# Frequency-Domain Lifting-Line Aerodynamic Modelling for Wing Aeroelasticity
## 2 Frequency-Domain LLT-like Modelling
### Aerodynamic Loads  
Dividing the wing into a prescribed number, \(N_{B}\), of finite sections, discretising the wake into \(N_{W}\) streamwise elements, adopting a frequency-domain representation for the wake normalwash induced by the wake vortices and recalling that the wake vorticity is convected downstream following the material points, it is possible to derive an expression of the following type relating the bound circulation distribution to the wake normalwash at the \(N_{B}\) sectional control points  
\[\hat{\mathbf{w}}^{i}=\mathbf{D}(k)\,\mathbf{\tilde{\Gamma}} \tag{1}\]  
where \(\hat{\mathbf{w}}^{i}\) denotes the vector collecting the wake-induced normalwash at the \(N_{B}\) control points and \(\mathbf{\Gamma}\) collects the \(N_{B}\) values of the discretised spanwise bound circulation distribution. The \([N_{B}\times N_{B}]\) matrix \(\mathbf{D}\), which is transcendental in the reduced frequency \(k=\omega b/V\) (with \(\omega\), \(V\) and \(b\) denoting, respectively, the angular frequency, the undisturbed air velocity and the semichord length), is obtained by application of the Biot-Savart law, with elements defined as  
\[D_{lm}(k)=\sum_{n=1}^{N_{W}}e^{-ik\tau_{n}}\tau_{ln}S_{nm} \tag{2}\]where \(\tau_{n}\) denotes the \"age\" of the \(n-\)th wake vortex (time delay), \(r_{ln}\) is the influence coefficient between the \(n-\)th wake vortex and the \(l-\)th collocation point and \(S_{nm}\) are the components of a topological matrix selecting the bound circulation associated to the considered wake strip. Next, let us consider the Theodorsen formulation for the aerodynamic loads of an airfoil [3]. For the lift force, it yields  
\[\tilde{L}=\tilde{L}_{c}+\tilde{L}_{nc}=2\pi\rho bVC(k)\tilde{\sigma}_{n}^{3/4}+ ik\pi\rho b^{2}\tilde{\sigma}_{n}^{1/2} \tag{3}\]  
where \(\rho\) is the air density, \(\tilde{L}_{c}\) and \(\tilde{L}_{nc}\) are the circulatory and noncirculatory parts of the lift, respectively, \(C(k)\) is the lift-deficiency function and \(\tilde{\sigma}_{n}^{1/2}\) and \(\tilde{\sigma}_{n}^{3/4}\) are the normalwash at the mid-chord and \(3/4\)-chord points, respectively. A similar expression is provided for the pitching moment in terms of the section kinematics [3]  
\[\tilde{M}=-ik\pi\rho b^{2}\tilde{\sigma}_{m}+b(1/2+a)\tilde{L}_{c} \tag{4}\]  
with \(a\) representing the dimensionless distance, with respect to the semichord \(b\), of the wing elastic axis from the mid-chord point, (e.g., if the airfoil shear centre is at the quarter-chord \(a=-1/2\), the pitching moment does not depend on the circulatory lift). See Appendix A for details about \(\tilde{\sigma}_{m}\). Expressing the section kinematics in terms of the wing elastic displacements, the circulatory lift at a given \(j\)-th wing section (and corresponding contribution to the pitching moment) can be expressed in the following way to take into account the three-dimensional wake vorticity effects  
\[\tilde{L}_{c_{j}}=2\pi\rho bV\,C(k)\,\left[\tilde{w}_{j}^{i}+\tilde{\sigma}_{n _{j}}^{3/4}(\tilde{w}_{j},\tilde{\sigma}_{j})\right] \tag{5}\]  
where \(\tilde{w}_{j}\) and \(\tilde{\sigma}_{j}\) denote, respectively, plunge displacement at the collocation point of the \(j-\)th section and pitch rotation of the same section while \(\tilde{w}_{j}^{i}\) is the wake-induced velocity at the same point.