# Research on the High Speed of Piston Pumps Based on Rapid Erecting of Launch Vehicles
## Methodology
### Axial Piston Pump Model  
The major functioning parts of an axial piston pump include a swash plate, cylinder block, piston, valve plate and spindle. Due to the effect of the fixed swash plate, the piston rotates with the cylinder body and conducts periodic reciprocating motion in the cylinder body. The volume of the sealing chamber composed of each piston and the cylinder body changes with time, and oil absorption and drainage are accomplished via the oil suction and discharge ports of the valve plate. Figure 4 depicts the operation of the piston pump. The instantaneous flow \(Q\) of a piston chamber is:  
\[Q_{i}=\pi r^{2}wR_{0}\tan(\beta)\sin(wt+i\times 2\pi/z) \tag{9}\]  
where \(w\) is the rotational speed of the piston chamber around the axis, \(R_{0}\) is the radius of the distribution circle of the piston chamber, \(\beta\) is the inclination angle of the swashplate of the axial piston pump, \(z\) is the number of piston chambers, \(t\) is time and \(r\) is the radius of  
Figure 4: Diagram of an axial piston pump. Figure 3: Hydraulic bearing capacity of erection. The blue line means that the hydraulic capacity bearing is equal to 0.  
the piston chamber. Assuming that at a certain time, the number of piston chambers in the oil discharge stage is \(m\), the theoretical flow of the axial piston pump \(Q_{t}\) is [29]:  
\[Q_{t}=\sum_{i}^{\rm m}Q_{i}=\pi r^{2}wR_{0}\tan(\beta)\sum_{i}^{m}\sin(wt+i\times 2 \pi/z) \tag{10}\]  
The above calculations of theoretical flow do not take cavitation into account, but in practice, cavitation occurs in the hydraulic oil in axial piston pumps. The full cavitation model includes the vapour equation, the free gas equation and the dissolved gas equation, where the vapour equation represents vaporous cavitation and the free gas equation and dissolved gas equation represent gaseous cavitation. This allows the model to accurately simulate cavitation phenomena in hydraulic machinery [30,31]. The vapour equilibrium equation is:  
\[\frac{\partial}{\partial t}\int_{\Omega(t)}\rho f_{\rm v}d\Omega+\int_{\sigma} \rho((\mathbf{v}-\mathbf{v}_{\sigma})\cdot\mathbf{n})f_{\rm v}d\sigma=\int_{ \sigma}\left(D_{\rm f}+\frac{\mu_{\rm t}}{\sigma_{\rm f}}\right)\cdot(\
abla f _{\rm v}\cdot\mathbf{n})d\sigma+\int_{\Omega}\left(R_{\rm e}-R_{\rm c}\right)d\Omega \tag{11}\]  
The vapour generation rate \(R_{\rm e}\) and the vapour dissipation rate \(R_{\rm c}\) are given by:  
\[R_{\rm e}=C_{\rm e}\rho_{\rm l}\rho_{\rm v}\left[\frac{2}{3}(P-P_{\rm v})/\rho _{\rm l}\right]^{1/2}(1-f_{\rm v}-g_{\rm f}) \tag{12}\]  
\[R_{\rm c}=C_{\rm c}\rho_{\rm l}\rho_{\rm v}\left[\frac{2}{3}(p-p_{\rm v})/\rho _{\rm l}\right]^{1/2}f_{\rm v} \tag{13}\]  
The equilibrium equations for the free gas and dissolved gas are, respectively:  
\[\frac{\partial}{\partial t}\int_{\Omega}\rho g_{\rm d}d\Omega+\int_{\sigma} \rho((\mathbf{v}-\mathbf{v}_{\sigma})\cdot\mathbf{n})g_{\rm f}d\sigma=\int_{ \sigma}\left(D_{\rm g}+\frac{\mu_{\rm t}}{\sigma_{\rm f}}\right)\cdot(\
abla g _{\rm f}\cdot\mathbf{n})d\sigma+\int_{\Omega}\left(\frac{\rho(g_{\rm d}-g_{ \rm dequil})}{\tau}\right)d\Omega \tag{14}\]  
\[\frac{\partial}{\partial t}\int_{\Omega}\rho g_{\rm d}d\Omega+\int_{\sigma} \rho((\mathbf{v}-\mathbf{v}_{\sigma})\cdot\mathbf{n})g_{\rm d}d\sigma=\int_{ \sigma}\left(D_{\rm g_{\rm d}}+\frac{\mu_{\rm t}}{\sigma_{\rm t}}\right)\cdot( \
abla g_{\rm d}\cdot\mathbf{n})d\sigma-\int_{\Omega}\left(\frac{\rho(g_{\rm d }-g_{\rm dequil})}{\tau}\right)d\Omega \tag{15}\]  
\[g_{\rm dequil}=\frac{P}{P_{\rm gdequilef}}\cdot g_{\rm dequilef} \tag{16}\]  
In the above equations, \(D_{\rm f}\) is the vapour diffusion coefficient, \(D_{\rm g}\) is the free gas diffusion coefficient, \(D_{\rm gd}\) is the dissolved gas diffusion coefficient, \(C_{\rm c}\) is the cavitation condensation coefficient and \(C_{\rm e}\) is the cavitation evaporation coefficient. As cavitation in axial piston pumps occurs in the high-speed flow region, \(C_{\rm c}\) is 0.01 and \(C_{\rm e}\) is 0.02 [30].