**3.3 Sound radiation from stiffened ABH shells**

Note that the embedded ABHs may obviously reduce the structural stiffness of the shell, therefore, the stiffeners (see **Figure 2d**) can be utilized to alleviate this issue. In this section, we first investigate the effects of the stiffener number, then study that of the stiffener width.

For simplicity, let us first consider three cases. That is, the stiffener number *N* ¼ f g 8, 16, 32 , while the width of each stiffener keeps *W* ¼ 4*huni*. As shown in **Figure 9a**, when adding eight stiffeners the vibration level is very similar to the ABH shell (without stiffener), except for the frequencies in the BGs. Increasing the stiffener number will further deteriorate the vibration in the BGs, but it seems that the MSV level is similar to that of the unstiffened one. This means that the combination of the ABH and stiffeners not only results in a more rigid structure (compared to the pure ABH one) but also maintains the overall damping effect. Similar results can be found for the radiated SWL (see **Figure 9b**). The existence of the stiffeners almost merely increases the SWL in the BGs, yet in the passbands, the stiffened ABH shell has a similar SWL to that of the unstiffened one. Furthermore, the radiation efficiency is checked (see **Figure 9c**). From the figure, we can see that after inserting stiffeners the radiation efficiency in the 2-nd (the 1-st BG is not shown here) is very small and, that for the case of 32 stiffeners the efficiency is the lowest over 700 Hz. In general, the stiffeners will not obviously degrade the reduction ability of the ABH shell.

#### **Figure 9.**

*(a) Mean square velocity (MSV), (b) sound power level (SWL), and (c) radiation efficiency, for the cylindrical shell having a different number of stiffeners with the same width W* ¼ 4*huni.*

Now we keep the 16 stiffeners in the circumferential direction for each cell but modify the width from *W* ¼ 4*huni* to *W* ¼ 8*huni*, then to *W* ¼ 16*huni*. Similar to the above results, from **Figure 10** it can be observed that increasing the stiffener width mainly intensifies the MSV and SWL in the BGs, while their values in the passbands almost do not change. Particularly, the radiation efficiency of the stiffened shells is obviously lower than the ABH shell at frequencies greater than 650 Hz. Even large stiffener width will enlarge the vibration and the radiated sound power at low frequencies (e.g., at 180 Hz), the high-frequency performance is still plausible and the structure is more rigid.

For the purpose of illustration, we have figured out the normal velocity and sound pressure distributions on the finite shells at 340 Hz (in passband). The former is graphed in **Figure 11** while the latter in **Figure 12**. From **Figure 11**, we can see that the vibration level of the uniform shell is very strong (**Figure 11a**), but the ABHs can help reduce the overall vibration, with only strong vibrations in the ABH portions where the damping layer is very effective (**Figure 11b**). After adding stiffeners (**Figure 11c1–d3**), the vibration in the ABH areas is intensified compared to **Figure 11b**. The sound pressure distributions in **Figure 12** display that the ABH and stiffened ABH shells can also effectively reduce the sound pressure, compared to the reference uniform shell. The existence of the stiffener makes the distributions no longer axially symmetrical, except for **Figure 12c3** where a lot of small and distributed stiffeners are imposed.

#### **Figure 10.**

*(a) Mean square velocity (MSV), (b) sound power level (SWL), and (c) radiation efficiency, for the cylindrical shell having 16 stiffeners with different widths.*

*Periodic Acoustic Black Holes to Mitigate Sound Radiation from Cylindrical Structures DOI: http://dx.doi.org/10.5772/intechopen.101959*

**Figure 11.**

*Surface normal velocity distributions at 340 Hz. (a) Uniform shell, (b) ABH shell without stiffener, (c1)-(c3) stiffened ABH shell with the different number of stiffeners of the same width, (d1)-(d3) stiffened ABH shell with 16 stiffeners having different widths. The red circles stand for the ring force.*
