## 3 Dynamic Force-Thermal Follow-Up Loading Method for Multi-Stage Telescopic Wings
### Method Comparison and Validation
#### 3.3.3 Continuous Process Analysis and Validation  
In practical load equivalence applications, the telescoping wing undergoes a continuous deformation process, requiring the equivalent aerodynamic loads to be applied throughout the entire process. During the equivalence, the entire process is discretized into a finite number of time instances, and the equivalent time-varying load curve for the whole process is fitted. To compare the dynamic process, the method proposed in this paper is contrasted with the traditional empirical method. Several typical working conditions during the entire process are selected for dynamic process comparison, as shown in Table 6. The selected working conditions include moments when the number of loading points changes and intermediate instances. For different operating conditions, comparative simulation models are established, with the same constraints as described in the previous section. The load data at different time instances are input into the simulation model, and the results are obtained. The wingtip deformation trajectories are extracted from the three sets of results to analyze the fitting effects of different equivalence methods under aerodynamic load conditions. Figure 26 shows the extracted wingtip deformation trajectories under the three loading conditions at different time instances. The RMSE and R\({}^{2}\) values are calculated using the wingtip deflection simulation trajectory data for each operating condition. The fitting error results for the dynamic multi-point distribution method and the area-based distribution method at each time instance are shown in Table 7. \begin{table}
\begin{tabular}{c c} \hline
**Time** & **Number of Loading Points** \\ \hline
0.1 s & 24 \\
0.54 s & 24 \\
1.11 s & 36 \\
1.77 s & 36 \\
2.46 s & 48 \\
3 s & 48 \\ \hline \end{tabular}
\end{table} 
Table 6: Typical operating conditions. \begin{table}
\begin{tabular}{c c c c c} \hline
**Time** & \multicolumn{2}{c}{**Dynamic Multi-Point Allocation Method**} & \multicolumn{2}{c}{**Equal-Area Distribution Method**} \\  & **RMSE** & **R\({}^{2}\)** & **RMSE** & **R\({}^{2}\)** \\ \hline
0.1 s & 0.2006 & 0.9720 & 0.2797 & 0.6252 \\
0.54 s & 0.7130 & 0.9808 & 0.7784 & 0.7907 \\
1.11 s & 0.7651 & 0.9903 & 0.7718 & 0.8894 \\
1.77 s & 0.7936 & 0.9983 & 2.5572 & 0.9689 \\
2.46 s & 2.6418 & 0.9988 & 3.7238 & 0.9812 \\
3 s & 2.2724 & 0.9992 & 3.6413 & 0.9909 \\ \hline \end{tabular}
\end{table} 
Table 7: Fitting errors at different moments. \begin{table}
\begin{tabular}{c c c c} \hline
**Time** & \multicolumn{2}{c}{**Dynamic Multi-Point Allocation Method**} & \multicolumn{2}{c}{**Equal-Area Distribution Method**} \\  & **RMSE** & **R\({}^{2}\)** & **RMSE** & **R\({}^{2}\)** \\ \hline
0 s & 0.1148 & 0.8491 & 0.1416 & 0.0583 \\ \hline \end{tabular}
\end{table} 
Table 5: Fitting error at the initial moment. Figure 27 visualizes the fitting error results at different time instances. From the figure, it can be observed that the dynamic multi-point distribution method proposed in this study outperforms the area-based distribution method in both RMSE and R\({}^{2}\) metrics at all time instances. In terms of RMSE, the dynamic multi-point method slightly outperforms the traditional method in the early stage and continues to maintain a lower error as the wing extends. Meanwhile, the R\({}^{2}\) value of the dynamic multi-point distribution method starts at 0.84 and rapidly approaches 1, maintaining a value close to 1 throughout the wing extension. This indicates that, in the entire equivalence process, the dynamic multi-point  
Figure 26: Wingtip deformation trajectories for various conditions.