## 2 Working Principles of the Piezoelectric-Hydraulic Pump Brake System
### The Force Applying on the Check Valve Operating in Fluid  
When the PZT actuator oscillates, the force acts on the check valve by the flow rate generated by the PZT actuator. However, the check valve will oscillate which in turn leads to the boundary conditions changing over time. Accordingly, the force acting on the check valve will be determined by an FSI interaction simulation by using the Ansys program. As mentioned above, the oscillation of the thin-plate spring check valve is in the liquid domain. The added liquid mass and added damping, therefore, are considered. The dynamic equation of the check valve operating in the fluid domain can be written as:  
\[M\ddot{y}+C\dot{y}+Ky=F(t) \tag{1}\]  
where \(M\), \(C\), \(K\) and \(F\) refer to the total effective mass of the valve in a fluid environment (\(M=M_{v}+M_{a}\), where \(M_{v}\) represents the effective mass of the check valve, and \(M_{a}\) is the added mass of liquid), the added damping, the stiffness of the check valve, and the driving force generated by the PZT actuator, respectively.  
In the piezoelectric-hydraulic pump, the liquid flow is driven by the oscillating actuator which opens and closes the check valve repeatedly. At the same time, the fluid reversely holds the main role in resistance to the check valve oscillation. The actuator oscillation, the check valve movement and the liquid are thus coupled. The fluid reaction force is referred to as an added mass and the added damping contributes to the dynamic response of the check valve without changing its stiffness. Assuming the fluid to be incompressible, the driving force generated by the actuator will be forced to the check valve the right way. Consequently, the external force, \(F(t)\), can be separated into two contributions:  
\[F(t)=F_{reaction}(t)+F_{drive}(t) \tag{2}\]  
the first term, \(F_{reaction}(t)\) in Equation (2), is a reaction loading component caused by the motion of the fluid around the check valve, while the second term, \(F_{drive}(t)\), refers to a driving force that excites the check valve. This force is usually a known function of time.  
Related to fluid-structure interaction, the hydrodynamic loading, which is mainly generated due to the pressure drag [17], has been separated by two uncoupled terms:  
\[F_{reaction}(t)=-M_{a}\ddot{y}-C_{a}\dot{y} \tag{3}\]  
where \(M_{a}\) represents the added mass per unit length and \(C_{a}\) refers to the added damping coefficient. The negative sign in Equation (3) demonstrates that the hydrodynamic forces always reverse the check valve motion. Substituting Equation (3) into Equation (2), we get an uncoupled equation of the check valve's motion:  
\[(M_{v}+M_{a})\ddot{y}+(C_{v}+C_{a})\dot{y}+Ky=F_{drive}(t) \tag{4}\]  
where the mass and damping coefficients contain the contributions from the check valve and the ambient liquid.  
The purpose of the numerical simulation, which considers the check valve and surrounding liquid, is to identify the response of the check valve in a fluid domain. It is possible to observe the simultaneous fluctuation between the check valve and piston during pump operation. From that simulation, it is possible to obtain some important characteristics such as the flow value, the flow velocity, the valve's oscillation, as well as the pressure in the chamber. The ANSYS program is also used to simulate the oscillation of the check valve and to provide internal flow field evolutions as a visualization for a better understanding of the pressure and flow fields inside the pump.