## 5 Hydraulic Stiffness of HA  
Figure 9c shows the pressure peaks during the step shift of the actuator piston, with the highest pressure peaks occurring at extreme positions of the actuator piston. This phenomenon is caused by the compressibility of the hydraulic oil in the actuator chambers. Due to their compressibility, hydraulic fluids behave like hydraulic springs. The stiffness of the actuator chamber depends on the stiffness of the hydraulic oil and is defined as follows:  
\[k_{1}=\frac{A^{2}}{C_{1}}\text{ and }k_{2}=\frac{A^{2}}{C_{2}} \tag{45}\]  
where \(k_{1},k_{2}\) are the hydraulic stiffness in the actuator chambers. The shift of the actuator piston causes a change in the volume in both chambers in proportion to the effective area \(A\) of the piston and inversely proportional to the hydraulic capacity \(C_{1}\) and \(C_{2}\), and thus the hydraulic stiffness of a symmetrical HA results from the series connection of the hydraulic oil springs as follows:  
\[k=k_{1}+k_{2}=\left(\frac{A^{2}}{C_{1}}+\frac{A^{2}}{C_{2}}\right)=K\,A^{2} \bigg{(}\frac{1}{V_{1}}+\frac{1}{V_{2}}\bigg{)} \tag{46}\]  
for \(0\leq s\leq S\),  
\[k(s)=K\,A^{2}\bigg{(}\frac{1}{V_{1}}+\frac{1}{V_{2}}\bigg{)}=K\,A^{2}\bigg{(} \frac{1}{A\,s}+\frac{1}{A\,(S-s)}\bigg{)}=K\,A\bigg{(}\frac{1}{s}+\frac{1}{S- s}\bigg{)} \tag{47}\]  
natural frequency \(f\) of the HA under mass load,  
\[f(s)=\frac{1}{2\,\pi}\sqrt{\frac{k(s)}{m}} \tag{48}\]  
For the simulation parameters adopted and step \(s=0.04\) m, the hydraulic stiffness HA is \(k(s)=1.4\times 10^{7}\) N/m and the natural frequency is \(f(s)=86\) Hz. Figure 10 shows the relative hydraulic stiffness \(k(s)/k_{m}(s\approx 0)\) and the relative natural frequency \(f(s)/f_{m}(s\approx 0)\) of the HA as a function of piston stroke. Note that the stiffness \(k(s)\) and the natural frequency \(f(s)\) for \(s=0\), \(s=S\) are infinite. After reaching the assumed DIHPS step position and closing the BV, the HA becomes stiff, as shown in Figure 11.  
For a given piston step \(s\), the mechanical model of stiff HA excited by force \(F\) is as follows:  
\[\left\{ \begin{array}{l}v=\frac{ds}{dt}\\ \frac{dv}{dt}=\frac{1}{m}[-b\,v+A\,(p_{1}-p_{2})\pm F]\\ \frac{dp_{1}}{dt}=\frac{K}{V_{1}}\,A\,v=\frac{K}{s}\,v\\ \frac{dp_{2}}{dt}=-\frac{K}{V_{2}}\,A\,v=-\frac{K}{S-s}\,v\end{array} \right. \tag{49}\]  
Figure 12 shows the dynamic responses of a rigid HA excited by a constant force. Figure 11: DIHPS diagram with a stiff HA loaded with an excitation force \(F\). Figure 12: Cont. Figure 10: Relative hydraulic stiffness and relative natural frequency of the HA as a function of the piston stroke. Figure 13 shows the resonance vibration parameters of a rigid HA excited by a cyclic force. Figure 12: Dynamic response of the stiff HA excited by a constant force \(F=1000\) N: (**a**) vibration of the piston, (**b**) fluctuation of the piston speed, and (**c**,**d**) pulsation pressure in the actuator chambers. Figure 13: Resonant vibration parameters of the stiff HA excited by a cyclic force: (**a**) piston vibration, (**b**) piston speed fluctuation, (**c**,**d**) pulsation pressure in the actuator chambers. The hydraulic stiffness of the HA in the stop position influences the dynamic behavior of the DIHPS. HA subjected to load disturbances such as excitation force is susceptible to vibration. In an HA, there can be a wide spectrum of pressure pulsation frequencies [35]. In the case of constant excitation force, the vibrations of the HA piston are suppressed, and the accompanying pressure pulsations disappear. When the HA is excited by a cyclic force, the vibrations are not dampened, and the accompanying pressure pulsations do not disappear. An excitation force limited to harmonic vibration was introduced to excite HA,  
\[F(t)=F_{0}\,\sin(2\,\pi\,f_{e}\,t) \tag{50}\]  
where \(F_{0}\) is a constant excitation force, and \(f_{e}\) is the excitation frequency.