# Dynamic Analysis of Bolted Cantilever Beam with Finite Element Method Considering Thread Relation
## 2 Materials and Methods  
Figure 1 depicts the considered bolted cantilever beam that consists of one cantilever beam in the left and a free lap beam in the right connected by a bolt. It is assumed that one single sine excitation force is applied in the right end of the structure, with being the excitation amplitude and being the excitation frequency. According to Figure 1, the x, y, and z represent the axial, lateral and transverse direction, respectively. Other parameters of the bolted cantilever beam, \(s,l,h\), and \(L\) denote the distance from the central axis of the bolt to the wall, the length and thickness of a single beam, the height of beam, and the length of the whole structure of the bolt connected beam, respectively. Commonly, scholars only simplify the bolt to the transverse and normal contact problems, and do not consider the physical characteristics of the bolt. Bolt connection is different from mortise joint and riveting joint. It has strict connection mode of thread pair. Therefore, a finite element model considering thread characteristics is established in this paper and dynamic response analysis is conducted under different external excitation. Figure 2 is a sectional view of the interaction between the screw and the nut. According to Figure 2, \(G_{v}\) represents the axial force on the screw, \(\beta_{v}\) represents the half angle of thread profile and \(\theta_{v}\) represents the half angle of groove. Obviously, the relationship between half angle of thread profile and half angle of groove is as follows, \(\theta_{v}=90^{\lx@math@degree}-\beta_{v}\). Screw pair is composed of screw and nut. It is a kind of spatial motion pair, and its contact surface is spiral surface. When the screw and nut threads are subjected to axial load, when the screw or nut is turned, the relative sliding will occur between the two threads, and there is friction between the  
Figure 1: Structural diagram of bolted cantilever beam. Figure 2: Schematic diagram of threaded connection. screw surfaces. When studying the friction in a screw pair, it is usually assumed that the axial load \(G_{v}\) between the screw and the nut acts on the helix with pitch diameter. Since the helix can be developed into an oblique straight line on the plane, the effect of the force in the screw pair is the same as that between the slider and the inclined plane, as shown in Figure 3. In this way, the friction problem in the spatial screw pair can be transformed into the friction problem in the grooved moving pair. Assume the half angle of the thread being \(\beta_{v}=30^{\lx@math@degree}\); thus, the groove semi-angle is \(\theta_{v}=60^{\lx@math@degree}\), and the corresponding equivalent friction coefficient and the equivalent friction angle of the groove semi-angle are shown below separately,  
\[f_{v}=\frac{f}{\sin\theta_{v}}=\frac{f}{\cos\beta_{v}} \tag{1}\]  
\[\varphi_{v}=\arctan(\frac{f}{\cos\beta_{v}}) \tag{2}\]  
Figure 3 shows a triangular screw pair with axial force and \(G_{v}\) acting on the nut (including the weight of the nut itself). \(F_{R}\) is the reaction force of screw on nut, and \(F\) is the circumferential force acting on thread pitch circle. Since the nut can be simplified as a slider, and the screw thread can develop into an inclined groove surface with an inclination angle \(\lambda_{v}\) of along the middle diameter, the relationship between the nut and the screw can be simplified as that of the slider sliding along the groove surface. The inclination angle of the inclined plane is the thread rising angle on the pitch diameter of the thread; thus,  
\[\tan\lambda_{v}=\frac{P_{\text{h}}}{\pi d_{v}}=\frac{nP}{\pi d_{v}} \tag{3}\]  
where \(P_{\text{h}}\) is the thread lead, \(n\) is the numbers of thread heads, \(d_{v}\) is pitch diameter and \(P\) is the pitch of threaded bolt. When tightening the nut, it is equivalent to the sliding block rising along the inclined plane.