## 3 Characteristic Analysis of EHPAAPP
### Mathematical Model of the Pump
#### 3.1.2 Model of the Pump Section  
Summing forces in the positive direction and setting them to be equal to the time-rate-of-change of linear momentum of the regulator valve spool, the following equation may be generally written to describe the motion of the regulator valve spool  
\[P_{s}A_{\mathrm{v}}+F_{0}=M_{\mathrm{v}}\frac{d^{2}x_{\mathrm{v}}}{dt^{2}}+f \frac{dx_{\mathrm{v}}}{dt}+K_{\mathrm{s}}(x_{\mathrm{v}}+x_{0})+F_{st}. \tag{3}\]  
Based on assumptions that the fluid pressure within the control chamber is homogeneous throughout, and that the fluid inertia and viscous effects within this chamber are negligible compared to the hydrostatic pressure effects of the fluid, the fluid pressure within the control actuator is modeled with the following equation  
\[\frac{V_{c}}{\beta}\frac{dp_{c}}{dt}=Q_{c}+A_{c}L\frac{d\alpha}{dt}. \tag{4}\]  
The volumetric flow-rate into the control actuator can be expressed as  
Figure 2: (**a**) Program of a pressure continuous regulation for EHPAAPP; (**b**) Schematic of a pressure continuous regulation for EHPAAPP. \[Q_{c}=\left\{ \begin{array}{ll}C_{d}W(u-x_{v})\sqrt{\frac{2}{\rho}P_{c}}&-x_{m}< x_{v}<-u\\ C_{d}W(u+x_{v})\sqrt{\frac{2}{\rho}(P_{s}-P_{c})}-C_{d}W(u-x_{v})\sqrt{\frac{2}{\rho} (P_{s}-P_{c})}&-u\ \leq x_{v}\leq u\\ C_{d}W(u+x_{v})\sqrt{\frac{2}{\rho}(P_{s}-P_{c})}&u<x_{v}<x_{m}\end{array}  \right.. \tag{5}\]  
The equation of motion for the swash-plate may be written as  
\[I\frac{d^{2}\alpha}{dt^{2}}=-c\frac{d\alpha}{dt}+F_{b}L-F_{c}L-\sum_{n=1}^{n}F _{n}L. \tag{6}\]  
The equation of motion for the control actuator can be written as follows  
\[M_{c}\frac{d^{2}x_{c}}{dt^{2}}=F_{c}\cos\alpha-P_{c}A_{c}, \tag{7}\]  
where the displacement of the control chamber \(x_{c}\) can be written as  
\[x_{c}=L\tan\alpha, \tag{8}\]  
and it can be obtained that  
\[F_{c}=M_{c}L\frac{d^{2}\alpha}{dt^{2}}+P_{c}A_{c}. \tag{9}\]  
The equation of motion for the nth piston as it moves in the x-direction may be written as  
\[M_{p}\frac{d^{2}x_{n}}{dt^{2}}=F_{n}\cos\alpha-P_{n}A_{p}, \tag{10}\]  
where \(x_{n}=r\tan\alpha\sin\theta\). So the following formula is obtained  
\[F_{n}=M_{p}r\sin\theta\frac{d^{2}\alpha}{dt^{2}}+2M_{p}r\cos\theta\omega\frac {d\alpha}{dt}-M_{p}r\sin\theta\omega^{2}\alpha+P_{n}A_{p}. \tag{11}\]  
Substituting the above-mentioned force expression into Equation (6), the following intermediate result is produced,  
\[\left[I+M_{c}L^{2}+M_{p}r^{2}\frac{N}{2}\right]\frac{d^{2}\alpha}{dt^{2}}+c \frac{d\alpha}{dt}-\left(k_{b}L^{2}+\frac{N}{2}M_{p}r^{2}\omega^{2}\right) \alpha=(F_{b}-P_{c}A_{c})-A_{p}r\sum_{n=1}^{N}P_{n}\sin\theta. \tag{12}\]  
The pressure of each piston has been studied numerically and has been shown to be fairly constant while the pistons are located directly over the intake and discharge ports, and that the pressure changes almost linearly as the pistons pass over the transition slots on the valve plate. Because this is so common, the fluid pressure within the nth piston bore is usually described using the pressure profile shown in Figure 3. In this figure the average transition-angle on the valve plate is shown by the symbol \(\gamma\), which is normally called the pressure carryover-angle. Using Figure 3, the following discrete representation for the fluid pressure within the nth piston bore may be written  
\[P_{n}=\left\{ \begin{array}{ll}P_{d}&-\frac{\pi}{2}+\gamma<\theta<\frac{\pi }{2}\\ P_{d}-\frac{(P_{d}-P_{l})}{\gamma}(\theta-\pi/2)&\frac{\pi}{2}<\theta<\frac{\pi }{2}+\gamma\\ P_{i}&\frac{\pi}{2}+\gamma<\theta<\frac{3\pi}{2}\\ P_{i}+\frac{(P_{d}-P_{i})}{\gamma}(\theta-3\pi/2)&\frac{3\pi}{2}<\theta<\frac{3 \pi}{2}+\gamma\end{array} \right.. \tag{13}\]Because the oscillation frequency of the force is much higher than for the pump system, it is sufficient to consider the average effect for the whole system.