## 2 Aero-Hydro-Servo-Elastic Model
### Tower Modelling  
The tower model used in this project is the same as previously used and described in [24] see this for more information. The towers and nacelles dynamics are described by a Finite Element Model (FEM) constructed in a MATLAB code. The FEM model is based on an assembly of 3D beam elements each with 6 DOF. The FEM of the structure of the tower consists of 20 nodes along the tower and 2 nodes for the nacelle. Each node has six DOFs, and between two nodes are an element. The governing equation for one element, can be described by a mass-spring-damper system  
\[\mathbf{m}_{tow}\ddot{\mathbf{z}}+\mathbf{\epsilon}_{tow}\dot{\mathbf{z}}+\mathbf{ k}_{tow}\mathbf{z}=\mathbf{f} \tag{12}\]  
where \(\ddot{\mathbf{z}}\in\mathbb{R}^{6},\dot{\mathbf{z}}\in\mathbb{R}^{6}\) and \(\mathbf{z}\in\mathbb{R}^{6}\) is the acceleration vector, velocity vector and position vector, respectively. \(\mathbf{m}_{tow}\in\mathbb{R}^{6\times 6}\) is the mass matrix of a single element, \(\mathbf{\epsilon}_{tow}\in\mathbb{R}^{6\times 6}\) is the damping matrix, \(\mathbf{k}_{tow}\in\mathbb{R}^{6\times 6}\) is the stiffness matrix, and \(\mathbf{f}\in\mathbb{R}^{6}\) is the vector describing the external force applied to each DOF. For the mass matrix of the elements, a combination of the lumped mass matrix and consistent mass matrix is chosen. The derivation can be found in several textbooks e.g., [31]. The combinational mass matrix is chosen as it tends to estimate the natural frequency better, than the use of a single one of the latter two mass matrix formulations [31]. The combinational mass matrix is written as  
\[\mathbf{m}_{tow}=\mathbf{m}_{Lump}\gamma+\mathbf{m}_{Con}(1-\gamma) \tag{13}\]  
where \(\mathbf{m}_{Lump}\) and \(\mathbf{m}_{Con}\) is the lumped mass matrix and consistent mass matrix, respectively. \(\gamma\) is a weighting parameter which takes a value between 0 and 1, in this paper a value of 0.5 is found suitable. The mass of the tower is formulated such that it is equally distributed along its longitudinal axis, whereas the mass of the nacelle and rotor is formulated to be placed in the center node of the nacelle. For the stiffness matrix of the elements a 3D beam formulation is utilized, for more information see [32]. The bending-transverse stiffness of the elements consist of two stiffness components; the bending stiffness and initial stress stiffness. The bending stiffness is described by Bernoulli-Euler beam theory, which is a good approximation for long slender structures, such as the tower of the AAUE-TLP. The initial stress stiffness arises due to the weight of the wind turbine and tower, this decreases the stiffness of the structure. The axial and twisting motion is assumed to small, and thereby their motion can assumed to be decoupled. Thereby the axial and twisting stiffness of the elements can be formulated as linear spring by [PERSON]'s law. The tower used in this paper have a symmetric cross section, this makes the decoupling of bending, axial and twisting motion a valid assumption [33]. For the damping matrix of the elements a classical Rayleigh damping formulation is used. Rayleigh damping is a viscous damping model, where the damping matrix is assumed to be a linear combination of the mass and stiffness matrices, which can be written as  
\[\mathbf{c}_{tow}=\alpha_{d}\mathbf{m}_{tow}+\beta_{d}\mathbf{k}_{tow} \tag{14}\]  
where \(\alpha_{d}\) and \(\beta_{d}\) are two weighting parameters. To determine the value of the weighting parameters in this paper, the damped natural frequency and the corresponding damping ratio are used. These are found by analyzing data from free decay tests of the tower mounted on the TLP, the mean damping ratio was found to be 0.025 and a damped natural frequency of 1.72 Hz. In this paper, the first natural frequency and damping ratio were found by free decay experiments.