## 7 DIHPS Dynamic Model with HPA  
The HPA was connected to the actuator chamber to effectively dampen piston vibrations and pressure peaks. The HPA is a gas-charged accumulator with a hydraulic fluid chamber and a gas (nitrogen) chamber separated by a gas-tight partition (piston, bladder, and diaphragm). A diaphragm (membrane) type HPA was used as a pressure-pulsation damper [36]. Figure 16 shows a diagram of the DIHPS with the HPA connected to the actuator chamber on the \(F\) side. The dynamic model (51) of the DIHPS supplemented with the flow rate balance in the actuator chamber is as follows:  
\[\left\{ \begin{array}{l}\frac{dp_{1}}{dt}=\frac{1}{C_{1}}(q_{v1}-A\,v-q_{v11} )\\ \frac{dp_{2}}{dt}=\frac{1}{C_{2}}(q_{v2}+A\,v-q_{v22}-q_{va})\end{array} \right. \tag{52}\]  
The HPA and the hydraulic tube connecting to the actuator chamber can be modeled with lumped elements connected in a series \(RLC\) (\(R\) is the hydraulic resistance, \(L\) is the hydraulic inductance, and \(C\) is the hydraulic conductance), according to the analogy between hydraulic and electric circuits [37; 38]. Figure 15: Dynamic response parameters of the DIHPS for a single-step shift of the actuator piston with and without constant excitation force: (**a**) single-step shift of the actuator piston, (**b**) speed of the actuator piston, and (**c**,**d**) pressure in the actuator chambers. The HPA model contains two first-order differential equations, the flow rate equation in the connected tube and the gas compressible equation, as follows:  
\[\left\{ \begin{array}{l}L_{h}\frac{d\eta_{aa}}{dt}+R_{h}q_{xu}=p_{2}-p_{a}\\ q_{va}=C_{a}\frac{dp_{a}}{dt}\end{array} \right. \tag{53}\]  
The notation of Equation (51) in numerical solutions is as follows:  
\[\left\{ \begin{array}{l}\frac{d\eta_{aa}}{dt}=\frac{1}{L_{h}}(-R_{h}q_{va}+p_ {2}-p_{a})\\ \frac{dp_{a}}{dt}=\frac{1}{C_{a}}q_{va}\end{array} \right. \tag{54}\]  
where \(q_{va}\) is the flow rate in the hydraulic tube, \(p_{2}\) is the fluid pressure in the actuator chamber, and \(p_{a}\) is the gas pressure in the HPA,  
\(C_{a}\) is the gas capacitance of the HPA,  
\[C_{a}=\frac{dV}{dp}=\frac{V_{0}}{n\,p_{0}}\left(1-\frac{\Delta V}{V_{0}} \right)^{\kappa+1}\approx\frac{V_{0}}{\kappa\,p_{0}} \tag{55}\]  
where \(p_{0}\) is the initial pressure (\(p_{0}\) = 4 MPa), \(V_{0}\) is the initial volume (\(V_{0}\) = 0.001 m\({}^{3}\)), and \(\kappa\) is the adiabatic exponent in the case of a diatomic gas as N\({}_{2}\) nitrogen (\(\kappa\) = 1.4). \(L_{a}\) is the hydraulic inductance of the hydraulic tube (hydraulic inductance describes the pressure difference required for a change in flow rate and is based on mass inertia),  
\[L_{h}\,=\frac{\rho\,l_{h}}{A_{h}} \tag{56}\]  
where \(\rho\) is the density of hydraulic oil, \(l_{h}\) is the length of the hydraulic tube (\(l_{h}\) = 0.05 m), and \(A_{a}\) is the cross-section of the hydraulic tube,  
\[A_{h}\,=\,\frac{\pi\,d_{h}^{2}}{4} \tag{57}\]  
where \(d_{h}\) is the diameter of the hydraulic tube (\(d_{h}\) = 0.005 m). In the HPA connecting tube during hydraulic shock, the kinetic energy is converted into potential energy; therefore, the hydraulic resistance caused by the pipe impedance refers to the wave resistance inside a tube,  
Figure 16: DIHIPS diagram with the HPA connected to the actuator chamber. \[R_{h}=\frac{\rho\,a}{A_{h}} \tag{58}\]  
where \(a\) is the wave propagation speed (speed of sound in the fluid),  
\[a\,=\sqrt{\frac{K}{\rho}} \tag{59}\]  
It is important to note that the hydraulic resistance caused by the flow resistance is smaller than the hydraulic resistance caused by the wave resistance. Figure 17 compares the undamped and damped DIHPS dynamic response parameters for a single-step shift of the actuator piston with a constant excitation force of \(F\) = 1000 N, representing the disturbance force. By comparing the dynamic responses of DIHPS shown in Figure 17, the effectiveness of HPA was demonstrated in damping piston vibrations and pressure peaks in the actuator chamber.