## 3 Dynamic Force-Thermal Follow-Up Loading Method for Multi-Stage Telescopic Wings  
### Design of the Force-Thermal Follow-Up Loading Platform  
To conduct force-thermal follow-up loading experiments under assessment conditions for deformable wings and enhance the experimental efficiency, this study proposes a design for a force-thermal follow-up loading platform for multi-stage telescopic wings of aircraft. As illustrated in Figure 11, the system comprises a multi-stage telescopic wing prototype, thermal load application units, force load application units, a supporting platform, and an associated control system. The telescopic wing prototype is mounted on the supporting platform, while the modular thermal and force load units are installed at predesignated positions on the wing surface, aligned with the prototype's movement direction. A dynamic multi-point distribution equivalence method based on grid reconstruction is proposed to simulate aerodynamic loads. This method equivalently transforms the varying aerodynamic loads during motion into time-varying loads at predefined loading points, enabling follow-up loading throughout the entire process. Both the thermal and force load units are equipped with sensors, such as load cells and thermocouples, to collect real-time loading state data. These sensors connect to the control system, which provides feedback regulation to ensure that the experimental loads accurately follow the predefined loading conditions. A load unit design analysis is conducted for the multi-stage telescopic wing. Given that the force distribution along the chordwise direction of the wing is nearly uniform, the telescopic wing can be equivalently modeled as a cantilever beam along the spanwise direction, as illustrated in Figure 12. In this equivalent model, the wing root is assumed to be fixed, while the other end remains free. Corresponding cantilever beam models are established to analyze the structural response under both aerodynamic and concentrated loads. When the equivalent model is subjected to a uniformly distributed pressure load \(\upmu\), the deflection at the free end (point B), denoted as \(\mathrm{w}_{\mathrm{BI}}\), is given by the following Equation (1):  
\[\omega_{\mathrm{BI}}=\frac{\upmu^{4}}{8\mathrm{EI}} \tag{1}\]  
where l represents the wingspan, E is the elastic modulus of the material, and I is the moment of inertia of the cross-section. Figure 11: Telescopic wing force–thermal follow-up loading platform. When the equivalent model is subjected to a concentrated load F, the deflection at the free end (point B), denoted as \(\mathrm{w_{B2}}\), is given by the following Equation (2):  
\[\omega_{\mathrm{B2}}=\frac{\mathrm{Fa^{2}}}{6\mathrm{EI}}(3\mathrm{I}-\mathrm{a}) \tag{2}\]  
where a represents the distance from the fixed end to the point of load application. As the number of concentrated loads increases, the accuracy of simulating the uniformly distributed load improves progressively. As shown in Figure 13, with an increasing number of concentrated loads, the distribution characteristics of shear force and bending moment in the equivalent model more accurately approximate those under a uniformly distributed load. This indicates that increasing the number of concentrated loads within a reasonable range enhances the accuracy of equivalent modeling, thereby providing a more precise representation of the effects of a uniformly distributed load on the structure. Figure 14 illustrates the variation in deflection error at the free end as the number of equivalent concentrated loads increases. The results indicate that when the number of spanwise concentrated load points reaches four or more, the error reduction rate falls below 2%. Additionally, within the range of [4,8] concentrated load points, the equivalent loading error remains below 10%. When the number of spanwise concentrated loads reaches four or more, further increasing the number of concentrated loads reduces the deflection fitting error by less than 2%.