## 1 Introduction
The deformation of the flexible diaphragm changes the liquid pressure in the pump cavity, which, through the inlet valve and the outlet valve, completes liquid suction and discharge. The displacement  
Figure 1: Flexible layout diagram of the HFAM: (**a**) the non-parallel force amplification transmission between the actuator and the load and (**b**) the orthogonal force amplification transmission between the actuator and the load. amplification ratio of the piezoelectric pump can reach 3.29. However, limited by the flat diaphragm mechanical property, the displacement amplification mechanism can only be used for small displacement conditions. [PERSON] et al. combined a piezoelectric actuator with a rigid piston hydraulic amplification mechanism, as shown in Figure 3[6]. They analyzed the displacement and mechanical characteristics of the piezoelectric actuator through finite element simulation with an electromechanical model and a Bouc-Wen hysteresis model. The amplification effect was analyzed through both the simulation and experiment methods. The maximum amplification ratio is approximately 77, and the error between the simulation model and the experiment is about 2.53%. Although the rigid piston-type hydraulic amplification mechanism can reach a large amplification ratio, the rigid piston occupies a larger volume and struggles to meet the design requirements of being lightweight and miniaturized. [PERSON] et al. utilized a flexible piston to analyze the characteristics of a flexible piston hydraulic displacement amplification mechanism based on small deflection theory and modal analysis, as shown in Figure 4[19]. They established a mechanical governing equation of the liquid-machine coupling model and used the incremental harmonic balance method for their analysis. The theoretical analyses were verified through finite element simulation. Owing to the design of flexible pistons, the hydraulic amplification mechanism has a compact structure and has a good amplification effect with a maximum ratio of 8.5. But, limited by the flexible positions' mechanical property, the effective motion range is within \(\pm\)0.25 mm, which struggles to meet the control valve requirement of a big motion range. The above research proves the potential of hydraulic amplification mechanisms in engineering applications. Among them, the diaphragm-type amplification mechanisms have the advantages of good sealing and compact structure, which are suitable for hydraulic control valves. Considering that the motion range and the amplification effect of the  
Figure 3: Structural diagram of the piezo hydraulic amplification mechanism [6]. Figure 2: Structural diagram of the piezoelectric micropump [18]. diaphrag-type hydraulic amplification mechanisms are related to the performances and characteristics of the diaphragms, to achieve a better amplification effect, the rolling diaphragms with larger motion ranges and lower elastic forces can be used to replace the flat diaphragm on the small-area side. In addition, the existing work on the hydraulic amplification mechanisms primarily focuses on structural design, mathematical modeling, finite element simulation, and dynamic analysis. However, there is a lack of theoretical models and experiments on the static and dynamic characteristics of the relationships between the forces and displacements of the hydraulic amplification mechanisms. The rest of this paper is organized as follows. In Section 2, the working principle of the proposed HFAM is introduced. Furthermore, the mathematical model and Simulink dynamic model are established, and the mechanical properties of the diaphragms are simulated. In Section 3, the proposed experimental system and the experimental methods are introduced. The experimental results are presented, focusing on the factors influencing the relationships between force and displacement in the static characteristics, the trend of the force amplification ratio (FAR), and the dynamic characteristics of the HFAM system. Finally, Section 4 offers the conclusions for the research.