## 2 Structure of Direct-Drive Valve  
In this study, a direct-drive valve (DDV) system actuated using a piezostack is developed to provide high precision, a fast response, and reliability. Figure 2a shows the schematic of the PADDV valve system. The valve system comprises a piezostack actuator, a displacement amplifier, a spool valve, a gap sensor, an adjust bolt, and a base structure. The small displacement, which is one of the drawbacks of the piezostack actuator, is overcome using a displacement amplifier. The amplifier, with the help of a lever-hinge mechanism, can be used to magnify the displacement generated by the piezostack. A spool valve is used to control the flow rate because the valve can linearly regulate the flow rate through the position of the spool. As shown in Figure 2a, the spool valve comprises a housing, a sleeve, a spool, and a return spring. The housing and sleeve have inlet and outlet ports. The inlet port is always open and has a circular shape to maintain a uniform input pressure, regardless of the spool displacement. However, the outlet port is normally closed and has a rectangular shape to maintain linearity between the spool displacement and the flow rate. The return spring is used to return the spool to the initial position because the considered piezostack is a unipolar piezoelectric actuator. The position of the spool is controlled using the piezostack actuator, with the help of the displacement amplifier. Figure 2b shows the manufactured PADDV valve system. Figure 2c shows a schematic of the mathematical model of the proposed valve system. The dynamic equations of the proposed valve system can be obtained with the displacement of the piezostack actuator (\(x_{p}\)) and spool (\(x_{s}\)) as follows:  
\[ \begin{split}& m_{pi}\ddot{x}_{p}+c_{pi}\dot{x}_{p}+k_{pi}x_{p}=k_{pi}d_{33}V _{pi}-F_{pi}\\ & m_{s}\ddot{x}_{s}+c_{s}\dot{x}_{s}+k_{s}x_{s}=F_{s}-F_{f}\end{split}  \tag{1}\]  
\(F_{s}\) can be obtained as follows:  
\[F_{s}=d_{r}{}^{-1}F_{pi}-\frac{I_{L}}{L_{2}}.\frac{\ddot{x}_{s}}{L_{2}} \tag{2}\]  
Figure 3 shows the hydraulic model of the spool valve. The pressure drop is generated between the input pressure (\(P_{i}\)) and output pressure (\(P_{o}\)) with jet angle (\(\theta\)). The pressure drop causes a force to push the spool, and this is called the flow force. \(F_{f}\) represents the steady flow force, which is calculated as follows:  
\[ \begin{split}& F_{f}=2\Delta PC_{d}^{2}A_{o}cos(\theta)\\ & A_{o}=2D_{s}x_{s}sin^{-1}(H_{s}/D_{s}),\Delta P=P_{i}-P_{o}\end{split}  \tag{3}\]  
Figure 1: Schematic configuration of a valve system. (**a**) Basic working principle of two-stage valve, (**b**) Basic working principle of Single-stage valve. Finally, the dynamic equation of the PADDV system can be written as follows:  
\[ \begin{split}&\Big{\{}m_{s}+m_{pi}{d_{r}}^{-2}+\frac{{I_{t}}}{{L_{2 }}^{2}}\Big{\}}\ddot{x}_{s}+\Big{\{}c_{s}+c_{pi}{d_{r}}^{-2}\Big{\}}\dot{x}_{s} \\ &+\Big{\{}k_{s}+k_{pi}{d_{r}}^{-2}+4\Delta PC_{d}^{2}D_{s}\cos( \theta)\sin^{-1}\left(\frac{{H_{s}}}{{D_{s}}}\right)\Big{\}}x_{s}={d_{r}}^{-1} k_{pi}{d_{33}}V_{pi}(t)\end{split}  \tag{4}\]  
The flow rate of the spool valve can be calculated as directly related to the open area as follows:  
\[Q_{s}=C_{d}A_{O}\sqrt{2\Delta P/\rho} \tag{5}\]  
In this study, VG 46 is used as the actuating fluid, having the density \(\rho=879\) kg/m\({}^{3}\). To select the spring coefficient of the return spring, the valve system is operated from 10 to 500 Hz. Four return springs with different stiffnesses are tested to determine the spring constant for the spool-valve system. Figure 4 shows the experimental results for the magnitude and phase delay in the frequency domain with difference stiffnesses of the return springs. The sinusoidal signal is applied, and the maximum and minimum displacement are measured to calculate the magnitude.