## 5 Results and Discussion  
### Convergence of the PSO Simulations and Mode Tracking  
As presented in Table 6, both approaches were performed using three different numbers of particles. The minimum objective function errors and calculation time of these simulations are listed in Table 10. It can be observed that the smallest error was obtained with the second approach, and overall, the errors of the second approach were better than those of the first one. Consequently, the assumption of using springs to simulate the constraint support stiffness is valid. In addition, based on the results, it can also be concluded that the increase of the number of particles inevitably leads to a significant increase in the calculation time, while the optimization results are not improved significantly. Thus, it is necessary to find a fine balance between calculation time and accuracy of the results.  
Figure 13 displays the convergence plots of the PSO simulations with different numbers of particles. Based on the results, the convergence of the PSO simulations can be divided into two stages: the fast convergence stage and the stable convergence stage. In the first stage, the fitness values of the particles are relatively scattered, while the mean value of the particle swarm and the minimum error converge rapidly. In the second stage, the particle swarm tends to concentrate towards the minimum error and their mean value decreases slowly. Furthermore, the increase in the number of particles does not significantly affect the convergence rate of the PSO simulation, but only increases the probability of the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{2}{c}{**First Approach**} & \multicolumn{2}{c}{**Second Approach**} \\ \cline{2-5}  & **Error (\%)** & **Time (h)** & **Error (\%)** & **Time (h)** \\ \hline
20 & 2.78 & 13.42 & 1.14 & 12.75 \\
30 & 1.40 & 21.60 & 1.05 & 19.13 \\
50 & 1.40 & 33.34 & 1.07 & 24.73 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Minimum objective function errors and calculation time of the PSO simulations.  
Figure 12: Switching range of the concerned mode orders. (**a**) First model-updating approach and (**b**) second model-updating approach.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Experimental Mode Order** & **FirstList Approach FE Mode Order** & **Second Approach FE Mode Order** \\ \hline
1 & 9 & 9, 11 \\
2 & 14 & 14, 16 \\
3 & 20 & 20, 22 \\
4 & 29 & 29, 31 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Mode orders to be tracked in the two model-updating approaches.  
particle swarm finding the optimal value. This is also one of the reasons why the accuracy of the results does not necessarily improve with the increase of the number of particles. From the results listed in Table 10, setting the number of particles to 30 is recommended in this work.  
Figure 13: Convergence plots of the PSO simulations for the second approach using different numbers of particles. (**a**) 20 particles, (**b**) 30 particles, and (**c**) 50 particles.  
During the process of optimization iteration, the mode-tracking results of a particle using the ISR index in the first and second approaches are presented in Figure 14. It can be observed that the ISR index could effectively identify the modes, and the order of the modes to be identified was within the previously-determined mode switching range. Moreover, the probability of mode switching in the second approach was higher than in the first approach, which leads to the conclusion that the constraint state is an important factor affecting the order of modes.