## 2 Working Principle and Modeling
### Diaphrag Mechanical Model  
A nonlinear coupling exists between the elastic force of the rubber and the pressure of the liquid when the diaphragm undergoes deformation along its axis. To scrutinize the interplay of the mechanical characteristics between the RD and the CD, finite element simulations were conducted. Due to the assumption that the liquid was incompressible, the pressure could not be transmitted within the sleeve chamber. The RD and the CD were subjected to finite element simulations, respectively. In accordance with the hypothesis, under identical conditions, the reduction in the RD volume chamber was equal to the increase in the CD volume chamber. At the same time, the liquid exerted the same pressure on the diaphragms. In the subsequent analyses, the two simulations could be combined through the same volume and pressure. Because the diaphragm is cylindrical, a two-dimensional axisymmetric space dimension was employed in the finite element simulation for the diaphragm analysis. The elastic modulus of the rubber is nonlinearly influenced by factors such as material characteristics and dimensions. The Mooney-Rivlin constitutive model is recognized for accurately depicting the hyperplastic characteristics of incompressible rubber materials under small and medium strains [22]. The strain energy density function is  
\[W=C_{10}(I_{1}-3)+C_{01}(I_{2}-3) \tag{16}\]  
where W is the strain energy, I\({}_{1}\) and I\({}_{2}\) are deformation tensors, and C\({}_{10}\) and C\({}_{01}\) are Mooney constants. The model effectively characterizes the mechanical characteristics of the rubber materials within a deformation range of less than 150%, thereby satisfying the computational demands applicable to the rubber materials in practice. The relationship between the elastic modulus \(E\) and the shear modulus \(G\) of rubber materials under the small strain condition is  
\[G=\frac{E}{2(1+\mu)} \tag{17}\]  
Assuming that rubber is incompressible, [PERSON]'s ratio \(\mu\) is 0.5, and Equation (17) can be simplified as  
\[G\approx\frac{E}{3}=2(C_{10}+C_{01}) \tag{18}\]  
The ratio of C\({}_{01}\) to C\({}_{10}\) is a constant, and the expression is  
\[k_{r}=\frac{C_{01}}{C_{10}} \tag{19}\]  
where k\({}_{r}\) is the mechanical characteristic coefficient of the constitutive model. The semi-empirical equation relating Shore hardness and [PERSON]'s modulus is [23]  
\[E=\frac{2.15s+15.75}{100-s} \tag{20}\]  
where s is the Shore hardness. In the research, the used diaphragm material is Hydrogenated Nitrile Butadiene Rubber (HNBR) with a Shore hardness of 70 HA. According to the research of [PERSON] [24] et al., the parameter k\({}_{r}\) equals 0.25. Combined with Equations (18)-(20), the parameters C\({}_{10}\) and C\({}_{01}\) are calculated as 0.7388 MPa and 0.1847 MPa, respectively. The rubber material parameters are shown in Table 1, while the other connectors' material is 304 stainless steel. Subsequently, finite element simulations were employed for the solid modeling and meshing of the HFAM. As shown in Figure 8, the established simulation models are composed of shafts, diaphragms, sleeve, and flanges. Considering that all these components have axis symmetry about the shafts' central axes, a two-dimensional axis symmetry model was used for the simulation model. The left ends of shaft 1 and shaft 2 in Figure 8 indicate the central axes of the actual shafts and set the symmetry axes. The boundary conditions of the simulation models are shown in Figure 8,b. In the simulations of the RD part, there was an acting force added to the upper end of shaft 1, and the force was regarded as being generated by the VCM. The sleeve and flange 1 were set as fixed constraints, and the contact surfaces of the different components were set as contact pairs. In the simulations of the CD part, there was an acting force added to the lower end of shaft 2, and the force was regarded as being generated by the external load.