## 5 Results and Discussion  
### Model Verification and External HPS Properties  
The nonlinear stiffness and damping characteristics of the integrated HPS with a gas-oil emulsion were obtained from the analytical model and the data acquired during the experiments, corresponding to the body temperature of 30 \(\pm\) 2, 40 \(\pm\) 2, and 50 \(\pm\) 2 \({}^{\circ}\)C and excitation frequency ranging from 0.5 to 8 Hz. The force-deflection relationship was used to characterize the equivalent stiffness property of the HPS, and the force-velocity relationship was used to characterize the equivalent damping property. The major stiffness and damping components of the considered HPS were further investigated by analyzing the gas pressure-deflection, pressure difference-velocity, seal friction-deflection, and seal friction-velocity relationships for the considered temperature and frequency ranges. The comparisons between the measured and simulated responses of the HPS at the excitation frequencies of 0.5, 2, and 8 Hz are demonstrated in Figures 4-9, and the results of other frequencies exhibit similar levels of agreement. The figures were analyzed and plotted using MATLAB software version R2021b. Figures 4-6 illustrate the comparisons between the measured and simulated force-deflection and force-velocity relationships at 0.5, 2, and 8 Hz, respectively. The seal friction was calculated from the measured total force and fluid pressures based on Equation (1). The comparisons of the experimental and simulated relationships between gas pressure/pressure difference between the two strut chambers/seal friction and the deflection/velocity at 0.5, 2, and 8 Hz are demonstrated in Figures 7-9, respectively. Figure 4: Comparisons of experimental and simulated total force-deflection (**a**) and total force-velocity (**b**) relationships at different temperatures (0.5 Hz). Figure 5: Comparisons of experimental and simulated total force-deflection (**a**) and total force-velocity (**b**) relationships at different temperatures (2 Hz). Figure 6: Comparisons of experimental and simulated total force-deflection (**a**) and total force-velocity (**b**) relationships at different temperatures (8 Hz). Figure 7: Comparisons of experimental and simulated gas pressure-deflection (**a**), pressure difference-velocity (**b**), seal friction-deflection (**c**), and seal friction-velocity (**d**) relationships at different temperatures (0.5 Hz). As shown in Figures 4-6, the increased temperature was found to increase the mean HPS force, which was mainly attributed to the increased gas pressure, as seen in Figures 7a, 8a and 9a. This is consistent with the reported studies in [38]. The hysteresis shown in the force-displacement and force-velocity relationships was mainly caused by the hysteresis in the seal friction, as shown in Figures 7-9. The increased excitation frequency also contributed to the hysteresis behavior, especially in the pressure difference between the two HPS chambers. The hysteresis in the gas pressure at 0.5 Hz or lower was caused by the hysteresis in the seal friction. The hysteresis in the seal friction-velocity is also caused by the hysteresis in the seal friction-velocity. by the heat transfer between the gas and its surroundings [17; 38], while the hysteresis in the gas pressure at a relatively higher frequency was mainly attributed to the considerable compressibility of the gas-oil emulsion. The Stribeck behavior, as modeled in Equation (3) and shown in Figure 2, was observed around the extremal deflection or zero velocity, which can be seen in Figures 4-9. The peak friction was observed near zero velocity and gradually decreased with the increase in the excitation velocity. The hydraulic damping property of the HPS is governed by the pressure difference between the two chambers, as shown in Figures 7b, 8b and 9b. The pressure difference was relatively smaller during the compression stroke (positive velocity) than the extension stroke (negative velocity) for the sake of vibration attenuation.