## 3 Coupled Modal Analysis of Ring-Stiffened Conical Shell with Bases
### Coupled Modal Analysis
#### 3.2.2 Frequency Response of the Ring-Stiffened Conical Shell  
To simulate the vibration of the actual engine, the sinusoidal load of 100 Pa is imposed normally on the plates of bases. The vibration responses of the shell in water are analyzed. The procedures are as follows:  
* The modal analysis of the structure is carried out firstly;
* The corresponding node information of the outer surface is obtained and imported into acoustic mesh;  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Order** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline Vacuum & 143.1 & 144.1 & 163.7 & 170.6 & 184.1 & 199.2 & 206.0 & 207.9 & 225.5 & 226.0 \\ Water & 55.4 & 55.6 & 76.0 & 81.1 & 86.1 & 86.6 & 90.0 & 92.5 & 115.6 & 121.4 \\ \hline
**Order** & **11** & **12** & **13** & **14** & **15** & **16** & **17** & **18** & **19** & **20** \\ \hline Vacuum & 248.0 & 249.0 & 249.2 & 259.0 & 261.8 & 262.3 & 266.3 & 266.4 & 267.3 & 268.0 \\ Water & 128.1 & 130.7 & 135.5 & 136.8 & 138.8 & 141.3 & 144.4 & 144.6 & 150.9 & 151.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Natural frequency in different medium/Hz.  
Figure 3: Coupled modal shape of the ring-stiffened conical shell in water: (**a**) 1 st order; (**b**) 2 nd order; (**c**) 3 rd order; (**d**) 4 th order; (**e**) 5 th order; (**f**) 6 th order; (**g**) 7 th order; (**h**) 8 th order.  
* The structural mesh and acoustic mesh are coupled by means of BEM for the acoustic-structure coupling analysis;
* The acoustic characteristic quantity of the structure is obtained.  
Due to the limitation of mesh and essence of FEM, the frequency range is 50-2950 Hz, which reaches the limitation of the computer's calculation ability. The step is 50 Hz. The three typical points on shell structure are selected to monitor the responses. As shown in Figure 4, Point 1 is located on the front end of the shell, Point 2 is located on the back end of the shell, and Point 3 is located in the middle part.  
Figure 5 shows the direct vibration displacement responses in vacuum and acoustic-structure coupling responses in water of three monitoring points. It can be seen from Figure 5 that the structural vibration responses have a similar trend in both vacuum and water: the vibration responses of Point 3 are a little larger than those of the other two points. The primary reason is that both ends of the structure are constrained, and the distance from Point 3 to the ends is the greatest, while Point 3 is just above the source. Although Points 1 and 2 are equally distant from the boundaries, Point 1 is closer to the source than Point 2; thus, the vibration responses of Point 1 are larger than those of Point 2. Moreover, since the vibration responses of Point 3 are the largest of them, Point 3 is selected for the comparison, as shown in Figure 6. The values of acoustic-structure coupling responses are lower than those of direct vibration responses. Meanwhile, with the increasing frequency, the acoustic-structure coupling responses decrease continuously, while the direct vibration responses have a relatively stable change. Thus, we can conclude that the acoustic medium has a notable influence on the vibration responses of structure. The acoustic medium not only reduces the whole vibration amplitudes, but also suppresses the high-frequency vibration, which is similar to the damping effect.  
Figure 4: Three monitoring points on the shell.