## 1 Introduction
### Test and Verification of the Torsional Stiffness of the Waist Type Laminated Membrane Coupling  
In order to evaluate the simulation accuracy, the static torsional experiment is carried out to measure the torsional stiffness of the laminated membrane coupling. To this end, a PNW-5 electro-hydraulic servo shaft torsional fatigue test machine (Science and Technology Company, Shandong, China) is employed. Figure 10 shows that the test machine is mainly composed of a pump station, test bench and a computer with WinQuick software. This testing machine is applied to carry out the load test on the laminated membrane coupling. Then the simulation results of the torsional stiffness of the coupling are verified. In order to perform the test, it is necessary to find out the zero point position of the hydraulic cylinder and the installation position of the test piece. To this end, the coupling should be installed in the shaft center. Figure 11 shows that the coupling is connected to the left and right connecting plates through the left and right connecting shafts, respectively. During the experiment, the applied torque is gently applied from the connecting disk at one end to 3000 N\(\cdot\)m. Simultaneously, the corresponding angular displacement and torque of the connecting  
Figure 8: Deformation of the lamination under the action of torque. Figure 9: Adding a flexible rotation probe. disk are recorded through the sensor. Figure 12 illustrates the distribution of the torque \(T\) against the angular displacement \(\varphi\). The horizontal line before \(0.05^{\circ}\) indicates that the angular displacement \(\varphi\) increases, while the torque \(T\) remains zero. This originates from the gap along the circumference of the test piece. Moreover, when the angular displacement \(\varphi\) exceeds \(0.05^{\circ}\), it gradually increases as the imposed torque \(T\) increases. It is found that there is a certain nonlinearity until the torque \(T\) reaches the set value of 3000 N-m. In order to calculate the torsional stiffness \(K_{T}\) at 3000 N-m, two points (0.141 \({}^{\circ}\), 2397 N-m) and (0.157 \({}^{\circ}\), 3000 N-m) are considered on the curve. Then the torsional stiffness can be calculated in Equation (2):  
\[K_{T}=\frac{180\times\Delta T}{\Delta\varphi\times\pi}=\frac{180\times(3000-239 7)}{(0.157-0.141)\times\pi}=1.88\times 10^{6}\mathrm{N}\cdot\mathrm{m}/ \mathrm{rad} \tag{2}\]  
Table 3 compares the torsional stiffness at 3000 N-m obtained from the simulation and the experiment and shows the relative error of simulation. It is observed that the calculated stiffness has excellent agreement with that of the experiment. This demonstrates the accuracy of the established model and settings of boundary conditions. Figure 11: Installation drawing of the coupling. Figure 12: Distribution of torque \(T\) applied on the laminated membrane coupling against the angular displacement \(\varphi\). Figure 10: Configuration of the torsional fatigue testing machine. Investigating the Influence of the Flange Elasticity on the Stiffness of the Laminated Membrane Coupling  
The main flexible components of laminated membrane couplings are laminated groups with multi-layer close fitting. It is worth noting that laminated groups are key components for transmitting the mechanical power and torque, compensating the displacement between the master and the slave, and compensating misalignments between the moving shafts. Scholars have performed many investigations on this issue. However, the majority of these investigations are focused on the stiffness of laminated groups and consider the flange as a rigid body. Studies show that the flange elasticity also affects the coupling stiffness. Therefore, it is intended to study the influence of the flange elasticity on the stiffness of the laminated membrane coupling through the comparative analysis of the coupling stiffness.