## 3 Finite Element Models of Anti-Loosening Bolted Connections  
The FEM built up in this paper is based on a typical bolted joint structure. The nominal diameter and pitch of bolt and nut were M10 and 1.5 mm, respectively. Other dimensions of the regular bolted joint are shown in Figure 6, where \(A\) = 6 mm, \(B\) = 33 mm, \(C\) = 12 mm, \(D_{1}\) = 10 mm, \(D_{2}\) = 11.2 mm, \(E\) = 15.7 mm, \(F\) = 18.1 mm, \(G\) = 7.5 mm, \(H_{1}\) = 9 mm, and \(H_{2}\) = 15 mm. Subsequently, a three-dimensional FEM of a regular bolted joint was built using HyperMesh 12.0 commercial preprocessor software, as shown in Figure 7. Using the method of [PERSON] and Nomura [38], the meshes of internal and external threads were generated accurately. This method takes the geometric structures of a bolted joint into account and generates three-dimensional hexahedral meshes. Compared with tetrahedral elements, hexahedral elements have higher calculation accuracy and better deformation resistance. In the process of modeling, the bolt head and hexagonal nut were both simplified as cylinders with a diameter of 15.7 mm. In ISO 16130-2015 [39] and German national standard DIN 65151 [40], the cyclic transversal vibration acts on the upper plate, and the lower plate is applied with a fixed constraint. Therefore, the upper plate is called a movable plate, and the lower plate is called a fixed plate. Since the friction effect between the two plates could be ignored, the fixed plate was omitted to simplify the calculation, and only the FEM of the movable plate was built. Figure 5: Schematic representation of the third type of anti-loosening bolted connections: (**a**) eccentric double nut; (**b**) double-thread bolt engaged with two nuts. Figure 6: Two-dimensional sketches of a regular bolted joint: (**a**) joint; (**b**) nut; (**c**) bolt. Subsequently, the three-dimensional FEMs of seven anti-loosening bolted connections were built, as shown in Figure 8. The meshes of a plain waser were generated via sweep, and all the meshes were hexahedrons. The mesh density was consistent with that of the moving plate. The inner diameter of the plain was consistent with that of the movable plate, and the outer diameter was slightly larger than that of the regular nut, as shown in Figure 8a. The meshes of the wedge washer were divided carefully, especially the radial tooth surfaces on both sides. The mesh density of radial teeth was set to a relatively high value in order to ensure the calculation accuracy of the contact analysis, while the other areas were meshed with low density, as shown in Figure 8b. The plain washer and wedge washer were both installed under a nut; the FEMs of the bolt, nut, and moving plate were the same as in the model of a regular bolted joint shown in Figure 7.  
The shapes or sizes and internal threads in a wedge-locking nut and variable-diameter nut were changed. Therefore, the mathematical expression of each portion on the thread profile along the axis within a pitch should be established first. The mathematical expression of the thread profile in a wedge-locking nut is a little complicated, as discussed in [34], while that in a variable-diameter nut is modified compared with a normal internal thread [38]. Next, the meshes of wedge-locking and variable-diameter nuts were both built using [PERSON] and [PERSON]'s meshing method. In the model of the wedge-locking nut, the wedge angle was 30\({}^{\circ}\). In the model of the variable-diameter nut, the internal and external threads were interfered. Figure 8c,d show the entire bolted joints with a wedge self-locking nut and a variable-diameter nut, respectively. The FEMs of the bolt and moving plate remained consistent with those of the bolted joint shown in Figure 7.  
In the finite element modeling process of bolted joints with regular double nuts, the length of the bolt and the number of thread teeth were increased, and the modeling method with two nuts was the same as that for a regular nut. Figure 8e presents the FEM result.