## 4 SEA Modeling
### Definition of SEA Subsystems and Estimation of Damping Loss Factors  
One of the main difficulties of SEA modeling is the definition of the subsystems in such a way that the conditions of applicability of SEA theory are respected. These conditions are the following [17]:  
* an adequate modal density for each subsystem (at least three modes per frequency band),
* a spatial mode covering (the modes used to calculate the modal density must affect all the subsystem), and
* a weak coupling between subsystems. The experimental measurements presented here make it possible to determine the SEA subsystems of the HB-2 structure in order that the conditions mentioned above should be respected. Furthermore, the damping loss factor of each subsystem, which is a key parameter of the model, is measured. The test setup is illustrated in Figure 14a. The entire structure was freely suspended and driven sequentially at 11 points, denoted \(E_{i}\) in Figure 14b, using a shaker attached to the excitation point. Analogously to the experimental setup for the vibrometry measurement shown in Figures 3 and 5, the structure is suspended horizontally for an excitation point on the bottom, and vertically for an excitation point on the envelope. For each excitation, the resulting vibration responses were recorded with 21 accelerometers, denoted as \(A_{i}\) in Figure 14b at randomly selected locations so that the spatial average could be estimated. The structure was excited with a white noise from 500 Hz to 16 kHz. The first clear division of the structure into subsystems is the separation of the bottom and the envelope. Then, the analysis of the FRF measurements at different locations enables the determination of the envelope's subsystems. The envelope is not homogeneous with stiffened zones and zones of different curvature. First, analysis of the measurements is performed by splitting the envelope where its curvature changes. As depicted in Figure 14b, three zones, designated ENV1, ENV2, ENV3, are defined. All the signals recorded by the accelerometers located in each zone are averaged together. Figure 15 shows the absolute value of the average point mobility \(Y_{ii}\) obtained in each zone. The mobility measured in the two zones ENV1 and ENV2 appear to be similar, thereby, revealing strong coupling between these two zones. The ENV3 zone appears to be less coupled to the others. Indeed, the envelope in this zone possesses more stiffeners. Therefore, the best choice of SEA subsystems for the HB-2 structure is the following: two subsystems for the envelope, designated SEA 2D 1 (for the ENV3 zone) and SEA 2D2 (for the ENV1-ENV2 zones grouped together), and one subsystem for the bottom, designated SEA 2D 3 (see Figure 14b). Figure 14: (**a**) The experimental setup. (**b**) HB-2 subsystem definition and the location of the excited and measured points. The blue dots indicate the points that were used for both excitation and response measurement. The red dots indicate the points that were used only for the acceleration response measurement. Figure 15: The average mobility measurements for ENV1 (blue line), ENV2 (red dotted line) and ENV3 (green dashed line) zones. The second objective of the experiments is the determination of the damping loss factors. The DLF of each subsystem are calculated with the decay-rate method [17]. They are related to the decay rate (\(DR\)) by:  
\[\eta_{ii}=\frac{DR}{27.3f}. \tag{11}\]  
For each accelerometer, the impulsive time response is measured using a fast process involving correlated time sequences with a low crest factor. One impulsive time response is plotted in Figure 16b. The signals obtained are time filtered for each third octave band. Then, as depicted in Figure 16a, the Energy Time Curve (ETC) is plotted as well as the Schroeder integration [37]. The Decay Rate (\(DR\)) is deduced from this integration. The \(DR\) corresponds to the time after which the residual integration dropped by \(-10\) dB. As illustrated in Figure 16a, the chosen starting point is when the integration drops by \(0.1\) dB.