## 3 Dynamic Model of Incremental Step Shift of DIHPS
### Outflow Gap Model  
Figure 4 shows the different distinctive positions of the actuator piston in relation to the outflow gaps 1 and 2. Two flow slots with negative overlaps are sharp flow edges (explained in detail in A in Figure 3). After opening the \(i\)-th BV, there was a pressure drop in the actuator chamber. Because of the pressure difference \(\Delta p=\left|p_{1}\!-\!p_{2}\right|\), the actuator piston moves rapidly to the flow position through the outflow gaps 1 and 2. For each distinguished position of the actuator piston, the flow rate \(q_{vij}\) is marked, where \(i\) is the number of actuator chambers, and \(j\) is the number of flow gaps.  
The shape of the edge between the actuator piston and the flow gap largely defines the change in the cross-sectional area and, therefore, the characteristics of the flow. Based on the different characteristic positions of the actuator piston, the static characteristics of  
Figure 4: Different distinctive positions of the actuator piston: (**a**) flows \(q_{v21}\) and \(q_{v22}\) from chamber 2, (**b**) flow \(q_{v22}\) from chamber 2, (**c**) flow \(q_{v11}\) from chamber 1 and flow \(q_{v22}\) from chamber 2, (**d**) flow \(q_{v11}\) from chamber 1, (**e**) flows \(q_{v11}\) and \(q_{v12}\) from chamber 1.  
Figure 3: Diagram of the DIHPS solution: 1, actuator piston; 2, actuator piston rod; 3, cylinder end caps; 4, inner sleeve with outflow gaps; 5, external sleeve with connections; 6, fixed TRVs; 7, block of BVs.  
the cross-sectional area \(A_{ij}\) of the flow through the two gaps were determined, as shown in Figure 5.  
From Detail A (Figure 3) and static characteristics (Figure 6), the range of opening \(x\) flow gaps is a result:  
\[\left\{\begin{array}{l}\pm x_{1}=w-|x_{0}|\\ \pm x_{2}=e+w-|x_{0}|\\ \pm x_{3}=e+2\,w-|x_{0}|\\ \end{array}\right. \tag{6}\]  
where \(x_{0}\) is the width of the outflow gap at the initial symmetric position of the actuator piston, \(e\) is the width of edge, and \(w\) is the width and length of the outflow gap.  
For the full opening of the outflow gap, the cross-sectional area of the flow is the following:  
\[A_{s}=\pi\,w^{2}+w\,c-w^{2} \tag{7}\]  
where \(c\) is the length of the outflow gap.  
In the simulation calculations, the equivalent cross-sectional area was assumed as:  
\[A_{s}\simeq A_{ij}\simeq w\,l\ \ \ \text{for}\ x=w \tag{8}\]  
\[A_{ij}(x)\simeq x\,l\ \ \text{for}\ 0\leq x\leq w \tag{9}\]  
where \(l\) is the equivalent gap length.  
Figure 5: Static characteristics of the cross-sectional flow area \(A_{ij}\) of the gap as a function of the width \(x\) of the outflow gap.  
Figure 6: Relationship between pressure drop \(p_{\
u}\) and flow rate \(q_{\
u}\) of BV: red line for the nominal parameters, blue line for the BV parameters.  
The general formula for the flow rate \(q_{v\bar{y}}\), from the \(i\)-th actuator chamber to the \(j\)-th outflow gap, is as follows:  
\[q_{vij}=C_{d}\,A_{ij}(x)\,\sqrt{\frac{2}{\rho}(p_{i}-p_{v})} \tag{10}\]  
where \(A_{ij}\) is the cross-sectional area, \(x\) is the opening of the gaps, \(p_{i}\) is the pressure in the \(i\)-th actuator chamber, \(p_{v}\) is the pressure in the outlet line to the on/off valve, \(\rho\) is the density of the hydraulic fluid, and \(C_{d}\) is the discharge coefficient, whose values for sharply edged orifices and turbulent flow can be theoretically calculated by [32]:  
\[C_{d}=\frac{\pi}{\pi+2}=0.611 \tag{11}\]  
The flow through the flow gaps takes the final form,  
\[q_{vij}=K_{q}\,A_{ij}(x)\,\sqrt{(p_{i}-p_{v})} \tag{12}\]  
where \(K_{q}\) is the flow gap coefficient, \(K_{q}=C_{d}\,\sqrt{\frac{2}{\rho}}\).  
The pressure drops in the flow gap result from (12):  
\[\Delta p_{ij}=p_{i}-p_{v}=\frac{1}{K_{q}^{2}}\left(\frac{1}{A_{ij}(x)}\right) ^{2}q_{v\bar{y}}^{2}=R_{ij}(x)\,q_{v}^{2} \tag{13}\]  
where \(R_{ij}\) is the hydraulic resistance of the on/off valve,  
\[R_{ij}(x)=\frac{1}{K_{q}^{2}}\left(\frac{1}{A_{ij}(x)}\right)^{2} \tag{14}\]