## 3 Materials and Methods
Moreover, the physical mass density and the effective bulk modulus of gas-oil emulsions are nonlinearly related to the gas volume fraction in hydraulic oil, the instantaneous fluid pressure, and the system temperature. The gas volume fraction of the gas-oil emulsion is defined as the ratio of the entrapped gas volume to the hydraulic oil volume (\(\gamma_{i}=V_{gi}/V_{hi};i=c,r\)). It could vary with the fluid pressure since both the entrapped gas and hydraulic oil are compressible. The entrapped gas volume \(V_{gi}\) and hydraulic oil volume \(V_{hi}\) of the gas-oil emulsion within chamber \(i\) can be derived as follows:  
\[ \begin{array}{c}V_{gi}=\left(\frac{P_{h}}{P_{i}}\right)^{\frac{1}{h}}V_{gi0 };\;i=c,r\\ \\ V_{hi}=\left(1-\frac{P_{i}-P_{h}}{\beta_{h}}\right)V_{hi0};\;i=c,r\end{array}  \tag{8}\]  
where \(V_{gi0}\) and \(V_{hi0}\) are the initial volumes of the entrapped gas and hydraulic oil within the emulsion, respectively; and \(\beta_{h}\) is the bulk modulus of pure hydraulic oil, which is affected by the HPS temperature. The effect of the temperature on the fluid bulk modulus was assumed to be linear in this study. The instantaneous gas volume fractions of the emulsions in the piston- and rod-side chambers can thus be obtained from  
\[ \begin{array}{c}\gamma_{i}=\frac{\left(\frac{P_{h}}{P_{i}}\right)^{\frac{1} {h}}V_{gi0}}{\left(1-\frac{P_{i}-P_{h}}{P_{h}}\right)V_{hi0}}=\frac{\left(\frac {P_{h}}{P_{i}}\right)^{\frac{1}{h}}}{1-\frac{P_{i}-P_{h}}{P_{h}}}\gamma o;\;i= c,r\\ \\ \beta_{h}=\beta_{h0}-\zeta\Delta T\end{array}  \tag{9}\]where \(\gamma_{0}\)\((V_{gi0}/V_{hi0};i=c,r)\) is the initial gas fraction of the emulsion in chamber \(i\), \(\beta_{h0}\) is the bulk modulus of hydraulic oil at room temperature, and \(\zeta\) is the thermal coefficient of hydraulic oil, which is approximately 8 Mpa/\({}^{\circ}\)C referring to [35]. Assuming the negligible mass of the entrapped gas compared with that of the hydraulic oil, the mass density of the emulsion in each chamber (\(\rho_{i};i=c,r\)) can thus be obtained, considering the temperature variation, as follows:  
\[ \begin{array}{c}\rho_{i}=\frac{\rho_{h}V_{hi0}}{V_{hi}+V_{gi}}=\frac{\rho_{h} }{\left(1-\frac{V_{i}-P_{i}}{P_{h}}\right)\left(1+\gamma_{i}\right)};i=c,r\\ \\ \rho_{h}=\frac{\rho_{h0}}{1+\alpha\Delta T}\end{array}  \tag{10}\]  
where \(\rho_{h}\) is the mass density of hydraulic oil, and \(\rho_{h0}\) is the mass density at room temperature of approximately 22 Celsius degrees. Similarly, the effective bulk modulus of the gas-oil emulsion within each chamber of the HPS (\(\beta_{i};i=c,r\)) also varies with the pressure and temperature parameters. Considering the compressibility of both the entrapped gas (\(-\frac{V_{gi}dP_{i}}{dV_{gi}}=nP_{i};i=c,r\)) and hydraulic oil (\(-\frac{V_{gi}dP_{i}}{dV_{hi}}=\beta_{h};i=c,r\)), the effective bulk modulus of the emulsion is formulated as follows:  
\[\beta_{i}=-\frac{V_{gi}+V_{hi}}{\frac{d}{dP_{i}}\left(V_{gi}+V_{hi}\right)}= \frac{n(1+\gamma_{i})P_{i}\beta_{h}}{nP_{i}+\gamma_{i}\beta_{h}};i=c,r \tag{11}\]  
The flow rates through bleed oriices (\(q_{b}\)), check valves (\(q_{c}\)), and clearance leakage (\(q_{l}\)) are derived by implementing the classical turbulent and laminar flow theory [2; 6], using the parameters of (\(C_{b}\), \(A_{b}\), \(n_{b}\)) and (\(C_{v}\), \(A_{v}\), \(n_{v}\)) as the discharge coefficient, the opening area, and the number of bleed oriices and check valves, respectively. The flow rates are derived as linear or quadratic relationships of the pressure difference between the piston-side and rod-side chambers, in the following form:  
\[ \begin{array}{c}q_{b}=C_{b}(n_{b}A_{b})\sqrt{\frac{2|P_{c}-P_{r}|}{\rho}}\cdot sgn (P_{c}-P_{r})\\ q_{c}=\left\{\begin{array}{cc}C_{v}(n_{v}A_{v})\sqrt{\frac{2(P_{c}-P_{r})}{ \rho}}&;P_{c}>P_{r}\\ 0&;P_{c}\leq P_{r}\\ q_{l}=k_{l}(P_{c}-P_{r})&\end{array}\right.\end{array}  \tag{12}\]  
where \(\overline{\rho}\) is considered as the average mass density of the emulsions within the two strut chambers.