## 3 Hybrid FEM/SEA Model and Interstage Response Analysis
### The Interstage Models  
The interstages IS01 (Figure 2) and IS12 are made of aluminium. The former is cylindrical and presents a nonuniform lateral surface thickness, 4 mm thicker in the area in which the openings are located, whereas the latter is constituted by two sections, having the form of a cone frustum connected through an internal interface. The third interstage, IS23, has different characteristics; it is realized by a composite material, and it is reinforced over the lateral surface by axial and circumferential stiffeners. The shape is still conical, split into two sections connected by an internal frame. Frames, reinforcing elements and skins are all constituted of a composite material. The simpler VEGA section among those analyzed, namely, interstage IS01, is taken as an example for illustrating the final part of the design verification described in this paper. Its main geometrical and dynamic characteristics are summarized in Table 1, along with the theoretical estimates of the ring and coincidence frequencies referred to the mean thickness. The various subsystems of the interstage model are shown in Figure 2: the SEA structural subsystem in green, the FEM structural subsystem in gold-brown, and the SEA acoustic subsystem in grey. In the same figure, the original FEM model used for deriving the hybrid model has been reported. \begin{table}
\begin{tabular}{l l} \hline \hline
**Property** & **Value** \\ \hline Length (m) & 1.18 \\ Diameter (m) & 3.4 \\ Ring freq. (Hz) & 500 \\ Coincidence freq. (Hz) & 1100 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical characteristics and theoretical estimates of dynamic frequencies for the IS01 interstage. Figure 2: From the top to the bottom: (**i**) finite-element method (FEM) model; (**ii**) hybrid model; and (**iii**) acoustic coupled model for the IS01 interstage. In particular, the hybrid FEM/SEA model has been built up by assembling six FEM structural subsystems (two large subsystems for the upper and lower rings, two large subsystems for the areas in which several equipment are located, and two small subsystems, each connected to a single equipment component), four large SEA structural subsystems (modeling the areas with lower thickness) and one SEA acoustic subsystem. A preliminary check of the model was performed by analyzing the modal density and radiation efficiency of the interstage: all SEA subsystems in the IS01 hybrid model presented a good agreement with the characteristic theoretical parameters reported in Table 1 and their asymptotic values, as shown in Figure 3, where the modal density and radiation efficiency are reported for each SEA subsystem of the hybrid model. For each SEA subsystem, the modal density displays how the ring frequency is close to the theoretical value. Asymptotic values also confirm theoretical modal density estimates, evaluated for the curved subsystems through the equivalent flat panel relation, valid above the ring frequency \(f_{r}\). This frequency, along with the structural modal density \(n_{s}(f)\), can be expressed as  
Figure 3: From the top to the bottom: (**i**) modal densities; and (**ii**) radiation efficiency for the statistical energy analysis (SEA) subsystems in the IS01 interstage hybrid model. \[f_{r} = \frac{1}{2\pi r}\sqrt{\frac{E}{\rho-(1-\
u^{2})}} \tag{9}\] \[n_{s}(f) = \frac{A}{2}\sqrt{\frac{12\rho_{s}(1-\
u^{2})}{Eh^{2}}}\,;A=\Delta \theta rL \tag{10}\]  
in which the structural properties (elasticity modulus \(E\), Poisson ratio \(\
u\), volume density \(\rho_{s}\), and thickness \(h\)) are recalled along with the inner ring radius \(r\), the axial length \(L\) and the arc angle \(\Delta\theta\)[19]. The radiation efficiency curves confirm the theoretical values for both the characteristic frequencies (ring and coincidence), as they superimpose almost perfectly with discrepancies limited to low frequencies. The congruency of the SEA subsystems is thus assured.