## 3 Materials and Methods
It is worth noting that these parameters would vary during compression and rebound stroke and with different system temperatures and excitation frequencies. The friction force within the linear transition band, \(\big{|}\dot{z}_{s}\big{|}<v_{hc}\) or \(\big{|}\dot{z}_{s}\big{|}<v_{hr}\) for 'compression' and 'extension' strokes, respectively, as seen in Figure 2 (the arrows indicate the directions of strut motion), describes the asymmetric hysteresis effect as the direction of velocity changes (\(z_{s}\dot{z}_{s}<0\)) and can be expressed as follows:  
\[ \begin{cases}f=\mu_{v}\dot{z}_{s}+F_{c}\big{(}\frac{2}{v_{hc}}\dot{z}_{s}-sgn \big{(}\dot{z}_{s}\big{)}\big{)}+F_{s}\big{(}\frac{1+e^{-k\mu_{hr}}}{v_{hc}} \dot{z}_{s}-sgn\big{(}\dot{z}_{s}\big{)}\big{)};\ z_{s}\dot{z}_{s}<0\ and\ \big{|}\dot{z}_{s}\big{|}<v_{hc}\\ f=\mu_{v}\dot{z}_{s}+F_{c}\big{(}\frac{2}{v_{hr}}\dot{z}_{s}-sgn\big{(}\dot{z} _{s}\big{)}\big{)}+F_{s}\big{(}\frac{1+e^{-k\mu_{hr}}}{v_{hr}}\dot{z}_{s}-sgn \big{(}\dot{z}_{s}\big{)}\big{)};\ z_{s}\dot{z}_{s}<0\ and\ \big{|}\dot{z}_{s}\big{|}<v_{hr} \end{cases}  \tag{3}\]  
When the HPS maintains the same direction of velocity within the linear transition band (\(z_{s}\dot{z}_{s}\geq 0\) and \(\big{|}\dot{z}_{s}\big{|}<v_{hc}\) or \(v_{hr}\)), the friction force is obtained from Equation (2). As illustrated in Figure 2, the dynamic seal friction force during the periodic excitations of the HPS thus exhibits obvious stick-slip, Stribeck effect, hysteresis, and asymmetric behavior. Considering the effect of system temperature and assuming a polytropic gas process for the considered excitation frequency, \(P_{c}\) can be obtained from the temperature variation, the initial gas pressure \(P_{0}\) and volume \(V_{g0}\), the instantaneous gas volume \(V_{g}\), and the temperature effect, such that  
\[ \begin{array}{l}P_{c}=P_{0}\big{(}V_{g0}/V_{g}\big{)}^{n}\\ V_{g0}=V_{gas}-\alpha\Delta T\cdot V_{oil}\end{array}  \tag{4}\]  
Figure 2: Dynamic seal friction model considering Stribeck, hysteresis, and asymmetric behavior. where \(n\) is the polytropic exponent; \(V_{gas}\) and \(V_{oil}\) are the charged volume of the nitrogen gas and hydraulic oil, respectively; \(\Delta T\) is the temperature variation; and \(\alpha\) is the thermal expansion coefficient, which is 0.00107/\({}^{\circ}\)C for the selected silicon oil. Applying the volume continuity equation with gas-oil emulsion compressibility, the instantaneous gas volume \(V_{g}\) in the piston-side chamber is derived in the form of  
\[A_{c}\dot{z}_{s}+\dot{V}_{g}=q_{b}+q_{c}+q_{l}+\frac{V_{c}}{\beta_{c}}\dot{p}_{c} \tag{5}\]  
where \(q_{b}\), \(q_{c}\), and \(q_{l}\) are the rates of fluid flow through the bleed orifices, the check valves, and the piston-cylinder clearance, respectively; and \(V_{c}\) and \(\beta_{c}\) are the instantaneous volume and effective bulk modulus of the gas-oil emulsion, respectively. Due to variations in pressure and temperature, the mass density of the emulsion within each chamber may be different, and for the continuity within the rod-side chamber, the mass density of the fluid flow between the two chambers should also be considered. The pressure of the fluid in the rod-side chamber, \(P_{r}\), is obtained from the volume continuity equation for the rod-side chamber of the HPS, which is derived as follows:  
\[(A_{c}-A_{r})\dot{z}_{s}=\frac{\rho_{c}}{\rho_{r}}(q_{b}+q_{c}+q_{l})-\frac{V_ {r}}{\beta_{r}}\dot{P}_{r} \tag{6}\]  
where \(\rho_{c}\) and \(\rho_{r}\) are the instantaneous mass densities of the emulsion within the piston- and rod-side chambers, respectively; and \(V_{r}\) and \(\beta_{r}\) are the volume and effective bulk modulus of emulsion within the rod-side chamber, respectively. The volumes of the emulsion in the two chambers of the HPS can be obtained from  
\[ \begin{array}{c}V_{c}=V_{c0}+V_{g0}-A_{c}z_{s}-V_{g}\\ V_{r}=V_{r0}+(A_{c}-A_{r})z_{s}\end{array}  \tag{7}\]  
where \(V_{c0}\) and \(V_{r0}\) is the initial volume of the emulsion within the piston- or rod-side chamber, respectively.