# A Model-Based Investigation of the Performance Robustness of the Deflector Jet Servo Valve
## 3 Methodology of Modeling Hydraulic Amplifier
### Deflector Jet and Pressure Recovery in Receivers
\[\frac{p}{p_{m}}=\leavevmode\
obreak\ \exp\Biggl{\{}-\biggl{(}\frac{\eta}{b_{p}} \biggr{)}^{2}\biggl{[}y^{2}\leavevmode\
obreak\ +\bigl{(}\beta^{\prime}z\bigr{)}^{2}\biggr{]}\Biggr{\}} \tag{21}\]  
Figure 7: Effective impact width of the receiver. where \(p_{m}\) is the pressure at the stagnation point on the shunt wedge, \(b_{p}\) is the eigen half-width, \(\eta=0.834\) is the distribution coefficient, and \(\beta^{\prime}=\frac{b_{p}^{\prime}}{2\pi}\) is the dimension factor. Since the jet thickness is much greater than the width of the deflector jet, it follows that the attenuation of the impact pressure along the \(z\)-axis can be ignored. Hence, Equation (21) can be simplified as follows. Figure 6.0  
where the eigen half-width \(b_{p}\) depends on specific jet scenarios. A high-speed jet formed at the deflector outlet impacts on the shunt wedge and the fluid surface of the receivers, and simulation results [16] indicate that the high-pressure region is mainly distributed on the shunt wedge. Thus, we approximately take half of the shunt wedge width as \(b_{p}\), and it can be ascertained that \(b_{p}=~{}0.05\) mm. Consider the outlet pressure of the deflector \(p_{out}\). According to simulations, this pressure is quite sensitive to the deflector jet distance \(H\). A fitting relationship is determined between \(p_{out}\) and \(H\) based on the simulation data of deflector outlet pressure at different jet distances, which can be approximately expressed by  
Figure 6.1  
where the units of \(p_{out}\) and \(H\) are Pa and meter, respectively. With the deflector outlet very close to the shunt wedge, it can be assumed that the kinetic energy of the fluid at the center of the outlet is all converted into pressure energy. With consideration of the back pressure, the pressure of the stagnation point can be computed as follows according to the Bernoulli equation. Figure 6.2  
with the result \(p_{m}=18.99\) MPa. Therefore, the pressure distribution on the solid impact surface can be determined by Equation (22), depicted in Figure 6.3 It is noteworthy that this distribution is effective only for the shunt wedge surface. Outside the shunt wedge is the effective impact area of the receiver, and its pressure is described by Assumption 7. **Assumption 8**. : _On both sides of the stagnation point, the same amount of momentum in the jet direction is depleted by the shunt wedge and the receivers; that is, each side is faced with half of the total momentum of the deflector jet._  
Figure 6.4 Pressure distribution on the solid impact surface. Hence, Equation (19) is rewritten to compute the pressures in the receivers as follows. Figure 6.5  
Figure 6.6  
where \(S_{f1}\) and \(S_{f2}\) are the solid impact surfaces on both sides of the stagnation point, which are determined by the deflector displacement \(x_{f}\), as depicted in Figure 6.7 When \(x_{f}=0\), it can be learned that \(p_{1}=p_{2}=p_{r}\), which is called the balance pressure; that is,  
Figure 6.8  
where \(S_{f0}\) is exactly half the solid impact surface. Further, the average pressure gain of the pilot valve of the DJV is  
Figure 6.9  
According to our theoretical calculations, a three-dimensional mathematical model of DJV considering the nonlinear expansion of the jet's external boundary is constructed. Additionally, the relationship between the balance pressure and the deflector displacement can be obtained based on the model.