# Effects of the Damping Parameters on the Opening and Closing Characteristics of Vent Valves
## 2 Model
### Membrane Assembly  
The membrane assembly is one of the pressure-sensitive elements in the vent valve. The membrane is a circular sheet made of rubber, which is the core deformation element in the membrane assembly. In the film assembly, the material of the diaphragm is rubber, and the rest of the structure is stainless steel, all of which are isotropic materials. Combined with the working principle of membrane assembly, the deformation can be described by [PERSON]'s law; that is, within the range of linear elasticity of a material, the unidirectional tensile deformation of a solid is proportional to the external force, and the mathematical model for this is shown in Equation (3). Rubber is a kind of hyperelastic material. The Mooney-Rivlin strain energy function is widely used to simulate the mechanical behavior of rubber. The constitutive equation is shown in Equation (4), and the typical binomial third-order expansion is shown in Equation (5). The membrane material is butadiene rubber, whose density \(\rho_{1}\) is 1200 kg/m\({}^{3}\), elastic modulus \(\mathrm{E}_{1}\) is 14.04 MPa, Poisson's ratio \(\mu_{1}\) is 0.499, friction coefficient \(\zeta\) is 0.2, constant \(\mathrm{C}_{1}\) is 1.87 and constant \(\mathrm{C}_{2}\) is 0.47 in the Mooney-Rivlin material model. The rest of the structure is structural steel, whose density \(\rho_{2}\) is 7850 kg/m\({}^{3}\), [PERSON]'s modulus \(\mathrm{E}_{2}\) is 2 \(\times\) 105 MPa and [PERSON]'s ratio \(\mu_{2}\) is 0.3. According to the actual  
Figure 2: Pressure displacement test kit for welded metal bellow bracket. working conditions, the membrane assembly exerts internal control pressure and adds fixed constraints on the outer annular surface:  
\[[K]\times\{\delta\}=\{F\}, \tag{3}\]  
\[\mathrm{W=\sum_{1+j=1}^{N}C_{\bar{ij}}(I_{1}-3)^{j}(I_{2}-3)^{i}+\sum_{1+j=1}^{N} \frac{1}{d_{k}}(I_{3}^{2}-1)^{2k},} \tag{4}\]  
\[\mathrm{W=C_{10}(I_{1}-3)+C_{01}(I_{2}-3)+\frac{1}{d}(J-1)^{2}.} \tag{5}\]  
For the same physical model, the smaller the grid size is, the more grids will be obtained, and the calculation results will be more accurate. However, too many grids will only increase the computing time of the computer, so grid independence analysis of the model is needed. Through the refinement of the grid and the simulation of 0.6-mm, 0.8-mm and 1.0-mm grid sizes, the results show that the change of the result was within 5% when the grid size was 0.8 mm, and the time was short. When the grid size was 0.8 mm, the grid division of the film assembly obtained is shown in Figure 3, with 262,246 nodes and 179,929 grids. An aircraft operates at different altitudes, and different altitudes correspond to different environmental pressures. Therefore, the terminal displacement of the membrane assembly caused by the changing control pressure was simulated under different environmental pressures. Figure 4 shows the partial deformation distribution of the membrane assembly with control pressure. According to the deformation distribution cloud diagram, the maximum deformation was on the diaphragm around the terminal and not the terminal itself, so the diaphragm was the main deformation element of the membrane assembly. According to the working characteristics of the membrane assembly, the maximum distance between the terminal and the nozzle was 0.8 mm. Therefore, the displacement of the deformation greater than 0.8 mm was actually 0.8 mm. The simulation results of the terminal displacement under different environmental pressures and control pressures were fitted. The terminal displacement under different environmental pressures was linearly correlated with the control pressures, and this result verifies that the membrane assembly conformed to the generalized Hooke's law. From, the unary linear equation obtained from the analysis and fitting, it could be found that the slope was the same, and there was a certain proportional relationship between the intercept and the environmental pressure. Finally, the general mathematical expression of the film module is obtained as Equation (6).