## 3 Analysis of Suspension Damping Coefficient
### 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):  
\[F=cv \tag{12}\]  
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 multi-body 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.  
Figure 5: Stiffness curve of air spring and position of motion joints. (**a**) The stiffness of the air spring at three pressure values. (**b**) The location of the added motion pair. Figure 6: Dynamics model of seat suspension. 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.  
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. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Frequency (Hz)**} & \multirow{2}{*}{\
\begin{tabular}{c} **Experimental** \\ **Value** \\ \end{tabular} } & \multirow{2}{*}{**Original Model**} & \multirow{2}{*}{**Error**} & \multicolumn{2}{c}{
\begin{tabular}{c} **With Friction** \\ **Damping** \\ \end{tabular} } & \multirow{2}{*}{**Error**} \\ \hline
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\% \\ \hline \hline \end{tabular}
\end{table}