## 2 Methodology
### Numerical Methodology
#### 2.2.1 Structural Model  
The structural model consists of a linear modal model that uses 12 normal modes obtained from a Nastran [16] finite element model (FEM). MSC Patran [17] was used for mesh generation and pre-processing of the FEM, and the model is shown in Figure 5, with the types and number of elements shown in Table 1. The underlying structure was modelled using beam elements, the tank using shell elements and beams for the side pillars (also see Figure 1), and the spring elements were used to model the flexibility of the spar-torque tube, root and tank connections. Multi-point constraints (MPC) of type RBE3 were used to distribute the lumped dry mass on the bottom of the tank and also, later, to distribute the effect of the liquid force. The Nastran FEM was calibrated for the structure without the tank (see [13]) and partially calibrated for the structure with the tank to achieve a very good match in the first two out-plane bending modes. This is sufficient for the present analysis, where the fluid has the most significant influence on the out-of-plane dynamics. Here, we describe how the Nastran normal modes solution is extracted and used in a way that provides a simple generalised framework to perform various transient fluid-structure coupled simulations. Figure 4: Example of residuals and summed squared residuals for the fitted functions shown in Figure 3: (**a**) residuals for the first bin; (**b**) summed squared residuals for all bins. Modal analysis of the FE model computes a matrix of normal mode shapes \(\psi\) and vector of the relevant structural frequencies \(\underline{\omega}\), which can be used to form the generalised mass and stiffness matrices. Assuming mass normalised mode shapes, the structural equations of motion take the form  
\[I\underline{\dot{q}}+C\underline{\dot{q}}+K\underline{q}=\underline{F}_{q}, \qquad K=\text{diag}\Big{(}\underline{\omega}^{2}\Big{)},\qquad C=\text{diag} \Big{(}2\underline{\zeta}\underline{\omega}\Big{)} \tag{4}\]  
where \(\underline{q}\) is the vector of modal displacements, \(I\) the identity matrix, \(K\) and \(C\) the generalised stiffness and modal damping matrix, respectively and \(\underline{F}_{d}\) the vector of applied forcing in the modal space. The damping ratios of individual modes of vibration are identified from ground vibration tests. Whilst this model is sufficient at capturing small magnitude vibrations (where the \(\underline{\zeta}\) values are identified), it is insufficient at representing the large amplitude vibration damping where the structure displays significant nonlinear characteristics. It is not straightforward to decide what type of nonlinear terms to include in this case. In recent years, there has been substantial analytical work conducted on \"generalised\" oscillators with noninteger order stiffness and damping. For example, in [18], the single degree of freedom model with nonlinear damping and nonlinear elastic forces of any order (integer or noninteger) was studied and then extended to multiple degrees of freedom in [19]. Possible applications are envisaged in the correct modelling of micro-electromechanical systems dynamics. Here, we use a similar approach where we consider the damping powers as optimisation parameters allowed to be nonintegers. The numerical model can thus capture the nonlinear experimental dry damping without much concern about the physical meaning of each term, which is not of direct interest in this current work.