## 5 Results and Discussion
### Variations in Internal Parameters
Increased temperature yielded a greater Coulomb friction, most probably due to the heat expansion of the rubber seals and thereby greater contact pressure at the seal-rod interface, as expected. The magnitude of the Coulomb friction at a given temperature and stroke direction varied only approximately \(\pm\)0.02 kN for the considered frequency range. This implied that the Coulomb friction component of the dynamic seal friction, as expressed in Equations (2) and (3), was only slightly affected by the excitation frequency/magnitude/velocity. The peak friction force during the compression and extension strokes was usually generated near zero velocity when the excitation direction was reversed, as shown in Figure 11b, and it was defined as the addition of Coulomb friction and the maximum Stribeck friction for the considered HPS system. For a given temperature and stroke direction, it was found that the peak friction generally tended to decrease with increasing excitation frequency. This implied a decrease in maximum Stribeck friction since the Coulomb friction only slightly varied with excitation frequency, as mentioned above. It  
Figure 11: Variations in seal friction properties: (**a**) Coulomb friction, (**b**) peak friction, (**c**) Stribeck coefficient, and (**d**) transition velocity. is further noted that the peak frictions at 30 \({}^{\circ}\)C and 40 \({}^{\circ}\)C exhibited comparable magnitude, irrespective of the excitation frequency. The average peak friction during extension decreased by approximately 20% when the excitation frequency increased from 0.5 Hz to 8 Hz, while the peak friction during compression stroke at the temperature of 50 \({}^{\circ}\)C was obviously greater than in other situations, as also noted for the Coulomb friction in Figure 11a. This would be attributed to the nonlinear effect of temperature on seal friction, which was not considered in the dynamic friction model in this study. Moreover, the identified Stribeck coefficients during compression and extension strokes, \(k\) in Equation (2), are shown in Figure 11c. The increased temperature and excitation frequency tended to yield a lower Stribeck coefficient, which implied a slighter change in the peak friction with increasing excitation velocity. This can also be noticed in Figures 7d, 8d and 9d. The maximum Stribeck coefficient occurred around 1\(\sim\)2 Hz for the considered temperature and frequency range in this study. Since the velocity range was the same for different scenarios, the frequency dependency of the Stribeck coefficient of the HPS might thus be correlated to the magnitude of deflection and the size of the seal-rod interface. The average Stribeck coefficient decreased by approximately 75% when the excitation frequency increased from 1 Hz to 8 Hz. The Stribeck coefficient during compression was generally greater than that during extension, especially at a relatively higher excitation frequency, which would also be attributed to the asymmetric structure of the rubber seal. It was also observed that the Stribeck effect was negligible during the extension at the excitation frequency of 8 Hz. As for the friction transition velocity, which indicated the width of seal hysteresis, the increase in temperature and excitation frequency generally led to a greater transition velocity, as shown in Figure 11d. A piecewise linear relationship was observed between the transition velocity and excitation frequency, and the critical frequency was approximately 2 Hz. The transition velocities at a low frequency of 0.5 Hz were almost identical, around 8 mm/s, irrespective of the temperature and stroke. When the excitation frequency was 8 Hz, the average transition velocity increased to approximately 28 mm/s.