## 1 Introduction
### Simulation of Torsional Stiffness of the Waist Type Laminated Membrane Coupling  
In fact, in a considerable number of work conditions, the friction caused by the contact between the laminations cannot be ignored. In the working process, the state of the coupling will be affected by the contact between the laminations and between the laminations and the gaskets. The characteristics of the coupling are the comprehensive performance of the interaction between the laminations and the laminations, as well as between the laminations and their connectors. Therefore, it is ideal and rough to calculate the stress level of the laminated coupling simply by analyzing a single laminated plate. Therefore, it is necessary to consider the contact and friction between the laminated plates and analyze the laminated coupling as a whole. The three-dimensional model of the coupling is established by CATIA software. The established 3D model is imported into the finite element software workbench and material properties are defined in the software according to the data provided in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Part** & **Quantity** & **Material** & \begin{tabular}{c} **Density** \\ \(\rho\)**(kg/m\({}^{3}\))** \\ \end{tabular} & \
\begin{tabular}{c} **Elastic** \\ **Modulus \(E\)**(Pa)** \\ \end{tabular} &
\begin{tabular}{c} **Poisson Ratio \(\mu\)** \\ \end{tabular} \\ \hline Flange plate & 2 & 30 CrNi4 MoA & 7885 & \(2.04\times 10^{11}\) & 0.307 \\ Spherical washer & 6 & 30 CrMnSiA & 7750 & \(1.96\times 10^{11}\) & 0.3 \\ Flange bushing & 6 & 30 CrMnSiA & 7750 & \(1.96\times 10^{11}\) & 0.3 \\ Bushing & 6 & 30 CrMnSiA & 7750 & \(1.96\times 10^{11}\) & 0.3 \\ Bolt & 6 & 30 Ni4 CrMoA & 7849 & \(2.04\times 10^{11}\) & 0.308 \\ Self-locking nut & 6 & 30 CrMoA & 7820 & \(2.09\times 10^{11}\) & 0.279 \\ Lamination & 16 & 301 stainless steel & 7930 & \(2.12\times 10^{11}\) & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantity, material and mechanical characteristics of main parts in the assembly. Figure 5: Main geometric parameters of laminations. The workbench simulation software is used for the simulation of the waist type laminated membrane coupling, and the multi-zone method is used to automatically divide the grid. By setting the element size, the density of the divided grid is adjusted. In the calculation, the grid size of the laminated sheet is set to 2 mm, the grid size of bolts, nuts, bushings, etc. is 4 mm, and the grid size of the two flanges is 8 mm. The number of component mesh nodes is 536,163 and the number of cells is 74,045. The final meshing effect is shown in Figure 6.  
It is worth noting that contacts between bolts and nuts, flange plates, lamination and flange bushing and washers are set as bonded, while contacts between the bushing and bolt, nut and washer are as no-separation. Meanwhile, contacts between laminations are set as frictional contact. It means that there is separation in normal direction and relative sliding along the tangent direction. According to the material and surface roughness of coupling, the friction coefficient between the laminations is set to 0.2. As shown in Figure 7, according to the relevant parameters provided by the coupling manufacturer, the pre-tightening force of the connecting bolts between driving and driven flanges are set to 15 kN, the preload is applied to each bolt. After fixing the driven flange, a torque of 3000 N-m is applied on the active flange, and then the torsional stiffness of the laminated membrane coupling is simulated. After calculation by finite element software, the deformation diagram of the outermost layer of the coupling model under the action of torque is obtained, as shown in Figure 8. In order to obtain an accurate deformation torsion angle \(\varphi\), a remote point is set on the cylindrical end face of the flange, and then a flexible rotation probe is added here, as shown in Figure 9.  
Figure 6: Generated mesh for the waist type laminated membrane coupling. Figure 7: Boundary condition setting under torque.