## 4 Laboratory Experiments and Parameter Identification
### Model Parameter Identification  
The parameters of the HPS system model were partly obtained from its design dimensions and nominal properties of hydraulic oil and nitrogen gas and partly identified from the experimental data. The experimental data included the measured total force, fluid pressure, displacement, and velocity, as well as the hydraulic pressure difference between the two coupled chambers and seal friction, which can be directly obtained from the measured data. The parameter identification was accomplished by minimizing the error between the experimental and simulated responses. Table 1 summarizes the constant parameters of the established model, which include the geometric parameters of the prototype, the polytropic exponent of nitrogen gas, and the mass density, bulk modulus, and thermal coefficients of hydraulic oil. These values were obtained partly from the structural design parameters and nominal material characteristics and partly from the parameter identification process described below.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(A_{c}\) (cm\({}^{2}\)) & Area of the piston & 44.179 \\ \(A_{r}\) (cm\({}^{2}\)) & Area of the rod & 19.635 \\ \(A_{b}\) (cm\({}^{2}\)) & Area of one orifice & 0.0707 \\ \(A_{v}\) (cm\({}^{2}\)) & Area of one check valve & 0.041 \\ \(\beta_{h}\) (Mpa) & Hydraulic oil bulk modulus & 1700 \\ \(\rho_{h}\) (kg/m\({}^{3}\)) & Hydraulic oil mass density & 850 \\ \(\alpha\) (1/\({}^{\circ}\)C) & Hydraulic oil thermal expansion coefficient & 0.00107 \\ \(\zeta\) (Mpa/\({}^{\circ}\)C) & Hydraulic oil thermal coefficient of bulk modulus & 8 \\ \(\mu_{v}\) (Ns/m) & Viscous friction coefficient & 50 \\ \(k_{l}\) (m\({}^{3}\)/Pa.s) & Leakage flow coefficient & \(5\times 10^{-12}\) \\ \(n\) & Polytropic exponent of gas & 1.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Constant parameters of the hydro-pneumatic suspension (HPS) prototype.  
Figure 3: Laboratory experiment of an integrated hydro-pneumatic suspension prototype including cross-beam, force transducer, thermocouple, pressure sensor #1, pressure sensor #2, RTD probe, LVDT, shaker, fan, ball valve, shield, DAQ system, pressure regulator, and nitrogen tank.  
The pressure-deflection data obtained from the low-excitation frequency of 0.5 Hz for different body temperatures were used to identify the polytropic exponent of the gas \(n\) in Equation (4). Since the width of hysteresis in pressure-deflection at a relatively higher frequency is related to the gas fraction within the emulsion, the initial gas volume fraction (\(\gamma_{0}\), Equation (9)) was determined while matching the pressure-deflection relationship. The gas volume fraction at a lower frequency of 0.5 Hz was set as an identical small value of 0.6%, due to the hysteresis caused by the heat transfer effect.  
The experimental pressure difference results were used to identify the fluid flow coefficients, namely, the leakage coefficient and discharge coefficients of the check valve and bleed orifice. The leakage flow coefficient was found to be nearly identical for the range of velocity considered. The experimental seal friction results were used to determine the seal friction parameters, namely, the viscous friction coefficient, the Coulomb friction, stiction force, Stribeck coefficient, and transition velocity, in Equations (2) and (3). The viscous friction component was found to be relatively small and the viscous friction coefficient was determined as a nearly constant value of 50 for the considered HPS.