## 2 Methodology
### The Structural Dynamics Model  
In order to account for the blades structural and dynamic response, hGAST [48; 49; 50; 51] is used. hGAST is a servo-elasto-dynamic solver based on a multi-body approach, where all flexible components are modeled as flexible \(1D\) full stiffness matrix [52] Timoshenko beams. Numerical discretization is applied through FEM approximation. In this way, apart from the blades, the flexible elements of the control chain of a helicopter main rotor (blade hub, pitch link, horn, etc.) or a wind turbine configuration (shaft, tower) may be included. Rigid body and elastic motion due to structural flexibility are concurrently accounted for within the multibody dynamics framework, alongside the non-linear inertial and structural effects and geometric couplings imposed by large deflections of highly flexible components (e.g., blades). These highly flexible components are approximated as an assembly of linear Timoshenko beam structures (sub-bodies). The interconnected linear sub-bodies are linked to each other through proper non-linear kinematic and dynamic compatibility equations. In this way, non-linear geometric and aeroelastic effects due to high deformation are accounted for using linear beam theory at the sub-body level, but considering non-linear effects at the connection points [51]. Following this approach, multiple load path joints can be introduced. Briefly, at the connection points between different sub-bodies, one of them provides the kinematic condition (position/velocity and orientation/angular velocity) to the others, while the rest have to contribute their reaction loads (see Figure 1). A local coordinate system \((O_{k},x_{k},y_{k},z_{k})\) is attached to the start point (P1) of every sub-body. Local elastic deformations resulting from the beam dynamic equations are defined with respect to this system. This local frame of every sub-body \(k\) is subjected to 3 translations \((\mathbf{q}_{t}^{k})\) and 3 rotations \((\mathbf{q}_{t}^{k})\), which include rigid body motion undergone by the sub-body and elastic motions induced by other sub-bodies connected to them. For the \(k\)th body, let \(\mathbf{q}^{k}=\{\mathbf{q}_{t}^{k}\mathbf{q}_{t}^{k}\}^{T}\) denote a set of (rigid body) displacements and rotations (time variant or invariant) that define the origin and orientation of the local system of the \(k\)th body in the undeflected state. Rigid body translations \((\mathbf{q}_{t}^{k})\) will place the component at \(\mathbf{R}^{k}\), while rigid body rotations \((\mathbf{q}_{t}^{k})\) lead to its rotation matrix \(\mathbf{T}^{k}\). Based on the above,the (global) position vector \(\mathbf{r}_{G}^{k}\) of any arbitrary point \(P\) over the deflected \(k\)th body with respect to the global inertial frame is expressed through the (local) position vector of \(P\) in the undeflected state \(\mathbf{r}_{l}^{k}\ =\ (x_{l}^{k},y_{l}^{k},z_{l}^{k})\) and the vector of the local elastic deflections (displacements and rotations) \(\mathbf{u}^{k}\ =\ (u^{k},v^{k},w^{k},\theta_{x}^{k},\theta_{y}^{k},\theta_{z}^{k})\) (both defined in sub-body local co-ordinates) and the rigid body motion \(\mathbf{q}^{k}\) as follows:  
\[\mathbf{r}_{G}^{k}=\mathbf{R}^{k}\Big{(}\mathbf{q}^{k};t\Big{)}+\mathbf{T}^{k }\Big{(}\mathbf{q}_{r}^{k};t\Big{)}\Big{(}\mathbf{r}_{l}^{k}+\mathbf{S}\mathbf{ u}^{k}(t)\Big{)}\]  
where \(\mathbf{S}\) matrix is given by:  
\[\mathbf{S}= \begin{bmatrix}1&0&0&0&z_{l}^{k}&0\\ 0&1&0&-z_{l}^{k}&0&x_{l}^{k}\\ 0&0&1&0&-x_{l}^{k}&0\end{bmatrix} .\]  
Kinematic constraints are applied to the end nodes of connected components. For example, in Figure 2, the \(P2\) node of component \(r\) is rigidly connected to the \(P1\) node of component \(m\).