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Acoustic absorbers at low frequency based on split-tube metamaterials
Physics Letters A 383 (2019) 2361–2366
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Acoustic absorbers at low frequency based on split-tube metamaterials Peng Wu a,1 , Qianjin Mu a,1 , Xiaoxiao Wu b , Li Wang a , Xin Li a , Yuqing Zhou a , Shuxia Wang a , Yingzhou Huang a,c,∗ , Weijia Wen b,d,∗∗ a
Chongqing Key Laboratory of Soft Condensed Matter Physics and Smart Materials, College of Physics, Chongqing University, Chongqing, 400044, China Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China c Chongqing University, Industrial Technology Research Institute, Chongqing, 400044, China d Materials Genome Institute, Shanghai University, Shanghai, 200444, China b
a r t i c l e
i n f o
Article history: Received 6 March 2019 Received in revised form 5 April 2019 Accepted 25 April 2019 Available online 3 May 2019 Communicated by M. Wu Keywords: Acoustic metamaterial Low-frequency absorption Square split tubes Helmholtz resonance absorption
a b s t r a c t The remarkable properties of acoustic metamaterials have attracted massive researches and applications, especially on low-frequency sound absorptions. Currently, most of the acoustic metamaterial absorbers employ resonances in plastic cavities, and their structural strengths are important in many circumstances, especially in harsh environment. However, studies of metamaterials including this point are very scarce. Here, we propose an acoustic metamaterial for low-frequency (<500 Hz) absorptions, composed of three nested square split tubes with inverted opening directions. The efficiency of the absorber is investigated both numerically and experimentally, and absorptions at the peeks are found to exceed 90% and the frequency can be effectively adjusted by tuning its geometric parameters. We further test its yield strength under compression and confirm its buckling behavior happens from the outmost layer. This tunable acoustic metamaterial with a fairly good mechanical strength may lead to broad applications in noise reduction. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Due to the weak dissipation of common materials in the low frequency region, the absorption of low frequency noises (<500 Hz) is technically challenging but necessary in acoustic researches due to their strong penetrating ability [1–3]. In recent years, the emergence of acoustic metamaterials has provided us with new ideas for solving this problem. Acoustic metamaterial absorbers which can break through the limits of natural materials, greatly enhances the low frequency absorption performance of the absorber with small mass and volume cost [4,5]. Under this concept, a variety of extraordinary acoustic metamaterial absorbers have been designed. Membrane-type acoustic metamaterials designed with impedance-matched decorated membranes do not produce reflections and exhibit good absorptions at low frequency resonances [6–8]. The perfect surface absorber with resonant design allows complete absorption of acoustic waves in a very low
frequency range, but requires a spiral channel of about a quarter wavelength to achieve efficient acoustic energy dissipation [9–12]. However, recently it has been shown that split-tube resonators do not have to employ channels with a quarter wavelength and can produce deep sub-wavelength acoustic bandgaps, achieving lowfrequency absorption with a much simpler structure [13]. In this study, a tunable absorber with good absorption performance in the low frequency band based on the split tube resonators is demonstrated. The sample can be manufactured using the 3D printing method. The simulation obtained through finite element numerical analysis is in the low frequency band and it shows good absorption performance, which is confirmed in the experiment. The geometric parameters which affect its performance have been identified, studied and experimentally proved, such that the absorber can be customized by adjusting these parameters. For more realistic applications, the load capacity of the absorber is theoretically analyzed. Further experiments on its buckling confirms that it indeed has a good load capacity, which is expected to be further promoted and applied in real engineering practice.
*
Corresponding author at: Chongqing Key Laboratory of Soft Condensed Matter Physics and Smart Materials, College of Physics, Chongqing University, Chongqing, 400044, China. Corresponding author at: Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. E-mail addresses: [email protected] (Y. Huang), [email protected] (W. Wen). 1 These authors contributed equally to this work.
**
https://doi.org/10.1016/j.physleta.2019.04.056 0375-9601/© 2019 Elsevier B.V. All rights reserved.
2. Simulation results Perspective schematic of the absorber is shown in Fig. 1(a), which consists of three cascaded square split tubes. Fig. 1(b) shows the sectional schematic of the absorber, with their geometric pa-
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P. Wu et al. / Physics Letters A 383 (2019) 2361–2366
Fig. 1. (a) Perspective schematic of the absorber. (b) Cross-sectional schematic of the absorber. (c) Schematic illustration of the experimental setup for measuring absorptions using the standard two-microphone method, k denotes the incident wave vector. (d) Simulations and experimental results when d = 4 mm and 5 mm, respectively. The black curve is the analog curve and the red curve is the experimental curve. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
rameters labeled accordingly. The length and width of the absorber are a = b = 60 mm, the thickness of walls is t = 3 mm, the total height H = 50 mm, the height of the split tube h = H − 2t = 44 mm, and the opening width is w = 6 mm. It should be noted that we will consider two slightly different samples in the subsequent study; both samples have identical geometric parameters, except in one sample split tube width d = 5 mm, and in the other d = 4 mm. In order to investigate the performance of the absorber, COMSOL Multiphysics is used for full-wave simulations, and the specific parameter settings in the simulations are mentioned detailedly in Appendix A [14,15]. In experiments, we measured the sound absorption performance using a circular impedance tube with a diameter 100 mm (Type 4206, Bruel & Kjaer, Denmark), corresponding to a plane wave cutoff frequency of ∼1600 Hz. The experimental setup is shown in Fig. 1(c), where the conduit with the loudspeaker has two microphones labeled as 1 and 2, which is consistent with the simulation. k represents the incident wave vector and since the rear of the wall is a hard wall condition, we can assume that there is no transmission. The standard twomicrophone method was used to measure the absorption coefficient A throughout the experiments, and this practice is consistent with the full-wave 3D simulations. It should be pointed out that the end of the impedance tube was sealed by an aluminum cover, so a zero transmission can be safely assumed. Fig. 1(d) shows both the simulated and experimental results for d = 4 mm and d = 5 mm, wherein the sample is made of polylactide (PLA) plastic by 3D printing technology, and the experimental absorption reaches maximum around 324 Hz and 384 Hz, respectively. The increase of d decreases the acoustic inductance of the split tubes and the acoustic capacitance of the interior cavity, which increases
the frequency of the absorption peak. An analysis of the acoustic impedance of the absorber is given in Appendix B. It is observed that the measured results agree well with the simulated results, except the fact that the maximum absorption is slightly lower than that in simulations. This discrepancy may be attributed to the fact that the samples are not as rigid as assumed in simulations. The assumption overestimates the pressure difference between inside cavities and the outside environment, which in turn leads to the slight overestimation of the absorption performance in simulations. In reality, the sample surfaces are relatively rough, which will also affect the friction losses near the wall. In addition, imperfect fixing of the samples in the impedance tube would also affect the experimental results non-negligibly. In order to understand its absorbing mechanism, the acoustic pressure and velocity intensity distributions around the absorber near the resonance frequency (380 Hz) and the central cross sections are plotted in Fig. 2(a) and Fig. 2(b) to be analyzed. In order to compare the intensity distribution, the intensity of the frequency slightly off the resonance peak at 420 Hz is plotted, as shown in Fig. 2(c) and Fig. 2(d). From the comparison of Fig. 2(a) and Fig. 2(c), it can be clearly seen that at 380 Hz the difference of acoustic pressure amplitude is 18 Pa between the interior cavity and outside environment, and the difference is reduced to 8 Pa at 420 Hz, indicated by the colors in Fig. 2(a) and Fig. 2(c). When resonance occurs, the two tubes between the three splits can be regarded as two discrete acoustic masses, and the difference in acoustic pressure amplitude between the three splits drives the airflows, and velocity increases significantly in the tubes. As shown in Fig. 2(b), the velocity of the acoustic wave at the entrance reaches 0.1 m/s, and the whole channel are strongly rubbed by the acoustic wave, so that the incident acoustic energy is rapidly attenuated
P. Wu et al. / Physics Letters A 383 (2019) 2361–2366
Fig. 2. Field maps of acoustic pressure (a) and velocity (b) at the center of the resonance peak (380 Hz), compared with the field maps at the frequency (420 Hz) away from the center of the resonance peak, acoustic pressure (c) and velocity (d).
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and converted to the thermal energy at resonance. Therefore, the principle of the absorber is similar to a lossy Helmholtz resonator. The coiled tubes, as long necks, contribute the acoustic inductance and also necessary acoustic resistance for an efficient absorption. Obviously, the function of the interior cavity is the tank of the Helmholtz resonator, contributing the acoustic capacitance. Because the airflow moves slowly in the cavity, no significant friction losses can occur there [16,17]. Velocity of the acoustic wave that deviates slightly from the resonant frequency, such as at 420 Hz, is shown in Fig. 2(d). The maximum speed will drop sharply to 0.03 m/s, hence the fricative dissipation caused by the viscosity of the rapidly moving airflow is at least nine times larger in the resonance state. To understand more of the relationship between structure and acoustic absorption, similar samples with various parameters are numerically investigated here. Fig. 3(a) shows the simulated absorptions with varying h, the geometric parameters of the innermost cavity are fixed. It is found that with the increase of h, the resonance peak of the absorber moves toward the higher frequency, and the reason for this phenomenon is that the increase of h leads to a reduction of the acoustic impedance, including the
Fig. 3. Simulated absorptions of the absorber when (a) h changes, while a3 = b3 = 22 mm is fixed. (b) Simulated absorptions of the absorber when w changes. (c) When t changes, simulated absorptions of the absorber. (d) Simulated absorptions of the absorber when t varies with a = b = 60 mm fixed. (e) Simulated absorptions of the absorber when d varies with a = b = 60 mm fixed. (f) Absorption at 375 Hz plotted as a function of a3 and H , indicated by the color scale between blue (0.600) and red (1.000).
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Fig. 4. (a) A front view of a sample placed in the impedance tube during the experimental test. These samples are made of PLA by 3D printing technology and the measured absorptions of them are shown in (b).
acoustic inductance and acoustic capacitance, of the absorber [18]. In Fig. 3(b), with the increase of the w, the absorption peak shifts to the right, and the movement is due to the reduction in acoustic inductance. In Fig. 3(c), when the geometry of the exterior acoustic absorber is constant, increasing the distance d between the inner and outer absorbers, the absorption peak to move to the right, it can be understood as being caused by the innermost acoustic capacity is reduced. In Fig. 3(d), when the geometry of the exterior absorber is constant at a = b = 60 mm, H = 50 mm and other conditions are maintained, with the thickness t of the absorber increasing, absorption peak moves to the right. The reason for the absorption peak shift can be understood as follows: the acoustic inductance of the split tubes unchanged, and the acoustic capacitance of the body is reduced. It can be seen in Fig. 2(a) that the inside and outside acoustic pressures of the absorber change greatly, and the acoustic attenuation is caused by the large friction between the acoustic wave and the channel, so the reduction of inner acoustic capacitance is the main cause and leads to the movement of the peak. It should also be pointed out that the thickness of the absorber should not be too thin, otherwise, the vibrations of the walls can not be ignored, and then they can no longer be regarded as rigid in simulations. In Fig. 3(e), as the value of d increases, the acoustic inductance of the split tubes decreases, and the acoustic capacitance also decreases to some extent, so the main peak moves to the right. In the experiment, we found that in a small range, the H and a3 were adjusted, and the absorption peak was shifted slightly, which leads to a certain degree of enhancement to the absorption rate. When changing a3 and H , the simulated result is shown as a contour map in Fig. 3(f), and the color changes from blue area to (absorption 0.600) to red area (absorption 1.000). Based on the result, we can choose adequate geometric parameters to enhance absorptions for specific frequencies. This is consistent with our expectations. The general increase of absorption when increasing H is caused by a better impedance match between the absorber and the outside environment, which minimizes the reflection of the sounds. 3. Experiment results During the change of the above geometric parameters, the absorption is still maintained at high efficiency when the corresponding absorption peak is sensitively shifted, which indicates that it is possible to manufacture the absorber for a specific frequency. Taking the air conditioners in real life as a typical example, the energy density of their annoying noise is usually concentrated between 125–500 Hz. Considering the influence of the above parameters on
Table 1 The specific geometric parameters of the three samples used in experiments. Sample
a (mm)
b (mm)
d (mm)
t (mm)
w (mm)
h (mm)
H (mm)
1 2 3
60 60 60
60 60 60
2 4 5
3 3 5
3.5 4 4.625
62 42 42
68 50 50
the main peak, we deliberately modulate the absorber such that its main peak is shifted to 230, 320, and 500 Hz, respectively, as summarized in Table 1. Fig. 4(a) shows a front view of a sample placed in the impedance tube during the experiments and the measured results are shown in Fig. 4(b). The absorption of the three samples all reached very high values, which confirms the feasibility of customizing the main peak of the absorber according to the simulations. Taking into account the engineering applications of the metamaterial absorber, stress analysis has been carried out to estimate its compressive strength, which is crucial for practical engineering applications. It is imaginable that when a fixed stress is applied from directly above, the exterior part is relatively fragile and easy to break since it in reality contains cracks and the thickness of the wall is less than 5 mm, so we set the outmost layer as the maximum force point under the safe range and carry out theoretical analysis [19–21]. Assuming S is the cross-sectional area, and the cross section is divided into three sections from the inside to outside, we have [22]
S 1 = 4(a − 4t − 4d)t − 4t 2 − t w S 2 = 4(a − 2t − 2d)t − 4t 2 − t w
(1)
2
S 3 = 4at − 4t − t w Since PLA is a brittle material, the compressive strength is comparable to the tensile strength [23,24]. Therefore, the yield strength can be used to estimate the compressive strength when plastic deformation is apparent under pressure:
σs = 60 MPa
(2)
Therefore, when a = b = 60 mm, H = 50 mm, d = 5 mm, t = 3 mm, w = 6 mm, h = H − 2t,
F max = σs × ( S 1 + S 2 + S 3 ) = 85.32 kN
(3)
Subsequently, we performed a pressure test using a compression material test system and the measured maximum strength in the safe range is F max = 67.85 kN. The experimental details of the
P. Wu et al. / Physics Letters A 383 (2019) 2361–2366
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Fig. 5. (a) Photograph of a real sample used in the compressive strength test, which finally buckled. (b) The measured deformation-force curve during the pressure test. The red dot denotes where the buckling happens.
material test are enclosed in Appendix C. The buckled sample after the pressure test is shown in Fig. 5(a), and Fig. 5(b) shows the deformation-force curve of the sample during the compression process. A photograph of the compression material test system loaded with the sample is shown in Fig. 5(c). It can be clearly seen that in the process of slowly increasing the pressure to the ultimate value, the exterior layer first buckles, and then, a small force causes the model to have a large-area fracture. The maximum strength from theoretical analysis is larger than the experimental one, which is as understandable since the maximum stress measured in the experiment is not only determined by the deformation under the maximum principal stress. Since the wall thickness t of the test sample is equal to 3 mm, it belongs to the unstable thin-walled structure. In the theoretical analysis, we have idealized the model sample, which is not completely compatible for our experimental conditions, which is the main factor that causes the deviation between theory and experiment. Nevertheless, it proves that the absorber has a good compressive strength and the buckling starts from the outmost layer. Further, in engineering practice, we can use steel materials, concrete structures and other building materials to fabricate the absorber, which have much better mechanical properties and fire performances, while maintaining the good absorption performance for low frequency sounds since our absorber does not depend on specific material properties. 4. Conclusion A tunable acoustic metamaterial absorber with excellent lowfrequency performance has been reported. Our results demonstrate that the acoustic properties of this metamaterial absorber composed of split tubes are highly parameter-dependent and can selectively absorb noises. Through the simulation results of acoustic pressure and air velocity distribution, the metamaterial can be regarded as a Helmholtz resonator, and experimental measurement confirms that it has a good load capacity. Therefore, it has broad prospects in applications, such as being used as building materials since it can be easily processed into sound absorbing bricks. Its high stability and load capacity would also boost its applications in harsh environment. Acknowledgements This work was supported Fundamental Research Funds for the Central Universities (2018CDXYWU0025, 2018CDJDWL0011), Key Technology Innovation Project in Key Industry of Chongqing (cstc2017zdcy-zdyf0338), Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No.
KJQN201800101), Areas of Excellence Scheme (AOE/P-02/12) from Research Grants Council (RGC) of Hong Kong, and Sharing Fund of Chongqing University’s Large-scale Equipment. Appendix A. Details of the simulation setup The full-wave simulations were performed in COMSOL Multiphysics, and the module “Pressure Acoustics, Frequency Domain” was employed. The node “Narrow Region Acoustics” was used to model the thermal dissipation in the nested square split tubes. In a simulation, the incident acoustic plane wave impinged on the sample from the −x direction, and its amplitude was set to 1 Pa, corresponding to a sound pressure level of 91 dB. The walls of the absorber were assumed as rigid in simulations because of the large impedance mismatch between air and solid materials. The speed of sound in air was set as c air = 343 m/s. The density, dynamic viscosity, and specific heat ratio of air were set as ρair = 1.293 kg/m3 , μdyn = 1.85 × 10−5 Pa·s, and γ = 1.40, respectively. The largest mesh element size was smaller than 1/10 of the shortest incident wavelength. Appendix B. Acoustic impedance of the metamaterial absorber The metamaterial absorber, when considering its Helmholtz resonance, can be analyzed using a lumped RLC circuit model. The three square split tubes contribute most acoustic inductance and resistance, and the interior cavity contributes most acoustic capacitance. Using formulae in the literature [25], it is found that the total acoustic inductance of the two tubes and three splits is
L=
=
ρair (a + b − 4t − 2d + a + b − 8t − 6d) dh 2ρair (a + b − 4d − 6t ) dh
+
3ρair t wh
+
3ρair t wh
(4)
.
The acoustic resistance of the split tube can be estimated as
R=
δ h
ωL =
where δ =
2ρair ω(a + b − 4d − 6t )δ
dh2 2μdyn
ωρ0
+
3ρair ωt δ wh2
,
(5)
is the thickness of the viscous boundary layer.
The acoustic capacitance of the interior cavity is
C=
(a − 4d − 6t )(b − 4d − 6t )h
γ p0
,
(6)
where p 0 is the pressure at equilibrium (atmospheric pressure). Since the inductance, resistance, and capacitance are connected in series, the total acoustic impedance Z of the absorber is hence
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where Z 0 = ρair c air /(π R 2 ) is the impedance of the circular impedance tube (radius R = 50 mm). The analytically calculated results are shown in Fig. A1, where the corresponding numerical and experimental results are also plotted, and a general agreement can be seen. Considering that we only use very rough estimations when deriving the acoustic impedance, the agreement, especially the position of the main absorption peak, has confirmed the correctness of our analytical model. Appendix C. Details of the compression material test system
Fig. A1. Comparison between absorptions obtained from simulation, experiment, and analytical expression when d = 4 mm. A general agreement can be observed.
The compressive strength tests were performed on a compression material test system, which is a commercial electromechanical universal testing machine (UTM), model CMT5105 (MTS, China). A compression plate kit of fixed type was used to load the sample and its maximum force capacity is 100 kN. Photographs taken during a compression test are shown in Fig. A2. References
Fig. A2. (a) Full view of the setup of a compression test. The sample (white plastic) is loaded on plate of the electromechanical UTM. (b) The buckling of the sample under compression during the test.
Z = iω L + R +
δ = ω(i + )[ h
+
1
iωC 2ρair (a + b − 6t − 4d)
γ p0
dh
i ω(a − 4d − 6t )(b − 4d − 6t )h
+
3ρair t wh
]
(7)
.
To verify the result, we calculate the absorption from the analytic acoustic impedance
A =1−|
Z − Z0 Z + Z0
|2 ,
(8)
[1] X. Wu, K.Y. Au-Yeung, X. Li, R.C. Roberts, J. Tian, C. Hu, Y. Huang, S. Wang, Z. Yang, W. Wen, Appl. Phys. Lett. 112 (2018) 103505. [2] J. Mei, G. Ma, M. Yang, Z. Yang, W. Wen, P. Sheng, Nat. Commun. 3 (2012) 756. [3] H. Ge, M. Yang, C. Ma, M.H. Lu, Y.F. Chen, N. Fang, P. Sheng, Nat. Sci. Rev. 5 (2018) 45–68. [4] G. Ma, M. Yang, S. Xiao, Z. Yang, P. Sheng, Nat. Mater. 13 (2014) 873–878. [5] H. Ge, M. Yang, C. Ma, M.-H. Lu, Y.-F. Chen, N. Fang, P. Sheng, Nat. Sci. Rev. 5 (2018) 159–182. [6] Z. Yang, J. Mei, M. Yang, N.H. Chan, P. Sheng, Phys. Rev. Lett. 101 (2008) 204301. [7] C.J. Naify, C.M. Chang, G. Mcknight, S. Nutt, J. Appl. Phys. 108 (2010) 204301. [8] L. Sun, K.Y. Au-Yeung, M. Yang, S.T. Tang, P. Sheng, AIP Adv. 6 (2016) 619–626. [9] K.M. Ho, Z. Yang, X.X. Zhang, P. Sheng, Appl. Acoust. 66 (2005) 751–765. [10] X. Cai, Q. Guo, G. Hu, J. Yang, Appl. Phys. Lett. 105 (2014) 339–356. [11] L. Yong, B.M. Assouar, Appl. Phys. Lett. 108 (2016) 204301. [12] Y. Li, B.M. Assouar, Appl. Phys. Lett. 108 (2016) 204301. [13] J. Li, J.H. Wu, G. Dong, N. Gao, J. Appl. Phys. 116 (2014) 184506. [14] X. Wu, C. Fu, X. Li, Y. Meng, Y. Gao, J. Tian, L. Wang, Y. Huang, Z. Yang, W. Wen, Appl. Phys. Lett. 109 (2016) e1501595. [15] N.I. Landy, S. Sajuyigbe, J.J. Mock, D.R. Smith, W.J. Padilla, Phys. Rev. Lett. 100 (2008) 207402. [16] H.V. Fuchs, Acta Acust. Acust. 86 (2000) 581–583. [17] K. Sakagami, M. Morimoto, W. Koike, Appl. Acoust. 67 (2006) 609–619. [18] C. Cai, C.M. Mak, Adv. Eng. Softw. 116 (2018) 60–66. [19] W.D. Pilkey, D.F. Pilkey, Peterson’s Stress Concentration Factors, John Wiley & Sons, New York, 2007, pp. 24–34. [20] H.L. He, R.Z. Li, Int. J. Solids Struct. 43 (2006) 6208–6219. [21] B. Grüber, W. Hufenbach, L. Kroll, M. Lepper, B. Zhou, Compos. Sci. Technol. 67 (2007) 1439–1450. [22] F.P. Beer, R. Johnston, J. Dewolf, D. Mazurek, Mechanics of Materials, McGrawHill, Boston, 2006, pp. 209–210. [23] N. Graupner, G. Ziegmann, J. Müssig, Compos. Sci. Technol. 149 (2017) S0266353816313811. [24] J.C. Chen, J.C. Lin, Int. J. Mod. Phys. B (2018) 1840084. [25] D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley & Sons, New York, 2001, pp. 144–162.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Acoustic absorbers at low frequency based on split-tube metamaterials Peng Wu a,1 , Qianjin Mu a,1 , Xiaoxiao Wu b , Li Wang a , Xin Li a , Yuqing Zhou a , Shuxia Wang a , Yingzhou Huang a,c,∗ , Weijia Wen b,d,∗∗ a
Chongqing Key Laboratory of Soft Condensed Matter Physics and Smart Materials, College of Physics, Chongqing University, Chongqing, 400044, China Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China c Chongqing University, Industrial Technology Research Institute, Chongqing, 400044, China d Materials Genome Institute, Shanghai University, Shanghai, 200444, China b
a r t i c l e
i n f o
Article history: Received 6 March 2019 Received in revised form 5 April 2019 Accepted 25 April 2019 Available online 3 May 2019 Communicated by M. Wu Keywords: Acoustic metamaterial Low-frequency absorption Square split tubes Helmholtz resonance absorption
a b s t r a c t The remarkable properties of acoustic metamaterials have attracted massive researches and applications, especially on low-frequency sound absorptions. Currently, most of the acoustic metamaterial absorbers employ resonances in plastic cavities, and their structural strengths are important in many circumstances, especially in harsh environment. However, studies of metamaterials including this point are very scarce. Here, we propose an acoustic metamaterial for low-frequency (<500 Hz) absorptions, composed of three nested square split tubes with inverted opening directions. The efficiency of the absorber is investigated both numerically and experimentally, and absorptions at the peeks are found to exceed 90% and the frequency can be effectively adjusted by tuning its geometric parameters. We further test its yield strength under compression and confirm its buckling behavior happens from the outmost layer. This tunable acoustic metamaterial with a fairly good mechanical strength may lead to broad applications in noise reduction. © 2019 Elsevier B.V. All rights reserved.
1. Introduction Due to the weak dissipation of common materials in the low frequency region, the absorption of low frequency noises (<500 Hz) is technically challenging but necessary in acoustic researches due to their strong penetrating ability [1–3]. In recent years, the emergence of acoustic metamaterials has provided us with new ideas for solving this problem. Acoustic metamaterial absorbers which can break through the limits of natural materials, greatly enhances the low frequency absorption performance of the absorber with small mass and volume cost [4,5]. Under this concept, a variety of extraordinary acoustic metamaterial absorbers have been designed. Membrane-type acoustic metamaterials designed with impedance-matched decorated membranes do not produce reflections and exhibit good absorptions at low frequency resonances [6–8]. The perfect surface absorber with resonant design allows complete absorption of acoustic waves in a very low
frequency range, but requires a spiral channel of about a quarter wavelength to achieve efficient acoustic energy dissipation [9–12]. However, recently it has been shown that split-tube resonators do not have to employ channels with a quarter wavelength and can produce deep sub-wavelength acoustic bandgaps, achieving lowfrequency absorption with a much simpler structure [13]. In this study, a tunable absorber with good absorption performance in the low frequency band based on the split tube resonators is demonstrated. The sample can be manufactured using the 3D printing method. The simulation obtained through finite element numerical analysis is in the low frequency band and it shows good absorption performance, which is confirmed in the experiment. The geometric parameters which affect its performance have been identified, studied and experimentally proved, such that the absorber can be customized by adjusting these parameters. For more realistic applications, the load capacity of the absorber is theoretically analyzed. Further experiments on its buckling confirms that it indeed has a good load capacity, which is expected to be further promoted and applied in real engineering practice.
*
Corresponding author at: Chongqing Key Laboratory of Soft Condensed Matter Physics and Smart Materials, College of Physics, Chongqing University, Chongqing, 400044, China. Corresponding author at: Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China. E-mail addresses: [email protected] (Y. Huang), [email protected] (W. Wen). 1 These authors contributed equally to this work.
**
https://doi.org/10.1016/j.physleta.2019.04.056 0375-9601/© 2019 Elsevier B.V. All rights reserved.
2. Simulation results Perspective schematic of the absorber is shown in Fig. 1(a), which consists of three cascaded square split tubes. Fig. 1(b) shows the sectional schematic of the absorber, with their geometric pa-
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P. Wu et al. / Physics Letters A 383 (2019) 2361–2366
Fig. 1. (a) Perspective schematic of the absorber. (b) Cross-sectional schematic of the absorber. (c) Schematic illustration of the experimental setup for measuring absorptions using the standard two-microphone method, k denotes the incident wave vector. (d) Simulations and experimental results when d = 4 mm and 5 mm, respectively. The black curve is the analog curve and the red curve is the experimental curve. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
rameters labeled accordingly. The length and width of the absorber are a = b = 60 mm, the thickness of walls is t = 3 mm, the total height H = 50 mm, the height of the split tube h = H − 2t = 44 mm, and the opening width is w = 6 mm. It should be noted that we will consider two slightly different samples in the subsequent study; both samples have identical geometric parameters, except in one sample split tube width d = 5 mm, and in the other d = 4 mm. In order to investigate the performance of the absorber, COMSOL Multiphysics is used for full-wave simulations, and the specific parameter settings in the simulations are mentioned detailedly in Appendix A [14,15]. In experiments, we measured the sound absorption performance using a circular impedance tube with a diameter 100 mm (Type 4206, Bruel & Kjaer, Denmark), corresponding to a plane wave cutoff frequency of ∼1600 Hz. The experimental setup is shown in Fig. 1(c), where the conduit with the loudspeaker has two microphones labeled as 1 and 2, which is consistent with the simulation. k represents the incident wave vector and since the rear of the wall is a hard wall condition, we can assume that there is no transmission. The standard twomicrophone method was used to measure the absorption coefficient A throughout the experiments, and this practice is consistent with the full-wave 3D simulations. It should be pointed out that the end of the impedance tube was sealed by an aluminum cover, so a zero transmission can be safely assumed. Fig. 1(d) shows both the simulated and experimental results for d = 4 mm and d = 5 mm, wherein the sample is made of polylactide (PLA) plastic by 3D printing technology, and the experimental absorption reaches maximum around 324 Hz and 384 Hz, respectively. The increase of d decreases the acoustic inductance of the split tubes and the acoustic capacitance of the interior cavity, which increases
the frequency of the absorption peak. An analysis of the acoustic impedance of the absorber is given in Appendix B. It is observed that the measured results agree well with the simulated results, except the fact that the maximum absorption is slightly lower than that in simulations. This discrepancy may be attributed to the fact that the samples are not as rigid as assumed in simulations. The assumption overestimates the pressure difference between inside cavities and the outside environment, which in turn leads to the slight overestimation of the absorption performance in simulations. In reality, the sample surfaces are relatively rough, which will also affect the friction losses near the wall. In addition, imperfect fixing of the samples in the impedance tube would also affect the experimental results non-negligibly. In order to understand its absorbing mechanism, the acoustic pressure and velocity intensity distributions around the absorber near the resonance frequency (380 Hz) and the central cross sections are plotted in Fig. 2(a) and Fig. 2(b) to be analyzed. In order to compare the intensity distribution, the intensity of the frequency slightly off the resonance peak at 420 Hz is plotted, as shown in Fig. 2(c) and Fig. 2(d). From the comparison of Fig. 2(a) and Fig. 2(c), it can be clearly seen that at 380 Hz the difference of acoustic pressure amplitude is 18 Pa between the interior cavity and outside environment, and the difference is reduced to 8 Pa at 420 Hz, indicated by the colors in Fig. 2(a) and Fig. 2(c). When resonance occurs, the two tubes between the three splits can be regarded as two discrete acoustic masses, and the difference in acoustic pressure amplitude between the three splits drives the airflows, and velocity increases significantly in the tubes. As shown in Fig. 2(b), the velocity of the acoustic wave at the entrance reaches 0.1 m/s, and the whole channel are strongly rubbed by the acoustic wave, so that the incident acoustic energy is rapidly attenuated
P. Wu et al. / Physics Letters A 383 (2019) 2361–2366
Fig. 2. Field maps of acoustic pressure (a) and velocity (b) at the center of the resonance peak (380 Hz), compared with the field maps at the frequency (420 Hz) away from the center of the resonance peak, acoustic pressure (c) and velocity (d).
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and converted to the thermal energy at resonance. Therefore, the principle of the absorber is similar to a lossy Helmholtz resonator. The coiled tubes, as long necks, contribute the acoustic inductance and also necessary acoustic resistance for an efficient absorption. Obviously, the function of the interior cavity is the tank of the Helmholtz resonator, contributing the acoustic capacitance. Because the airflow moves slowly in the cavity, no significant friction losses can occur there [16,17]. Velocity of the acoustic wave that deviates slightly from the resonant frequency, such as at 420 Hz, is shown in Fig. 2(d). The maximum speed will drop sharply to 0.03 m/s, hence the fricative dissipation caused by the viscosity of the rapidly moving airflow is at least nine times larger in the resonance state. To understand more of the relationship between structure and acoustic absorption, similar samples with various parameters are numerically investigated here. Fig. 3(a) shows the simulated absorptions with varying h, the geometric parameters of the innermost cavity are fixed. It is found that with the increase of h, the resonance peak of the absorber moves toward the higher frequency, and the reason for this phenomenon is that the increase of h leads to a reduction of the acoustic impedance, including the
Fig. 3. Simulated absorptions of the absorber when (a) h changes, while a3 = b3 = 22 mm is fixed. (b) Simulated absorptions of the absorber when w changes. (c) When t changes, simulated absorptions of the absorber. (d) Simulated absorptions of the absorber when t varies with a = b = 60 mm fixed. (e) Simulated absorptions of the absorber when d varies with a = b = 60 mm fixed. (f) Absorption at 375 Hz plotted as a function of a3 and H , indicated by the color scale between blue (0.600) and red (1.000).
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Fig. 4. (a) A front view of a sample placed in the impedance tube during the experimental test. These samples are made of PLA by 3D printing technology and the measured absorptions of them are shown in (b).
acoustic inductance and acoustic capacitance, of the absorber [18]. In Fig. 3(b), with the increase of the w, the absorption peak shifts to the right, and the movement is due to the reduction in acoustic inductance. In Fig. 3(c), when the geometry of the exterior acoustic absorber is constant, increasing the distance d between the inner and outer absorbers, the absorption peak to move to the right, it can be understood as being caused by the innermost acoustic capacity is reduced. In Fig. 3(d), when the geometry of the exterior absorber is constant at a = b = 60 mm, H = 50 mm and other conditions are maintained, with the thickness t of the absorber increasing, absorption peak moves to the right. The reason for the absorption peak shift can be understood as follows: the acoustic inductance of the split tubes unchanged, and the acoustic capacitance of the body is reduced. It can be seen in Fig. 2(a) that the inside and outside acoustic pressures of the absorber change greatly, and the acoustic attenuation is caused by the large friction between the acoustic wave and the channel, so the reduction of inner acoustic capacitance is the main cause and leads to the movement of the peak. It should also be pointed out that the thickness of the absorber should not be too thin, otherwise, the vibrations of the walls can not be ignored, and then they can no longer be regarded as rigid in simulations. In Fig. 3(e), as the value of d increases, the acoustic inductance of the split tubes decreases, and the acoustic capacitance also decreases to some extent, so the main peak moves to the right. In the experiment, we found that in a small range, the H and a3 were adjusted, and the absorption peak was shifted slightly, which leads to a certain degree of enhancement to the absorption rate. When changing a3 and H , the simulated result is shown as a contour map in Fig. 3(f), and the color changes from blue area to (absorption 0.600) to red area (absorption 1.000). Based on the result, we can choose adequate geometric parameters to enhance absorptions for specific frequencies. This is consistent with our expectations. The general increase of absorption when increasing H is caused by a better impedance match between the absorber and the outside environment, which minimizes the reflection of the sounds. 3. Experiment results During the change of the above geometric parameters, the absorption is still maintained at high efficiency when the corresponding absorption peak is sensitively shifted, which indicates that it is possible to manufacture the absorber for a specific frequency. Taking the air conditioners in real life as a typical example, the energy density of their annoying noise is usually concentrated between 125–500 Hz. Considering the influence of the above parameters on
Table 1 The specific geometric parameters of the three samples used in experiments. Sample
a (mm)
b (mm)
d (mm)
t (mm)
w (mm)
h (mm)
H (mm)
1 2 3
60 60 60
60 60 60
2 4 5
3 3 5
3.5 4 4.625
62 42 42
68 50 50
the main peak, we deliberately modulate the absorber such that its main peak is shifted to 230, 320, and 500 Hz, respectively, as summarized in Table 1. Fig. 4(a) shows a front view of a sample placed in the impedance tube during the experiments and the measured results are shown in Fig. 4(b). The absorption of the three samples all reached very high values, which confirms the feasibility of customizing the main peak of the absorber according to the simulations. Taking into account the engineering applications of the metamaterial absorber, stress analysis has been carried out to estimate its compressive strength, which is crucial for practical engineering applications. It is imaginable that when a fixed stress is applied from directly above, the exterior part is relatively fragile and easy to break since it in reality contains cracks and the thickness of the wall is less than 5 mm, so we set the outmost layer as the maximum force point under the safe range and carry out theoretical analysis [19–21]. Assuming S is the cross-sectional area, and the cross section is divided into three sections from the inside to outside, we have [22]
S 1 = 4(a − 4t − 4d)t − 4t 2 − t w S 2 = 4(a − 2t − 2d)t − 4t 2 − t w
(1)
2
S 3 = 4at − 4t − t w Since PLA is a brittle material, the compressive strength is comparable to the tensile strength [23,24]. Therefore, the yield strength can be used to estimate the compressive strength when plastic deformation is apparent under pressure:
σs = 60 MPa
(2)
Therefore, when a = b = 60 mm, H = 50 mm, d = 5 mm, t = 3 mm, w = 6 mm, h = H − 2t,
F max = σs × ( S 1 + S 2 + S 3 ) = 85.32 kN
(3)
Subsequently, we performed a pressure test using a compression material test system and the measured maximum strength in the safe range is F max = 67.85 kN. The experimental details of the
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Fig. 5. (a) Photograph of a real sample used in the compressive strength test, which finally buckled. (b) The measured deformation-force curve during the pressure test. The red dot denotes where the buckling happens.
material test are enclosed in Appendix C. The buckled sample after the pressure test is shown in Fig. 5(a), and Fig. 5(b) shows the deformation-force curve of the sample during the compression process. A photograph of the compression material test system loaded with the sample is shown in Fig. 5(c). It can be clearly seen that in the process of slowly increasing the pressure to the ultimate value, the exterior layer first buckles, and then, a small force causes the model to have a large-area fracture. The maximum strength from theoretical analysis is larger than the experimental one, which is as understandable since the maximum stress measured in the experiment is not only determined by the deformation under the maximum principal stress. Since the wall thickness t of the test sample is equal to 3 mm, it belongs to the unstable thin-walled structure. In the theoretical analysis, we have idealized the model sample, which is not completely compatible for our experimental conditions, which is the main factor that causes the deviation between theory and experiment. Nevertheless, it proves that the absorber has a good compressive strength and the buckling starts from the outmost layer. Further, in engineering practice, we can use steel materials, concrete structures and other building materials to fabricate the absorber, which have much better mechanical properties and fire performances, while maintaining the good absorption performance for low frequency sounds since our absorber does not depend on specific material properties. 4. Conclusion A tunable acoustic metamaterial absorber with excellent lowfrequency performance has been reported. Our results demonstrate that the acoustic properties of this metamaterial absorber composed of split tubes are highly parameter-dependent and can selectively absorb noises. Through the simulation results of acoustic pressure and air velocity distribution, the metamaterial can be regarded as a Helmholtz resonator, and experimental measurement confirms that it has a good load capacity. Therefore, it has broad prospects in applications, such as being used as building materials since it can be easily processed into sound absorbing bricks. Its high stability and load capacity would also boost its applications in harsh environment. Acknowledgements This work was supported Fundamental Research Funds for the Central Universities (2018CDXYWU0025, 2018CDJDWL0011), Key Technology Innovation Project in Key Industry of Chongqing (cstc2017zdcy-zdyf0338), Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No.
KJQN201800101), Areas of Excellence Scheme (AOE/P-02/12) from Research Grants Council (RGC) of Hong Kong, and Sharing Fund of Chongqing University’s Large-scale Equipment. Appendix A. Details of the simulation setup The full-wave simulations were performed in COMSOL Multiphysics, and the module “Pressure Acoustics, Frequency Domain” was employed. The node “Narrow Region Acoustics” was used to model the thermal dissipation in the nested square split tubes. In a simulation, the incident acoustic plane wave impinged on the sample from the −x direction, and its amplitude was set to 1 Pa, corresponding to a sound pressure level of 91 dB. The walls of the absorber were assumed as rigid in simulations because of the large impedance mismatch between air and solid materials. The speed of sound in air was set as c air = 343 m/s. The density, dynamic viscosity, and specific heat ratio of air were set as ρair = 1.293 kg/m3 , μdyn = 1.85 × 10−5 Pa·s, and γ = 1.40, respectively. The largest mesh element size was smaller than 1/10 of the shortest incident wavelength. Appendix B. Acoustic impedance of the metamaterial absorber The metamaterial absorber, when considering its Helmholtz resonance, can be analyzed using a lumped RLC circuit model. The three square split tubes contribute most acoustic inductance and resistance, and the interior cavity contributes most acoustic capacitance. Using formulae in the literature [25], it is found that the total acoustic inductance of the two tubes and three splits is
L=
=
ρair (a + b − 4t − 2d + a + b − 8t − 6d) dh 2ρair (a + b − 4d − 6t ) dh
+
3ρair t wh
+
3ρair t wh
(4)
.
The acoustic resistance of the split tube can be estimated as
R=
δ h
ωL =
where δ =
2ρair ω(a + b − 4d − 6t )δ
dh2 2μdyn
ωρ0
+
3ρair ωt δ wh2
,
(5)
is the thickness of the viscous boundary layer.
The acoustic capacitance of the interior cavity is
C=
(a − 4d − 6t )(b − 4d − 6t )h
γ p0
,
(6)
where p 0 is the pressure at equilibrium (atmospheric pressure). Since the inductance, resistance, and capacitance are connected in series, the total acoustic impedance Z of the absorber is hence
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where Z 0 = ρair c air /(π R 2 ) is the impedance of the circular impedance tube (radius R = 50 mm). The analytically calculated results are shown in Fig. A1, where the corresponding numerical and experimental results are also plotted, and a general agreement can be seen. Considering that we only use very rough estimations when deriving the acoustic impedance, the agreement, especially the position of the main absorption peak, has confirmed the correctness of our analytical model. Appendix C. Details of the compression material test system
Fig. A1. Comparison between absorptions obtained from simulation, experiment, and analytical expression when d = 4 mm. A general agreement can be observed.
The compressive strength tests were performed on a compression material test system, which is a commercial electromechanical universal testing machine (UTM), model CMT5105 (MTS, China). A compression plate kit of fixed type was used to load the sample and its maximum force capacity is 100 kN. Photographs taken during a compression test are shown in Fig. A2. References
Fig. A2. (a) Full view of the setup of a compression test. The sample (white plastic) is loaded on plate of the electromechanical UTM. (b) The buckling of the sample under compression during the test.
Z = iω L + R +
δ = ω(i + )[ h
+
1
iωC 2ρair (a + b − 6t − 4d)
γ p0
dh
i ω(a − 4d − 6t )(b − 4d − 6t )h
+
3ρair t wh
]
(7)
.
To verify the result, we calculate the absorption from the analytic acoustic impedance
A =1−|
Z − Z0 Z + Z0
|2 ,
(8)
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