# Frequency-Domain Lifting-Line Aerodynamic Modelling for Wing Aeroelasticity
## 4 Conclusions  
A novel frequency-domain LLT-like approach for wing unsteady aerodynamics has been presented. It includes two different versions of the Kutta-Joukowski theorem for the definition of the relationship between the bound circulation and circulatory lift and two different approaches for the evaluation of the sectional loads (namely the Theodorsen theory and the Glauert theory). The magnitude and phase of the aerodynamic transfer functions provided by the proposed solvers have been calculated for two sets of shape functions describing the elastic displacements and three wings with different aspect ratios. The results from the LLT-like tools have been compared to those from a three-dimensional BEM solver for potential flows, extensively validated in the past. The predictions provided by the U-T and S-T solvers (based on the sectional Theodorsen theory), for which only trailed vortices affect the normalwash distribution along the wingspan, are very close and in very good agreement with the three-dimensional BEM solution for a slender wing. Their  
Figure 14: Comparison between computed and approximated transfer function \(E_{12}(k)\).  
Figure 13: Comparison between computed and approximated transfer function \(E_{11}(k)\).  
accuracy decreases as the aspect ratio of the wing decreases, and significant discrepancies with respect to the BEM simulation appear for a very small aspect ratio. The solvers based on the [PERSON] theory and wake normalwash given by both the shed and trailed vortices (namely the U-G and S-G solvers) provide similar, but different predictions for any wing aspect ratio, particularly in terms of the transfer function magnitude. Their accuracy is satisfactory for high values of the aspect ratio and, akin to the U-T and S-T tools, decreases significantly for a very low aspect ratio. For these solvers, the presence of the shed vorticity in the evaluation of the wake effects on the normalwash produces evident differences between the results based on the steady and the unsteady versions of the Kutta-Joukowski theorem. In addition, it is interesting to observe that the presence of the shed vortices to be considered for the evaluation of the wake-induced velocity makes the U-G and S-G solvers computationally more expensive than the U-T and S-T ones. Nonetheless, all the LLT-like computational tools examined are significantly faster than the higher-fidelity BEM solver applied for comparison, and hence they are particularly suitable for the aerodynamic matrix frequency sampling aimed at aeroelastic investigations for preliminary design and aeroservoelastic purposes. Finally, for the slender wing solved by the U-T approach, the aerodynamic matrix has been successfully approximated through a rational matrix form, from which the aerodynamic reduced-order model to be effectively coupled with the structural dynamics one can be directly derived.  
Conceptualisation: [PERSON], [PERSON], M.G.; methodology: [PERSON], [PERSON], M.G.; software: [PERSON], [PERSON]; validation, [PERSON]; writing--original draft preparation: [PERSON]; writing--review and editing: [PERSON], [PERSON], M.G. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The following abbreviations are used in this manuscript: