Low-Frequency Bandgaps by Topological Acoustic Black Holes

Abstract

Nowadays, acoustic black holes (ABHs) are very popular for producing efficient vibration reduction at high frequencies in combination with some damping mechanisms. However, its low-frequency performance is hard to improve since the ABH effect principally occurs beyond its cut-on frequency. Fortunately, periodic ABH configuration offers some bandgaps below that frequency for wave attenuation. In this chapter, a topological ABH structure is suggested to produce a new bandgap at very low frequencies, by taking a supercell and decreasing the ABH distance. The wave and Rayleigh-Ritz method (WRRM) is adopted to compute the complex dispersion curves. Examinations of the dispersion curves and transmissibilities confirm the efficiency of the low-frequency vibration reduction capability of the proposed topological ABHs.

1. Introduction

Acoustic black holes (ABHs) in mechanics are very efficient for wave manipulation in structures. By decreasing structural thickness following a power law, the impinging wave can be significantly retarded and concentrated as it propagates to the tip. Usually, in the vicinity of the ABH tip, some damping mechanisms [1, 2] are adopted; thus, the vibration energy is converted to, for example, heat, resulting in highly efficient vibration attenuation. However, there exists a cut-on frequency that relies on the ABH size. It is generally thought that the ABH effect can only be efficient above that frequency. While in practical engineering situations the tough task is related to low frequencies. Therefore, it is imperative to ameliorate the low-frequency performance of ABH structures.

Periodic distribution of ABHs [3, 4, 5], a new type of phononic crystal/metamaterial, could produce some bandgaps below the cut-on frequency, constituting a promising approach for low-frequency vibration reduction. This concept is first investigated in Refs. [6, 7, 8], where the purpose is to modulate the wavefront of bending waves in plates. Later, a periodic ABH beam is proposed [9], reporting that many locally resonant low-frequency bandgaps are formed below the cut-on frequency. Recently, it has been unveiled that these bandgaps are caused by Bragg scattering [10]. Interestingly, a compound ABH beam structure (also called double-leaf ABH beam) has been extensively studied [11, 12, 13], due to that many broad bandgaps can be found. In Ref. [14], very low bangaps are obtained by modifying the ABH parameters. Nevertheless, it is very tricky to obtain complete bandgaps in plates. In Ref. [15], a circular double-leaf ABH structure is proposed, showing that complete and sub-wavelength bandgaps are found. While in Ref. [16], a strip ABH is proposed, providing broad bandgaps and showing promising application for wave manipulation in plates. On the other hand, periodically placing resonators on the ABH plate has also been investigated [17, 18], where a very low bandgap is formed to suppress the first formant of the plate. Not only that, nonlinear effect has been applyed for ABHs, taking the advantage of energy transfer from low to high frequencies [19]. Recently, as an alternative of embedded ABHs, the additive ABHs have been widely explored, showing that the added mass could somehow facilitate low-frequency wave reduction [20, 21, 22, 23].

To further improve the low-frequency performance of embedded ABHs, we must resort to the combination of different physics. Inspired by topological metamaterials, in the current chapter, we propose topological ABHs to generalize a low-frequency bandgap, by adjusting the ABH distance in a supercell, as illustrated in Figure 1. It is worthwhile mentioning that the concept of topology has been introduced in previous works [24, 25]. However, the purpose of those works is concentrated on the presence of robust edge or surface states within the bandgap. Very differently, in the current chapter, we exploit the low-frequency bandgap generalized by the topological ABHs, to further enhance the low-frequency performance of ABHs. Please note that it is easy to prove the topological property by showing, for example, interface states or topological symmetry. We will overlook that because the focus is only placed on generating low-frequency bandgaps.

Figure 1.

Illustration of (a) a unit ABH cell with detailed geometry parameters, (b) periodic ABHs, and (c) topological ABHs.

To unveil the properties of the proposed topological ABHs, we adopt our previously established wave and Rayleigh-Ritz method (WRRM, see [18, 26]) to recover the complex dispersion curves, where the imaginary part stands for the wave attenuation (including damping and bandgaps). Furthermore, the transmissibilities of finite structures will also be tested, to prove the existence of low-frequency topological bandgap.

2. Wave and Rayleigh-Ritz method

In this section, we will place the focus on the wave and Rayleigh-Ritz (WRRM) to compute the complex dispersion curves. For the ease of exposition, we take the Euler-Bernoulli assumption, such that the bending displacement of the beam can be expanded by a series of basis functions,

$$w\left(x,t\right)=\sum _{i=1}^n{a}_i\left(t\right){\phi }_i\left(x\right)≔{\mathit{\phi }}^{\top }\left(x\right)\mathit{a}\left(t\right),$$E1

where $\mathit{a}$ is the unknown coefficient vector to be determined and $\mathit{\phi }$ signifies a Gaussian function vector (see the Gaussian expansion method, GEM [27, 28]).

One can continue in the Rayleigh-Ritz framework to reach the following equation of motion,

$$\left(\mathit{K}-{\omega }^2\mathit{M}\right)\mathit{A}=\mathbf{0},$$E2

where $\mathit{M}$ and $\mathit{K}$ respectively denote the mass and stiffness matrices. Please note that in Eq. (2) we have taken $\mathit{a}=\mathit{A}exp\left(\mathrm{i}\mathit{\omega t}\right)$, where $\omega$ represents the angular frequency.

The following step is to impose the Bloch-Floquet periodic conditions at the cell boundaries,

$$w_1=\lambda w_2,$$E3

$${\partial }_x{w}_1=\lambda {\partial }_x{w}_2,$$E4

where $w_1$ and ${\partial }_x{w}_1$ respectively represent the displacement and rotational angle at the left end, and $w_2$ and ${\partial }_x{w}_2$ correspond to the right one. $\lambda =exp\left(\mathrm{i}\mathit{kL}\right)$ stands for the propagation constant with $k$ being the wavenumber.

Now inserting Eq. (1) into Eqs. (3) and (4) we have

$$\left[\begin{array}{cccc}-\lambda & 0& 1& 0\\ 0& -\lambda & 0& 1\end{array}\right]\left[\begin{array}{c}{\mathit{\phi }}_2^{\top }\\ {\partial }_x{\mathit{\phi }}_2^{\top }\\ {\mathit{\phi }}_1^{\top }\\ {\partial }_x{\mathit{\phi }}_1^{\top }\end{array}\right]\mathit{A}=\left[\begin{array}{c}0\\ 0\end{array}\right]≕\tilde{\mathbf{\Lambda }}{\mathbf{\Phi }}^{\top }\mathit{A}=\mathbf{0},$$E5

where we have identified $\tilde{\mathbf{\Lambda }}$ and $\mathbf{\Phi }$.

The key step here is to attain the nullspace basis prescribed by Eq. (5). With the indication in Ref. [29], we can get the basis $\mathit{Z}$ for $\mathcal{N}\left(\tilde{\mathbf{\Lambda }}{\mathbf{\Phi }}^{\top }\right)$,

$$\mathit{Z}=\left[{\mathit{Z}}_{\phi },{\mathcal{C}}_{\mathit{Z}}\right]=\left[{\mathit{Z}}_{\phi },\mathcal{C}\right]\left[\begin{array}{cc}\mathit{I}& \mathbf{0}\\ \mathbf{0}& {\mathit{Z}}_r\end{array}\right],$$E6

where

$${\mathit{Z}}_r≔\left[\begin{array}{cc}1& 0\\ 0& 1\\ \lambda & 0\\ 0& \lambda \end{array}\right]=\left[\begin{array}{c}\mathit{I}\\ \lambda \mathit{I}\end{array}\right],$$E7

$${\mathit{Z}}_{\phi }=\mathcal{N}\left({\mathbf{\Phi }}^{\top }\right),$$E8

and

$$\mathbf{C}={\left({\mathbf{\Phi }}^{\top }\right)}^{+}.$$E9

Assuming the final response $\mathit{A}$ is linearly constituted by the basis of $\mathit{Z}$, that is, $\mathit{A}=\mathit{Z}k$, we arrive at

$$\left(\mathit{K}-{\omega }^2\mathit{M}\right)\mathit{Zk}=\mathbf{0},$$E10

with $\mathit{k}$ being the unknown vector to be solved. Eq. (10) becomes the equation of motion satisfying the periodic boundary conditions in Eqs. (3) and (4).

To simplify the computation, we define ${\mathit{Z}}_c\equiv \left[{\mathit{Z}}_{\phi },\mathcal{C}\right]$ and

$${\mathit{Z}}_l\equiv \left[\begin{array}{cccc}1& 0& {\lambda }^{-1}& 0\\ 0& 1& 0& {\lambda }^{-1}\end{array}\right]\equiv \left[\mathit{I}{\lambda }^{-1}\mathit{I}\right].$$E11

Pre-multiplying Eq. (10) by $diag\left(\mathit{I},{\mathit{Z}}_l\right){\mathit{Z}}_c^{\top }$ we get

$$\left[\begin{array}{cc}\mathit{I}& \mathbf{0}\\ \mathbf{0}& {\mathit{Z}}_l\end{array}\right]\left(\overline{\mathit{K}}-{\omega }^2\overline{\mathit{M}}\right)\left[\begin{array}{cc}\mathit{I}& \mathbf{0}\\ \mathbf{0}& {\mathit{Z}}_r\end{array}\right]\mathit{k}=\mathbf{0},$$E12

where ${\overline{\mathit{K}}}_{n\times n}≔{\mathit{Z}}_c^{\top }{\mathit{KZ}}_c$ and ${\overline{\mathit{M}}}_{n\times n}≔{\mathit{Z}}_c^{\top }{\mathit{MZ}}_c$ are square matrices.

Now, it is possible to compute the real dispersion curves once a wavenumber $k$ is given. However, our purpose is to get the complex dispersion curves, where $\omega$ is fixed and $k$ is to be determined. To do so, we define $\mathit{D}\equiv \overline{\mathit{K}}-{\omega }^2\overline{\mathit{M}}$ and partition it as

$${\mathit{D}}_{n\times n}=\left[\begin{array}{cc}{\mathit{D}}_{11,\left(n-4\right)\times \left(n-4\right)}& {\mathit{D}}_{12,\left(n-4\right)\times 4}\\ {\mathit{D}}_{12,4\times \left(n-4\right)}^{\top }& {\mathit{D}}_{22,4\times 4}\end{array}\right].$$E13

Now, Eq. (12) becomes

$$\begin{array}{c}{\mathit{D}}_{11}{\mathit{k}}_1+{\mathit{D}}_{12}{\mathit{Z}}_r{\mathit{k}}_2=\mathbf{0},\\ {\mathit{Z}}_l{\mathit{D}}_{12}^{\top }{\mathit{k}}_1+{\mathit{Z}}_l{\mathit{D}}_{22}{\mathit{Z}}_r{\mathit{k}}_2=\mathbf{0},\end{array}$$E14

where $\left[{\mathit{k}}_1^{\top },{\mathit{k}}_2^{\top }\right]\equiv \mathit{k}$. Clearing ${\mathit{k}}_1$ we have

$${\mathit{Z}}_l\left[{\mathit{D}}_{22}-{\mathit{D}}_{12}^{\top }{\mathit{D}}_{11}^{-1}{\mathit{D}}_{12}\right]{\mathit{Z}}_r{\mathit{k}}_2=\mathbf{0}.$$E15

The Schur complement of ${\mathit{D}}_{11}$ can be designated as $\mathit{G}≔{\mathit{D}}_{22}-{\mathit{D}}_{12}^{\top }{\mathit{D}}_{11}^{-1}{\mathit{D}}_{12}$, which can be further partitioned as

$$\mathit{G}=\left[\begin{array}{cc}{\mathit{G}}_{11}& {\mathit{G}}_{12}\\ {\mathit{G}}_{12}^{\top }& {\mathit{G}}_{22}\end{array}\right].$$E16

Recalling the definition of ${\mathit{Z}}_l$ and ${\mathit{Z}}_r$, we can obtain the following equation,

$$\left[\begin{array}{cc}\mathit{I}& {\lambda }^{-1}\mathit{I}\end{array}\right]\left[\begin{array}{cc}{\mathit{G}}_{11}& {\mathit{G}}_{12}\\ {\mathit{G}}_{12}^{\top }& {\mathit{G}}_{22}\end{array}\right]\left[\begin{array}{c}\mathit{I}\\ \lambda \mathit{I}\end{array}\right]{\mathit{k}}_2=\left({\mathit{G}}_{11}+{\mathit{G}}_{22}+{\mathit{G}}_{12}^{\top }{\lambda }^{-1}+{\mathit{G}}_{12}\lambda \right){\mathit{k}}_2=\mathbf{0},$$E17

which can be converted into a quadratic eigenvalue problem for $\lambda$. Therefore, the wavenumber $k\in \mathrm{ℂ}$ can be solved for a given $\omega \in \mathbb{R}$.

3. Numerical results

Once built the computational model, we start to show some numerical results. As shown in Figure 1, the lattice constant is selected as $L=0.1$ m, and the beam thickness at the uniform part is $h_{\mathit{uni}}=3$ mm. The ABH profile is defined as $h\left(x\right)=\epsilon x^m+h_c$, where $m=2$, $h_c=0.6$ mm, $\epsilon =\left(h_{\mathit{uni}}-h_c\right)/r_{\mathit{abh}}^m=6{\mathrm{m}}^{-1}$, and $r_{\mathit{abh}}=2$ cm. The beam is made of steel, with Young modulus being $E=210$ GPa, density $\rho =7800\mathrm{kg}/{\mathrm{m}}^3$, and loss factor $\eta =0.005$. On the other hand, the damping layer has size $r_d=2$ cm and $h_d=1.8$ mm, and the Young modulus $E_d=5$ GPa, density ${\rho }_d=950\mathrm{kg}/{\mathrm{m}}^3$, and loss factor ${\eta }_d=0.5$.

In the following, we will first examine the complex dispersion curves, then the transmissibilities are inspected to ensure the existence of topological low-frequency bangap.

3.1 Complex dispersion curves

To start with, we do not consider any damping (no damping layer and $\eta =0$), and a standard periodic cell is considered. As illustrated in Figure 2a and b, where the real and imaginary parts of the complex dispersion curves have been computed. It is observed that there are two bandgaps appearing in 175–555 Hz and 1635–2330 Hz, respectively. According to the analysis in Ref. [10], those bandgaps are induced by the Bragg scattering effect. Now, we take the supercell (as depicted in Figure 1b), with $d=\mathit{\alpha L}$, where $\alpha =1$. Please note that in this case the bandgaps should occur at the same frequency range. As demonstrated in Figure 2c and d, this is true, indicating that our WRRM is accurate to recover the complex dispersion curves. Particularly, in Figure 2d we found that the evanescent wave is doubled compared to that in Figure 2b. This is reasonable because two cells provide more wave attenuation. Please note that there are two passbands and two stopbands in the frequency range of interest (see Figure 2a), computed from the periodic cells. However, when using supercell structure, each passband has been divided into two ones (see $f=50$ Hz and $f=1040$ Hz), which gives us the possibility to open them in the following content.

Figure 2.

(a) Real and (b) imaginary part of the complex dispersion curves of a standard periodic ABH cell. (c) Real and (d) imaginary part of the complex dispersion curves of a standard periodic supercell. Please note that in this case we have not included the damping layer and beam intrinsic loss.

Now, we shrink the distance of the ABHs in a supercell (see Figure 1c), and set $\alpha =0.5$. The complex dispersion curves are illustrated in Figure 3. It is seen that the passbands are opened at $f=50$ Hz and $f=1040$ Hz, and there emerge two new bandgaps in 45–80 Hz and 895–1100 Hz. Since we are more interested in low frequencies, the new first bandgap (45–80 Hz) is much lower than the former one (175–555 Hz). Particularly, we have only changed the ABH distance and the structure has the same weight as before.

Figure 3.

(a) Real and (b) imaginary part of the complex dispersion curves of a topological supercell with $\alpha =0.5$. Please note that in this case we have not included the damping layer and beam intrinsic loss.

At this stage, we include the damping layers and the intrinsic loss of the beam ($\eta =0.005$). The complex dispersion curves of a supercell with $\alpha =1$ have been demonstrated in Figure 4a and b, which are nothing but corresponding to standard periodic lattice. In such a case we see that bandgaps no longer exist in the real dispersion curves but in the imaginary part (see Figure 4b). Also, in the passbands we also have some wave attenuation because of damping (again see Figure 4b).

Figure 4.

(a) Real and (b) imaginary part of the complex dispersion curves of a standard periodic supercell. (c) Real and (d) imaginary part of the complex dispersion curves of a topological supercell with $\alpha =0.5$. Please note that in this case we have included the damping layer and beam intrinsic loss.

Once we decrease the ABH distance and set $\alpha =0.5$ (see Figure 4c and d), as expected, we can get a new bandgap at very low frequency $f=95$ Hz. Please note that at very high frequencies the ABH effect starts to take control, so the imaginary part is very smooth and steadily becomes larger over $f=2000$ Hz.

3.2 Transmissibilities

We next shed light on the wave transmission in finite beams. As usual, we first exclude any damping. In total, 12 cells have been taken into account; thus, the beam length equals to 1.2 m. For reference, we have computed the transmissibillity, $T=10lg\left[w^2\left(1.2\right)/w^2\left(0\right)\right]$, of a uniform beam (UNI) where a point force is exerted at $x=0$, see the brown line in Figure 5a. For the better inspection of low frequencies, we have chosen logarithmic scale for the frequency axis. Not only that, the $T$ of standard periodic ABH beam has also been computed, see the black line in Figure 5a. We can find that two bandgaps take places at the frequency ranges predicted by the dispersion curves in Figure 2. Once we have set $\alpha =0.5$ (see the red curve in Figure 5a), we can get a new bandgap centred at 70 Hz.

Figure 5.

Transmissibilities of finite beams with 12 cells. (a) Without damping layer nor the beam intrinsic loss. (b) With damping layer and the beam intrinsic loss.

Once the damping layer and intrinsic loss $\eta =0.005$ are taken into account, it is seen that vibration peaks are substantially smoothed, especially at high frequencies, where the ABH effect starts working. However, the most intriguing phenomenon is the presence of the low-frequency topological bandgap at $f=100$ Hz, confirming that the proposed supercell ABH with shrinking distance is very promising for low-frequency vibration reduction.

4. Conclusions

In this chapter, we have proposed a topological ABH metamaterial to generalize a low-frequency bandgap for vibration reduction. Based on a standard periodic ABH beam, we have taken a supercell and reduced the ABH distance, opening new bandgaps, especially at low frequencies.

To characterize such a feature, we have adopted the wave and Rayleigh-Ritz method (WRRM) to compute the complex dispersion curves, which relies on solving the quadratic eigenvalue problem of $k\in \mathbb{C}$ for a fixed $\omega \in \mathbb{R}$. After that, we have analyzed impact of changing the ABH distance on the bandgaps. Before including the damping, the bandgaps can be both appreciated in the real and imaginary parts of the complex dispersion curves. However, after considering the damping, bandgaps can only be inspected in the imaginary part.

Both the analyses on the dispersion curves and tranmissibilities indicate that the topological ABHs can indeed produce a low-frequency bandgap, which constitutes a promising low-frequency wave manipulation technique.

Acknowledgments

J. Deng acknowledges the support received by the National Natural Science Foundation of China (52301386) and the support of the Beatriu de Pins postdoctoral program of the Department of Research and Universities of the Generalitat of Catalonia (2022 BP 00027).

Conflict of interest

The author declares no conflict of interest.