# Frequency-Domain Lifting-Line Aerodynamic Modelling for Wing Aeroelasticity
## 2 Frequency-Domain LLT-like Modelling
### Relation between Bound Circulation and Circulatory Lift  
As stated in Equation (1), the definition of wake vorticity requires the knowledge of the bound circulation spanwise distribution that, in lifting-line theories, has to be related to the spanwise distribution of the circulatory lift. Here, this is accomplished considering both the widely used Kutta-Joukowski theorem for steady flows and through its extension to unsteady linear aerodynamics recently proposed by some of the authors [25], whose mathematical formulation moves from the work of [PERSON] for the solution of the velocity potential for circulatory flows around thin rectilinear airfoils in harmonic motion [3]. Specifically, the extended Kutta-Joukowski theorem is expressed in terms of a frequency-response function between bound circulation and circulatory lift, \(G(k)\), which is a transcendental function of the reduced frequency through a combination of Bessel functions [25], and reads as  
\[\bar{L}_{c}(k)=\rho VG(k)\bar{\Gamma}(k) \tag{12}\]  
Therefore, in this work, two models for the definition of the circulatory lift bound circulation relation are considered:  
* The Kutta-Joukowski theorem for steady flows with \[G(k)=1\] (13)
* The extension of the Kutta-Joukowski theorem to unsteady flows with \[G(k)=ik\,e^{ik}\,K_{1}(ik)\] (14) where \(K_{1}\) denotes the first-order modified Bessel function of the second kind. From Equation (12), it is possible to relate the bound circulation at each wing section to its sectional circulatory lift and obtain the following compact matrix form  
\[\tilde{\mathbf{\Gamma}}=\mathbf{H}(k)\,\tilde{\mathbf{L}}_{c} \tag{15}\]  
where \(\mathbf{H}\) is a diagonal transcendental \([N_{B}\times N_{B}]\) matrix, whose elements depend on the Kutta-Joukowski model and, for a rectangular wing, is given by  
\[\mathbf{H}=G^{-1}(k)/\rho V\mathbf{I} \tag{16}\]  
Equation (6), combined with Equations (1) and (15), yields the following matrix expression relating the distribution of the wing sectional degrees of freedom to the circulatory lift distribution  
\[\tilde{\mathbf{L}}_{c}=[\mathbf{I}-\mathbf{C}(k)\,\mathbf{D}(k)\,\mathbf{H}( k)]^{-1}\mathbf{C}(k)\,\mathbf{W}_{n}^{1}(k)\,\tilde{\mathbf{g}} \tag{17}\]  
Furthermore, combining Equation (17) with Equations (9) and (10), provides the following \([2N_{B}\times 1]\) vector, collecting all the sectional aerodynamic loads along the wing \[\tilde{\mathbf{f}}_{s}= \begin{cases}\tilde{\mathbf{L}}\\ \tilde{\mathbf{M}}\end{cases} =\mathbf{E}_{g}(k)\,\tilde{\mathbf{g}} \tag{18}\]  
where the \([2N_{B}\times 2N_{B}]\) matrix, relating the sectional aerodynamic loads to the sectional plunge and pitch motions, is defined as  
\[\mathbf{E}_{g}=\begin{bmatrix}[\mathbf{I}-\mathbf{C}(k)\,\mathbf{D}(k)\, \mathbf{H}(k)]^{-1}\mathbf{C}(k)\,\mathbf{W}_{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf  
such that  
\[\tilde{\mathbf{f}}=\mathbf{E}(k)\,\tilde{\mathbf{q}} \tag{26}\]  
The matrix \(\mathbf{E}\) is a frequency-domain transcendental aerodynamic operator. The presence of the transcendental contributions is due to three sources: \((i)\) the Biot-Savart law used to evaluate the normalwash generated by the wake vorticity (see Equation (1)), \((ii)\) the Theodorsen theory used to evaluate the sectional aerodynamic loads (see Equation (8)) and \((iii)\) the application of the unsteady Kutta-Joukowski condition (see Equation (15)). This implies that, in this form, the aerodynamic matrix is unsuitable for aeroelastic stability analysis and control applications.