## 2 Methodology
### Coupling of Fluid and Structure
The kinematics spatial coordinates and respective time derivatives are transformed into modal coordinates using  
\[\{x(t)\}_{n\times 1}=[\Phi]_{n\times n}\{\overline{x}(t)\}_{n\times 1} \tag{6}\]and pre-mutiplying the resulting equation with \([\Phi]_{n\times n}^{T}\), leading to  
\[\left[\overline{M}\right]_{n\times n}\{\overline{x}(t)\}_{n\times 1}+\left[ \overline{B}\right]_{n\times n}\{\overline{x}(t)\}_{n\times 1}+\left[ \overline{K}\right]_{n\times n}\{\overline{x}(t)\}_{n\times 1}=\left\{\overline{F}(t) \right\}_{n\times 1} \tag{7}\]  
where \(\left[\overline{M}\right]_{n\times n^{\prime}}\left[\overline{B}\right]_{n \times n}\) and \(\left[\overline{K}\right]_{n\times n}\) are the modal mass, stiffness and damping matrices (all diagonal), respectively. It will be assumed that the following condition holds:  
\[\left[\overline{B}\right]_{n\times n}\left[\overline{M}\right]_{n\times n}^{-1 }\left[\overline{K}\right]_{n\times n}=\left[\overline{K}\right]_{n\times n} \left[\overline{M}\right]_{n\times n}^{-1}\left[\overline{B}\right]_{n\times n} \tag{8}\]  
ensuring the fact that the damped system will possess the same mode shapes as its undamped counterpart. Consequently, the eigenvectors will be real and the equation will contain \(n\) uncoupled systems of equations, representing an SDoF-analogy for each modal component and leading to a scalar system for each, individually. Modal reduction. Systematically decreasing the involved modal components, one chooses \(m\leq n\) modes to represent a subset \(\left[\Phi\right]_{n\times m}=\left[\{\Phi_{\dagger}\}_{n\times 1},\{\Phi_{ \dagger}\}_{n\times 1},\ldots,\{\Phi_{n}\}_{n\times 1}\right]_{m}^{T}\) for \(i=1,m\) of the full modal matrix. Carrying out the transformation considering only this selection of modes, the following equation holds:  
\[\left[\overline{M}\right]_{m\times m}\left\{\overline{\overline{x}}(t)\right\} _{m\times 1}+\left[\overline{B}\right]_{m\times m}\{\overline{\overline{x}}(t) \}_{m\times 1}+\left[\overline{K}\right]_{m\times m}\{\overline{x}(t)\}_{m\times 1}= \left\{\overline{F}(t)\right\}_{m\times 1} \tag{9}\]  
representing the EOM in reduced modal space. Consequently, the response in the initial coordinate system is made up only by the contribution (mathematically a linear superposition) of a reduced subset of (decoupled) modes. \[[M]_{n\times n}\{\ddot{x}(t)\}_{n\times 1}+[B]_{n\times n}\{\dot{x}(t)\}_{n \times 1}+[K]_{n\times n}\{x(t)\}_{n\times 1}=\{F(t)\}_{n\times 1} \tag{3}\]0  
It should be noted that transferring from continuous space to finite space is implicitly already a model-order reduction, where the number of modes is reduced from \(\infty\) to \(n\), further reduced here to \(m\). For large systems, determining all mode shapes and frequencies is practically prohibitive. Analytical solutions are not viable for such cases. Thus, numerical methods (such as the Rayleigh-Ritz) exist, and they approximate a certain subset of eigenforms and values, mostly starting at the lowest frequency (in civil/structural engineering applications). Similarly, experimental mode shape analysis only enables the determination of a limited number of lower-frequency modes. The result is, from either procedure, that one only has a limited subset of eigenforms corresponding to the lower eigenfrequencies. The current study uses a parametric FEM formulation of the Timoshenko beam, such that all eigenforms and values are assessed for a (relatively) highly detailed spatial model. The spatial and modal resolution is systematically reduced such that the effect of a lower number of FE elements or the mode truncation can be properly measured.