## 2 Mechanical Modeling of the ISU System
### Thermodynamic Modeling of Nitrogen Gas Spring  
The selection of nitrogen as the working medium in the HSU system is based on its unique physical and chemical characteristics. Unlike air, nitrogen is chemically inert, avoiding the risk of oxidation to hydraulic fluids and metal components, which is critical for military and off-road vehicles that operate in harsh environments for long periods of time [16]. Recent studies predominantly employ the ideal gas equation of state (\(PV=nRT\)) to characterize hydropneumatic suspension systems [17; 18; 19]. In such models, the pressure-volume relationship of nitrogen gas springs is simplified as \(P_{N}=P_{ni}(\frac{V_{ni}}{V_{n}})^{n}\), where the polytropic index \(n\) is empirically set to 1.0 for isothermal processes or 1.4 for adiabatic conditions. However, rapid compression cycles can lead to sharp temperature changes in nitrogen within tracked vehicles, which has a significant impact on the dynamic behavior and stiffness characteristics of the hydraulic gas spring, and the calculation of the vehicle dynamic characteristics using the ideal gas model will give a significant error [14]. This study employed the Beattie-Bridgman equation of state to model the real gas behavior of nitrogen, which extends the classical van der Waals model by introducing empirical correction terms for intermolecular forces and volume deviations. Although multiple real gas models with specific applicable ranges are available, previous studies have shown that the Benedict-Webb-Rubin (BWR) equation is particularly well-suited for the typical pressure and temperature conditions of the hydropneumatic suspension systems [20]. The BWR equation can more reliably predict the behavior of nitrogen under dynamic conditions compared to the ideal gas law, and thus improve the calculation accuracy of the dynamic characteristics of the HSP systems during rapid temperature changes [21]. The BWR equation of state is expressed as [22]:  
\[P_{R}=\frac{R(T_{0}+273.15)(1-\epsilon)}{\overline{v}^{2}}(\overline{v}+B)- \frac{A}{\overline{v}^{2}} \tag{24}\]  
\[\overline{v}=\frac{M(V_{ni}-A_{g}\Delta Z)}{m} \tag{25}\]  
\[\epsilon=\frac{c}{\overline{v}\,\overline{T}_{0}^{3}} \tag{26}\]  
\[A=A_{0}\Big{(}1-\frac{a}{\overline{v}}\Big{)} \tag{27}\]\[B=B_{0}\left(1-\frac{b}{\overline{v}}\right) \tag{28}\]  
where \(P_{R}\) denotes real gas pressure, \(\overline{v}\) represents the molar volume of nitrogen gas, \(\epsilon\) corresponds to the temperature correction factor, and \(\Delta Z\) is the displacement of the cylinder piston \(P_{R}\). Assuming that hydraulic oil is incompressible, the displacement of the cylinder piston \(\Delta Z\) can be calculated as:  
\[\Delta Z=\left(\frac{\Delta\omega}{\mathrm{Ag}}\right)\Delta Y \tag{29}\]  
where \(\Delta Y\) denotes the displacement of the hydraulic cylinder (main) piston. Thus, the elastic force \(F_{k}\) and the stiffness load \(K_{s}\) of the hydropneumatic spring can be expressed by the following relationship:  
\[F_{k}=P_{R}A_{O} \tag{30}\]  
The predicted gas chamber pressures using three thermodynamic models, including isothermal ideal gas (\(K=1.0\)), adiabatic ideal gas (\(K=1.4\)), and Beattie-Bridgeman real gas models, displayed noticeable differences across various vertical wheel displacements, as shown in Figure 7a. These discrepancies amplified significantly as the temperature increased. The predictions from all three models were relatively close within small displacement ranges at an ambient temperature of 20 \({}^{\circ}\)C (see Figure 7). However, the differences gradually increased with higher compression levels. Conversely, the deviation between the real gas and the ideal gas models became more pronounced across all displacement ranges, particularly in regions of larger compression when the temperature rose to 120 \({}^{\circ}\)C (refer to Figure 7b). The elastic force \(F_{\mathrm{zrk}}\) acting on the wheel can be calculated based on the moment equilibrium principle using the following expression.