## 1 Introduction
### Angular Stiffness Analysis of the Laminated Membrane Coupling and the Lamination Group  
In this analysis, the pre-tightening force of connecting bolts between driving and driven flanges are set to 15 kN. After fixing the driven flange, a rotational angle of 0.5\({}^{\circ}\) is applied on the active flange. Then the load sub-step method, which is divided into five sub-steps, is applied to analyze the angular stiffness of the laminated membrane coupling and the lamination group.  
In the software, the end face of the driven flange is set as a fixed support constraint. The loading process is divided into two load steps, load on the sides of six bolts, set a pre-tightening force of 15 Kn in the first step, and lock in the second. An angular displacement load in the X direction is applied to the end face of the active flange. Through the tabular data tool, the first step is set to 0\({}^{\circ}\), the second step is set to 0.5\({}^{\circ}\), the second load step is divided into five sub-steps, the values are 0.1\({}^{\circ}\), 0.2\({}^{\circ}\), 0.3\({}^{\circ}\), 0.4\({}^{\circ}\) and 0.5\({}^{\circ}\)respectively.  
It can be seen from the structure of the laminated group that the effect of the angular displacement in the X and Z direction on the coupling is different, so the same angular displacement in the Z direction is also applied for analysis.  
Through the simulation analysis, the reaction moment of the laminated membrane coupling and the lamination group at the rotational angles of 0.1\({}^{\circ}\), 0.2\({}^{\circ}\), 0.3\({}^{\circ}\), 0.4\({}^{\circ}\) and 0.5\({}^{\circ}\)can be obtained. In this case, the angular stiffness \(K_{\alpha}\) of the laminated membrane coupling and the lamination group can be expressed as:  
\[K_{\alpha}=\frac{\text{d}M}{d\theta\times\pi/180}\approx\frac{180\Delta M}{ \pi\Delta\theta} \tag{5}\]  
Figure 17 shows the reaction moment and the corresponding stiffness of different rotational angles along the X and Z directions of the laminated membrane coupling and the coupling lamination group in each load sub-step.  
Figure 17 indicates that for a small displacement range (i.e., rotational angles within the range of 0-0.5\({}^{\circ}\)), there is a nonlinear correlation between the bending moment and displacement of the laminated membrane coupling along the X- and Z-directions. The same is true for the coupling lamination group. Moreover, it is found that the angular stiffness along the X-direction is slightly greater than that along the Z-direction. In other words, for the laminated membrane coupling, the angular stiffness along the X-direction is about \(4.50\times 10^{3}\) N-m/rad, while that along the Z-direction is about \(4.48\times 10^{3}\) N-m/rad. For the coupling lamination group, the stiffness along the X-direction  
Figure 16: Influence of the flange elasticity on the reaction force and radial stiffness along X- and Z-directions.  
varies within the range of \(6.11\times 10^{3}\)-\(6.21\times 10^{3}\) N-m/rad, while that along the Z-direction varies within the range of \(6.03\times 10^{3}\)-\(6.18\times 10^{3}\) N-m/rad. Anyway, both stiffness values are relatively small so that the laminated membrane coupling along the circumferential direction can be assumed as flexible.