## 3 Dynamic Force-Thermal Follow-Up Loading Method for Multi-Stage Telescopic Wings
### Dynamic Multi-Point Distribution Equivalence Method for Aerodynamic Loads  
Existing research on load equivalence methods for loading experiments primarily relies on empirical methods, such as equal-area averaging. These methods divide the wing surface into multiple equal-area regions and distribute the aerodynamic loads across these regions, approximating them as concentrated loads to satisfy static equivalence conditions. In contrast, this study adopts the commonly used multi-point distribution method for mapping aerodynamic loads to structural grid points. This method is extended to calculate load equivalence at loading points by incorporating dynamic grid reconstruction, thus proposing the dynamic multi-point distribution equivalence method, which achieves full-process equivalence of time-varying loads. The basic concept of the multi-point distribution method is as follows: in finite element analysis, after obtaining the aerodynamic loads acting on the wing structure, these loads are mapped from the aerodynamic grid to the structural grid. This is necessary because the grids used for aerodynamic analysis and those for static structural analysis do not perfectly correspond. When the number of structural grid points is sufficiently small, they can be regarded as predefined equivalent loading points. As shown in Figure 16, given the position coordinates and corresponding loads of i \(\times\) k aerodynamic grids on a certain wing surface, the goal is to equivalently transfer all aerodynamic grid loads to predefined loading points(\(\mathrm{(x_{k},\,y_{k}),\,k=1,\,2,\,3,\,\ldots,\,n}\). Taking a particular aerodynamic grid as an example, consider the aerodynamic load po acting on the grid at point \(\mathrm{o(x_{u},\,y_{q})}\). It is assumed that all the predefined loading points and the aerodynamic grid o are connected by a virtual beam (of length \(\mathrm{l_{k}}\)), and they both act as cantilever beams with the fixed end at point o. The deformation energy when the load pk is applied to the loading point is as follows:  
\[\mathrm{U_{k}}=\frac{\mathrm{P_{k}^{2}i_{k}^{3}}}{6\mathrm{EJ}} \tag{3}\]  
where EJ represents the bending stiffness of the virtual beam element. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node** & **Pressure (mm)** & **12 Points (mm)** & **24 Points (mm)** & **36 Points (mm)** & **48 Points (mm)** \\ \hline
1 & 1.0931 & 0.9964 & 1.0306 & 1.0350 & 1.0671 \\
2 & 2.2776 & 1.9997 & 2.1437 & 2.2006 & 2.1201 \\
3 & 2.5636 & 2.2490 & 2.4356 & 2.4332 & 2.3832 \\
4 & 2.4112 & 2.0646 & 2.2909 & 2.2030 & 2.1830 \\
5 & 1.3784 & 1.1628 & 1.1219 & 1.1804 & 1.1999 \\
6 & 0.3978 & 0.3215 & 0.2930 & 0.3154 & 0.2994 \\
7 & 0.5435 & 0.5430 & 0.5619 & 0.5566 & 0.5292 \\
8 & 0.5347 & 0.6311 & 0.6319 & 0.5454 & 0.5450 \\
9 & 0.5574 & 0.5517 & 0.5583 & 0.5237 & 0.5298 \\
10 & 0.5516 & 0.5176 & 0.5224 & 0.5314 & 0.5407 \\ Error & 0 & 10.78\% & 9.35\% & 7.49\% & 7.16\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitting errors under different numbers of concentrated loads. The loads distributed to each structural point should satisfy the following static equilibrium condition:  
\[\left\{ \begin{array}{l}\sum\limits_{k=1}^{n}P_{k}=P_{o}\\ \sum\limits_{k=1}^{n}P_{k}x_{k}=P_{o}x_{o}\\ \sum\limits_{k=1}^{n}P_{k}y_{k}=P_{o}y_{o}\end{array} \right. \tag{4}\]  
where k represents the kth loading point, with its value ranging from 1 to n, while n denotes the total number of loading points.