## 1 Introduction  
Laminated membrane coupling is an all-metal dry flexible coupling, which can compensate relative displacements between the driving and driven shafts through the elastic deformation of the laminated group. Superior characteristics of these couplings, including the compact structure, reliable connection, simple maintenance and excellent environmental adaptability, have led to the widespread use of these components in diverse industrial applications such as petroleum and chemical industry, aviation, ships and wind turbines [1, 2]. Different types of laminated membrane couplings have been proposed so far. These couplings can be divided into the ring type, spoke type, multilateral type, connecting rod type and the waist type. Among these types, the waist type lamination can achieve the optimal stress distribution on the premise of the least material without a remarkable impact on its performance [3]. Studies show that the waist type lamination has reasonable elasticity. However, the manufacturing process of this type of coupling is relatively complicated. Laminated membrane coupling plays a key role in mechanical power transmission. Its working reliability is directly related to the safety of the power transmission device and even the whole system. On the other hand, the stiffness of the laminated membrane coupling reflects its ability to compensate misalignment displacements, which is the basic index that must be considered for evaluating the coupling characteristics. Accordingly, the systematic study of the stiffness characteristics of the laminated membrane coupling is of significant importance to analyze the shaft vibration deeply and accurately. Then a reliable guarantee can be provided for the safe operation of the power plant. Reviewing the literature indicates that many scholars have carried out investigations on laminated membrane coupling. [PERSON] studied the correlation between the low cycle fatigue properties and the adhesive properties of laminations [4]. [PERSON] studied the stress of laminated membrane coupling subjected to diverse torques and axial loads, and emphatically compared the influence of different transmitted powers and axial displacements on the laminated stress [5]. Furthermore, [PERSON] et al. applied MATLAB software to optimize laminated parameters of the coupling [6]. [PERSON] proposed a heuristic adaptive truncation criterion based on linear search and internal iteration of truncated Newton method. The effectiveness and robustness of the adaptive criterion were verified by numerical experience of unconstrained optimization problems, which has important reference significance for optimization problems [7; 8]. [PERSON] and [PERSON] performed a finite element analysis and found that the structural performance of the flexible tapered diaphragm coupling, which is applied in the aerospace industry, is mainly related to the outer diameter of lamination and the stress intensity at the joint of the rim support structure. Then they designed an asymmetric tapered diaphragm structure [9]. [PERSON] carried out an efficient and accurate nonlinear finite element analysis and proposed an optimization design method for thin-walled structures. He modified the composite laminated plate element to consider the bending coupling effect of the diaphragm [10]. [PERSON] et al. proposed an effective surface shell element to analyze the geometric nonlinearity of composite laminated structures by the common rotational method, and then analyzed some laminated composite structures with nonlinear geometries [11]. [PERSON] et al. proposed an effective surface shell element to analyze the geometric nonlinearity of laminated structures with materials by the common rotation method. They showed that the proposed method can be applied to analyze the limit load of laminated structures [12]. Moreover, [PERSON] studied the stress characteristics of the square laminated membrane coupling and then analyzed the influence of the guard plate on the laminated stress [13]. [PERSON] et al.