Geometry and material parameters of the ABH cylindrical shell.

## Abstract

Acoustic black holes (ABHs) achieved by progressively diminishing structural thickness have been proved a very efficient approach for wideband vibration reduction, sound suppression, energy harvesting, and wave manipulation. In this chapter, the focus is placed on mitigating the sound emitted from cylindrical shells with embedded ABHs. In the applications of aeronautics, astronautics, and underwater vehicles, cylindrical shells are very common yet the vibroacoustic problems in such structures are very challenging. Even the researches on ABHs for straight beams and flat plates are boosting in recent years, the ABH effect is unclear for curved structures thus it deserves further investigations to push forward their applications. Since cylindrical shells are usually long in, for example, airplanes and rockets, periodic ABHs are designed to alleviate the acoustic emission from them. The Gaussian expansion method (GEM) is employed to recover the vibration field on the cylinder and, based on that, the sound radiation model is developed to determine the emitted sound power level (SWL). The band gaps (BGs) are shown for infinite periodic ABH shells, followed by the vibroacoustic level for a finite periodic shell. Particularly, axial stiffeners are introduced and the influences of their quantity and width are carried out.

### Keywords

- acoustic black holes
- acoustic radiation
- cylindrical shells
- band gaps
- stiffeners

## 1. Introduction

An acoustic black hole (ABH) is usually realized by reducing structural thickness following a power law

It has been shown that the ABHs are very efficient to reduce vibration from straight beams [5, 6] and flat plates [7, 8]. The shapes and lengths of the damping layers have been extensively investigated [4, 9]. Also, different ABH designs have been proposed for the purpose of enhancing energy consumption [7, 10, 11, 12]. Thanks to the vibration reduction because of highly efficient damping, the sound radiation from ABH structures is accordingly reduced [13]. Not only that, recent studies have shown that the ABHs can also impair the sound radiation efficiency because of the thickness reduction [14, 15]. Particularly, for cavity noise, the ABH profile can destroy the coupling strength between structural and acoustic modes, which is the third underlying mechanism of the ABHs for reducing room noise [16, 17]. Furthermore, periodic and gradient-index ABHs are investigated for steering waves [18, 19, 20]. It is also worthwhile mentioning that ABHs can also enhance energy harvesting due to wave focalization [21], using piezoelectric layers rather than viscoelastic ones.

The state-of-the-art reviewed above are mainly centered on flat structures. However, in aeronautics, astronautics, and underwater vehicles, cylindrical shells are very common. There, the vibroacoustic problems are very critical to determine their comfortability and safety, thus it is very demanding to apply ABH features on them. Our previous efforts have been focused on the vibration of cylindrical beams [22] and shells [23, 24], together with the sound radiation from a finite cylinder [25]. In this chapter, we continue this topic but analyze the sound radiation from periodic ABH shells.

As shown in Figure 2a, an infinite periodic ABH shell is considered, with each unit cell having radius

## 2. Vibration characteristics of infinite periodic ABH shells

### 2.1 Gaussian expansion for the vibration of infinite periodic ABH shells

The goal of this section is that of developing a semi-analytical model for characterizing the vibration of the infinite periodic ABH shell. Let us consider three variables,

where

are the coefficient vectors to be determined, while

where

in which

Unlike finite and flat plates, bear in mind that the displacements, on the one hand, must be continuous in the circumferential direction (

On the other hand, Bloch-Floquet periodic boundary conditions must be imposed in the axial direction for a unit cell (

where

Provided the kinetic energy,

where

represents the assembled undetermined time-dependent vector related to admissible shape functions.

whose solution permits calculating the dispersion curves and eigenmodes for infinite periodic ABH shells.

### 2.2 Numerical results

#### 2.2.1 Dispersion curves and band gaps

The dispersion curves of an ABH cell, whose geometry and material are detailed in Table 1, have been carried out and plotted in Figure 3a. For the purpose of validation, the result from a reference FEM model has also been included in Figure 3a. From the figure, it is seen that the two results are very close at each wavenumber, indicating the correctness of the present GEM model. Most importantly, four-band gaps (BGs) are observed within 1000 Hz, in which the first one is very small. Using the GEM without axial periodic boundary conditions for a finite shell having five cells, we can compute the transmission from one end to the other. As shown in Figure 3b, the transmission is very low at BG frequencies. For the shell without damping layers (undamped), the transmission is strong in the passbands. However, this situation can be ameliorated after implementing the damping layers, with a maximum reduction of up to

Geometry parameters | Material parameters |
---|---|

To reveal the mechanism of the BGs, we have computed the first six eigenmodes of the unit cell, at wavenumber

#### 2.2.2 Parametric analysis: effects of the ABH order, central thickness, and radius

The ABH profile is generally controlled by three parameters,

Let us first look at the influence of the ABH order. As shown in Figure 5a, the 2-nd BG starts to gradually decrease as

Different from the ABH order, the BGs are very sensitive to the central thickness. As graphed in Figure 5b, generally, the BGs become very narrow as the central thickness increases from

Finally, the effects of the ABH radius are computed and illustrated in Figure 5c. We consider the ABH radius varying from 0 to

## 3. Sound radiation from finite periodic ABH shells

### 3.1 Radiation theory for cylindrical shells

Once the radial displacement

where

For numerical estimation, the cylindrical shell can be segmented into

where

here

Next, we can write the sound power as

where the superscript

where

### 3.2 Sound radiation from unstiffened ABH shells

Now we investigate the sound radiation from a finite unstiffened shell containing five ABH cells. Based on the reconstructed GEM presented in Section 2.1, we cancel the axial periodic conditions, namely Eqs. (12) and (13), such that the vibration field of the finite ABH shell can be characterized. It is well-known that there is a cut-on frequency for an ABH, which is mainly determined by its size. Only beyond this frequency (wavelength smaller than the ABH size,

As plotted in Figure 6a, the radial mean square velocity (MSV) on the surface of the ABH shell is compared to that of the uniform (UNI) shell having the same damping layer configuration. From the figure, it can be seen that in the BGs, the vibration of the ABH shell is very low. In the passbands, the vibration beyond

To clearly manifest the characteristics of the ABH shell, we have further calculated the vibration field at each frequency. As illustrated in Figure 7, compared to the uniform shell whose distribution of vibration nodes is very regular to location and frequency (over the ring frequency 173 Hz), the vibration in the BGs is obviously isolated as propagating in the axial direction. For frequencies outside of the BGs, the amplitude of the displacement is also clearly reduced to the right direction. Specifically, we choose 287 Hz and 340 Hz as two representative frequencies in the BG and the passband, respectively, for demonstrating the effectiveness of the ABH shell. In Figure 8a, it is clearly seen that the local resonance effect in the ABHs is very strong, such that vibration can be substantially stopped when compared to the uniform shell. While in Figure 8b we can see that the wave can be transmitted to the whole shell, but the amplitude is very small because of the highly efficient damping effect by the ABH + damping layer configuration.

### 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

Now we keep the 16 stiffeners in the circumferential direction for each cell but modify the width from

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.

## 4. Conclusions

In the current chapter, the acoustic black hole (ABH) effect is concerned and applied to cylindrical structures. By reducing thickness following the power law, the wave velocity is substantially slowed and the wavenumber is increased when it propagates to the ABH center, where the damping layer is very efficient to consume vibrational energy. The focus is placed on reducing the sound emission from cylindrical shells, which can be found in many fields, via embedding periodic ABHs. First, the reconstructed Gaussian expansion method (GEM) is presented to characterize infinite periodic ABH shells. The band gaps (BGs), induced by the locally resonant effect in the ABH area, are investigated, together with the influence of the ABH parameters. Next, the sound radiation model for finite periodic cylindrical shells is developed. Numerical results show that the periodic ABHs can both reduce the vibration and sound power, relying on two mechanisms—(i) the BGs for isolating vibrations and (ii) the damping effects for energy consumption. The inclusion of stiffeners not only strengthens the structural stiffness but also keeps the vibration and sound power level as the pure ABH one in the passbands.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant number 11704314) and the China Postdoctoral Science Foundation (grant numbers 2018M631194 and 2020T130533).

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