Research on the Vibration Characteristics of Air Spring Suspension Seats Considering Friction Damping

Abstract

Good seat comfort can bring a pleasant experience to commercial vehicle drivers. Therefore, it is necessary to study the vibration characteristics of commercial vehicle seats. This study focuses on commercial vehicle seats with air spring suspension. The friction damping expression of the suspension system was derived. Comprehensive simulation and experimental investigations were conducted on the vertical vibration transmission characteristics of the seat. A multi-objective optimization framework was established by integrating the NSGA-II algorithm with a BP neural network. Specifically, a nonlinear mathematical model was developed using the GA-BP neural network algorithm, with four design parameters as optimization variables: air spring stiffness (K₁), damper damping coefficient (C₁), cushion equivalent stiffness (K₂), and cushion equivalent damping coefficient (C₂). The optimization objective was defined as minimizing the maximum seat transmissibility (T_R) at the resonance frequency (f). Through the NSGA-II, Pareto optimal solutions were systematically explored, and an optimal parameter combination was identified to enhance the dynamic comfort of the commercial vehicle seat.

1. Introduction

With the improvement of living standards, more and more people are paying attention to the riding comfort of vehicles. Because it directly affects riding experience, it is increasingly valued by researchers. When people travel long distances, drivers are exposed to vibrations transmitted through the seats for a prolonged period of time, which can make them feel uncomfortable, tired, and even injured [1,2,3,4,5]. This phenomenon is more pronounced in commercial vehicles, making targeted vibration-reduction design for seats particularly important.

Ride comfort is an important performance index of vehicle design, which is a key consideration in the design of vibration reduction systems in vehicle development. As the vibration reduction link that connects with the human body, the seat suspension of commercial vehicles directly affects the ride comfort of the driver. Among various suspension technologies, air spring-based seats have gained prominence due to their nonlinear stiffness characteristics, adaptive damping properties, and ability to maintain posture stability across varying payloads [6,7,8,9].

Therefore, it is of great significance to study the influence of seat suspension parameters on the riding comfort of commercial vehicles. The vibration characteristics of the seat will directly affect its riding comfort [10,11]. The vibration characteristics of the seat refer to its dynamic response under vibrational or impact loads, which are quantified by parameters such as vibration type, amplitude, velocity, and acceleration. By optimizing the seat suspension parameters, the dynamic comfort of the seat can be significantly improved.

2. Introduction to the Air Spring Suspension and Seat Dynamic Comfort

2.1. Air Spring Suspension

The air spring suspension of the commercial vehicle seat investigated in this study is shown in Figure 1a. The air spring suspension system is primarily composed of three key components: a damper, an air spring, and a scissor mechanism. The scissor mechanism, shown in Figure 1b, is located between the upper plate and the bottom plate. The damper is located between the lower bar and the center bar at a certain angle, while the air spring is located between the bottom plate and the airbag support frame. When the bottom of the suspension is subjected to vibration, the air spring and vibration-damping damper improve the dynamic comfort of the seat by attenuating the motion through suspension excitation [12].

2.2. Seat Dynamic Comfort

The riding comfort of a vehicle encompasses two primary aspects: dynamic characteristics and static characteristics. Static comfort primarily focuses on the structural design of the seat, the relationship between geometric parameters and the physiological attributes of human seating posture, body pressure distribution, and issues related to adjustment mechanisms. In contrast, dynamic comfort emphasizes the design of an effective suspension system that isolates and absorbs vibrations and impacts transmitted to the driver. Ensuring stable operation during dynamic conditions is crucial for enhancing ride comfort. The dynamic performance of the seat suspension system largely depends on its stiffness and damping properties.

The main parameters for evaluating the dynamic comfort of commercial vehicle seats are vibration transmissibility (T_R), resonance frequency (f), and the effective amplitude transfer coefficient (SEAT value) [13].

3. Analysis of Suspension Damping Coefficient

3.1. Calculation of Equivalent Damping Coefficient

Based on the above introduction to the seat suspension mechanism, the complexity of the seat suspension structure can be simplified. To facilitate the study of this seat type, a simplified air suspension model is developed and presented in Figure 2.

The simplified mathematical structure and the overall force analysis of the simplified model of the seat with scissor suspension springs are shown in Figure 3, where 1 is the seat upper plate, 2 and 3 are the scissor bars, 4 is the seat base plate, 5 is the air spring, and 6 is the damper.

The upper plate of the seat and the bottom plate are connected by scissor rods, which are hinged to each other at point O. The air spring force and the damping from the dampers act through the scissor rods. The air spring is arranged between the scissor bar and the base plate, and the air spring force and the damping force of the damping damper act on the seat upper plate through the scissor bars. The scissor bar 2 intersects with the seat upper plate at D, while the scissor bar 3 intersects with the seat upper plate at A. The lower end of the scissor bar B can be slid along the track in the linear groove on the right side of the seat base plate. The air spring 5 is arranged between the scissor bar and the base plate, and vibration damper 6 is arranged above the crossbar in the middle of the base plate and the scissor frame.

Firstly, the air spring force $F_k$, damper force $F_{by}$, and friction force $F_{fb}$ are calculated, and the calculation results are as follows. $F_{fb}$ is the friction force caused by the contact between the moving parts, which can cause damping. In the following formula, $k$ is air spring stiffness, $c$ is the damping coefficient of the damper, $y_0$ is the sinusoidal excitation, and $y$ is vibration response.

$$F_k={\displaystyle \frac{k(y-y_0)l_2}{l_1+l_2}}$$

(1)

$$F_{fb}=\mu F_{by}$$

(2)

$$F_d=c(\dot{y}-{\dot{y}}_0)$$

(3)

The force equilibrium equations of the scissor frame are established as follows:

$$\begin{array}{l}(\mathrm{a}):{\displaystyle \sum X=0,F_{cx}+F_{bx}-F_d\cos \theta }=0\\ (\mathrm{b}):{\displaystyle \sum Y=0,}{F}_{cy}+F_{by}-m\ddot{y}+F_d\sin \theta -F_k=0\\ (\mathrm{c}):{\displaystyle \sum M_c=0,2F_{by}{l}_2\cos \alpha -m\stackrel{..}{y}{l}_2\cos \alpha -F_k{l}_2\cos \alpha =0}\end{array}$$

(4)

where (a) is the force balance equation in the x-direction; (b) is the force balance equation in the y-direction; and (c) is the overall moment balance equation of the suspension model. $F_{bx}$ and $F_{cx}$ are the horizontal forces at point b and c, and $F_{by}$ and $F_{cy}$ are the vertical forces at point b and c. The vertical force at point B can be obtained from Equation (5):

$$F_{by}={\displaystyle \frac{1}{2}}[m\stackrel{..}{y}+{\displaystyle \frac{k(y-y_s)}{l_1+l_2}}{l}_2]$$

(5)

The equilibrium equations are established for the local force analysis of the scissor frame, as shown in Figure 4. Figure 4a shows the force diagram for rod1, and Figure 4b shows the force diagram for rod2.

The forces acting on rod1 and rod2 are analyzed as follows:

$$\begin{array}{l}(\mathrm{a}):{\displaystyle \sum X=0,} F_{ox}-F_{ax}^{\prime }+F_{bx}=0\\ (\mathrm{b}):{\displaystyle \sum Y=0,} F_{by}+F_{oy}^{\prime }-F_{ay}=0\\ (\mathrm{c}):{\displaystyle \sum M_O=0,F_{ax}{l}_1\sin \alpha +F_{ay}{l}_1\cos \alpha +F_{by}{l}_2\cos \alpha +F_{by}{l}_2\sin \alpha =0}\end{array}$$

(6)

where (a) is the force balance equation in the x-direction; (b) is the force balance equation in the y-direction; and (c) is the force balance equation at point O of rod1 in the suspension model. $F_{ox}$ and $F_{dx}$ are the horizontal forces at points o and d, and $F_{oy}$ and $F_{dy}$ are the vertical forces at points o and d. The forces acting on rod2 are analyzed as follows:

$$\begin{array}{l}(\mathrm{a}):{\displaystyle \sum X=0,} F_{cx}-F_{ox}+F_{dx}=0\\ (\mathrm{b}):{\displaystyle \sum Y=0,} F_{cy}-F_{dy}-F_{oy}-F_k=0\\ (\mathrm{c}):{\displaystyle \sum M_O=0,F_{cx}{l}_2\sin \alpha -F_{cy}{l}_2\cos \alpha -F_{dx}{l}_1\sin \alpha -F_{dy}{l}_1\cos \alpha =0}\end{array}$$

(7)

The following formula is obtained by sorting through Equations (1)–(9)

$$\begin{array}{l}\{{\displaystyle \frac{1}{l_1+l_2}}[{\displaystyle \frac{1}{\cos \alpha }}\sin (\alpha +\theta )l_1+{\displaystyle \frac{\mu \tan \alpha -1}{\mu \tan \alpha +1}}\sin (\alpha -\theta )l_2]mc\}\dot{y}+{\displaystyle \frac{l_2^{2}k}{{(l_1+l_2)}^2}}y\\ =\{{\displaystyle \frac{1}{l_1+l_2}}[{\displaystyle \frac{1}{\cos \alpha }}\sin (\alpha +\theta )l_1+{\displaystyle \frac{\mu \tan \alpha -1}{\mu \tan \alpha +1}}\sin (\alpha -\theta )l_2]mc\}{\dot{y}}_0+{\displaystyle \frac{l_2^{2}k}{{(l_1+l_2)}^2}}{y}_0\end{array}$$

(8)

The simplified form of the differential equation of motion for the vibration system caused by harmonic excitation is shown in Equations (9) and (10), where $\zeta$ is the damping ratio, ${\omega }_0$ is the excitation frequency, and $m$ is the system mass.

$$\ddot{y}+2\zeta {\omega }_0\dot{y}+{\omega }_0^{2}y=2\zeta {\omega }_0{\dot{y}}_0+{\omega }_0^2{y}_0$$

(9)

$$c_s=2m\zeta {\omega }_0$$

(10)

The expression for the equivalent damping $c_s$ is derived from Equations (8)–(10) as follows:

$$c_S={\displaystyle \frac{1}{l_1+l_2}}[{\displaystyle \frac{1}{\cos \alpha }}\cos (\alpha +\theta )l_1+{\displaystyle \frac{\mu \tan \alpha -1}{\mu \tan \alpha +1}}\sin (\theta -\alpha )l_2+1]c$$

(11)

From the expression for the calculated equivalent damping, it can be seen that the equivalent damping of the scissor frame is related to the lengths $l_1$ and $l_2$ of the scissor bars of the suspension, the damping coefficient $c$ of the damper, the damping angle $\theta$ of the damper, the scissor bar angle $\alpha$, and the coefficient of friction $\mu$ at the suspension balance position.

3.2. Analysis of Vibration Characteristics of Air Spring Suspension

To verify the correctness of the above conclusions, the suspension dynamics model is established based on dynamic simulation software. The construction of a dynamic model of seat air spring suspension includes the modeling of the air spring, damper, and the addition of motion pairs for each component.

The core function of the damper is to work in conjunction with the spring damping unit. The core function of the damper is to dissipate vibration energy (related to velocity) through damping force. ADAMS’ spring damping unit can define the damping coefficient ($c$), and its force model is expressed as shown in Equation (12):

$$F=cv$$

(12)

The MBS model focuses on global dynamic characteristics (such as vertical vibration of the seat and vehicle body transfer rate) rather than the details of fluid mechanics inside the shock absorber (such as oil flow and valve plate opening and closing). The spring damping unit can provide sufficient accuracy at a low computational cost. Moreover, complex damper models would significantly increase the computational load. The spring damping element only needs to solve algebraic equations, making it suitable for coupling with multi-rigid body systems.

Therefore, the damper within the seat suspension system is modeled in the multibody dynamics environment using an ideal damping element (spring-damper element in ADAMS), where stiffness is set to 0 and the damping coefficient is assigned the fitted value for the damper. The air spring model is built using the AKISPL interpolation function. The stiffness curve of the air spring and the moving pair are added between the moving parts, as shown in Figure 5. Figure 5a shows the stiffness of the air spring at three pressure values. Figure 5b shows where each motion pair is added. The above settings ensure that the established suspension dynamics model can imitate the actual motion conditions.

The suspension multi-body dynamics model established through the above steps is shown in Figure 6. An experimental system was constructed to investigate the vibration isolation characteristics of the suspension, aiming to provide a comparative analysis with the experimental data. The experimental system built is shown in Figure 7.

The vibration transmissibility curve of the seat suspension was obtained by applying a sinusoidal excitation with a frequency range of 1–10 Hz and an amplitude of 5 mm to the seat floor. According to the derivation of the friction damping expression in the previous section, the friction damping coefficient of the suspension was calculated and used in the dynamic simulation. Simulation data under different conditions, along with the experimental data, are shown in

Table 1. Comparison results between experimental data and simulated vibration transmissibility data.

Frequency (Hz)	Experimental Value	Original Model	Error	With Friction Damping	Error
1	1.16	1.04	12.9%	1.08	6.8%
2	1.26	1.13	11.6%	1.21	4.8%
3	1.12	0.85	24.1%	1.05	6.2%
4	0.83	0.67	19.2%	0.80	3.6%
5	0.68	0.56	17.6%	0.62	8.8%
6	0.61	0.52	14.7%	0.56	8.1%
7	0.51	0.48	5.9%	0.49	3.9%
8	0.43	0.36	16.3%	0.41	4.6%
9	0.38	0.33	13.1%	0.36	5.2%
10	0.27	0.24	11.1%	0.26	3.7%

.

Figure 8 compares the vibration transmissibility curves of the seat suspension dynamic model under two configurations (with/without friction damping) against the experimental suspension transmissibility data.

Analysis of the aforementioned data reveals that adding friction damping into the dynamic model reduces the maximum error rate between simulation and experimental data by 17.9%, compared to the setting without friction damping. This also shows that adding friction damping can improve the accuracy of the dynamic model and verify the correctness of the friction damping expression derived above.

4. Research on Dynamic Seat Comfort

4.1. Dynamic Modeling and Simulation of the Vehicle Seat Systems

Based on the establishment of the seat suspension dynamics model, this section establishes a dynamics model of the vehicle seat. The position, size, hinge point coordinates, and other parameters of the seat are obtained from the geometric drawings and actual measurements of the seat, and the quality parameters of each part of the seat are measured by dismantling the seat. The multi-body dynamics model of the vehicle seat is shown in Figure 9. The established seat dynamics model includes four rigid bodies in addition to the seat suspension: the backrest, cushion, seat frame, and counterweight. These components are interconnected through fixed joints in ADAMS.

The vertical vibration of the seat is simulated by using the dynamics model of the seat. The vibration condition of the cab floor used in the simulation process is 0~20 Hz sine curve scanning, which is realized by the SWEEP function. The driving function of the simulation input signal is set as (13):

Function = sweep (time, 5.0, 0.0, 0.1, 20.0, 20.0, 0.01)

(13)

The input and output signals obtained before and after simulation are time-domain signals. Frequency-domain signals are obtained after the FFT transformation of time-domain signals, and their curves are shown in Figure 10a,b.

The dynamic simulation software ADAMS 2020 is used in the simulation part of this research. Due to GSTIFF’s efficient numerical stability and accuracy, as well as its variable-order algorithm suitable for stiff systems, the GSTIFF solver is selected in ADAMS for this simulation. During the experiment, the simulation duration is set to 10 s, and the step size is set to 500 steps.

To investigate the effects of air spring stiffness and damper damping coefficient on the seat’s transmissibility characteristics, dynamic simulations were conducted under two conditions as follows:

(1)

Fixed air spring stiffness with varying damper damping coefficient;

(2)

Fixed damper damping coefficient with varying air spring stiffness.

The relationship between the transmissibility curves obtained under these two conditions and the air spring stiffness and damper damping coefficient is illustrated in Figure 11.

Figure 11a,b illustrates the relationship between transmissibility, resonant frequency, and air spring stiffness under Condition 1. Figure 11c,d illustrates the relationship between transmissibility, resonant frequency, and damping coefficient under Condition 2.

From the above simulation data analysis, it can be seen that with the increase of air spring stiffness in the suspension, the resonance frequency and maximum transmissibility of the seat increase synchronously. However, the effect of air spring stiffness on the resonance frequency is more significant. On the other hand, with the increase of the damper damping coefficient in the air spring suspension, the resonance frequency and the maximum transmissibility of the seat will start to increase after reaching a valley value, as the damper coefficient increases.

4.2. Experimental Validation of the Dynamic Model

To further investigate the vibration transmission characteristics of the seat suspension and validate the accuracy of the developed dynamic model, an experimental system of seat vibration transmission characteristics was constructed utilizing an MTS hydraulic servo system. The test photo of the transmission characteristics of the seat is shown in Figure 12. The vibration test platform is controlled via a hydraulic servo system, enabling the generation of sinusoidal sweep vibrations with variable amplitude and constant intensity. The frequency range of the vibration spans from 0.5 Hz to 30 Hz, which encompasses the critical frequency band to which the human body exhibits heightened sensitivity in terms of vibrational perception.

The seat dynamic comfort tests were performed in accordance with ISO 17025 [14] requirements, with laboratory temperature maintained at 20–25 °C and air relative humidity controlled at 40–60%. The experimental protocol employed sinusoidal sweep excitation signals within the frequency range of 0.5–20 Hz, and the platform acceleration was maintained at a root mean square RMS value of 2 m/s². An air compressor was used to change the air pressure in the air spring to make the seat reach different heights. The seat was fixed at the center position of the excitation table, a weight was placed at the middle position of the cushion, and two acceleration sensors were used to measure the response of the excitation table and the vibration response on the surface of the cushion [15]. A total of nine operating conditions were tested in this study. For each condition, three independent trials were conducted, and the results were averaged to obtain the final experimental data for that condition. To ensure result consistency, all three trials for each operating condition showed less than 5% variation in key metrics.

The data were input to a PC through an acquisition system, and the vibration transmission characteristic curve of the seat was obtained after processing. By applying the excitation signal to the system and processing and analyzing the data collected by the acceleration sensor, the vibration transmission characteristics of the seat were obtained. In the experiment, three levels of damping values and three levels of seat heights were set to carry out crossover experiments (e.g., RMS acceleration, vibration transmissibility). This repeatability demonstrates the reliability of the experimental data.

Dynamic simulations were carried out according to the above experimental conditions, and the experimental and simulation results were compared, as shown in

Table 2. Results of comparison between experimental and simulation data.

		Max Transmissibility			Resonant Frequency
Damping Level	Seat Height	Simulation Value	Experimental Value	Error	Simulation Value	Experimental Value	Error
Min	Min	1.187	1.156	1.11%	1.616	1.831	11.74%
	intermediate	1.221	1.162	5.07%	1.822	2.051	11.20%
	Max	1.366	1.426	4.21%	2.155	2.344	9.10%
Medium	Min	1.312	1.182	10.99%	1.642	1.831	10.32%
	intermediate	1.383	1.227	12.71%	1.822	2.051	11.16%
	Max	1.332	1.208	11.30%	2.234	2.271	1.62%
Max	Min	1.263	1.296	2.60%	3.627	3.589	1.05%
	intermediate	1.273	1.254	11.40%	3.295	3.589	8.57%
	Max	1.484	1.324	12.08%	3.592	3.662	1.91%

.

From the overall analysis above, it can be seen that the error between the seat dynamic comfort experiment and the simulation is controlled within 15%, which proves that the established dynamic model is highly reliable.

The sources of the error between the experimental and simulation data were analyzed, and the reasons for the median error may be as follows:

(1) During the dynamic modeling of the seat, the secondary structures of the seat were overlooked, and these components may contribute to vibration transmission in the experiment.

(2) The super elastic or viscoelastic behavior of the seat foam material was simplified as a linear model in the simulation, resulting in errors between the damping characteristics and the experiment.

(3) In the simulation, it is assumed that the components have an ideal contact with fixed constraints or a constant coefficient of friction, but in reality, there exist complex contact behaviors such as minor slippage and gap changes.

(4) Boundary condition deviation: the installation stiffness of the seat and the vibration table in the experiment was inconsistent with the ideal fixed constraint in the simulation, which affected the natural frequency of the system.

To reduce these errors, the suggestions we put forward for future research on the dynamic comfort of seats are as follows:

(1) Conduct a stiffness test experiment of the foam material on the surface of the cushion to obtain the stiffness curve of the cushion, and apply this curve in the ADAMS simulation to improve the accuracy of the simulation model.

(2) When conducting dynamic modeling for each pair of components in mutual motion, use friction pairs for connections. Although this connection method enhances model accuracy, it significantly increases computational cost and time, and may even result in simulation failure under certain working conditions.

(3) In the experimental setup, install sensors at the center of the vibration table surface to better obtain vibration signals. When installing the seat, increase the preload of the bolts at the connection between the seat base and the vibration table surface to achieve an ideal fixed restraint state to the greatest extent.

5. Multi-Objective Optimization of Seat Dynamic Comfort Parameters

5.1. Construction of GA-BP Neural Network Model

The GA-BP neural network synergizes Genetic Algorithm (GA) and Backpropagation (BP) to overcome traditional limitations. GA provides global optimization by escaping local minima through population-based search, while BP refines solutions via gradient descent for high precision [16,17,18]. The neural network model constructed in this chapter is shown in Figure 13. There are four inputs (K₁, C₁, K₂, C₂) and two outputs (T_R, f) in the model.

Thirty data sets randomly selected from seat dynamic simulation results were used as the training dataset for the GA-BP neural network. The neural network parameters are listed in

Table 3. Genetic algorithm parameters.

GA Parameters	Value
Hidden layer neurons	5
BP training epochs	1000
Learning rate	0.01
GA population size	50
GA generations	100
Crossover probability	0.8
Mutation probability	0.1

.

The comparative results of the maximum transmissibility and resonant frequency predictions obtained by the trained model are presented in Figure 14.

Figure 14a shows that the root mean squared error (RMSE) between the predicted value and the actual value of the maximum transmissibility reaches 0.04940. Figure 14b shows that the root mean squared error (RMSE) between the predicted value and the actual value of the resonant frequency reaches 0.09165.

The RMSE analysis demonstrates that the established GA-BP neural network achieves close agreement between the predicted and actual values, indicating high predictive accuracy. To ensure the stability of the optimization results, we validated the model’s robustness through 10 independent training runs and quantified its performance using three key metrics: the coefficient of determination (R2), mean absolute error (MAE), and root mean square error (RMSE). The results are shown in

Table 4. The accuracy of the prediction model.

Number	Max Transmissibility			Resonant Frequency
	R2	MAE	RMSE	R2	MAE	RMSE
1	0.933	0.213	0.246	0.955	0.162	0.287
2	0.909	0.268	0.220	0.986	0.196	0.344
3	0.932	0.248	0.193	0.924	0.238	0.270
4	0.962	0.157	0.215	0.935	0.305	0.286
5	0.976	0.119	0.205	0.916	0.219	0.272
6	0.935	0.180	0.286	0.882	0.224	0.271
7	0.897	0.334	0.229	0.916	0.196	0.321
8	0.953	0.241	0.194	0.876	0.215	0.276
9	0.787	0.442	0.218	0.945	0.235	0.285
10	0.871	0.313	0.184	0.898	0.254	0.272
Average value	0.916	0.252	0.193	0.923	0.224	0.288

.

5.2. Multi-Objective Optimization Using the NSGA-II Algorithm

NSGA-II is a powerful multi-objective optimization algorithm, renowned for efficiently balancing solution quality and diversity. It employs non-dominated sorting to prioritize Pareto-optimal solutions and uses crowding distance metrics to ensure a uniform spread across the Pareto front while maintaining computational efficiency and robustness in complex engineering and machine learning applications [19,20,21].

This chapter establishes a nonlinear mathematical model with four design parameters as optimization variables: air spring stiffness (K₁), damper damping coefficient (C₁), equivalent stiffness of the seat cushion (K₂), and equivalent damping coefficient of the seat cushion (C₂). The optimization objectives are the maximum transmissibility ratio (TR) and the resonance frequency (f) of the seat surface response. Multi-objective optimization is performed using the Non-dominated Sorting Genetic Algorithm II (NSGA-II).

The mathematical model of the optimization objectives can be expressed as (14) and (15):

$$\min f_{T_R}(K_1,C_1,K_2,C_2)$$

(14)

$$\min f_f(K_1,C_1,K_2,C_2)$$

(15)

According to the range of mechanical parameters provided by the manufacturer of the seats studied in this paper, the optimization constraints can be expressed as (16):

$$\left\{\begin{array}{c}20 \mathrm{N}/\mathrm{mm}\le K_1\le 90 \mathrm{N}/\mathrm{mm}\\ 1 \mathrm{N}\times \mathrm{s}/\mathrm{mm}\le C_2\le 15 \mathrm{N}\times \mathrm{s}/\mathrm{mm}\\ 10 \mathrm{N}/\mathrm{mm}\le K_2\le 50 \mathrm{N}/\mathrm{mm}\\ 0.1 \mathrm{N}\times \mathrm{s}/\mathrm{mm}\le C_2\le 3 \mathrm{N}\times \mathrm{s}/\mathrm{mm}\end{array}\right.$$

(16)

The NSGA-II multi-objective optimization algorithm was implemented based on the established GA-BP neural network model, with the initial population size set to 50, 100 iterations, a crossover probability of 0.90, and a mutation probability of 0.10. The optimized parameters for the seat suspension and cushion were obtained, with the resulting Pareto front shown in Figure 15. The optimal target parameter selected from the Pareto front is T_R = 1.142, f = 1.32 Hz.

The corresponding optimized seat structural parameters are listed in

Table 5. Optimized seat parameters.

Seat Parameter	Value
K1	45.74 N/mm
C1	4.83 N·s/mm
K2	36.73 N/mm
C2	1.16 N·s/mm

.

To validate the optimization results, dynamic simulations were conducted using the seat parameters corresponding to the selected Pareto optimal solutions. The optimized vibration response curves of the seat are illustrated in Figure 16.

Figure 16a,b depict the displacement response curves and acceleration input–output curves, respectively. The calculated results indicate that the maximum transmissibility is 1.18 and the resonance frequency is 1.26 Hz. These metrics demonstrate significant improvements in vibration isolation performance compared to the baseline configuration, aligning with the design objectives of minimizing resonant amplification and enhancing ride comfort.

To validate the effectiveness of the dynamic comfort parameters optimized by NSGA-II, three groups of data with the most comprehensive optimal dynamic comfort metrics were selected from the sample dataset for comparison, as illustrated in Figure 17 and

Table 6. Data comparison of pre- and post-optimization performance.

	Pareto	No. 3	No. 9	No. 21	Max Reduction Rate
Max transmissibility	1.14	1.20	1.32	1.25	13.6%
Resonant frequency	1.32	1.43	1.21	1.20	7.6%

.

The calculations revealed that the maximum transmissibility was reduced by 13.6% and the resonance frequency decreased by 7.6% compared to the pre-optimized configurations. The optimized vibration transmissibility of the seat is significantly reduced, the resonance frequency is shifted outside the human-sensitive frequency range, and the dynamic comfort of the seat is significantly improved. These results highlight the enhanced vibration isolation performance achieved through the multi-objective optimization framework.

These optimized parameters can also be applied to industrial production. The stiffness and damping parameters of the optimized seat cushion can guide manufacturers in selecting more suitable seat cushion materials, thereby enhancing the comfort of drivers. The optimized stiffness of the air spring and the damping coefficient of the shock absorber can guide seat manufacturers in designing the volume and height of the air spring airbag, as well as the type of damper.

However, when manufacturing seats, the selected seat cushion material should ideally possess the optimal stiffness and damping characteristics obtained through the above optimization. This type of material may be relatively expensive. Considering the manufacturing costs for enterprises, seats using such materials may be difficult to produce on a large scale. Additionally, due to the limited seat suspension space, the height of the airbag designed based on the optimized air spring stiffness may exceed the suspension space, potentially affecting the movement of the seat suspension.

6. Conclusions

(1)

A mathematical model and vibrational differential equations for the scissor-type seat suspension were established, and expressions for the equivalent damping coefficient and frictional damping coefficient under actual working conditions were derived.

(2)

Through dynamic simulations and experimental studies of a commercial vehicle air-spring seat system, this work investigates the influence of seat parameters on transmissibility characteristics and validates the accuracy of the developed dynamic model. The results demonstrate that the proposed model effectively captures the system’s vibration response, providing a reliable foundation for further optimization of seat suspension performance.

(3)

Seat parameters were optimized using the NSGA-II algorithm combined with the GA-BP neural network, resulting in significant improvements in dynamic comfort metrics. Compared to the pre-optimized configuration, the maximum transmissibility was reduced by 7.1% and the resonance frequency decreased by 8.2%.