# Dynamic Aeroelastic Response of Stall-Controlled Wind Turbine Rotors in Turbulent Wind Conditions
## 2 The Numerical Model  
In this section, we shall provide a brief description of a novel member of the BEM model family used in our research, which is called the Dynamic Rotor Deformation-Blade Element Momentum model (DRD-BEM) and was introduced in [PERSON] et al. [14]. This model accounts for the aerodynamic effects of the misalignments at every blade section. This is achieved by transforming the velocity and force vectors across different coordinate systems--from the coordinate system of the free-stream wind to the coordinate system of the blade section. These coordinate transformations are performed with a set of orthogonal matrices that account for all forms of misalignments at each instance. These include misalignments brought about by blade deformation, mechanical inputs such as pitching and yawing, or even pre-conformed misalignments, such as pre-bending of the original blade. The DRD-BEM is coupled with an advanced technique used to simulate the structural response of the blade that is based on the Generalized Timoshenko Beam Model (GTBM). The GTBM, a dimensional reduction method, has the same parameters as the traditional Timoshenko model and can work for complex beams that may have twisted or curved shapes. In this model, originally introduced by Prof. [PERSON] [15,16], the beam section does not remain planar after deformation, and a 2-D finite element mesh is used to interpolate the warping of the deformed section. A mathematical formulation is then used to write the 3-D strain energy in terms of the classical variables of the 1-D Timoshenko model. The complexities of the 3-D model and geometry of the blades are reduced into a stiffness matrix for the corresponding 1-D beam. This 1-D beam can then be solved along a reference line representing the beam's axis on the original configuration, and it can curve in any direction after deformation. A new coordinate system is introduced to represent the dynamic and kinematic variables along the reference line. This system follows the blade deformation into the instantaneous configuration. The blade sections in the chord-normal, chord-wise, and span-wise directions stay aligned with the intrinsic system. Therefore, accurate position tracking can be achieved by using this technique, even in cases of large displacements or rotations of the blade section. The GTBM technique uses a fully populated 6 \(\times\) 6 stiffness matrix instead of only six stiffness coefficients, as in the classical Timoshenko theory, which gives it the capacity to represent the effects of coupled modes of deformation, such as bend-twist. More details about our implementation of the GTBM and its coupling with DRD-BEM can be found in [PERSON] [17] and [PERSON] et al. [14]. The inertia properties are also dimensionally reduced to produce a 6 \(\times\) 6 inertia matrix for the equivalent beam along the reference line \(L\). This inertia matrix contains the moments of inertia of both first and second order for blade sections along the span. These are acquired from a 2-D integration that operates over the area of each blade section and also considers the distribution of material properties and the details of the blade section's shape. These matrices operate on the linear and angular velocities, which give the linear and angular momentum of the vibrational motion, as well as the inertia forces and moments associated with them. We can also calculate the inertia forces of the rotation of the main shaft and of mechanisms such as pitch and yaw. Therefore, all parameters, such as the angular, linear, centrifugal, and acceleration effects, are taken into account in a full 3-D representation. The aforementioned interaction of structural and aerodynamic modules can maximize the capabilities of this advanced model. The equations of motion of the equivalent beam modeled as a 1-D finite element problem are solved by means of a non-linear adaptive ODE algorithm.