The geometric parameters of the acoustic black hole (ABH) structure are changed in power exponent, and this feature can be used to control the flexural wave to achieve energy concentration, vibration attenuation, or noise reduction. However, in practice, the ABH structure often has a truncation due to the limitation of manufacturing, which will cause the reflection coefficient to increase significantly and seriously affect the ABH effect. In this paper, a semianalytical model of the sandwichtruncated ABH beam structure with aluminum in the middle layer and steel in the upper and lower layers is constructed based on the energy principle. The ABH effect of the sandwich beam under the clampedfree boundary condition is analyzed. Meanwhile, the effects of damping layer parameters, middle layer material, and thickness on the vibrational acceleration response of the ABH region and the uniform beam region of the sandwich beam are also studied. It is observed that, for the sandwich ABH beam structure, the influence of damping layer thickness on the acceleration response peak values of both the ABH region and the uniform region is very obvious in middle and high frequencies and the peaks at about 9 kHz are completely suppressed when the damping layer thickness reaches 3 mm. It also reveals that the use of aluminum as the middle layer material can bring a vibration attenuation at around 9 kHz both for the ABH region and the uniform beam region compared with using steel as the middle layer material. Experiments are carried out to verify the accuracy of simulation analysis.
In the past two decades, the acoustic black hole (ABH) structure has attracted a lot of attention as a new method of vibration attenuation and noise reduction, and it can also bring positive effects to structural lightweighting [
In the actual manufacturing, a truncation at the tip end of the ABH region is inevitable. Even a small thickness of truncation can significantly increase the reflection coefficient, seriously affecting the effectiveness of energy concentration and vibration attenuation [
To analyze the performance of the ABH structure and the effect of damping layers or other attachments on the dynamic property of the entire system, we need to resort to some analytical or numerical approach. The analysis methods for ABH beam structures are geometrical acoustic method, impedance method, semianalytical method, and finite element method (FEM). The geometrical acoustic method is based on the assumption that the thickness of the damping layer is much smaller than the thickness of the ABH structure. For ABH structures with a certain thickness of the damping layer, the calculation accuracy is greatly affected and the influence of variation of the thickness and length of the damping layer on the dynamic characteristics of the ABH structure could not be studied [
In this paper, a sandwich ABH beam structure with truncation at the tip of the ABH region is proposed. Unlike previous onedimensional ABH structures, the ABH beam investigated in this paper is a sandwich beam composed of three layers. The middle layer material of the proposed ABH beam is aluminum, and the upper and lower layers are steel. The sandwich ABH beam is an elongated loadbearing structure composed of three layers of materials bonded together, and its length is much larger than the width and thickness. The Euler–Bernoulli beam theory can analyze the dynamics of slender laminated beams accurately [
As shown in Figure
An Euler–Bernoulli sandwich beam with truncated ABH.
The thickness of the bare ABH beam is given by
The entire system is symmetric about the
In order to analyze the dynamic properties of the sandwich ABH beam, the following assumptions are made:
Shear deformation of the cross section perpendicular to the neutral axis is ignored
The movement of the beam only occurs in the
Only the normal stress of the damping layer is considered, regardless of the shear stress of the damping layer
Ignore the moment of inertia effect of the sandwich ABH beam
According to the Euler–Bernoulli beam theory, the displacement field of the ABH beam can be written as follows:
The kinetic energy of the system can be obtained by
The work done by the external force
The Lagrangian of the entire system can be expressed as
According to the Hamiltonian equation, the following Lagrangian equation can be obtained after a series of transformation:
The mass matrix of the entire system can be written as
For each part of the sandwich ABH beam, the mass matrices are expressed as
In (
The stiffness matrix of the entire system is given by
For each part of the sandwich ABH beam, the stiffness matrices are computed by
Similar to the expression of the mass matrix, the upper and lower limits of the integral in equations (
Since the stiffness matrix
To fit the displacement filed of nonuniform beams or plates, the polynomial functions have been used to solve linear or nonlinear thickness variation of the nonuniform beams [
The MHW function is the second derivative of the Gaussian distribution function. It can be seen that, from equation (
MHW values under different scaling and translation parameters.
The key to fitting the shape function of the semianalytical model using MHW is to determine the value ranges of
The geometric parameters and material parameters of the sandwich ABH beam are shown in Table
Geometrical and material parameters of the sandwich ABH beam.
Geometrical parameters  Material parameters 















Model used in FEM validation.
To verify the computational accuracy of the established model, the first 30 natural frequencies of the sandwich ABH beam under the clampedfree boundary condition without considering damping layers are calculated, as shown in Table
Comparison of the first 30 natural frequencies obtained by the present model and FEM.
Natural frequency  Present model  FEM  Error (%) 


121.65  121.74  −0.070 

142.21  142.56  −0.247 

751.74  748.92  0.376 

869.24  865.82  0.392 

1765.8  1767.2  −0.078 

2050.4  2044.6  0.286 

2331.9  2321.5  0.447 

3493.5  3469.3  0.408 

3895.3  3875.3  0.515 

4298.9  4301.5  −0.060 
…  …  …  … 

13285  13216  0.521 

14816  14723  0.634 

16024  15922  0.640 

16016  15929  0.544 

17386  17295  0.528 

17936  17837  0.554 

19141  19019  0.639 

19131  19352  −1.143 

21105  21023  0.390 

21224  21118  0.501 
Modal shapes of the bare ABH beam calculated with the present model and FEM method: (a) first mode; (b) sixth mode; (c) twelfth mode; (d) twentyninth mode.
According to the geometric and material parameters in Table
Acceleration responses of the bare ABH beam and the uniform beam: (a) ABH region (
To further investigate the ABH effect, the flexural wave focalization effect of the bare ABH beam is studied in the time domain with the commercial software COMSOL Multiphysics. The position and amplitude of the driving force is the same as above, and the frequency of the driving force is 100 kHz. The displacement of the bare ABH beam along the
Displacement of bare ABH beam at different times: (a)
To conclude above analysis, for a sandwichtruncated ABH beam structure without damping layers, the ABH region produces a certain energy concentration effect in middle and high frequencies, but due to the existence of the truncation, the acceleration attenuation of the uniform beam region is not obvious and even get increased at some frequencies. In the actual manufacturing, it is impossible to achieve an ABH area without truncation. So that other measures should be taken to improve the vibration reduction effect of the sandwich ABH beam in application.
To evaluate the vibration reduction performance of the sandwich ABH beam, the influence of the damping layer parameters on the dynamic responses of the ABH region and uniform region of the sandwich ABH beam will be analyzed. The damping layers coupled with the sandwich ABH beam have a thickness of 2 mm and a length of 20 cm (along the
Effects of the damping layers on the acceleration responses: (a) ABH region (
To summarize, in order to achieve an obvious vibration reduction effect, it is important to attach damping layers on the ABH region of the sandwich ABH beam with truncation even if the length and thickness of the damping layer are much smaller than the sandwich ABH beam.
Figure
Effects of the damping loss factor of the damping layers on acceleration responses: (a) ABH region (
The influence of the thickness of the damping layer on the vibration acceleration response of the sandwich ABH beam structure is also studied. The results are shown in Figure
Effects of the damping layer thickness on acceleration responses: (a) ABH region (
The mode shapes of ABH beams with different damping layer thickness at around 9 kHz are presented in Figure
Mode shapes of ABH beams with different damping layer thickness at around 9 kHz: (a)
Modal loss factor of ABH beams with different damping layer thickness: (a) calculation frequency from 0 to 10 kHz; (b) calculation frequency from 8500 to 9500 Hz.
To summarize, it can be seen that the influence of the thickness variation of the damping layers on the acceleration response of the sandwich ABH beam is much greater than the variation of the damping loss factor. Therefore, in the application of the sandwich ABH beam, the selection of a damping layer of appropriate thickness can significantly improve the vibration attenuation effect of the entire system.
Unlike the traditional singlelayer ABH beam structures, the sandwich ABH beam with aluminum in the middle layer and steel in the upper and lower layers is studied in this paper. The effect of aluminum or steel as the middle layer material on vibration acceleration response of the sandwich ABH beam is shown in Figure
Effects of the middle layer material on acceleration responses: (a) ABH region (
The influence of the thickness variation of the aluminum layer on the vibration acceleration of the sandwich ABH beam structure will be further investigated. The thickness of the middle aluminum layer is 4 mm, 6 mm, and 8 mm, respectively. The thickness of the steel layers changes with the thickness of the middle aluminum layer, so as to ensure that the total thickness of the sandwich ABH beam remains 20 mm. Figure
Effect of the aluminum layer thickness on acceleration responses: (a) ABH region (
An experiment system was conducted to validate the accuracy of the numerical simulation. The geometric parameters of the sandwich ABH beam are the same as in Table
Experimental setup: (a) photograph of the experiment system; (b) schematic of the experiment system.
The acceleration response of the ABH region of the sandwich ABH beam structure was measured and compared with the numerical simulation results, as shown in Figure
Comparison between the numerical simulation and the experimental results.
In this paper, a sandwich acoustic black hole beam has been proposed and a semianalytical model of the sandwich ABH beam with truncation has been established. The displacement field of the proposed ABH beam is fitted by the Mexican wavelet function, and the accuracy of the model of the sandwich ABH beam has been verified by the FEM method.
We have investigated the vibrational acceleration responses of the ABH region and the uniform region of the sandwich ABH beam with the middle layer being aluminum and the upper and lower layers being steel. Simulations reveal that, for the bare ABH beam, there is a certain energy concentration effect in the ABH region in middle and high frequencies. The damping layers attached on the sandwich ABH beam can significantly improve the vibration suppression effect of both the ABH region and the uniform beam region. In the highfrequency domain of 9 kHz, the ABH region and the uniform beam region have attenuation effects of 18.8 dB and 10.9 dB when the damping layers are applied, and the attenuation effect being more than 7 dB in middle and lowfrequency domains. It has been shown that, for every 0.1 increase in the damping loss factor of the damping layers, the acceleration peak values of the ABH region and the uniform beam region can be attenuated by about 1 dB. Numerical results also indicate that the influence of the damping layer thickness on the dynamic response of the sandwich ABH beam is very significant. When the thickness of the damping layer is increased to 3 mm, the highfrequency acceleration peaks at 9 kHz can be completely suppressed. It has been concluded that the material and thickness of the middle layer of the sandwich ABH beam can bring a vibration attenuation at around 9 KHz both for the ABH region and the uniform beam region in the present model.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This study was supported by the Science and Technology Support Program of Jiangsu, China (no. BE2014133), the Prospective Joint Research Program of Jiangsu, China (no. BY201412701), and the Achievement Transformation Project of Nanjing, China (no. 201701213). The financial support is gratefully acknowledged.