## 3 Materials and Methods  
The considered hydro-pneumatic suspension (HPS) system design in this study comprised only two chambers coupled via bleed orifices and check valves, as illustrated in Figure 1a. Wear-resistant composite-material guiding rings were placed between the cylinder and the piston (Figure 1b) and between the cylinder and the rod (Figure 1c), in order to reduce structural wear during operation and enhance the robustness and service life. Additional high-pressure fluid seals and dust seals, as seen in Figure 1c, were implemented only between the cylinder and the rod, for a simplified and low-cost design. The leakage flow between the two chambers should thus be considered, due to the clearance between the guiding rings and the cylinder. Before operation, the piston-side chamber was charged with both pressurized nitrogen gas and hydraulic oil, while the annular rod-side chamber was only filled with hydraulic oil. During operation, dynamic excitation motions would lead to the entrapment of gas within the oil, which results in the gas-oil emulsion in both chambers, as seen in Figure 1a. Therefore, the physical mass density and effective bulk modulus of the generated gas-oil emulsion would be reduced compared with hydraulic oil. Nonlinear relationships are dependent on the excitation frequency and strut temperature, as well as the volume fraction of the entrapped gas [31; 36; 37]. The average gas volume fractions of the emulsion within the two chambers might not be identical. The rates of fluid flow between the piston-side chamber and the rod-side chamber are related to the pressures within the two chambers, as well as velocity and the properties of the gas-oil emulsion, which are strongly affected by the system temperature and excitation frequency. The stiffness of the HPS prototype is mainly related to the pressure and volume of the gas and emulsion fluid. Variations in the pressure difference between the two chambers and the seal friction force at different velocities determine the HPS damping property. These are further affected by the temperature and excitation frequency in a nonlinear manner. In order to investigate the nonlinear function between HPS characteristics and the design parameters, fluid properties, and external excitation, a model of the HPS system was developed. The additional nonlinearities due to variations in system temperature and excitation frequency, as well as the asymmetric seal friction, were also further formulated in this study. For the integrated design, the total force \(F\) of the HPS can be intuitively expressed, using the force balance, as follows:  
\[F=P_{c}A_{c}-P_{r}(A_{c}-A_{r})+f \tag{1}\]  
Figure 1: Mechanism of the integrated hydro-pneumatic suspension system (**a**) including the guiding rings (**b**) and rubber seals (**c**). where \(P_{c}\) and \(P_{r}\) are the pressures of fluid within the piston- and rod-side chambers, respectively; \(A_{c}\) and \(A_{r}\) are the effective areas of the piston and the rod, respectively, as shown in Figure 1a; and \(f\) is the dynamic friction force. Due to the 'Y' shape design of fluid and dust seals, as seen in Figure 1c, the dynamic seal friction exhibits notable asymmetric hysteresis behavior during compression (denoted by '\(hc\)') and rebound (denoted by '\(hr\)') stroke [34]. When the system velocity exceeds the hysteresis transition velocity \(v_{h}\), which could be \(v_{hc}\) or \(v_{hr}\) for different suspension strokes, as seen in Figure 2, the friction force can be expressed as follows:  
\[f=\mu_{v}\dot{z}_{s}+F_{c}sgn\big{(}\dot{z}_{s}\big{)}+F_{s}e^{-k\dot{z}_{s}|} sgn\big{(}\dot{z}_{s}\big{)};\ \big{|}\dot{z}_{s}\big{|}\geq v_{h} \tag{2}\]  
where \(z_{s}\) and \(\dot{z}_{s}\) are displacement and velocity of the external excitation, respectively; \(F_{c}\) is the Coulomb friction component; \(F_{s}\) is the stiction force; \(k\) is the Stribeck coefficient; \(\mu_{v}\) is the viscous friction coefficient; and \(sgn\) represents the sign function.