# Frequency-Domain Lifting-Line Aerodynamic Modelling for Wing Aeroelasticity
## 1 Introduction  
One of the main topics of aircraft design has always been the prediction of aerodynamic loads. The first attempts in this field were made at the beginning of the last century by [PERSON] [1], [PERSON] [2], [PERSON] [3], [PERSON] [4] and several other researchers, who developed theoretical methods to calculate analytically the lift, drag and pitching moment of airfoils and finite wings, both for steady and unsteady flows. Later, technological progress led to the definition of aerodynamic computational solvers based on numerical methods, which permit to study complex bodies in a wide range of flight conditions. Among these methods, the lifting-line theory (LLT) has proven to be a very effective tool for the preliminary design of aircraft aerodynamics that, although less accurate than CFD solvers, provides a reliable aerodynamic loads prediction at a significantly lower computational cost. Initially developed by [PERSON] for the study of steady flows [2], it consists of considering the wing represented by a single lumped vortex representing the bound circulation placed at the quarter-chord line, combined with a system of trailedvorticity that takes into account the spanwise variation in the wing circulation. These trailed vortices induce a distributed normalwash over the wing that alters the local angle of attack. The application of the widely known Kutta-Joukowski theorem for airfoils in steady flows relates the circulation to the local wing lift. Limited to a narrow range of wing geometries, the original [PERSON] model was later improved by several researchers. For instance, [PERSON] adapted the LLT to swept wings [5], and his method was simplified by [PERSON] [6]. [PERSON] and [PERSON] extended the LLT to generic curved wings [7], inspiring the further work of [PERSON] and [PERSON] on morphing wings of arbitrary curvature, chord distribution and airfoil cross-sections [8]. The model proposed by [PERSON] was later modified by [PERSON] to include nonlinear aerodynamics [9], and this type of application has become an object of interest for many research activities (see, for example, [10; 11; 12]. Next, the LLT was modified in order to include the analysis of unsteady flows, with the introduction of the wake shed vorticity effects. [PERSON] [13] developed one of the first unsteady lifting-line theories (ULLTs), based on the separation of the solution in an outer one, confined in the far-field wake and taking into account the three-dimensional effects (a Prandtl-like theory), and an inner one, due to the portion of the wake near to the wing, where the flow can be assumed to be two dimensional (a [PERSON] theory). This model was valid only for rectangular and elliptic wings, while [PERSON] et al. [14] extended the method presented by [PERSON] and [PERSON] [7] to curved wings in oscillatory motion. One of the most interesting works dealing with the application of a ULLT approach is that presented by [PERSON] [15], where an approximate extension of the Kutta-Joukowski theorem to unsteady flows was introduced. [PERSON]'s model provides an unsteady-flow version of the Kutta-Joukowski theorem derived from the integration of the Bernoulli theorem under the hypotheses of small, low-frequency flow perturbations (a similar extension to unsteady flows of the Kutta-Joukowski theorem is also presented in [16] for application to the Vortex Lattice Method, VLM). Born for the study of simple wings, the LLT was applied to several different lifting aerodynamics configurations. For example, [PERSON] and [PERSON] [17] developed a ULLT approach suitable for horizontal-axis wind turbines (HAWTs), including the dynamic stall model of Beddoes-Leishman, while [PERSON] and [PERSON] [18] derived a free-wake model for the study of offshore floating wind turbines. [PERSON] et al. [19], instead, presented an LLT approach for the ice formation on wings, whereas [PERSON] et al. [20] modified the Prandtl LLT using a finite element method approach to study nonplanar wings. The works of [PERSON] and [PERSON] on ULLTs inspired many authors.