# New Hydraulic High-Pressure Impulse Generator for Long-Duration Impulse Tests
## 3 Mathematical Models  
Since the impulse generator part and the booster part are connected, the low-pressure chamber plays a critical role in this device. Assuming that the booster piston is located at the right end of the stroke for the initialization, the low-pressure chamber has the minimum volume, and at the moment depicted in Figure 2, the rotary spool starts rotating. The instantaneous pressure in the chamber is governed by the following pressure build-up equation [19]. \[\frac{dp_{D}}{dt}=-\beta_{e}\frac{\frac{dV_{D}}{dt}+Q}{V_{D}}, \tag{1}\]  
where \(p_{D}\) is the instantaneous pressure in the low-pressure chamber; \(t\), the time since the rotation started; \(\beta_{e}\), the bulk modulus of the hydraulic oil; \(V_{D}\), the volume of the low-pressure chamber; \(Q\), is the flow rate passing through the throttle valve as shown in Equation (4). Further, the volume of the low-pressure chamber and its time variation are shown in Equations (2) and (3), respectively. \[V_{D}=V_{D0}+\frac{\pi}{4}{D_{D}}^{2}.\times_{B} \tag{2}\]  
where \(V_{D0}\) is the minimum volume of the low-pressure chamber; \(D_{D}\) is the diameter of the right end of the booster piston as shown in Figure 2; \(x_{B}\) is the stroke of the booster piston. \[\frac{dV_{D}}{dt}=\frac{\pi}{4}{D_{D}}^{2}.\frac{dx_{B}}{dt} \tag{3}\]  
\[Q=C_{d}A_{o}\sqrt{\frac{2|p_{D}-p_{in}|}{\rho}}sign(p_{D}-p_{in}) \tag{4}\]  
where \(C_{d}\) is the orifice coefficient [19]; \(A_{o}\), the cross-sectional area of the throttle valve; \(\rho\), the oil density; \(p_{in}\), the pressure of the inner space of spool and can be described as follows. \[\frac{dp_{in}}{dt}=-\beta_{e}\frac{Q_{P}+Q_{R}+Q_{T}-Q}{V_{in}} \tag{5}\]  
where \(Q_{P}\), \(Q_{R}\), and \(Q_{T}\) are three flow rates related to the peak pressure source, rated pressure source, and back pressure, respectively, and \(V_{in}\) is the volume of the inner space of the spool. As seen in Figures 3 and 4b, the inner space of spool can only be connected to one pressure source at any time, especially under the assumption of no leakage among the  
Figure 3: Schematic assembling diagram of the sleeve and the spool. The blue rectangle represents the back pressure window; the red rectangle, the peak pressure window; the yellow rectangle, the rated pressure window; the dark rectangle, the slot on the rotary spool. three pressure sources. Therefore, the three flow rates are directly determined by the cross-sectional areas and are given as shown in Equation (6a-c). \[Q_{P} =2C_{d}A_{P}\sqrt{\frac{2|p_{in}-p_{P}|}{\rho}}sign(p_{in}-p_{P}) \tag{6a}\] \[Q_{R} =2C_{d}A_{R}\sqrt{\frac{2|p_{in}-p_{R}|}{\rho}}sign(p_{in}-p_{R})\] (6b) \[Q_{T} =2C_{d}A_{T}\sqrt{\frac{2|p_{in}-p_{T}|}{\rho}}sign(p_{in}-p_{T}), \tag{6c}\]  
where \(p_{P}\), \(p_{R}\), and \(p_{T}\) are the pressure values of the peak pressure source and rated pressure source and the back pressure, respectively. They are all considered as constants. \(A_{P}\), \(A_{R}\), and \(A_{T}\) (Figure 4b), are the cross-sectional areas related to the three pressure sources. These areas are defined by the overlaps of the windows on the sleeve and the slots on the rotary spool as shown in Equation (7a-c). \[A_{P} = \begin{cases}\frac{1}{2}L_{P}\theta d&0\leq\theta\leq\frac{2H_{P}}{ d}\\ L_{P}H_{P}&\frac{2H_{P}}{d}<\theta\leq\frac{2H_{S}}{d}\\ \frac{1}{2}L_{P}\Big{(}\frac{2H_{1}}{d}-\theta\Big{)}d&\frac{2H_{S}}{d}<\theta \leq\frac{2H_{1}}{d}\end{cases}  \tag{7a}\] \[A_{R} = \begin{cases}\frac{1}{2}L_{R}\Big{(}\theta-\frac{2H_{1}}{d} \Big{)}d&\frac{2H_{1}}{d}<\theta\leq\frac{2H_{1}}{d}+\frac{2H_{S}}{d}\\ L_{R}H_{S}&\frac{2(H_{1}+H_{S})}{d}<\theta\leq\frac{2H_{1}}{d}+\frac{2H_{R}}{d} \\ \frac{1}{2}L_{R}\Big{(}\frac{2H_{2}}{d}-\theta\Big{)}d&\frac{2(H_{1}+H_{R})}{ d}<\theta\leq\frac{2H_{2}}{d}\end{cases} \] (7b) \[A_{T} =