# The Effect of Clamping Force on the Wear Behavior of a Steam Generator Tube
## 4 Results and Discussion
### Dynamic Behavior Analysis Based on the Finite Element Method  
Wear volume is proportional to clamping force and sliding distance, according to the [PERSON] equation [20]. Many scholars have carried out fitting wear experiments on the basis of [PERSON]'s research, and experimental results have well proved this conclusion [21; 22]. However, clamping force affects the sliding distance of the tube. The mutual influence between these two variables is taken into account in this section, and the friction energy at the contact position is used to measure the impact of the clamping force on the wear behavior of the tube.  
For ease of explanation, we define the normal direction of AVB as y-direction, the axial direction of the tube as z-direction, and another as x-direction. To simulate the dynamic response of the tube under different clamping forces, we should determine how much force the normal displacement applied at the AVB corresponds to first. Therefore, in the static finite element analysis, the displacement in y-direction was applied to the AVB-1, and the clamped supported boundary condition was applied to AVB-2, as shown in Figure 8a. After the solution, the reaction force in y-direction at the contact position was obtained, representing the clamping force generated by tightening the bolts at AVB in the experiment. As a result, the relationship between the clamping force and the displacement constraint of AVB-1 was determined, as shown in Figure 8b. It can be seen intuitively in the figure that the clamping force observes a linear relationship with the displacement of AVB-1 in the y-direction, which is related to the elastic material property of the tube being set.  
Figure 7: Natural frequencies of the double-span tube under different clamping forces.  
A node at the center of the contact surface is selected as an analysis object. The displacement and friction of the node along x-direction under different clamping forces are extracted. Simulation results under 45 excitation cycles are selected in the present work. The friction energy in each excitation cycle is calculated according to Equation (2).  
\[E=\sum_{i}^{n}(\frac{F_{x,i}+F_{x,i+1}}{2})\times(s_{i+1}-s_{i})\qquad(1\leq i \leq n), \tag{2}\]  
where \(F_{x}\) is the friction, \(s_{i}\) is the tube displacement in the x-direction, and \(E\) is the friction energy.  
The friction energy in each cycle under different clamping forces is shown in Figure 9a. Considering the monotonically increasing trend of friction energy during the first four cycles, which may be caused by the system being in an unsteady state, the data of the first four cycles are not shown in the figure. Moreover, the average friction energy of a single cycle and the maximum sliding distance under different clamping forces are shown in Figure 9b.  
Obviously, as the clamping force increases from 53 N to 126 N, the maximum sliding distance of the tube at contact position decreases correspondingly, and the friction energy at the contact position becomes lower under increasing clamping force. Furthermore, when the value of the clamping force is at a high level, the friction energy decreases rapidly.