# The Effect of Clamping Force on the Wear Behavior of a Steam Generator Tube
## 4 Results and Discussion  
### Natural Frequencies under Different Clamping Forces  
The natural frequency of the single-span tube measured in the experiment and the results of modal analysis based on the finite element method are shown in Figure 5. Among them, the experimental results are represented by a solid line, the simulated results of the tube with simply supported boundary conditions are represented by dotted lines, and the clamped constraints are represented by dashed lines. The first five natural frequencies of the tube are selected in Figure 5. It is not difficult to find that the value of the natural frequency of the tube obtained in the experiment is within range of the simulation results in different boundary conditions. That indicates that the constraint at both ends of the tube is not simply a supported or clamped supported constraint, but a constraint between the simply supported and the clamped supported constraints. \begin{table}
\begin{tabular}{c c c} \hline  & **Parameter** & **Value** \\ \hline \multirow{3}{*}{Material property} & [PERSON]’s modulus & \(E=211\) GPa \\  & [PERSON]’s ratio & \(\Upsilon=0.3\) \\  & Density & \(P=8190\) kg/m\({}^{3}\) \\ \hline \multirow{3}{*}{Geometric parameter} & Length & 2985 mm \\  & Outer diameter & 17.48 mm \\ \cline{1-1}  & Thickness & 1.01 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of modeling. Figure 5: The natural frequency of the single-span tube. The response amplitudes of the first three natural frequencies are greater than that of other orders in Figure 5. Consequently, the values of the first three frequencies are used to calculate the relative error given by Equation (1),  
\[\delta_{ave}=\sqrt{\frac{1}{3}\times\sum_{i=1}^{3}{(\frac{f_{Si}-f_{Ei}}{f_{Ei}})^ {2}}}\qquad(i=1,\ 2,\ 3), \tag{1}\]  
where \(f_{Si}\) is the frequency obtained by simulation, \(f_{Ei}\) is the frequency obtained in the experiment, and \(\delta_{ave}\) is the relative error. The relative error of the first three natural frequencies under the simply supported boundary condition is greater than that under the clamped supported boundary condition, as shown in Table 2. Therefore, in finite element modeling, the clamped supported boundary conditions are adopted to simplify the calculation. In addition, a small peak next to the large peak in the third-order natural frequency is shown in Figure 5. Considering that its amplitude is only about 18% of the nearby peak, combined with the modal analysis results of the tube under simply supported or clamped supported boundary conditions, it is inferred that the small wave peak is the interference peak caused by noise. The state of the double-span tube is changed by applying different clamping forces at the AVB. The first-order natural frequencies of the tube under different clamping forces are shown in Figure 6. According to experimental results, when the AVB is in contact with the tube without clamping force being applied, the natural frequency of the tube is 27.344 Hz. In order to make it more intuitive, we performed a quadratic polynomial fit to the data points. It is easy to see that as the clamping force increases, the first-order natural frequency of the tube increases slightly, and the end of the curve tends to be flat. The first seven natural frequencies of the tube under different clamping forces are drawn in the form of waterfall diagrams, as shown in Figure 7. It can be seen from the figure that when the tube is subjected to a clamping force below 10 N, the response amplitudes of \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **First-Order** & **Second-Order** & **Third-Order** & \(\delta_{ave}\) \\ \hline Experimental result & 9.766 Hz & 28.564 Hz & 57.373 Hz & / \\ Clamped supported & 11.821 Hz & 32.567 Hz & 63.800 Hz & 0.160 \\ Simply supported & 5.2165 Hz & 20.860 Hz & 46.915 Hz & 0.328 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The first three natural frequencies of the tube. Figure 6: Natural frequencies of the first order under different clamping forces (double-span tube).