## 2 Flexible Aircraft with Sloshing Integrated Modeling
The linearised relation between \(\Delta\eta\) in Equation (6) and \(\Delta\
u\) is expressed as  
\[\frac{\mathrm{d}\mathbf{h}_{\mathrm{ G}}}{\mathrm{d}t} = \mathbf{m}_{\mathrm{ G}} \tag{1}\]1  
with  
\[{\rm T}_{1}^{\rm*}=\left[ \begin{array}{ccc}0_{3\times 3}&-\hat{\rm V}_{ \rm G_{e}}&0_{3\times N_{m}}\\ 0_{3\times 3}&0_{3\times 3}&0_{3\times N_{m}}\\ 0_{N_{m}\times 3}&0_{N_{m}\times 3}&0_{N_{m}\times N_{m}}\end{array} \right]\ \,\ \ \hat{\rm V}_{\rm G_{e}}=\left[ \begin{array}{ccc}0&0&0\\ 0&0&-U_{\infty}\\ 0&U_{\infty}&0\end{array} \right] \tag{8}\]  
where \(0_{\rm*\times\bullet}\) are zero matrices with suitable number of rows and columns. Equation (8) allows one to highlight the link between the variables expressed in the PMAs and those defined in the inertial reference system (being \({\rm T}_{1}^{\rm*}\) a square matrix with dimension \(N=6+N_{m}\)). Taking into account the external action of aerodynamics and sloshing dynamics, the system of Equation (2) can be rewritten as follows:  
\[{\rm M}\,\Delta\dot{\
u}+{\rm D}\,\Delta\
u+{\rm K}\,\Delta\eta\ =\,{\rm e}+{ \rm g}+{\rm f}^{\rm(ext)} \tag{9}\]  
where \({\rm M}\), \({\rm D}\) and \({\rm K}\) are, respectively, the modal mass, damping and stiffness (diagonal) matrices of the flexible aircraft, whereas \({\rm e}=\left[e_{1},e_{2},\ldots,e_{N}\right]^{\rm T}\) and \({\rm g}=\left[g_{1},g_{2},\ldots,g_{N}\right]^{\rm T}\) are, respectively, the generalized _aerodynamic_ and _sloshing_ forces induced by the aircraft motion. These forces are generally defined as a projection of physical applied force fields (aerodynamics and sloshing) on the assumed rigid and flexible mode shapes. In the present paper, the wing body is assumed to be locally rigid in the neighborhood of the tank (see Refs. [16; 32]) and therefore, the force-field projection becomes a scalar product (see later Equation (20)). The \({\rm f}^{\rm(ext)}\) is the vector of the current external forcing terms such as the gust excitation. A description of the aircraft motion in the PMA non-inertial reference requires accounting for the projection of the weight force on the aircraft body reference. Under the assumption of small perturbations with respect to the trimmed configuration, such a contribution was modelled as an additional stiffness term included inside \({\rm K}\). The generalised aerodynamic force vector is generally computed as a function of the reduced frequency \(k=\omega b/U_{\infty}\) (with \(b\) semi-chord and \(U_{\infty}\) free stream velocity) and Mach number \(M_{\infty}\) domain (see Ref. [28]) as:  
\[\bar{\rm e}=q_{D}\,{\rm Q}(k,M_{\infty})\,\Delta\bar{\eta} \tag{10}\]  
where \({\rm Q}(k,M_{\infty})\) is the generalised aerodynamic forces matrix, \(q_{D}\) is the dynamic pressure and the symbol - is used to represent the Laplace/Fourier transforms. For a fixed value of \(M_{\infty}\), the following rational function approximation for the unsteady aerodynamics can be considered (Ref. [29])  
\[\mathsf{Q}(k)\approx\mathsf{A}_{0}+jk\mathsf{A}_{1}-k^{2}\mathsf{A}_{2}+jk \mathsf{C}\big{(}jk\mathsf{I}+\mathsf{P}\big{)}^{-1}\mathsf{B} \tag{11}\]  
Introducing the aerodynamic states vector \(\mathsf{a}\) and exploiting the definition of inverse Fourier transform, the aerodynamic forces can be recast in time domain as:  
\[\mathsf{e} = q_{D}\,\mathsf{A}_{0}\,\Delta\eta+q_{D}\,\frac{b}{U_{\infty}}\, \mathsf{A}_{1}\,\Delta\upsilon+q_{D}(\frac{b}{U_{\infty}})^{2}\,\mathsf{A}_{2 }\,\Delta\upsilon+q_{D}\,\mathsf{C}\,\mathsf{a} \tag{12}\] \[\dot{\mathsf{a}} = \frac{U_{\infty}}{b}\,\mathsf{P}\,\mathsf{a}+\mathsf{B}\,\Delta\upsilon \tag{13}\]  
In this work, the effects of two different sloshing dynamics are considered. The first is present when small lateral displacements and rotations are imposed on the tanks. In these cases, the dynamics can be considered linear (Ref. [32]) and give rise to sloshing generalised loads referred to as \(\mathsf{g}_{l}\). The second is nonlinear, and its action \(\mathsf{g}_{nl}\) is triggered by large vertical perturbations imposed on the tanks.