the paper. Figure 5 shows the schematic diagram of the mechatronic inerter. The rotary motor serves as a passive device, generating power within the acceptable range. An outside electrical network consisting of inductor(s), resistor(s), and capacitor(s) is employed to match the impedance of the mechanical network consisting of spring(s), damper(s), and inductor(s). The mechatronic inerter consists of a rotary motor, a ball-screw inerter, and an external electrical network.

L_e is the equivalent inductance of the motor and R_e represents the corresponding resistance of the motor. In this paper, the effects of corresponding inductance and resistance of the motor are ignored.

The reasons for choosing this scheme are: (1) The rotary motor is small in size, which can effectively save the installation space of the seat suspension; (2) The ball-screw inerter has many advantages such as strong adaptability, simple and compact structure, long service life, and low friction force, so this paper chooses this mechatronic inerter.

The working principle of the mechatronic inerter is as follows: In the mechatronic inerter, the flywheel’s rotational inertia and the lead of the ball screw determine the inertance. The ball-screw type mechatronic inerter can convert the vertical motion into the rotating motion of the lead screw. The lead screw is connected to the rotor of the rotating motor through a coupler. The outer electrical network is connected with the stator winding coil to form a closed current loop. In the process of vibration, a current is continuously generated in the closed loop. The current passing through the mechatronic inerter creates an electromagnetic damping force. This force is used in the seat suspension system to achieve passive vibration reduction.

4. Study on Dynamic Performance of Seat Mechatronic ISD Suspension

4.1. Performance Analysis with Random Pavement Input

This paper assumes that a vehicle is traveling at a speed of 36 km/h on a C-level road surface. To demonstrate the performance advantages of the seat mechatronic ISD suspension, a filtered white noise model is employed as the input model of the road surface. The mathematical Formulation for this model is shown below [46]:

$${\dot{z}}_r(t)=-0.111\left[u{z}_r(t)+40\sqrt{G_q\left(n_0\right)u}w(t)\right]$$

(15)

where u represents the traveling speed, z_r(t) represents the vertical input displacement, G_q(n₀) represents the coefficient of road unevenness, and w(t) represents white noise with a mean of zero.

The input signal following body suspension filtering is designated as the input signal for the seat ISD suspension. The dynamic response of the seat, pelvis, upper trunk, viscera, and head is then compared and analyzed under such working conditions. The filter transfer function of vehicle suspension is shown in Formula (16):

$$\frac{z_0}{z_r}=\frac{(cs+k)k_t}{m_s{m}_u{s}^4+c\left(m_s+m_u\right)s^3+\left[\left(k_t{m}_s+k{m}_s+k{m}_u\right)\right]s^2+cks+k{k}_t}$$

(16)

where z_r represents the input of the vertical displacement of road surface, z₀ represents the simulation input of seat base, m_s is vehicle body mass, m_u is the unsprung mass of the vehicle, k_t is the equivalent tire stiffness, and k and c are the stiffness and damping coefficient of vehicle body suspension, respectively. This paper employs the seat mechatronic ISD suspension model depicted in Figure 1b as the foundation for simulation analysis. Table 1 presents the parameters of the seat ISD suspension model. Simultaneously, the suspension model with only one spring and one damper in parallel is selected as the comparison model, which is called the traditional passive seat suspension model [26]. The fundamental premise of comparative analysis is that the fundamental parameters of the seat mechatronic ISD suspension are consistent with the traditional passive seat suspension model. This includes the input of the seat suspension, the spring stiffness of the seat suspension, and the human body vibration model. Subsequently, the two models are subjected to simulation and analysis. Figure 6 illustrates the response characteristic curve of the seat and various parts of the human body in response to random road input.

As illustrated in Figure 6, the seat’s acceleration and various parts of the body are effectively improved due to the introduction of the inerter into the seat suspension. Further, the vibration isolation performance of the seat suspension is evaluated by calculating the RMS of the time domain diagram. Subsequently, the determination of the improvement percentage in the performance index was carried out and is presented in

Table 5. Comparison of two seat suspension indexes.

	Traditional Passive Seat Suspension	Seat Mechatronic ISD Suspension	Improvement
RMS of head (m/s2)	0.5982	0.4855	18.33%
RMS of upper trunk (m/s2)	0.5820	0.4152	27.59%
RMS of viscera (m/s2)	0.7330	0.4817	34.24%
RMS of pelvis (m/s2)	0.5500	0.3533	36.36%
RMS of seat (m/s2)	0.4851	0.4099	16.33%
T	0.7598	0.6420	15.78%

.

Table 5 shows that RMS of seat acceleration of seat mechatronic ISD suspension decreases by 16.33%, seat vibration transmissibility decreases by 15.78%, RMS of upper trunk acceleration decreases by 27.59%, RMS of pelvis acceleration decreases by 36.36%, RMS of viscera acceleration decreases by 34.24%, and RMS of head acceleration decreases by 18.33%. The results of the simulation suggest that the seat mechatronic ISD suspension demonstrates effective enhancement in the ride comfort of commercial vehicle seats when compared to traditional passive seat suspensions. In addition, the introduction of the inerter into the human seat suspension model can change the vibration characteristics of the model, resulting in different degrees of improvement in the acceleration of the human equivalent mass module in the model.

The time domain response characteristics of Figure 6 are analyzed using Fourier transform. This analysis allows the power spectral density (PSD) plot to be generated for further investigation of the frequency response characteristics of the ISD seat suspension. The acceleration PSD of the seat and various parts of the human body is shown in Figure 7.

The following results can be derived from the information shown in Figure 7: Under the class c random road input, the passive seat suspension has a great peak near 2 Hz, which will seriously affect the seat’s riding comfort. The seat ISD suspension proposed in this paper using the mechatronic inerter can effectively improve the ride comfort at 2 Hz. This is effectively verified by the acceleration power spectral density of the seat and different body parts.

4.2. Performance Analysis with Sinusoidal Wave Pavement Input

In order to study the vibration isolation performance of the seat mechatronic ISD suspension, the sinusoidal wave is used to simulate the input of the wheel under the long wave road surface. Considering the RMS of head and viscera acceleration provides a more realistic reflection of the seat ride comfort. Therefore, the dynamic response characteristics of equivalent mass of head and viscera under different sinusoidal frequency inputs are analyzed. The results are shown in Figure 8 and Figure 9.

It can be observed from Figure 8a and Figure 9a that the seat mechatronic ISD suspension can greatly decrease the acceleration of the head and viscera at a constant frequency (2 Hz). It can be observed from Figure 8b and Figure 9b that the seat mechatronic ISD suspension significantly reduces the acceleration of the head and viscera in the low-frequency resonance region. The low frequency resonance peak of the head acceleration of the seat ISD suspension is reduced by 46.8% compared with the passive seat suspension, and the low frequency resonance peak of the viscera acceleration of the seat ISD suspension is decreased by 43.1%. In conclusion, the proposed seat ISD suspension is effective in enhancing ride comfort of vehicle.

4.3. Parameter Sensitivity Analysis of Seat Mechatronic ISD Suspension

In the paper, the main parameters of the model include the stiffness coefficient of the spring, the inertance b₀ of the mechatronic inerter, and the coefficients A₀, A₁, A₂, B₀, B₁, and B₂ of the biquadratic impedance transfer function. As the main structure of the suspension system, the spring plays the role of supporting the seat; if there is no spring, the suspension system will not work, so the spring type affects the riding comfort of the important parts. The main research idea of this paper is to introduce the inerter into the seat suspension system, and realize the complex impedance transfer function through the external electrical network, so the inerter is an important part of this paper. To investigate the impact of spring stiffness and inertance on ride comfort, a sensitivity analysis was conducted on the seat suspension stiffness coefficient k₁ and inertance b₀. In this section, the RMS of head acceleration is employed as the evaluation index of parameter sensitivity analysis. Subsequently, the sensitivity analysis was conducted on the k₁ and b₀ parameters of the seat suspension system. The vibration characteristics of the head under different stiffness and inertance of the inerter are analyzed. The 3D diagram of the simulation analysis is shown in Figure 10.

As shown in Figure 10, when the stiffness coefficient of the spring is 5000 N/m, the RMS value of the seat acceleration decreases with the decrease of the inertance. When the spring stiffness coefficient is 15,000 N/m, the RMS of seat acceleration first decreases and then increases with the decrease of inertance. Therefore, it can be concluded that when the spring stiffness of the seat suspension changes, the inertance of the inerter should be redesigned to ensure that the seat ISD suspension can effectively suppress vibration. When the seat suspension stiffness is a definite value, there is always an optimal value of inertance that minimizes RMS of head acceleration.

In addition, the paper gives two-dimensional graphs under different spring stiffness and inertance, as shown in Figure 11. The following conclusion can be drawn from the information presented in Figure 11a: when b₀ is 10 kg, the RMS of head acceleration increases rapidly with stiffness; when b₀ is 70 kg, the RMS of head acceleration decreases slightly at first and then increases slowly with the increase of stiffness. Therefore, it can be concluded that the larger the b₀ is, the smaller the influence of spring stiffness on the RMS of head acceleration. However, considering the cost problem, the appropriate b₀ should be selected. According to Figure 11b, the following conclusion can be drawn: when the spring stiffness increases, the inerter with a small inertance can no longer meet the vibration damping performance requirements, so the inertance needs to be adjusted appropriately.

5. Conclusions

In this paper, a five-DOF human seat ISD suspension model based on a positive real transfer function is built. The model parameters of the seat ISD suspension were obtained by using particle swarm optimization under the conditions of positive real constraint and suspension performance constraint. Then, the mechanical network structure corresponding to the transfer function was obtained by network synthesis theory. Based on the mechanical–electrical similarity theory, the parameters of the external electrical network elements of the mechatronic were obtained. This paper forms a design scheme of seat ISD suspension based on mechatronic by optimizing design.

In order to verify the vibration isolation performance of the proposed seat ISD suspension, the seat ISD suspension model under random road input is simulated and analyzed. The results indicate that the RMS of seat acceleration decreased by 16.33%, the seat vibration transmissibility decreased by 15.78%, and the RMS of upper trunk acceleration decreased by 27.59%. Additionally, the RMS of pelvis acceleration decreased by 36.36%, the RMS of viscera acceleration decreased by 34.24%, and the RMS of head acceleration decreased by 18.33%. The simulation results show that compared with the passive suspension, the vibration damping performance of the seat ISD suspension designed in this paper is effectively improved. At the same time, the vibration isolation performance of seat ISD suspension under different frequency sinusoidal inputs is analyzed. The outcomes illustrate a notable decline in the acceleration of the simulated mass representing the head and viscera within the model as the frequency of the sine wave approaches the resonance frequency. Finally, the parameter sensitivity analysis of the mechatronic ISD suspension is further analyzed. The results show that the inertance should be adjusted with the change of the spring stiffness of the seat suspension in the design process of ISD suspension.