## 2 Materials and Methods
### Fluid Model Characterization
#### 2.2.2 Fluid Action of the Chamber  
The nonlinear behavior of air springs is influenced by various factors, including geometric nonlinearity, material nonlinearity, and contact nonlinearity [40]. Furthermore, the pressure variation within the chamber also affects the nonlinear behavior of air springs. The fluid element based on surface modeling shares a node with the gas spring structure, and by linking it to a reference point, it transforms into a bulk element.  
The following are the node coordinates that define the fluid element situated within the chamber of the air spring [41].  
\[X(x,y)=\sum_{i}N_{i}(\xi,\eta)X_{i}(x,y) \tag{18}\]  
where \(N_{i}\) is a fluid element-shaped function, which is expressed by the isoparameter coordinates \(\xi,\eta,\) and \(X_{i}(x,y)\) are the reference point coordinates.  
The Jacobian determinant of this fluid is as follows.  
\[\begin{cases}\frac{\partial X}{\partial\xi}=\sum_{i}\frac{\partial N_{i}}{ \partial\xi}X_{i}\\ \frac{\partial X}{\partial\eta}=\sum_{i}\frac{\partial N_{i}}{\partial\eta}X _{i}\end{cases} \tag{19}\]  
The following equation is obtained from the fluid element normal vector \(\mathbf{n}\) and the microelement area \(dA\).  
\[\mathbf{nd}A=\left(\frac{\partial X}{\partial\xi}\frac{\partial X}{\partial\eta} \right)d\xi d\eta \tag{20}\]  
From the microelement area, the microelement volume can be deduced as follows:  
\[dV=\frac{1}{3}(X_{i}-X)\mathbf{nd}A \tag{21}\]  
The fluid element volume can be further obtained by the following equation through differential integration.  
\[\overline{V}^{e}=\int dV=\int_{-1}^{1}\int_{-1}^{1}\frac{1}{3}(X_{i}-X)(\frac {\partial X}{\partial\xi}\frac{\partial X}{\partial\eta})d\xi d\eta \tag{22}\]  
Since the temperature of all cells in the chamber is the same, the fluid volume of each cell can be calculated using the following formula:  
\[\overline{V}^{e}=\overline{V}(P,\theta,m^{e}) \tag{23}\]  
The volume of gas can be expressed as  
\[V=\sum_{e}\overline{V}^{e}=\sum_{e}\overline{V}(P,\theta,m^{e})=\frac{m}{ \rho(P,\theta)} \tag{24}\]  
From this equation, the fluid density \(\rho(P,\theta)\) in the chamber can be obtained.  
\[\rho(P,\theta)=\rho_{R}\frac{(\theta_{R}-\theta_{A})(P+P_{A})}{(\theta- \theta_{A})(P_{R}+P_{A})} \tag{25}\]  
where \(P_{A}\) is the temperature at the reference point density \(\rho_{R}\), \(P_{R}\) is the pressure when the reference point density is \(\rho_{R}\), \(\theta_{A}\) is the ambient temperature, and \(P_{A}\) is the ambient pressure.  
The gas pressure volume [21] in the air spring chamber can be obtained from the combination of Formula (24) and Formula (25).  
\[\frac{dV}{dP}=-\frac{md\rho}{\rho^{2}dP}=-\frac{m}{\rho_{R}}\frac{(\theta_{R }-\theta_{A})(P+P_{A})}{(\theta-\theta_{A})(P_{R}+P_{A})^{2}} \tag{26}\]Through the above fluid-structure coupling theory, the fluid cavity model, fluid model, and adiabatic process of air spring chamber gas were established, and the coupling relationship between the air spring binding structure and chamber gas was obtained. Combined with the principle of virtual work and the fluid action of the chamber, the changes in parameters such as the deformation, volume, pressure, and temperature of the air spring under the action of pressure gas were finally obtained. By setting parameters to the values shown in Tables 1 and 2 below, a numerical simulation model suitable for the nonlinear simulation of air spring was established. The nonlinear static and dynamic load characteristics of air springs were simulated and analyzed.