## 3 Result and Discussion  
### Modal Analysis and Model Validation of the Premium Connection  
This section analyzes only the first five natural frequencies and mode shapes of the tubing with the premium connection. Therefore, only the first five natural frequencies are extracted from the finite element software, and higher-order natural frequencies obtained from the experiments are not discussed in this section. As shown in Figure 11, the modal assurance criterion (MAC) parameters from the post-processing of the software are used to assess the reliability of the experimental results. A higher MAC value indicates greater similarity between mode shapes, with a MAC value of 1 indicating identical mode shapes and a MAC value of 0 indicating significant differences between mode shapes. The analysis of the MAC parameters reveals some deviation in the mode shape analysis for the first natural frequency. However, the mode shape test results for the remaining natural frequencies are satisfactory, indicating that the modal test results are reliable and provide a high level of reference. Considering the potential errors introduced during the test impact and the inherent uncertainties in the finite element analysis, this section compares and analyzes the modal results by integrating the finite element analysis results with the experimental results. Table 3 presents the natural frequencies of the tubing obtained through software post-processing analysis. It is observed that the first-order natural frequency obtained from the  
Figure 11: Modal assurance criterion parameters. Figure 10: Critical sealing index of the premium connection. (**a**) The presence of thread sealant; (**b**) The absence of thread sealant. experiment differs significantly from the finite element analysis results, with an error of up to 27% when using experimental data as the baseline. This discrepancy is attributed to the hardness of the hammer used during the test, which results in better excitation effects for higher-order modes but poorer excitation effects for lower-order modes. Consequently, the error in the first-order modal analysis results is relatively large. Aside from the significant difference in the first-order natural frequency, the results for the other natural frequencies are generally satisfactory. Figure 12 shows the frequency response function of the premium connection under the optimal makeup torque during the experiment. As shown in Figure 13, the experimental results indicate that the third natural frequency is 621.93 Hz, which is close to the fourth natural frequency at 795.21 Hz. However, the fourth natural frequency observed in the experimental results is not present in the finite element analysis, leading to doubts about its authenticity. To address this issue, the modal shapes from the fourth to the sixth natural frequencies were analyzed and compared, as shown in Figure 9. The comparison revealed that the third natural frequency (621.93 Hz) obtained from the experimental post-processing has three peaks, which is consistent with the results for a simply supported beam's third mode shape. On the other hand, the mode shape for the fourth natural frequency (795.21 Hz) obtained from the experimental post-processing appears to be \"incomplete\". Therefore, it is concluded that the frequency of 795.21 Hz falls between the third and fourth natural frequencies, and that the experimental result for the fourth natural frequency is likely due to testing errors. Figure 14 shows the comparison between the experimental results of the tubing vibration mode under the optimal make-up torque for the premium connection with the finite element analysis (FEA) results. It is observed that the FEA results align well with the experimental analysis results (the magnification factor between the FEA and experimental results is 10).