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Acoustic Black Holes in Structural Design for Vibration and Noise Control

Acoustic Black Holes in Structural Design for Vibration and Noise Control acoustics Review Acoustic Black Holes in Structural Design for Vibration and Noise Control Chenhui Zhao and Marehalli G. Prasad * Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA; czhao1@stevens.edu * Correspondence: mprasad@stevens.edu; Tel.: +1-201-216-5571 Received: 2 October 2018; Accepted: 28 January 2019; Published: 25 February 2019 Abstract: It is known that in the design of quieter mechanical systems, vibration and noise control play important roles. Recently, acoustic black holes have been effectively used for structural design in controlling vibration and noise. An acoustic black hole is a power-law tapered profile to reduce phase and group velocities of wave propagation to zero. Additionally, the vibration energy at the location of acoustic black hole increases due to the gradual reduction of its thickness. The vibration damping, sound reduction, and vibration energy harvesting are the major applications in structural design with acoustic black holes. In this paper, a review of basic theoretical, numerical, and experimental studies on the applications of acoustic black holes is presented. In addition, the influences of the various geometrical parameters and the configuration of acoustic black holes are presented. The studies show that the use of acoustic black holes results in an effective control of vibration and noise. It is seen that the acoustic black holes have a great potential for quiet design of complex structures. Keywords: acoustic black hole; structure design; noise and vibration control 1. Introduction It is known that with the development of high-speed machinery, the control of unwanted vibration and noise are very important for their stability and reliability, as well as the environmental noise impact [1]. The two well-known methods for passive control of structural vibrations which also results in a reduction of noise are constrained layer damping and tuned dynamic absorbers [2]. The first method is based on using a viscoelastic layer attached to the structure and the second method needs an attachment of additional weight [3] to the target structure. Additionally, the active vibration control devices are also used for vibration damping [4]. However, these active methods require consistent input energy and more complex electro-mechanical design. Thus, for the reasons of limitation of size, budget, or weight, sometimes it is not possible, and also is not desirable, to use these above methods. There is always a need for an effective design of structures for vibration and noise control [5]. In recent times micro-devices, such as portable electronics and wireless remote sensors, are developed and widely used [6]. Most of these low-power electronics are powered by battery. However, even for the long-lasting batteries, they still need to be replaced because of their limited lifecycle. For some applications, such as sensors deployed in remote locations or inside the human body [7], it is challenging and costly, or even impractical. Energy harvesting is the process of capturing and converting ambient energy in the environment into usable electrical energy to extend the life of batteries, which makes the devices self-sustainable and environmental-friendly. Piezoelectric vibration energy harvesting (PVEH) is one of the typical energy harvesting methods. In the design of portable micro devices, the challenge is to reduce the weight and size of the host structure. Thus, approaches to increase the energy harvested from the vibrations of the host structures are desirable [8]. Acoustics 2019, 1, 220–251; doi:10.3390/acoustics1010014 www.mdpi.com/journal/acoustics Acoustics 2018, 1, x FOR PEER REVIEW 2 of 29 structure. Thus, approaches to increase the energy harvested from the vibrations of the host structures are desirable [8]. Acoustics 2019, 1 221 Recently, an approach for passive vibration control, acoustic black holes (ABH), has been developed. An ABH is usually a power-law tapered profile built on structures, such as beams, plates, Recently, an approach for passive vibration control, acoustic black holes (ABH), has been and shells, where the vibration energy is concentrated due to the reduction of wave speed [9] (as developed. An ABH is usually a power-law tapered profile built on structures, such as beams, shown in Figure 1). Therefore, due to this concentration effect of the ABH, the vibration energy can plates, and shells, where the vibration energy is concentrated due to the reduction of wave speed [9] be absorbed by attaching small amount of damping material at the ABH location, which also results (as shown in Figure 1). Therefore, due to this concentration effect of the ABH, the vibration energy can in reduced sound radiation. Additionally, the performance of energy harvesting is enhanced by be absorbed by attaching small amount of damping material at the ABH location, which also results in attaching piezoelectric material at the ABH location [10]. An ABH is a tailing method which cuts reduced sound radiation. Additionally, the performance of energy harvesting is enhanced by attaching material away from the host structure, and it also decreases the usage of damping layer, so it piezoelectric material at the ABH location [10]. An ABH is a tailing method which cuts material away decreases the weight of host structures. Therefore, it is a good option for vibration and noise control from the host structure, and it also decreases the usage of damping layer, so it decreases the weight of of lightweight structures. host structures. Therefore, it is a good option for vibration and noise control of lightweight structures. The ABH effect was first discovered by Pekeris in 1946 [11]. He exploited the central physical The ABH effect was first discovered by Pekeris in 1946 [11]. He exploited the central physical principle of ABH, namely the phase velocity of sound waves that propagate in a stratified fluid are principle of ABH, namely the phase velocity of sound waves that propagate in a stratified fluid are progressively decreased to zero with increasing depth. In 1988 Mironov determined that a flexural progressively decreased to zero with increasing depth. In 1988 Mironov determined that a flexural wave propagates in a thin plate slows down and needs infinite time to reach a tapered edge [12]. wave propagates in a thin plate slows down and needs infinite time to reach a tapered edge [12]. Later, Krylov first used the name “acoustic black hole” to this effect [13], and applied ABH on beams Later, Krylov first used the name “acoustic black hole” to this effect [13], and applied ABH on and plates, also indicating that the ABH approach results in an increased amount of energy to be beams and plates, also indicating that the ABH approach results in an increased amount of energy absorbed by adding a small amount of material attenuation near the ABH locations [14–17]. Then to be absorbed by adding a small amount of material attenuation near the ABH locations [14–17]. Conlon developed further numerical and experimental work to analyze the ABH effect on vibration Then Conlon developed further numerical and experimental work to analyze the ABH effect on and sound radiation of thin plates [18–20]. Later, researchers from different countries around the vibration and sound radiation of thin plates [18–20]. Later, researchers from different countries around world worked on the ABH effect on structural vibration control, sound radiation, and vibration the world worked on the ABH effect on structural vibration control, sound radiation, and vibration energy harvesting. Recently, a review on mechanics problem of the ABH structure was presented by energy harvesting. Recently, a review on mechanics problem of the ABH structure was presented by Ji et al. [21]. This study systematically introduced the theoretical study on mechanics for 1D and 2D Ji et al. [21]. This study systematically introduced the theoretical study on mechanics for 1D and 2D structures and a summary of applications of ABH. Another review on the applications of ABH on structures and a summary of applications of ABH. Another review on the applications of ABH on vibration damping and sound radiation was conducted by Chong et al. [22]. Due to the increase of vibration damping and sound radiation was conducted by Chong et al. [22]. Due to the increase of complexity of structure design with ABH and the limitation of traditional manufacturing methods, complexity of structure design with ABH and the limitation of traditional manufacturing methods, such as milling [23], 3D printing technology is applied. In the study of Chong et al. [22], a numerical such as milling [23], 3D printing technology is applied. In the study of Chong et al. [22], a numerical and experimental study on vibration response of the 3D-printed ABH beams was also developed. and experimental study on vibration response of the 3D-printed ABH beams was also developed. Furthermore, a series of studies on dynamic and static properties [24] and applications in vibration Furthermore, a series of studies on dynamic and static properties [24] and applications in vibration damping [25] and energy harvesting [8,26] of 3D-printed structures embedded with ABH was also damping [25] and energy harvesting [8,26] of 3D-printed structures embedded with ABH was also investigated by other researchers. investigated by other researchers. This paper presents a review of recent studies on the use of ABH in structural design. The paper This paper presents a review of recent studies on the use of ABH in structural design. The paper presents studies on the applications of ABH in structural vibration control, noise reduction, and presents studies on the applications of ABH in structural vibration control, noise reduction, and vibration energy harvesting. In addition, the review particularly focusses on the influence of vibration energy harvesting. In addition, the review particularly focusses on the influence of geometrical parameters of 1D ABH and the layout of the 2D ABH on the structural response in order geometrical parameters of 1D ABH and the layout of the 2D ABH on the structural response in to make the ABH features more efficient in structural design. order to make the ABH features more efficient in structural design. Figure 1. ABH concentration effect, the amplitude of the incident wave increases significantly when Figure 1. ABH concentration effect, the amplitude of the incident wave increases significantly when propagating to the end of the ABH wedge. propagating to the end of the ABH wedge. 2. Theoretical Analysis 2. Theoretical Analysis Mironov [12] showed that the bending wave speed goes to zero for beams and plates whose Mironov [12] showed that the bending wave speed goes to zero for beams and plates whose thickness decreases according to: thickness decreases according to: h(X) = aX , (1) ( ) (1) ℎ 𝑋 =𝑎𝑋 , where h is the thickness, a is constant, m is exponent of the power-law curve, and X is the distance from the tip of the ideal power-law curve. Acoustics 2018, 1, x FOR PEER REVIEW 3 of 29 Acoustics 2019, 1 222 where h is the thickness, 𝑎 is constant, m is exponent of the power-law curve, and X is the distance from the tip of the ideal power-law curve. However, in reality, due to the limitation of manufacturing technics, it is impossible to build a However, in reality, due to the limitation of manufacturing technics, it is impossible to build a zero zero thickness, so there will be a residual thickness ℎ at the free end. Then the equation of the power- thickness, so there will be a residual thickness h at the free end. Then the equation of the power-law law curve (1D ABH) becomes: curve (1D ABH) becomes: h(x) = #x + h , (2) ( ) ℎ 𝑥 =𝜀𝑥 +ℎ , (2) where the exponent m is a positive rational number and m  2, parameter # is a constant, x is the where the exponent m is a positive rational number and 𝑚 2 , parameter 𝜀 is a constant, x is the distance from the tip of the power-law curve with residual thickness, and the scheme is shown in distance from the tip of the power-law curve with residual thickness, and the scheme is shown in Figure 2. Figure 2. Figure 2. Schematic of a 1D ABH [27]. Figure 2. Schematic of a 1D ABH [27]. For the specific thickness of the beam T, Parameter 𝜀 affect the length of ABH: For the specific thickness of the beam T, Parameter # affect the length of ABH: 𝑇− ℎ T h (3) 𝐿 = 1 , L = , (3) ABH 𝜀 thus: thus: T h 𝑇− ℎ # = , (4) 𝜀= L , (4) ABH then the ABH power-law curve is: then the ABH power-law curve is: T h 𝑇− ℎ m h(x) = x + h , (5) ( ) ℎ 𝑥 = 𝑥 +ℎ , (5) ABH where T is the thickness of the beam, h is the residual thickness of the ABH part, L is the length of where T is the thickness of the beam, ℎ is the residual thickness of the ABH part, 𝐿 is the length 1 ABH the ABH, and m is the exponent of the power-law profile [27]. of the ABH, and m is the exponent of the power-law profile [27]. The phase velocity C and group velocity C are given by: The phase velocity 𝐶 p and group velocity 𝐶 g are given by: (6) 𝐶 = 𝜔ℎ(𝑥) , C = wh(x), (6) 3𝜌(1 − 𝜈 ) 3r(1 n ) 4E 4𝐸 C = wh(x), (7) (7) 𝐶 = 𝜔ℎ(𝑥) , 3r 1 n ( ) 3𝜌(1 − 𝜈 ) where E is the Young’s modulus, n is the Poisson’s ratio, r is the mass density, h(x) is the varying where E is the Young’s modulus, ν is the Poisson's ratio, ρ is the mass density, h(x) is the varying thickness at ABH of the beams, and ! is the angular frequency of the flexural wave. When h = 0 and thickness at ABH of the beams, and ω is the angular frequency of the flexural wave. When ℎ =0 x ! 0 , the phase velocity and group velocity tend to zero [9,15]. and 𝑥→ 0 , the phase velocity and group velocity tend to zero [9,15]. Propagation time T from x to x is shown in Figure 2: 0 ABH 0 Propagation time 𝑇 from 𝑥 to 𝑥 is shown in Figure 2: x 2 1 4 12r(1 n ) 1 1m/2 1m/2 12𝜌(1 − 𝜈 ) 1 T = dx = x L , (8) 0 0 ⁄ 1 2 ABH (8) 𝑇 = c 𝑑𝑥 = Ew 2 m 𝑥 −𝐿 , g 𝑐 ABH 𝐸𝜔 2−𝑚 when x tends to 0, h = 0 then T tends to infinity, only if m  2 [12]. 0 1 0 when 𝑥 tends to 0, ℎ =0 then 𝑇 tends to infinity, only if 𝑚 2 [12]. Equations (6)–(8) indicate that the ABH can alter wave speed to decrease and that also result in Equations (6)–(8) indicate that the ABH can alter wave speed to decrease and that also result in the concentration of vibration energy at the ABH location when m  2. the concentration of vibration energy at the ABH location when m ≥ 2. Acoustics 2019, 1 223 If the host structure has a non-zero-loss factor, the reflection coefficient and wave number can be presented as below: (2 Im k(x)dx) ABH R = e , (9) k(x) = 12 , (10) h(x) 2 2 r 1 n w k = , (11) where E is the Young’s modulus, n is the Poisson’s ratio, r is the mass density, h is the thickness of the plate, and w is the angular frequency of flexural wave [28,29]. Equations (9)–(11) indicate that when x tends to zero, if the residual thickness h equal to zero, wave numbers k(x) tend to infinity at ABH locations, and the reflection coefficient tends to zero, which indicates that no wave can escape from the ABH location [30]. When the ABH is partially covered with a damping layer, the reflection coefficient at any point in the damping area can be expressed as: 8 9 r r #x < = m m #x #x h R (x) = + + 1  exp K q (12) 0 2 : ; h h #x 1 1 + 1 1/4 1/2 3 12 k u E d K = (13) 1/2 4# E h 1/4 1/2 12 k h K = , (14) 1/2 2# where u is the loss factor of the material of the absorbing layer, h is the loss factor of the wedge material. d is the layer thickness and E /E is the ratio of Young’s moduli of the absorbing layer and the plate, 2 1 respectively [31,32]. A 2D ABH can be seen as a rotation of the 1D ABH by 360 degrees. When a wave propagates in the 2D ABH, it deviates into the center of the ABH [17,32]. O’Boy et al. developed the theoretical analysis for the thin plate [33]. A systematic summary of theoretical analysis containing a 2D ABH is studied by Ji et al. [21]. When attaching the damping material on the plate, the structure can be seen as a composite structure. The loss factor can be expressed as: E h h h D D D D h 4 + 6 + 3 E h(r) h(r) h(r) h (r) = (15) comp E h h h D D D D 1 + 4 + 6 + 3 E h(r) h(r) h(r) where E is the Young’s modulus of damping, h is the loss factor of the damping material. E is the D D w elasticity modulus of the plate in plural form. h(r) is the thickness of the plate and h is the thickness of damping layer. r is the distance to the center of ABH [33]. When r decreases, the loss factor increases which means more vibration energy is absorbed. The reflection coefficient of ABH becomes: ( ) 1/4 1/4 1/2 2 3 12 w r 1 n R = exp 2 h (r) dr (16) 0 comp 1/2 1/4 4# E R where R is the radius of whole ABH and R is the truncation length which means the radius inner i t hole [33]. The theoretical analysis of the flexural wave propagation in ABH with damping being developed and more new mathematical models are currently under investigation. For example, a Acoustics 2018, 1, x FOR PEER REVIEW 5 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 5 of 29 semi- Acoustics analyt2019 ical mode , 1 l to analyse an Euler-Bernoulli beam with ABH and its full coupling wit 224h the semi-analytical model to analyse an Euler-Bernoulli beam with ABH and its full coupling with the damping layers coated over its surface is presented by Tang et al. [34]. damping layers coated over its surface is presented by Tang et al. [34]. semi-analytical model to analyse an Euler-Bernoulli beam with ABH and its full coupling with the 3. Applications to Structural Design with Acoustic Black Holes damping layers coated over its surface is presented by Tang et al. [34]. 3. Applications to Structural Design with Acoustic Black Holes 3.1. Ap 3. Applications plication of ABHs to Vibration to Structural Design Control with Acoustic Black Holes 3.1. Application of ABHs to Vibration Control 3.1. Application of ABHs to Vibration Control 3.1.1. Vibration Control of Beams with ABHs 3.1.1. Vibration Control of Beams with ABHs 3.1.1. Vibration Control of Beams with ABHs In 2003, Krylov first used the name “acoustic black hole” [13]. The theoretical and numerical In 2003, Krylov first used the name “acoustic black hole” [13]. The theoretical and numerical work [29] on beams with a one-dimensional acoustic black hole (1D ABH), as shown in Figure 3, was In 2003, Krylov first used the name “acoustic black hole” [13]. The theoretical and numerical work [29] on beams with a one-dimensional acoustic black hole (1D ABH), as shown in Figure 3, was developed. Figures 4 and 5 show the effect of various truncation length x0 and thickness of absorbing work [29] on beams with a one-dimensional acoustic black hole (1D ABH), as shown in Figure 3, developed. Figures 4 and 5 show the effect of various truncation length x0 and thickness of absorbing was developed. Figures 4 and 5 show the effect of various truncation length x and thickness of film δ on the reflection coefficient of the ABH. First, it can be observed from the behavior of the film δ on the reflection coefficient of the ABH. First, it can be observed from the behavior of the absorbing film d on the reflection coefficient of the ABH. First, it can be observed from the behavior of reflection coefficient for the uncovered wedge (solid curve) that a small truncation can result in a reflection coefficient for the uncovered wedge (solid curve) that a small truncation can result in a the reflection coefficient for the uncovered wedge (solid curve) that a small truncation can result in large increase of the refection coefficient. Second, it can be observed that the reflection coefficient large increase of the refection coefficient. Second, it can be observed that the reflection coefficient a large increase of the refection coefficient. Second, it can be observed that the reflection coefficient increases with the increase of truncation length. Third, damping layers with higher relative stiffness increases with the increase of truncation length. Third, damping layers with higher relative stiffness increases with the increase of truncation length. Third, damping layers with higher relative stiffness can decrease the reflection coefficient further. Finally, the reflection coefficient decreases with the can decrease the reflection coefficient further. Finally, the reflection coefficient decreases with the can decrease the reflection coefficient further. Finally, the reflection coefficient decreases with the increase of excitation frequency. These two figures indicate that the presence of thin absorbing layers incre incr ase o ease f ex of citation excitation frefr qequency uency. The . These se tw two o figure figures s indicate indicate tha that the t the presence of presence of thin thi absorbing n absorblayers ing layers on the surfaces of ABH wedges can result in very low reflection coefficients of flexural waves from on the surfaces of ABH wedges can result in very low reflection coefficients of flexural waves from on the surfaces of ABH wedges can result in very low reflection coefficients of flexural waves from their edges. their edges. their edges. Figure 3. Truncated quadratic wedges covered by thin damping layers. x0 is the truncation length. Figure 3. Truncated quadratic wedges covered by thin damping layers. x0 is the truncation length. Figure 3. Truncated quadratic wedges covered by thin damping layers. x is the truncation length. Reprinted with permission from [29]. Copyright Elsevier, 2004. Reprinted with permission from [29]. Copyright Elsevier, 2004. Reprinted with permission from [29]. Copyright Elsevier, 2004. Figure 4. Reflection coefficient for the wedge covered by a thick absorbing film. The solid curve Figure 4. Reflection coefficient for the wedge covered by a thick absorbing film. The solid curve Figure 4. Reflection coefficient for the wedge covered by a thick absorbing film. The solid curve corresponds to an uncovered wedge, and dotted and dashed curves correspond to wedges covered corresponds to an uncovered wedge, and dotted and dashed curves correspond to wedges covered by corresponds to an uncovered wedge, and dotted and dashed curves correspond to wedges covered by thin absorbing thin absorbing film films s with t with the he valu valueses of rel of relative ative s stiffness tiffnes E s /E E2/= E12/30 =2/30 and and E E / 2E /E1= =2/3 2/3, , resp respectively; ectively; the 2 1 2 1 by thin absorbing films with the values of relative stiffness E2/E1=2/30 and E2/E1=2/3, respectively; the the film material loss factor n is 0.2, and the film thickness d is 5 m. Reprinted with permission film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. from [29]. Copyright Elsevier, 2004. Copyright Elsevier, 2004. Copyright Elsevier, 2004. Acoustics Acoustics 2019 2018,, 1 1, x FOR PEER REVIEW 6 of 225 29 Acoustics 2018, 1, x FOR PEER REVIEW 6 of 29 Figure 5. Frequency dependence of the reflection coefficient R: x0 = 1.5 and 2.5 cm (thicker and thinner Figure 5. Frequency dependence of the reflection coefficient R: x = 1.5 and 2.5 cm (thicker and thinner Figure 5. Frequency dependence of the reflection coefficient R: x0 = 1.5 and 2.5 cm (thicker and thinner curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to uncovered wedges; the film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted uncovered wedges; the film material loss factor n is 0.2, and the film thickness d is 5 m. Reprinted uncovered wedges; the film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. Copyright Elsevier, 2004. with permission from [29]. Copyright Elsevier, 2004. with permission from [29]. Copyright Elsevier, 2004. A numerical study of beams with a spiral ABH was developed by Jeon et al. [35,36]. The spiral A num A numerical erical st study udy of b of beams eams wit with h a sp a spiral iral ABH w ABH was as dev developed eloped b by y Jeon et Jeon et al al. . [[3 35 5,,36 36]. The sp ]. The spiral iral ABH is a compact and curvilinear shape by using an Archimedean spiral with a uniform gap-distance ABH ABH is is a com a compact pact and and curv curvili ilin near ear sh shape ape by us by using ing an anA Ar rchimede chimedean an sspiral piral w with ith a au uniform niform gap gap-distance -distance between adjacent baselines of the spiral as shown in Figure 6. Figure 7 shows the driving point bet between ween adj adjacent acent base baselines lines of t of the he spir spiral al a as s shown shown iin n Fig Figur ure e 6 6.. F Figur iguree 7 show 7 shows s tthe he driv driving ing point point mobility of the beam with a 720 mm spiral ABH compared with reference uniform beam (black line). mobilit mobility y of of tthe he be beam am w with ith a a 7 720 20 mm mm sp spiral iral AB ABH H compared compared wit with h re re ffer erence enceuni uniform form beam beam (b (black lack line line). ). The beam with the ABH is 10% lighter than the reference beam, but it reduces the resonant peak The bea The beam m with the ABH is 10 with the ABH is 10% % li lighter ghter tha than n the re the refer ference beam, ence beam, but it reduce but it reduces s the reson the resonant ant peak peak levels to 90% without additional damping. This indicates that the spiral ABH has great potential for llevels evels to 9 to 90% 0% wi without thout a additional dditional da damping. mping. Thi This s iindicates ndicates tha that t the spi the spiral ral ABH has grea ABH has great t potent potential ial ffor or vibration damping. vibrat vibration ion da damping. mping. Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with permission from [36]. Copyright Acoustic Society of America, 2017. permission from [36]. Copyright Acoustic Society of America, 2017. permission from [36]. Copyright Acoustic Society of America, 2017. Figure 7. Driving point mobility of the beam with a 720 mm spiral ABH (grey line) and the reference Figure 7. Driving point mobility of the beam with a 720 mm spiral ABH (grey line) and the reference uniform beam (black line). Reprinted with permission from [36]. Copyright Acoustic Society of uniform beam (black line). Reprinted with permission from [36]. Copyright Acoustic Society of America, 2017. America, 2017. Acoustics 2018, 1, x FOR PEER REVIEW 6 of 29 Figure 5. Frequency dependence of the reflection coefficient R: x0 = 1.5 and 2.5 cm (thicker and thinner curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to uncovered wedges; the film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. Copyright Elsevier, 2004. A numerical study of beams with a spiral ABH was developed by Jeon et al. [35,36]. The spiral ABH is a compact and curvilinear shape by using an Archimedean spiral with a uniform gap-distance between adjacent baselines of the spiral as shown in Figure 6. Figure 7 shows the driving point mobility of the beam with a 720 mm spiral ABH compared with reference uniform beam (black line). The beam with the ABH is 10% lighter than the reference beam, but it reduces the resonant peak levels to 90% without additional damping. This indicates that the spiral ABH has great potential for vibration damping. Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with Acoustics 2019, 1 226 permission from [36]. Copyright Acoustic Society of America, 2017. Figure Figure 7. 7. Driving Driving point mobility of the beam point mobility of the beam with a 720 mm spiral A with a 720 mm spiral ABH BH (gr (grey ey line) and th line) and the e reference reference uniform uniform beam beam (bla (black ck l line). ine). Repr Reprinted wit inted with h perm permission ission f frrom om [36]. Copyright Acousti [36]. Copyright Acoustic c S Society ociety of of America, America, 2017. 2017. Acoustics 2018, 1, x FOR PEER REVIEW 7 of 29 Recently, Zhou and Cheng proposed a numerical and experimental work to develop an ABH that Recently, Zhou and Cheng proposed a numerical and experimental work to develop an ABH featured a resonant beam damper, as shown in Figure 8 (ABH-RBD) [37]. Figure 9 shows the measured that featured a resonant beam damper, as shown in Figure 8 (ABH-RBD) [37]. Figure 9 shows the driving and cross point mobility of the host beam with and without ABH-RBD. It can be observed that measured driving and cross point mobility of the host beam with and without ABH-RBD. It can be a significant vibration reduction is obtained with mounting the ABH-RBD. This study shows a great observed that a significant vibration reduction is obtained with mounting the ABH-RBD. This study damping treatment of ABH-RBD to control the vibration of the host structure. shows a great damping treatment of ABH-RBD to control the vibration of the host structure. Figure 8. Schematics of the host beam with an ABH-RBD. Reprinted with permission from [37]. Figure 8. Schematics of the host beam with an ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. Copyright Elsevier, 2018. (a) (b) Figure 9. Measured (a) driving and (b) cross-point mobility of the primary host beam with and without ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. 3.1.2. Vibration Damping of Plates with ABH The vibration performance of the plates with various 1D ABH slots was studied by Bowyer [38]. One of the samples is shown in Figure 10a. From the experimental result of this sample, it can be observed that a substantial reduction of acceleration is obtained in comparison with the reference plate. The experimental results of other steel plates and composite plates with various slots also show acceptable damping performance. The composite plates have good damping performance even without the damping layer attached due to the large material loss factor. Acoustics 2018, 1, x FOR PEER REVIEW 7 of 29 Recently, Zhou and Cheng proposed a numerical and experimental work to develop an ABH that featured a resonant beam damper, as shown in Figure 8 (ABH-RBD) [37]. Figure 9 shows the measured driving and cross point mobility of the host beam with and without ABH-RBD. It can be observed that a significant vibration reduction is obtained with mounting the ABH-RBD. This study shows a great damping treatment of ABH-RBD to control the vibration of the host structure. Figure 8. Schematics of the host beam with an ABH-RBD. Reprinted with permission from [37]. Acoustics 2019, 1 227 Copyright Elsevier, 2018. (a) (b) Figure 9. Measured (a) driving and (b) cross-point mobility of the primary host beam with and Figure 9. Measured (a) driving and (b) cross-point mobility of the primary host beam with and without without ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. 3.1.2. Vibration Damping of Plates with ABH 3.1.2. Vibration Damping of Plates with ABH The vibration performance of the plates with various 1D ABH slots was studied by Bowyer [38]. The vibration performance of the plates with various 1D ABH slots was studied by Bowyer [38]. One of the samples is shown in Figure 10a. From the experimental result of this sample, it can be One of the samples is shown in Figure 10a. From the experimental result of this sample, it can be observed that a substantial reduction of acceleration is obtained in comparison with the reference observed that a substantial reduction of acceleration is obtained in comparison with the reference plate. The experimental results of other steel plates and composite plates with various slots also show plate. The experimental results of other steel plates and composite plates with various slots also acceptable damping performance. The composite plates have good damping performance even show acceptable damping performance. The composite plates have good damping performance even Acoustics 2018, 1, x FOR PEER REVIEW 8 of 29 without the damping layer attached due to the large material loss factor. without the damping layer attached due to the large material loss factor. (a) (b) Figure 10. (a) Carbon fiber composite sample and longitudinal cross-section of the sample; and (b) Figure 10. (a) Carbon fiber composite sample and longitudinal cross-section of the sample; Acceleration for the plate with ABH slot (solid line) and reference plate (dashed line). Reprinted from and (b) Acceleration for the plate with ABH slot (solid line) and reference plate (dashed line). Reprinted [38] under a CC BY 4.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2016. from [38] under a CC BY 4.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2016. Experimental investigation on damping flexural vibrations using two-dimensional acoustic Experimental investigation on damping flexural vibrations using two-dimensional acoustic black holes (2D ABHs) was developed by Bowyer et al. [3]. Figure 11 shows the schematics of the black holes (2D ABHs) was developed by Bowyer et al. [3]. Figure 11 shows the schematics of experimental samples. The forced excitation was applied to the center of the plate via shaker over a the experimental samples. The forced excitation was applied to the center of the plate via shaker over frequency range of 0–9 kHz. The experimental results show that, in comparison with the reference plate, the plate embedded with single 2D ABH with a damping layer provides little damping below 3 kHz. In the region of 3.8–9 kHz, damping varies between 3–8 dB, and maximum damping occurs at 6.6 kHz. Due to the long wave length, the damping performance of the ABH is not evident at low frequency, but it has better damping performance for higher frequencies. It is also indicated by this study that the plates with multiple holes in the current random layout does not significantly improve the damping performance of the 2D ABHs in comparison with the plate with a single 2D ABH. (a) (b) Figure 11. (a) A singular 2D ABH with a center hole, and (b) three 2D ABH with central holes. Adapted from [3] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2010. Later, another experimental investigation of damping flexural vibrations in plates was also developed by Bowyer et al. [39]. Figure 12 shows the steel plate containing an array of six 2D ABH. Figure 13 clearly shows that, compared with the reference plate without ABH, the acceleration of the plate with six ABHs attaching damping layers sharply decreases. Additionally, it shows that the minimum frequency of effective damping performance is about 1.5 kHz. These two experimental investigations indicate that ABHs with damping layers can decrease the vibration of plates and the number and configuration of ABHs affect the performance of vibration damping. The increase in the number of ABHs can expand the effective frequency of damping performance of the plate with a 2D ABH and damping layer. Acoustics 2018, 1, x FOR PEER REVIEW 8 of 29 (a) (b) Figure 10. (a) Carbon fiber composite sample and longitudinal cross-section of the sample; and (b) Acceleration for the plate with ABH slot (solid line) and reference plate (dashed line). Reprinted from [38] under a CC BY 4.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2016. Experimental investigation on damping flexural vibrations using two-dimensional acoustic Acoustics 2019, 1 228 black holes (2D ABHs) was developed by Bowyer et al. [3]. Figure 11 shows the schematics of the experimental samples. The forced excitation was applied to the center of the plate via shaker over a frequency range of 0–9 kHz. The experimental results show that, in comparison with the reference a frequency range of 0–9 kHz. The experimental results show that, in comparison with the reference plate, the plate embedded with single 2D ABH with a damping layer provides little damping below plate, the plate embedded with single 2D ABH with a damping layer provides little damping below 3 kHz. In the region of 3.8–9 kHz, damping varies between 3–8 dB, and maximum damping occurs 3 kHz. In the region of 3.8–9 kHz, damping varies between 3–8 dB, and maximum damping occurs at 6.6 kHz. Due to the long wave length, the damping performance of the ABH is not evident at low at 6.6 kHz. Due to the long wave length, the damping performance of the ABH is not evident at low frequency, but it has better damping performance for higher frequencies. It is also indicated by this frequency, but it has better damping performance for higher frequencies. It is also indicated by this study that the plates with multiple holes in the current random layout does not significantly improve study that the plates with multiple holes in the current random layout does not significantly improve the damping performance of the 2D ABHs in comparison with the plate with a single 2D ABH. the damping performance of the 2D ABHs in comparison with the plate with a single 2D ABH. (a) (b) Figure 11. (a) A singular 2D ABH with a center hole, and (b) three 2D ABH with central holes. Figure 11. (a) A singular 2D ABH with a center hole, and (b) three 2D ABH with central holes. Adapted Adapted from [3] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2010. from [3] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2010. Later, another experimental investigation of damping flexural vibrations in plates was also Later, another experimental investigation of damping flexural vibrations in plates was also developed by Bowyer et al. [39]. Figure 12 shows the steel plate containing an array of six 2D ABH. developed by Bowyer et al. [39]. Figure 12 shows the steel plate containing an array of six 2D ABH. Figure 13 clearly shows that, compared with the reference plate without ABH, the acceleration of the Figure 13 clearly shows that, compared with the reference plate without ABH, the acceleration of plate with six ABHs attaching damping layers sharply decreases. Additionally, it shows that the the plate with six ABHs attaching damping layers sharply decreases. Additionally, it shows that the minimum frequency of effective damping performance is about 1.5 kHz. These two experimental minimum frequency of effective damping performance is about 1.5 kHz. These two experimental investigations indicate that ABHs with damping layers can decrease the vibration of plates and the investigations indicate that ABHs with damping layers can decrease the vibration of plates and the number and configuration of ABHs affect the performance of vibration damping. The increase in the number and configuration of ABHs affect the performance of vibration damping. The increase in the number of ABHs can expand the effective frequency of damping performance of the plate with a 2D number of ABHs can expand the effective frequency of damping performance of the plate with a 2D ABH and damping layer. ABH and damping layer. Acoustics 2018, 1, x FOR PEER REVIEW 9 of 29 Figure 12. Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission Figure 12. Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission from [39]. Copyright Elsevier, 2013. Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central ABH with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The point- and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared with the reference plate. A numerical and experimental study of the acoustic black hole effect for vibration damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with ABH and without ABH were tested. An elliptical plate with disks of resin placed at the location of the ABHs and an elliptical plate completely covered by resin were also tested. Figure 14 shows velocity fields of plates with ABH (b) and without ABH (a). The excitation force was applied to the left focus whereas the ABH is in the right one. It can be observed that the plate with ABH (b) has a lower amplitude of vibration. Figure 15 shows the point mobility measured at the left focus of the plate. It can be observed that the point mobility of the plate with ABH with the damping material at its location was reduced over 2 kHz. (a) (b) Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Hz. Reprinted with permission from [41]. Copyright Elsevier, 2011. Acoustics 2018, 1, x FOR PEER REVIEW 9 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 9 of 29 Acoustics Figure 12. 2019, 1 Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission 229 Figure 12. Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central ABH ABH with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The ABH with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The point- point- and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared point- and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared with the with the reference plate. A numerical and experimental study of the acoustic black hole effect for with the reference plate. A numerical and experimental study of the acoustic black hole effect for reference plate. A numerical and experimental study of the acoustic black hole effect for vibration vibration damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with vibration damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with ABH and ABH and without ABH were tested. An elliptical plate with disks of resin placed at the location of ABH and without ABH were tested. An elliptical plate with disks of resin placed at the location of without ABH were tested. An elliptical plate with disks of resin placed at the location of the ABHs and the ABHs and an elliptical plate completely covered by resin were also tested. Figure 14 shows the ABHs and an elliptical plate completely covered by resin were also tested. Figure 14 shows an elliptical plate completely covered by resin were also tested. Figure 14 shows velocity fields of plates velocity fields of plates with ABH (b) and without ABH (a). The excitation force was applied to the velocity fields of plates with ABH (b) and without ABH (a). The excitation force was applied to the with ABH (b) and without ABH (a). The excitation force was applied to the left focus whereas the ABH left focus whereas the ABH is in the right one. It can be observed that the plate with ABH (b) has a left focus whereas the ABH is in the right one. It can be observed that the plate with ABH (b) has a is in the right one. It can be observed that the plate with ABH (b) has a lower amplitude of vibration. lower amplitude of vibration. Figure 15 shows the point mobility measured at the left focus of the lower amplitude of vibration. Figure 15 shows the point mobility measured at the left focus of the Figure 15 shows the point mobility measured at the left focus of the plate. It can be observed that the plate. It can be observed that the point mobility of the plate with ABH with the damping material at plate. It can be observed that the point mobility of the plate with ABH with the damping material at point mobility of the plate with ABH with the damping material at its location was reduced over 2 kHz. its location was reduced over 2 kHz. its location was reduced over 2 kHz. (a) (a) (b) (b) Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Hz. Hz. Reprinted with permission from [41]. Copyright Elsevier, 2011. Hz. Reprinted with permission from [41]. Copyright Elsevier, 2011. Reprinted with permission from [41]. Copyright Elsevier, 2011. Acoustics 2018, 1, x FOR PEER REVIEW 10 of 29 Acoustics 2019, 1 230 Acoustics 2018, 1, x FOR PEER REVIEW 10 of 29 Figure 15. Measured point mobilities of the elliptical plate. The black line shows the plate with ABH Figure 15. Measured point mobilities of the elliptical plate. The black line shows the plate with ABH by Figure 15. by attaching damping materi Measured point mobiliti al at its es of the e locatilon, liptica the dashe l plate. T dh line e black l shows t ine sh he plate witho ows the plate u with ABH t ABH by attaching damping material at its location, the dashed line shows the plate without ABH by attaching by attaching damping materi attaching damping at the same locat al at its ion, and location, the gr the dashe een line shows the plate d line shows the plate witho without ABH b ut ABH by y covering damping at the same location, and the green line shows the plate without ABH by covering the attaching damping at the same locat the damping material over the whole p ion, and late. R the gr eprinte een line shows the plate d with permission from without ABH b [41]. Copyrigy h covering t Elsevier, damping material over the whole plate. Reprinted with permission from [41]. Copyright Elsevier, 2011. the damping material o 2011. ver the whole plate. Reprinted with permission from [41]. Copyright Elsevier, Not only were plates with ABH with damping material, but also ABH with dynamic vibration Not only were plates with ABH with damping material, but also ABH with dynamic vibration Not only were plates with ABH with damping material, but also ABH with dynamic vibration absorbers (DVA) were studied by Jia et al., as shown in Figure 16 [42]. It can be observed from the absorbers (DVA) were studied by Jia et al., as shown in Figure 16 [42]. It can be observed from the absorber simulatis (D on results that VA) were st there is udied b a reduction of ov y Jia et al., as shown er 10 in dB Fig at maj ure 1 or r 6 [4 esponse 2]. It ca peak n be observed s over 1 kH fro z for the m the simulation results that there is a reduction of over 10 dB at major response peaks over 1 kHz for simulation results that there is a reduction of over 10 dB at major response peaks over 1 kHz for the plate with ABH with DVA. This result shows great potential of combining ABHs and DVAs for the plate with ABH with DVA. This result shows great potential of combining ABHs and DVAs for plat vibrat e wit ion con h ABtH wit rol. h DVA. This result shows great potential of combining ABHs and DVAs for vibration control. vibration control. (b) (b) (c) (a) (a) (c) Figure 16. Cross-section of the plate structure considered in the FE model and the experiments: (a) a Figure 16. Cross-section of the plate structure considered in the FE model and the experiments: (a) a Figure 16. top-view of p Cros la s-sec te structure wit tion of the plate h two ABH structu s, re con (b) a plate sidere embe d in the F dded with ABH and dampin E model and the experim g layer, and ents: (a) a top-view of plate structure with two ABHs, (b) a plate embedded with ABH and damping layer, and (c) top-view of p (c) a plate em la bedded te structure wit with ABh H and DVA two ABHs, . ( A bd ) a plate apted w embe ith pe dded with ABH and dampin rmission from [42]. Copyright American g layer, and a plate embedded with ABH and DVA. Adapted with permission from [42]. Copyright American (c Society ) a plate of Me embedded chanical Engine with ABH and DVA ers, 2015. . Adapted with permission from [42]. Copyright American Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Since the structures with multiple ABHs have great damping effect for vibration and sound, Since the structures with multiple ABHs have great damping effect for vibration and sound, Since the structures with multiple ABHs have great damping effect for vibration and sound, such structures can be considered as metamaterials [43]. Semperlotti and Zhu developed a meta- such structures can be considered as metamaterials [43]. Semperlotti and Zhu developed a such structures can be co structure based on the consider ncept of ed as ABmeta H [44] ma . This terials [ load-be 43]. Semper aring thlotti and Zh in-wall struu develope cture element d a meta- enables meta-structure based on the concept of ABH [44]. This load-bearing thin-wall structure element structure based on the concept of ABH [44]. This load-bearing thin-wall structure element enables propagation characteristics comparable with resonant metamaterials without the fabrication enables propagation characteristics comparable with resonant metamaterials without the fabrication pcompl ropagat exiion ty. The experi characterist menta ics com l work shows tha parable with reso t the ABH trea nant meta tm ments ca aterials wit n signi hout fica t ntlhy improve the e fabrication complexity. The experimental work shows that the ABH treatments can significantly improve the complexity. The experimental work shows that the ABH treatments can significantly improve the damping effect. damping effect. damping A waveg effect. uide is designed to observe travelling waves by Foucaud et al. [45], which is inspired A waveguide is designed to observe travelling waves by Foucaud et al. [45], which is inspired by A waveguide is designed to observe travelling waves by Foucaud et al. [45], which is inspired by artificial cochlea. The experimental study (shown in Figure 17) uses a varying width plate artificial cochlea. The experimental study (shown in Figure 17) uses a varying width plate immersed by art immeirsed ficia in flu l cochlea id and . The te experiment rminated wit alh st an ud ABH. y (sho It wn in F shows itgu hat re a 17 n ) AB use H use s a d varyin as ang wi anecho dth plat ic ened in fluid and terminated with an ABH. It shows that an ABH used as an anechoic end improves the immersed in fluid and terminated with an ABH. It shows that an ABH used as an anechoic end improves the quality of measurements and the accuracy of tonotopic maps due to the attenuation of quality of measurements and the accuracy of tonotopic maps due to the attenuation of reflected waves. improves the reflected waves. quality of measurements and the accuracy of tonotopic maps due to the attenuation of reflected waves. Acoustics 2019, 1 231 Acoustics 2018, 1, x FOR PEER REVIEW 11 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 11 of 29 Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Copyright Elsevier, 2014. Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Copyright Elsevier, 2014. Copyright Elsevier, 2014. 3.1.3. Vibration Damping of Turbofan Blades with ABH 3.1.3. Vibration Damping of Turbofan Blades with ABH 3.1.3. Vibration Damping of Turbofan Blades with ABH An experimental work of damping of flexural vibrations in turbofan blades using ABH was An experimental work of damping of flexural vibrations in turbofan blades using ABH was An experimental work of damping of flexural vibrations in turbofan blades using ABH was studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a blade blade with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has blade with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has about 50% about 50% reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with about 50% reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with appropriate appropriate damping layers are efficient in the reduction of airflow-excited vibrations of the fan appropriate damping layers are efficient in the reduction of airflow-excited vibrations of the fan damping layers are efficient in the reduction of airflow-excited vibrations of the fan blades. This study blades. This study shows the great potential of the ABH in jet engine design to reduce flexural blades. This study shows the great potential of the ABH in jet engine design to reduce flexural shows the great potential of the ABH in jet engine design to reduce flexural vibration in the blades, vibration in the blades, thus reducing internal stresses in the blades and increasing their fatigue life vibration in the blades, thus reducing internal stresses in the blades and increasing their fatigue life thus reducing internal stresses in the blades and increasing their fatigue life cycle. cycle. cycle. (a) (a) (b) (b) Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted from Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Acoustics 2018, 1, x FOR PEER REVIEW 11 of 29 Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Copyright Elsevier, 2014. 3.1.3. Vibration Damping of Turbofan Blades with ABH An experimental work of damping of flexural vibrations in turbofan blades using ABH was studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a blade with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has about 50% reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with appropriate damping layers are efficient in the reduction of airflow-excited vibrations of the fan blades. This study shows the great potential of the ABH in jet engine design to reduce flexural vibration in the blades, thus reducing internal stresses in the blades and increasing their fatigue life cycle. (a) (b) Acoustics 2019, 1 232 Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Acoustics 2018, 1, x FOR PEER REVIEW 12 of 29 Figure 19. Experimental setup of closed-circuit wind tunnel and samples of the test. Experimental Figure 19. Experimental setup of closed-circuit wind tunnel and samples of the test. Experimental samples: Flow visualization diagram for: (a) reference fan blade, (b) fan blade with a power-law samples: Flow visualization diagram for: (a) reference fan blade, (b) fan blade with a power-law wedge, ( wedge, (c c) fan ) fan blade having a blade having a power-law wedge power-law wedge with a with a sing single le dam damping ping lay layer er, and ( , and (d d) fan ) fan blade hav blade having ing a power-law w a power-law wedge edge with a with a shaped damping layer. shaped damping layer. Reprint Reprinted ed from [46] u from [46] under nder a CC BY a CC BY3.0 license 3.0 license. . Copy Copyright right Bowy Bowyer er, E. , E.P P. . and and K Krylov rylov, V.V., 201 , V.V., 2014.4. Figure 20. Acceleration for a twisted reference blade (dashed line) compared to a twisted blade with Figure 20. Acceleration for a twisted reference blade (dashed line) compared to a twisted blade with a a 1D ABH and damping layer (solid line). Reprinted from [46] under a CC BY 3.0 license. Copyright 1D ABH and damping layer (solid line). Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Bowyer, E.P. and Krylov, V.V., 2014. 3.2. Application of ABH to Sound Reduction 3.2. Application of ABH to Sound Reduction An experimental work was developed by Bowyer and Krylov [47]. A 300 × 400 mm × 5-mm thick An experimental work was developed by Bowyer and Krylov [47]. A 300 400 mm 5-mm thick plate embedded with six 2D ABHs (Figure 21) with damping at the center of each ABH was plate embedded with six 2D ABHs (Figure 21) with damping at the center of each ABH was suspended suspended vertically. The excitation force was applied centrally on the plate. The results compare the vertically. The excitation force was applied centrally on the plate. The results compare the sound sound radiation power level of a plate containing six 2D ABHs with a damping layer with the sound radiation power level of a plate containing six 2D ABHs with a damping layer with the sound radiation radiation power level of the reference plate, which are shown in Figure 22. Below 1 kHz there is little power level of the reference plate, which are shown in Figure 22. Below 1 kHz there is little reduction reduction in the sound power level. Between 1 and 3 kHz, the sound power level is reduced by 10– in the sound power level. Between 1 and 3 kHz, the sound power level is reduced by 10–18 dB in 18 dB in comparison with the reference plate, and the maximum reduction in the sound radiation comparison with the reference plate, and the maximum reduction in the sound radiation occurs at 1.6 occurs at 1.6 kHz. Above 3 kHz, almost all responses in sound radiation are absorbed. This indicates kHz. Above 3 kHz, almost all responses in sound radiation are absorbed. This indicates that the plate that the plate with ABHs having damping layers effectively reduce the sound radiation of the steel with ABHs having damping layers effectively reduce the sound radiation of the steel plate. plate. Figure 21. Sample of the plate with six 2D ABHs. Reprinted from [47] under a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. Acoustics 2018, 1, x FOR PEER REVIEW 12 of 29 Figure 19. Experimental setup of closed-circuit wind tunnel and samples of the test. Experimental samples: Flow visualization diagram for: (a) reference fan blade, (b) fan blade with a power-law wedge, (c) fan blade having a power-law wedge with a single damping layer, and (d) fan blade having a power-law wedge with a shaped damping layer. Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Figure 20. Acceleration for a twisted reference blade (dashed line) compared to a twisted blade with a 1D ABH and damping layer (solid line). Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. 3.2. Application of ABH to Sound Reduction An experimental work was developed by Bowyer and Krylov [47]. A 300 × 400 mm × 5-mm thick plate embedded with six 2D ABHs (Figure 21) with damping at the center of each ABH was suspended vertically. The excitation force was applied centrally on the plate. The results compare the sound radiation power level of a plate containing six 2D ABHs with a damping layer with the sound radiation power level of the reference plate, which are shown in Figure 22. Below 1 kHz there is little reduction in the sound power level. Between 1 and 3 kHz, the sound power level is reduced by 10– 18 dB in comparison with the reference plate, and the maximum reduction in the sound radiation occurs at 1.6 kHz. Above 3 kHz, almost all responses in sound radiation are absorbed. This indicates Acoustics 2019, 1 233 that the plate with ABHs having damping layers effectively reduce the sound radiation of the steel plate. Figure 21. Sample of the plate with six 2D ABHs. Reprinted from [47] under a CC BY license. Figure 21. Sample of the plate with six 2D ABHs. Reprinted from [47] under a CC BY license. Copyright Copyright Bowyer E.P. and Krylov, V.V., 2012. Bowyer E.P. and Krylov, V.V., 2012. Acoustics 2018, 1, x FOR PEER REVIEW 13 of 29 Figure 22. Sound power level comparison for a plate containing six indentations of power-law profile Figure 22. Sound power level comparison for a plate containing six indentations of power-law profile with a damping layer (black line) compared to a reference plate (grey line). Reprinted from [47] under with a damping layer (black line) compared to a reference plate (grey line). Reprinted from [47] under a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. Feurtado and Conlon developed a numerical and experimental investigation of sound power of Feurtado and Conlon developed a numerical and experimental investigation of sound power a plate with an array of ABH [48]. A 4 × 5 array of 10-cm diameter 2D ABHs, which is shown in Figure of a plate with an array of ABH [48]. A 4  5 array of 10-cm diameter 2D ABHs, which is shown in 23, was machined into a 6.35-mm thick, 61 cm × 91 cm aluminum plate. The ABH with various Figure 23, was machined into a 6.35-mm thick, 61 cm  91 cm aluminum plate. The ABH with various diameters of damping layers was tested to assess the effects of the amount of damping layer on ABH diameters of damping layers was tested to assess the effects of the amount of damping layer on ABH performance. The experimental setup is shown in Figure 24. The plate was mounted on a frame and performance. The experimental setup is shown in Figure 24. The plate was mounted on a frame and excited with band-limited white noise. Figure 25 shows one-third octave band radiated sound power excited with band-limited white noise. Figure 25 shows one-third octave band radiated sound power tested by an intensity probe for a uniform plate and ABH plate with various diameters of damping tested by an intensity probe for a uniform plate and ABH plate with various diameters of damping material. It can be observed that an ABH with a damping layer effectively reduced the radiated sound material. It can be observed that an ABH with a damping layer effectively reduced the radiated sound power over 1.5 kHz compared with the reference plate (blue line). It can also be observed that sound power over 1.5 kHz compared with the reference plate (blue line). It can also be observed that sound radiation of the plate with the diameters of the damping layer of 6.75 cm and 10 cm have almost the radiation of the plate with the diameters of the damping layer of 6.75 cm and 10 cm have almost the same performance, namely higher radiated sound power. The 3.5 cm diameter damping layer shows same performance, namely higher radiated sound power. The 3.5 cm diameter damping layer shows comparable performance to the larger damping diameters. comparable performance to the larger damping diameters. (a) (b) Figure 23. (a) FEA model and (b) cross-section of an aluminum plate (green) with a 2D ABH attached damping layer (yellow). Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Acoustics 2018, 1, x FOR PEER REVIEW 13 of 29 Figure 22. Sound power level comparison for a plate containing six indentations of power-law profile with a damping layer (black line) compared to a reference plate (grey line). Reprinted from [47] under a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. Feurtado and Conlon developed a numerical and experimental investigation of sound power of a plate with an array of ABH [48]. A 4 × 5 array of 10-cm diameter 2D ABHs, which is shown in Figure 23, was machined into a 6.35-mm thick, 61 cm × 91 cm aluminum plate. The ABH with various diameters of damping layers was tested to assess the effects of the amount of damping layer on ABH performance. The experimental setup is shown in Figure 24. The plate was mounted on a frame and excited with band-limited white noise. Figure 25 shows one-third octave band radiated sound power tested by an intensity probe for a uniform plate and ABH plate with various diameters of damping material. It can be observed that an ABH with a damping layer effectively reduced the radiated sound power over 1.5 kHz compared with the reference plate (blue line). It can also be observed that sound radiation of the plate with the diameters of the damping layer of 6.75 cm and 10 cm have almost the same performance, namely higher radiated sound power. The 3.5 cm diameter damping layer shows Acoustics 2019, 1 234 comparable performance to the larger damping diameters. (a) (b) Figure 23. (a) FEA model and (b) cross-section of an aluminum plate (green) with a 2D ABH attached Figure 23. (a) FEA model and (b) cross-section of an aluminum plate (green) with a 2D ABH attached damping layer (yellow). Reprinted with permission from [48]. Copyright American Society of damping layer (yellow). Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Mechanical Engineers, 2015. Acoustics 2018, 1, x FOR PEER REVIEW 14 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 14 of 29 Figure 24. Aluminum plate with a 4 × 5 array of embedded ABHs with full-diameter damping layers Figure 24. Aluminum plate with a 4  5 array of embedded ABHs with full-diameter damping layers Figure 24. Aluminum plate with a 4 × 5 array of embedded ABHs with full-diameter damping layers in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with varying diameters of damping material. Reprinted with permission from [48]. Copyright American varying diameters of damping material. Reprinted with permission from [48]. Copyright American varying diameters of damping material. Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on sound radiation of engine cover [49].The ABH were machined on two plates and then bonded into sound radiation of engine cover [49].The ABH were machined on two plates and then bonded into the engine cover with glue as shown in Figure 26. Figure 27 shows the sound radiation from a the engine cover with glue as shown in Figure 26. Figure 27 shows the sound radiation from a reference engine cover (dashed line) in comparison with the engine cover attaching plates with ABH reference engine cover (dashed line) in comparison with the engine cover attaching plates with ABH (black line) at 2100 rpm with the bonnet closed. A total average reduction from the reference specimen (black line) at 2100 rpm with the bonnet closed. A total average reduction from the reference specimen of 6.5 dB was recorded. This indicates that engine covers with ABHs can decrease the sound radiation of 6.5 dB was recorded. This indicates that engine covers with ABHs can decrease the sound radiation from the vehicle engine. from the vehicle engine. (b) (a) (b) (a) Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted from Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 14 of 29 Figure 24. Aluminum plate with a 4 × 5 array of embedded ABHs with full-diameter damping layers in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with varying diameters of damping material. Reprinted with permission from [48]. Copyright American Acoustics 2019, 1 235 Society of Mechanical Engineers, 2015. A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on sound A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on radiation sound rof adi engine ation of cover engine [49 cov ].The er [ ABH 49].Th wer e AB e machined H were mon achined two plates on twand o plat then es an bonded d then b into onde the d int engine o cover the engine c with glue over with glue as shown as shown in Figure 26. in Figur Figure 27e shows 26. Figure the sound 27 shows the sound ra radiation from adia refer tioence n from a engine reference engine cover (dashed line) in comparison with the engine cover attaching plates with ABH cover (dashed line) in comparison with the engine cover attaching plates with ABH (black line) at (black line) at 2100 rpm with the bonnet closed. A total average reduction from the reference specimen 2100 rpm with the bonnet closed. A total average reduction from the reference specimen of 6.5 dB of 6.5 dB was recorded. This indicates that engine covers with ABHs can decrease the sound radiation was recorded. This indicates that engine covers with ABHs can decrease the sound radiation from the from the vehicle engine. vehicle engine. (a) (b) Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted from Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 15 of 29 Figure 27. Sound radiation from a reference engine cover (dashed line) compared with the engine Figure 27. Sound radiation from a reference engine cover (dashed line) compared with the engine cover attaching plates with ABH (black line) at 2100 rpm with the bonnet closed. Reprinted from [49] cover attaching plates with ABH (black line) at 2100 rpm with the bonnet closed. Reprinted from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. An experimental study on sound absorption in air of ABH based inhomogeneous acoustic An experimental study on sound absorption in air of ABH based inhomogeneous acoustic waveguides [50] was developed by Azbaid et al. [51,52]. The inner radius of the structures as shown waveguides [50] was developed by Azbaid et al. [51,52]. The inner radius of the structures as shown in in Figure 28 are built following the linear function (exponent m = 1) and power-law function Figure 28 are built following the linear function (exponent m = 1) and power-law function (exponent m (exponent m = 2), respectively. Using two microphone transfer function methods, the experiment = 2), respectively. Using two microphone transfer function methods, the experiment results show a results show a substantial reduction in the reflection coefficient. The adding of absorbing porous substantial reduction in the reflection coefficient. The adding of absorbing porous materials results in materials results in a further reduction of the reflection coefficient. a further reduction of the reflection coefficient. (b) (a) Figure 28. Photo of a linear ABH (exponent m = 1) and a quadratic ABH (exponent m = 2). Reprinted from [51] under a CC BY-NC-ND 4.0 license. Copyright Azbaid El Ouahabi, A., Krylov, V.V. and O'Boy, D.J., 2015. A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption performance was developed by Zhou and Semperlotti [53]. The HR cavity shown in Figure 29 is partitioned by a flexible plate-embedded ABH and the numerical results show a great increase in the energy absorption of the HR system. Figure 29. Schematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Foundation, 2016. Acoustics 2018, 1, x FOR PEER REVIEW 15 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 15 of 29 Figure 27. Sound radiation from a reference engine cover (dashed line) compared with the engine cover Figure 27. attaching plate Sound radiation s with ABH (bla from a reference engine co ck line) at 2100 rpm wit ver h the b (dashed line) compared onnet closed. Reprinte with the d from [49] engine under a CC BY cover attaching plate -NC-ND 4.0 s with license. Copyri ABH (black line) ght Bowyer E.P. a at 2100 rpm wit nd Krylov, V.V h the bonnet closed. Reprinte ., 2015. d from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. An experimental study on sound absorption in air of ABH based inhomogeneous acoustic An experimental study on sound absorption in air of ABH based inhomogeneous acoustic waveguides [50] was developed by Azbaid et al. [51,52]. The inner radius of the structures as shown in Fi waveg gure uide 28 a s [r5e buil 0] was deve t followi loped by Azb ng the linea aid r et al functi . [on ( 51,52]. The inner exponent m rad = 1) ius o and power- f the structure law functi s as shown on in Figure 28 are built following the linear function (exponent m = 1) and power-law function (exponent m = 2), respectively. Using two microphone transfer function methods, the experiment resu (exponent lts show a subst m = 2), a resp ntial reduction ectively. Us in the ing two m reflecti icrop on ho coe ne t fficient. The addin ransfer function methods, the experi g of absorbing porous ment Acoustics 2019, 1 236 results show a substantial reduction in the reflection coefficient. The adding of absorbing porous materials results in a further reduction of the reflection coefficient. materials results in a further reduction of the reflection coefficient. (b) (a) (b) (a) Figure 28. Photo of a linear ABH (exponent m = 1) and a quadratic ABH (exponent m = 2). Reprinted Figure 28. Photo of (a) a linear ABH (exponent m = 1) and (b) a quadratic ABH (exponent m = 2). from [51] unde Figure 28. Phot r a CC o of a l BY-NC inear A -ND 4 BH (exponent .0 license. Copy m = 1) and a right Azbaid El Ouah quadratic ABH ( abi, exponent A., Kryl mo = 2). Reprinted v, V.V. and Reprinted from [51] under a CC BY-NC-ND 4.0 license. Copyright Azbaid El Ouahabi, A., Krylov, V.V. O'Boy, D.J from [51] unde ., 2015. r a CC BY-NC-ND 4.0 license. Copyright Azbaid El Ouahabi, A., Krylov, V.V. and and O’Boy, D.J., 2015. O'Boy, D.J., 2015. A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption performance was developed by Zhou and Semperlotti [53]. The HR cavity shown in Figure 29 is performance was developed by Zhou and Semperlotti [53]. The HR cavity shown in Figure 29 is part performa itioned by a nce wa flexibl s developed by Zhou e plate-embedded ABH and and Semperlotti the numer [53] ica . The HR l results show cavity shown i a great incre n Fi ase gure in t29 he is partitioned by a flexible plate-embedded ABH and the numerical results show a great increase in the partitioned by a flexible plate-embedded ABH and the numerical results show a great increase in the energy absorption of the HR system. energy absorption of the HR system. energy absorption of the HR system. Figure 29. Schematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Foundation, 20 Figure 29. Sch 16. ematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Figure 29. Schematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Foundation, 2016. Foundation, 2016. 3.3. Application of ABH to Vibration Energy Harvesting Zhao and Conlon developed a numerical study to investigate the structures tailored with ABHs to enhance vibration energy harvesting under both steady state and transient excitation [10]. Figure 30 shows the schematic of the plate with five equally spaced 1D ABH grooves attached with transducers. The plate is excited on the right side by a 400 N force sweeping from 0 to 10 kHz. The external resistance is 1 W. The numerical results of this study show the performance of the energy harvesting under steady state excitation and transient excitation. It can be observed that, by comparing with the flat plate, the normalized energy ratio of the plate with ABH increases drastically up to 80% in the 5–10 kHz frequency band at all the five ABH locations, and it increases most at Location 2. This study shows that the structure with ABH can drastically increase the efficiency of the energy harvesting. Later, another numerical and experimental investigation of a plate with three 2D ABHs (as shown in Figure 31) was also studied by Zhao and Conlon [54]. The ABHs effectively focus broadband energy to the center of the ABH. This also indicates that the focusing ability of the ABH is independent of the spectral and spatial characteristics of the external mechanical load. Acoustics 2018, 1, x FOR PEER REVIEW 16 of 29 3.3. Application of ABH to Vibration Energy Harvesting Zhao and Conlon developed a numerical study to investigate the structures tailored with ABHs Acoustics 2018, 1, x FOR PEER REVIEW 16 of 29 to enhance vibration energy harvesting under both steady state and transient excitation [10]. Figure 30 shows the schematic of the plate with five equally spaced 1D ABH grooves attached with 3.3. Application of ABH to Vibration Energy Harvesting transducers. The plate is excited on the right side by a 400 N force sweeping from 0 to 10 kHz. The Zhao and Conlon developed a numerical study to investigate the structures tailored with ABHs external resistance is 1 Ω. The numerical results of this study show the performance of the energy to enhance vibration energy harvesting under both steady state and transient excitation [10]. Figure harvesting under steady state excitation and transient excitation. It can be observed that, by 30 shows the schematic of the plate with five equally spaced 1D ABH grooves attached with comparing with the flat plate, the normalized energy ratio of the plate with ABH increases drastically transducers. The plate is excited on the right side by a 400 N force sweeping from 0 to 10 kHz. The up to 80 exte% i rnal r n e the 5– sistance is 10 kHz 1 Ω f . The numeric requency ba al re nd a sults of t t all th he fi is study sho ve ABH wloca the perfor tions, mance of the and it increa energy ses most at harvesting under steady state excitation and transient excitation. It can be observed that, by Location 2. This study shows that the structure with ABH can drastically increase the efficiency of comparing with the flat plate, the normalized energy ratio of the plate with ABH increases drastically the energy harvesting. Later, another numerical and experimental investigation of a plate with three up to 80% in the 5–10 kHz frequency band at all the five ABH locations, and it increases most at 2D ABHs (as shown in Figure 31) was also studied by Zhao and Conlon [54]. The ABHs effectively Location 2. This study shows that the structure with ABH can drastically increase the efficiency of Acoustics 2019, 1 237 focus broadband energy to the center of the ABH. This also indicates that the focusing ability of the the energy harvesting. Later, another numerical and experimental investigation of a plate with three ABH is independent of the spectral and spatial characteristics of the external mechanical load. 2D ABHs (as shown in Figure 31) was also studied by Zhao and Conlon [54]. The ABHs effectively focus broadband energy to the center of the ABH. This also indicates that the focusing ability of the ABH is independent of the spectral and spatial characteristics of the external mechanical load. Figure 30. A schematic of the plate with five equally spaced 1D ABH grooves, and one of the ABH Figure 30. A schematic of the plate with five equally spaced 1D ABH grooves, and one of the ABH Figure 30. A schematic of the plate with five equally spaced 1D ABH grooves, and one of the ABH with the surface mounted piezo-transducer. Adapted from [10]. Copyright IOP Publishing, 2014. with the surface mounted piezo-transducer. Adapted from [10]. Copyright IOP Publishing, 2014. with the surface mounted piezo-transducer. Adapted from [10]. Copyright IOP Publishing, 2014. Figure 31. A schematic of the plate with three equally-spaced 2D ABHs. Adapted from [54]. Copyright FigureFigure 31. 31. A schematic A schematic of of the the plate plate with thre with threee equally-s equally-spaced paced 2D 2D ABHs. ABHs. Adapted from Adapted fr[54] om. Copyright [54]. Copyright IOP Publishing, 2015. IOP Publishing, 2015. IOP Publishing, 2015. An experimental study on vibration energy harvesting using a cantilever beam with a modified An exper ABH caviitme y was ntal deve study lope on d by vibration Zhao an ene d Pras rgy h ad [8 a]. rvest The cant ing using ilever be a cam antilever is designed beam wit with a modified h an ABH An experimental study on vibration energy harvesting using a cantilever beam with a modified cavity near the fixed end due to the presence of higher strain energy. The experimental setup of a ABH cavity was developed by Zhao and Prasad [8]. The cantilever beam is designed with an ABH cantilever beam with modified ABH cavity attaching a piezo sensor is shown in Figure 32. Figure 33 ABH cavity was developed by Zhao and Prasad [8]. The cantilever beam is designed with an ABH cavity near the fixed end due to the presence of higher strain energy. The experimental setup of a shows the energy concentration effect of the ABH cavity at 2000 Hz. Figure 34 shows the experimental cavity near the fixed end due to the presence of higher strain energy. The experimental setup of a cantilever beam with modified ABH cavity attaching a piezo sensor is shown in Figure 32. Figure 33 results of the voltage power spectra. The vertical axis presents the decibel value referring to 1 volt. It shows the energy concentration effect of the ABH cavity at 2000 Hz. Figure 34 shows the experimental cantilever beam with modified ABH cavity attaching a piezo sensor is shown in Figure 32. Figure 33 can be observed that the beam with ABH cavity (red line) has a higher voltage output level within results of the voltage power spectra. The vertical axis presents the decibel value referring to 1 volt. It shows the energy concentration effect of the ABH cavity at 2000 Hz. Figure 34 shows the experimental the frequency range from 900–500 Hz, the voltage output is 10 dB higher than the beam without the can be observed that the beam with ABH cavity (red line) has a higher voltage output level within ABH (blue line). The increases due to the ABH cavity are substantial, even without considering the results of the voltage power spectra. The vertical axis presents the decibel value referring to 1 volt. the frequency range from 900–500 Hz, the voltage output is 10 dB higher than the beam without the neutralization of the piezo material. Embedding a modified ABH cavity uses less piezoelectrical It can be observed that the beam with ABH cavity (red line) has a higher voltage output level within ABH (b material lue line). The and decreases the increase weight of th s due to the host stru e ABH cacture. Tuna vity are subst bility ca antial, even wit n be achieved by a hout conside djusting the ring the the frequency range from 900–500 Hz, the voltage output is 10 dB higher than the beam without length of the ABH cavity. Thus, ABH cavity design into structures has good potential in increasing neutralization of the piezo material. Embedding a modified ABH cavity uses less piezoelectrical the ABH (blue the energy line). harvested. The increases due to the ABH cavity are substantial, even without considering material and decreases the weight of the host structure. Tunability can be achieved by adjusting the the neutralization of the piezo material. Embedding a modified ABH cavity uses less piezoelectrical length of the ABH cavity. Thus, ABH cavity design into structures has good potential in increasing material the energy and decr harvested. eases the weight of the host structure. Tunability can be achieved by adjusting the length of the ABH cavity. Thus, ABH cavity design into structures has good potential in increasing the Acoustics 2018, 1, x FOR PEER REVIEW 17 of 29 energy harvested. (a) (b) Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached piezo sensor. (b) Schematic of a 3D-printed beam with an ABH cavity [8]. piezo sensor. (b) Schematic of a 3D-printed beam with an ABH cavity [8]. (a) (b) Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity and (b) with an ABH cavity at 2000 Hz [8]. Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity [8]. A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the numerical result of power harvested. The results show that the beam with ABH cavities are more effective for vibration energy harvesting than uniform structures. Acoustics 2018, 1, x FOR PEER REVIEW 17 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 17 of 29 (a) (b) (a) (b) Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached Acoustics piezo sensor. ( 2019, 1 b) Schematic of a 3D-printed beam with an ABH cavity [8]. 238 piezo sensor. (b) Schematic of a 3D-printed beam with an ABH cavity [8]. (a) (b) (a) (b) Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity and (b) Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity and (b) with an ABH cavity at 2000 Hz [8]. and (b) with an ABH cavity at 2000 Hz [8]. with an ABH cavity at 2000 Hz [8]. Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity [8]. the ABH cavity and red line is for the beam with the ABH cavity [8]. the ABH cavity and red line is for the beam with the ABH cavity [8]. A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the numerical result of power harvested. The results show that the beam with ABH cavities are more the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the numerical result of power harvested. The results show that the beam with ABH cavities are more effective for vibration energy harvesting than uniform structures. numerical result of power harvested. The results show that the beam with ABH cavities are more effective for vibration energy harvesting than uniform structures. Acoustics 2018, 1, x FOR PEER REVIEW 18 of 29 effective for vibration energy harvesting than uniform structures. Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from [26]. the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from Copyright IEEE, 2018. [26]. Copyright IEEE, 2018. Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 Kω. Reprinted with permission from [26]. Copyright IEEE, 2018. 3.4. Discussion Applications of ABH for vibration control, sound radiation and vibration energy harvesting are presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with damping materials or piezo transducers are the major methods for the design of structures to enhance the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. 4. Design of Structures Using Acoustic Black Holes 4.1. Design of 1D ABH The length of the ABH (LABH), the exponent m, and the residual thickness h1 are the three geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform thickness (no ABH). LABH affects the length of ABH part when m is fixed as shown in Figure 37b. For 2D ABH, LABH represents the radius of outer circle abstracting the radius of inner hole. (a) (b) Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with various LABH(in). Acoustics 2018, 1, x FOR PEER REVIEW 18 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 18 of 29 Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from Acoustics 2019, 1 239 the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from [26]. Copyright IEEE, 2018. [26]. Copyright IEEE, 2018. Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 Kω. Reprinted Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 K!. Reprinted Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 Kω. Reprinted with permission from [26]. Copyright IEEE, 2018. with permission from [26]. Copyright IEEE, 2018. with permission from [26]. Copyright IEEE, 2018. 3.4. Discussion 3.4. Discussion 3.4. Discussion Applications of ABH for vibration control, sound radiation and vibration energy harvesting are Applications of ABH for vibration control, sound radiation and vibration energy harvesting are Applications of ABH for vibration control, sound radiation and vibration energy harvesting are presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with damping materials or piezo transducers are the major methods for the design of structures to enhance damping materials or piezo transducers are the major methods for the design of structures to enhance damping materials or piezo transducers are the major methods for the design of structures to enhance the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. 4. Design of Structures Using Acoustic Black Holes 4. Design of Structures Using Acoustic Black Holes 4. Design of Structures Using Acoustic Black Holes 4.1. Design of 1D ABH 4.1. Design of 1D ABH 4.1. Design of 1D ABH The length of the ABH (LABH), the exponent m, and the residual thickness h1 are the three The length of the ABH (L ), the exponent m, and the residual thickness h are the three ABH 1 The length of the ABH (LABH), the exponent m, and the residual thickness h1 are the three geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform thickness (no ABH). LABH affects the length of ABH part when m is fixed as shown in Figure 37b. For thickness (no ABH). L affects the length of ABH part when m is fixed as shown in Figure 37b. ABH thickness (no ABH). LABH affects the length of ABH part when m is fixed as shown in Figure 37b. For 2D ABH, LABH represents the radius of outer circle abstracting the radius of inner hole. For 2D ABH, L represents the radius of outer circle abstracting the radius of inner hole. ABH 2D ABH, LABH represents the radius of outer circle abstracting the radius of inner hole. (a) (a) (b) (b) Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with various LABH(in). various LABH(in). various L (in). ABH In the study by Krylov, which is mentioned in Section 3.1.1 [29], the concept of “truncation” is applied. For a given value of exponent m, the truncation length is controlled by two variables, the length of the ABH L and the residual thickness h . In this numerical study, it can be observed ABH 1 that when the truncation length is larger than about 0.01 m the reflection coefficient of the beam with ABH (the solid line in Figure 4) increases sharply. This indicates that even a small truncation length results in a large increase in the reflection coefficient, which weakens the ABH effect. Thus, a sharp Acoustics 2018, 1, x FOR PEER REVIEW 19 of 29 In the study by Krylov, which is mentioned in Section 3.1.1 [29], the concept of “truncation” is applied. For a given value of exponent m, the truncation length is controlled by two variables, the length of the ABH LABH and the residual thickness h1. In this numerical study, it can be observed that when the truncation length is larger than about 0.01 m the reflection coefficient of the beam with ABH Acoustics 2019, 1 240 (the solid line in Figure 4) increases sharply. This indicates that even a small truncation length results in a large increase in the reflection coefficient, which weakens the ABH effect. Thus, a sharp end of end of an ABH is critical for structural design. It is noted that a small amount of damping material can an ABH is critical for structural design. It is noted that a small amount of damping material can effectively restrain the increase of reflection coefficient. effectively restrain the increase of reflection coefficient. A numerical and experimental study on sound radiation of a beam with a 1D ABH is developed A numerical and experimental study on sound radiation of a beam with a 1D ABH is developed by Li and Ding [55]. In Figure 38, it can be seen that the increase of the truncation thickness results by Li and Ding [55]. In Figure 38, it can be seen that the increase of the truncation thickness results in in decrease of radiated sound power in the frequency from 43 to 160 Hz and increase from 626 Hz decrease of radiated sound power in the frequency from 43 to 160 Hz and increase from 626 Hz to 6 to 6 kHz. The reason for this phenomenon was discussed and explained. The increase of truncation kHz. The reason for this phenomenon was discussed and explained. The increase of truncation thickness causes added mass and equivalent stiffness effects, which makes an enhancement of the thickness causes added mass and equivalent stiffness effects, which makes an enhancement of the low frequency bandwidth. On the other hand, an increase of truncation thickness causes an increase low frequency bandwidth. On the other hand, an increase of truncation thickness causes an increase of the reflection coefficient and a decrease of the degree of wave concentration in the ABH, therefore of the reflection coefficient and a decrease of the degree of wave concentration in the ABH, therefore weakening noise suppression at high frequencies. This indicates that an optimization of the truncation weakening noise suppression at high frequencies. This indicates that an optimization of the thickness is required for the best performance of noise reduction. truncation thickness is required for the best performance of noise reduction. Figure 38. Sound radiation from the beams with various truncation thicknesses. Reprinted with Figure 38. Sound radiation from the beams with various truncation thicknesses. Reprinted with permission from [55]. Copyright Elsevier, 2019. permission from [55]. Copyright Elsevier, 2019. A wavelet-decomposed energy model is developed and investigated by Tang and Cheng [56]. A wavelet-decomposed energy model is developed and investigated by Tang and Cheng [56]. A new type of beam structure to achieve broad attenuation bands in relatively low frequencies is A new type of beam structure to achieve broad attenuation bands in relatively low frequencies is proposed. The model can be used to predict the frequency bounds of the band gaps. A parametric proposed. The model can be used to predict the frequency bounds of the band gaps. A parametric analysis is also illustrated. The increase of m and decrease of the truncation thickness h0 would analysis is also illustrated. The increase of m and decrease of the truncation thickness h would decrease the lower band gaps so as to enhance the ABH effect. decrease the lower band gaps so as to enhance the ABH effect. A numerical study on the influence of geometrical parameters of ABHs on vibration of cantilever A numerical study on the influence of geometrical parameters of ABHs on vibration of cantilever beams was studied by Zhao and Prasad [57]. Figure 39 shows the 3rd mode shape simulation results beams was studied by Zhao and Prasad [57]. Figure 39 shows the 3rd mode shape simulation results of the beam without ABH [58] and beams with 1D ABH with various m values. It can be observed of the beam without ABH [58] and beams with 1D ABH with various m values. It can be observed that the amplitude of displacement increases at the ABH location (with the increase of m value) and that the amplitude of displacement increases at the ABH location (with the increase of m value) and decreases at other locations of nodes. decreases at other locations of nodes. The numerical study on the sound radiation from vibrating cantilever aluminum beams with The numerical study on the sound radiation from vibrating cantilever aluminum beams with various exponent m, which are 10 inches long, 1 inch wide, and 1/8 inch thick, with various shapes various exponent m, which are 10 inches long, 1 inch wide, and 1/8 inch thick, with various shapes of ABHs at the free end, and the residual thickness h1 equal to 1/64 inch is developed by Zhao and of ABHs at the free end, and the residual thickness h equal to 1/64 inch is developed by Zhao and Prasad [27]. The results show that, in this group of beams with given residual thickness and length Prasad [27]. The results show that, in this group of beams with given residual thickness and length of of the ABH, the concentration effect performs better when m is less than 5. However, when m is larger the ABH, the concentration effect performs better when m is less than 5. However, when m is larger than 5, the ABH loses its concentration effect. Figure 40b shows how the logarithmic ratio of near than 5, the ABH loses its concentration effect. Figure 40b shows how the logarithmic ratio of near field sound radiation at the free end and the other three anti-node locations changes with various m field sound radiation at the free end and the other three anti-node locations changes with various m values, which essentially shows how the m values affect the concentration effect. This indicates that values, which essentially shows how the m values affect the concentration effect. This indicates that the m value has a maximum limitation. Another theoretical and numerical study on vibration energy the m value has a maximum limitation. Another theoretical and numerical study on vibration energy concentration of the 1D ABH was developed by Li and Ding [59]. The length of the edge part with concentration of the 1D ABH was developed by Li and Ding [59]. The length of the edge part with comparatively large deflection deformation Le, as show in Figure 41, was investigated. With the given comparatively large deflection deformation L , as show in Figure 41, was investigated. With the given parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy are trapped within a small area (L ) from the wedge tip, which shows the energy concentration effect. For a e Acoustics 2018, 1, x FOR PEER REVIEW 20 of 29 parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy are trapped within a small area (Le) from the wedge tip, which shows the energy concentration effect. Acoustics 2018, 1, x FOR PEER REVIEW 20 of 29 For a fixed value of Le, the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. This study also indicates that the exponent m has an optimum value to provide the best concentration parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy Acoustics 2019, 1 241 effect. are trapped within a small area (Le) from the wedge tip, which shows the energy concentration effect. Acoustics 2018, 1, x FOR PEER REVIEW 20 of 29 For a fixed value of Le, the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. fixed value of L , the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. This study This study also indicates that the exponent m has an optimum value to provide the best concentration parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy also indicates that the exponent m has an optimum value to provide the best concentration effect. effect. are trapped within a small area (Le) from the wedge tip, which shows the energy concentration effect. For a fixed value of Le, the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. This study also indicates that the exponent m has an optimum value to provide the best concentration effect. Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free (a) (a) end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. (b) (a) (b) (b) Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at free end and other three anti-nodes along with the various m values in Group 1 with excitation of 1600 Hz [57]. Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at free end and o free endther three anti-nodes and other three anti-nodes along w along with ith the various m values the various m values in Gr in G oupr1oup 1 with excitation o with excitation of f free end and other three anti-nodes along with the various m values in Group 1 with excitation of 1600 Hz [57]. 1600 Hz [57]. 1600 Hz [57]. Figure 41. Schematic in 1D shows the ABH length of the edge part with a comparatively large deflection deformation Le. Adapted from [59]. Copyright SAGE Publications, 2018. One possible explanation of the phenomenon is that when the m value is larger, the ABH part Figure Figure 41. 41. Schematic Schematic in 1D in 1D shows shows the ABH length of the ABH length of the edge the edge part w part with a comparatively ith a comparatively large large deflection becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration deformation deflection defo L . rm Adapted ation Le. Adapted from [59 from [59]. Copyright ]. Copyright SAGE Publications, 2018 SAGE Publications, 2018. . Figure 41. Schematic in 1D shows the ABH length of the edge part with a comparatively large systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a deflection deformation Le. Adapted from [59]. Copyright SAGE Publications, 2018. untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 One possible explanation of the phenomenon is that when the m value is larger, the ABH part One possible explanation of the phenomenon is that when the m value is larger, the ABH part generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration One possible explanation of the phenomenon is that when the m value is larger, the ABH part [9] applied another assumption. In this study, it is shown that the violation of smoothness criterion systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a is a significant design problem for ABHs. Smoothness criterion is an assumption that the change of becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon [9] [9] applied another assumption. In this study, it is shown that the violation of smoothness criterion untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 is a significant design problem for ABHs. Smoothness criterion is an assumption that the change of generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon [9] applied another assumption. In this study, it is shown that the violation of smoothness criterion is a significant design problem for ABHs. Smoothness criterion is an assumption that the change of Acoustics 2019, 1 242 Acoustics 2018, 1, x FOR PEER REVIEW 21 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 21 of 29 applied another assumption. In this study, it is shown that the violation of smoothness criterion is a the local wave number is small enough over distances to make the ABH perform better, which is significant design problem for ABHs. Smoothness criterion is an assumption that the change of the the local wave number is small enough over distances to make the ABH perform better, which is stated by Mironov [12]. Wave number variation (NWV) is investigated to assess the validity of the local wave number is small enough over distances to make the ABH perform better, which is stated by stated by Mironov [12]. Wave number variation (NWV) is investigated to assess the validity of the smoothness criterion. Equation (17) shows that NWV is a function of frequency and position and Mironov [12]. Wave number variation (NWV) is investigated to assess the validity of the smoothness smoothness criterion. Equation (17) shows that NWV is a function of frequency and position and need to be much less than 1. criterion. Equation (17) shows that NWV is a function of frequency and position and need to be much need to be much less than 1. less than 1. / (𝑥) 1   1 𝐸  1 𝑑ℎ 1/4 (𝑥) 1 1 𝐸 1 𝑑ℎ (17) dk(x) 1 = 1 E 1 dh ≪1 , (17) = = ( ) ≪1  1, , (17) 𝑑𝑥 𝑘 2 𝜌𝜔 12 1−𝜐 ℎ 𝑑𝑥 2 2 2 1/2 𝑑𝑥 𝑘 2 𝜌𝜔 12(1−𝜐 ) ℎ 𝑑𝑥 dx k 2 rw 12(1 u ) dx where E is the Young’s modulus, ν is the Poisson’s ratio, ρ is the density, h is the varying thickness at where where EE is th is the e Young’s mo Young’s modulus, dulus, νn is the is the Poisson’ Poisson’s ratio, s ratio,ρ is the den is the density sity, , hh is t is the he v varyin arying t g thickness hickness at at the ABH of the beams, and ω is the angular frequency of the flexural wave. According to this theory, the ABH of the beams, and w is the angular frequency of the flexural wave. According to this theory, the ABH of the beams, and ω is the angular frequency of the flexural wave. According to this theory, an increased m value increases the NWV and the smoothness criterion is violated for a greater range a an n iincr ncrea eased sed m val m value ue iincr ncrea eases ses the NWV the NWV a and nd the the ssmoothness moothness ccriterion riterion iis s vi violated olated ffor or a g a gr rea eater ter ra range nge of frequencies. For m = 10, the NWV is above one for the frequency range of study, which is shown of frequencies. For m = 10, the NWV is above one for the frequency range of study, which is shown in of frequencies. For m = 10, the NWV is above one for the frequency range of study, which is shown in Figure 42. Further work investigates that NWV < 0.4 is a good rule for designing ABHs [60]. in Figur Figu ere 42 42. Further . Further work worinvestigates k investigate that s that NWV NWV < 0. < 0.4 is 4 is a good a good rule rule for for designing designinABHs g ABHs [6 [60].0]. (a) (b) (a) (b) Figure 42. (a) Reflection coefficients and NWV for 5-cm long ABHs with m = 2, 4, 10. (b) Increased m Figure 42. (a) Reflection coefficients and NWV for 5-cm long ABHs with m = 2, 4, 10. (b) Increased m Figure 42. (a) Reflection coefficients and NWV for 5-cm long ABHs with m = 2, 4, 10. (b) Increased increases the NWV and violates the smoothness criterion. Reprinted with permission from [9]. increases the NWV and violates the smoothness criterion. Reprinted with permission from [9]. m increases the NWV and violates the smoothness criterion. Reprinted with permission from [9]. Copyright Acoustic Society of America, 2014. Copyright Acoustic Society of America, 2014. Copyright Acoustic Society of America, 2014. In the study of Denis et al. which investigated ABH as a vector for energy transfer from the low In the study of Denis et al. which investigated ABH as a vector for energy transfer from the low In the study of Denis et al. which investigated ABH as a vector for energy transfer from the frequencies to high frequencies in the framework of nonlinear vibration, a parametric study on a frequencies to high frequencies in the framework of nonlinear vibration, a parametric study on a low frequencies to high frequencies in the framework of nonlinear vibration, a parametric study on a number of beams shown in Figure 43 was developed [61]. Longer additional termination, which number of beams shown in Figure 43 was developed [61]. Longer additional termination, which number of beams shown in Figure 43 was developed [61]. Longer additional termination, which means means longer LABH, can improve the efficiency of energy transfer. It is also noted that a 2D ABH can means longer LABH, can improve the efficiency of energy transfer. It is also noted that a 2D ABH can longer L , can improve the efficiency of energy transfer. It is also noted that a 2D ABH can present present larger LABH with minimal thickness and obtains larger modal density in the low-frequency ABH present larger LABH with minimal thickness and obtains larger modal density in the low-frequency larger L with minimal thickness and obtains larger modal density in the low-frequency range. range. ABH range. Figure 43. Beams with various lengths of ABHs and various lengths of damping. Reprinted with Figure 43. Beams with various lengths of ABHs and various lengths of damping. Reprinted with Figure 43. Beams with various lengths of ABHs and various lengths of damping. Reprinted with permission from [61]. Copyright Elsevier, 2017. permission from [61]. Copyright Elsevier, 2017. permission from [61]. Copyright Elsevier, 2017. The effect of the length of ABH is also investigated by Zhao and Prasad [8,27]. For a given beam The ef The effect fect o of f tthe he len length gth of of ABH ABHis is also also iinvestigated nvestigated b by y Z Zhao hao and andP Prasad rasad [[8 8,,2 27 7]]. . F For or a a g given iven b beam eam at a given excitation frequency there are specific optimum values of LABH. The simulation results of a at t a a gi given ven exci excitation tation ffr req equency uency ther there e ar are specific e specific optimum values optimum values of of L LABH. The si . The simulation mulation resu results lts of of ABH total energy at the ABH location of beams with the ABH cavity (shown in Figure 32b) is shown in total energy at the ABH location of beams with the ABH cavity (shown in Figure 32b) is shown in Figure 44. It is seen that the beam with LABH of 19 mm has maximum energy output. This indicates Figure 44. It is seen that the beam with LABH of 19 mm has maximum energy output. This indicates that the ABH is tunable by changing LABH to obtain a higher concentration of vibration energy [8]. that the ABH is tunable by changing LABH to obtain a higher concentration of vibration energy [8]. 𝑑𝑘 𝑑𝑘 Acoustics 2019, 1 243 total energy at the ABH location of beams with the ABH cavity (shown in Figure 32b) is shown in Figure 44. It is seen that the beam with L of 19 mm has maximum energy output. This indicates ABH that the ABH is tunable by changing L to obtain a higher concentration of vibration energy [8]. ABH Acoustics 2018, 1, x FOR PEER REVIEW 22 of 29 Figure 44. Simulation results of total energy output of 3D-printed beams with ABH cavities with Figure 44. Simulation results of total energy output of 3D-printed beams with ABH cavities with various LABH [8]. various L [8]. ABH From previous studies in this section, it can be seen that, all the three geometrical parameters From previous studies in this section, it can be seen that, all the three geometrical parameters need an optimization design to get a best performance on vibration damping and sound reduction. need an optimization design to get a best performance on vibration damping and sound reduction. An investigation of the optimization design and position of an embedded 1D ABH is studied by An investigation of the optimization design and position of an embedded 1D ABH is studied by McCormick [62]. This study is on a thin simply-supported beam with a 1D ABH at some position McCormick [62]. This study is on a thin simply-supported beam with a 1D ABH at some position along along the beam. The relevant design variables are shown in Figure 45, which are the length of the the beam. The relevant design variables are shown in Figure 45, which are the length of the ABH L , ABH ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point the portion of the damped taper L , the thickness of the damping layer h , the drive point location D, d d location D, and the offset between the center of the beam and the center of the ABH B. The and the offset between the center of the beam and the center of the ABH B. The optimization result optimization result is shown in Figure 46. Comparing with the uniform beam, the simple damped is shown in Figure 46. Comparing with the uniform beam, the simple damped beam can decrease beam can decrease the surface-averaged velocity, but increases the mass by 5%. The optimized design the surface-averaged velocity, but increases the mass by 5%. The optimized design can decrease the can decrease the total mass of the beam by 15% and decreases the total surface-averaged velocity total mass of the beam by 15% and decreases the total surface-averaged velocity response by 12 dB, response by 12 dB, which has more reduction than a simple damped beam. The aim of the study is which has more reduction than a simple damped beam. The aim of the study is to figure out the to figure out the ABH design that minimizes the total mass of the beam and, at the same time, ABH design that minimizes the total mass of the beam and, at the same time, minimizes the total minimizes the total surface-averaged velocity response. This benefits the design of ABHs for surface-averaged velocity response. This benefits the design of ABHs for vibration reduction without vibration reduction without adding mass. adding mass. As shown in the studies on vibration of beams and plates with ABH in Section 3.1, there is a As shown in the studies on vibration of beams and plates with ABH in Section 3.1, there is a minimum effective frequency, below which the ABH damping performance is not evident. A minimum effective frequency, below which the ABH damping performance is not evident. A numerical numerical and experimental study on the investigation of disappearance of the ABH effect is and experimental study on the investigation of disappearance of the ABH effect is presented by Tang presented by Tang and Cheng [63]. It shows that cut-on frequency bands close to the low-order local and Cheng [63]. It shows that cut-on frequency bands close to the low-order local resonant frequencies resonant frequencies of the beam exist and the ABH effect failure in these frequency bands. The of the beam exist and the ABH effect failure in these frequency bands. The failure frequencies of the failure frequencies of the beam are delimited by the excitation point in order to avoid the beam are delimited by the excitation point in order to avoid the phenomenon in the structural design. phenomenon in the structural design. Figure 45. Schematic of the simple supported beams with the relevant design variables: Length of ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point location D, and the offset between the center of the beam and the center of the ABH B. Reprinted with permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. Acoustics 2018, 1, x FOR PEER REVIEW 22 of 29 Figure 44. Simulation results of total energy output of 3D-printed beams with ABH cavities with various LABH [8]. From previous studies in this section, it can be seen that, all the three geometrical parameters need an optimization design to get a best performance on vibration damping and sound reduction. An investigation of the optimization design and position of an embedded 1D ABH is studied by McCormick [62]. This study is on a thin simply-supported beam with a 1D ABH at some position along the beam. The relevant design variables are shown in Figure 45, which are the length of the ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point location D, and the offset between the center of the beam and the center of the ABH B. The optimization result is shown in Figure 46. Comparing with the uniform beam, the simple damped beam can decrease the surface-averaged velocity, but increases the mass by 5%. The optimized design can decrease the total mass of the beam by 15% and decreases the total surface-averaged velocity response by 12 dB, which has more reduction than a simple damped beam. The aim of the study is to figure out the ABH design that minimizes the total mass of the beam and, at the same time, minimizes the total surface-averaged velocity response. This benefits the design of ABHs for vibration reduction without adding mass. As shown in the studies on vibration of beams and plates with ABH in Section 3.1, there is a minimum effective frequency, below which the ABH damping performance is not evident. A numerical and experimental study on the investigation of disappearance of the ABH effect is presented by Tang and Cheng [63]. It shows that cut-on frequency bands close to the low-order local resonant frequencies of the beam exist and the ABH effect failure in these frequency bands. The failure frequencies of the beam are delimited by the excitation point in order to avoid the Acoustics 2019, 1 244 phenomenon in the structural design. Figure 45. Schematic of the simple supported beams with the relevant design variables: Length of Figure 45. Schematic of the simple supported beams with the relevant design variables: Length of ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point ABH L , the portion of the damped taper L , the thickness of the damping layer h , the drive point ABH d d location D, and the offset between the center of the beam and the center of the ABH B. Reprinted with location D, and the offset between the center of the beam and the center of the ABH B. Reprinted with Acoustics 2018, 1, x FOR PEER REVIEW 23 of 29 permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. Figure 46. Approximate Pareto front (black curve) after 10,000 evaluations. Objective function Figure 46. Approximate Pareto front (black curve) after 10,000 evaluations. Objective function evaluations for a sample of designs are denoted by red dots. The black × shows the function evaluation evaluations for a sample of designs are denoted by red dots. The black shows the function evaluation for an unmodified beam. The black circle shows the optimization. The black + shows the evaluation for an unmodified beam. The black circle shows the optimization. The black + shows the evaluation of of a uniform beam with the same damping. Reprinted with permission from [62]. Copyright a uniform beam with the same damping. Reprinted with permission from [62]. Copyright McCormick, McCormick, C.A.; Shepherd, M.R., 2018. C.A.; Shepherd, M.R., 2018. 4.2. Design of a 2D ABH 4.2. Design of a 2D ABH For a 2D ABH, it is to be noted that, besides geometrical parameters, the spatial layout of the For a 2D ABH, it is to be noted that, besides geometrical parameters, the spatial layout of the ABH, the material of the host structure and damping material, the diameter of center hole, and the ABH, the material of the host structure and damping material, the diameter of center hole, and the number of ABH also affect the performance of the ABH. number of ABH also affect the performance of the ABH. A numerical analysis of flexural ray trajectories in 2D ABH is developed by Huang et al. [32]. A numerical analysis of flexural ray trajectories in 2D ABH is developed by Huang et al. [32]. Figure 47 shows that for a given value of exponent m, the 2D ABH with thinner residual thickness Figure 47 shows that for a given value of exponent m, the 2D ABH with thinner residual thickness (h1 = 0.0002 m) absorbs more propagation energy than the one with residual thickness of h1 = 0.001 m. (h = 0.0002 m) absorbs more propagation energy than the one with residual thickness of h = 0.001 m. 1 1 In this study, the results also show the ratio of the rays converging to the center is 99.69% in the ideal In this study, the results also show the ratio of the rays converging to the center is 99.69% in the ideal ABH. The wave focalization can be enhanced with a larger m and a small h1. ABH. The wave focalization can be enhanced with a larger m and a small h . (a) (b) Figure 47. Numerical analysis of flexural ray trajectories in 2D ABH with various geometrical analysis (a) m = 2; and (b) m = 3. Reprinted with permission from [32]. Copyright Elsevier, 2018. Acoustics 2018, 1, x FOR PEER REVIEW 23 of 29 Figure 46. Approximate Pareto front (black curve) after 10,000 evaluations. Objective function evaluations for a sample of designs are denoted by red dots. The black × shows the function evaluation for an unmodified beam. The black circle shows the optimization. The black + shows the evaluation of a uniform beam with the same damping. Reprinted with permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. 4.2. Design of a 2D ABH For a 2D ABH, it is to be noted that, besides geometrical parameters, the spatial layout of the ABH, the material of the host structure and damping material, the diameter of center hole, and the number of ABH also affect the performance of the ABH. A numerical analysis of flexural ray trajectories in 2D ABH is developed by Huang et al. [32]. Figure 47 shows that for a given value of exponent m, the 2D ABH with thinner residual thickness (h1 = 0.0002 m) absorbs more propagation energy than the one with residual thickness of h1 = 0.001 m. Acoustics 2019, 1 245 In this study, the results also show the ratio of the rays converging to the center is 99.69% in the ideal ABH. The wave focalization can be enhanced with a larger m and a small h1. (a) (b) Figure 47. Numerical analysis of flexural ray trajectories in 2D ABH with various geometrical analysis Figure 47. Numerical analysis of flexural ray trajectories in 2D ABH with various geometrical analysis (a) m = 2; and (b) m = 3. Reprinted with permission from [32]. Copyright Elsevier, 2018. (a) m = 2; and (b) m = 3. Reprinted with permission from [32]. Copyright Elsevier, 2018. Acoustics 2018, 1, x FOR PEER REVIEW 24 of 29 A study on various layout of ABH on glass fiber composite plates with 1D and 2D ABH was A study on various layout of ABH on glass fiber composite plates with 1D and 2D ABH was studied by Bowyer and Krylov [16]. In this study, the glass fiber composite honeycomb sandwich studied by Bowyer and Krylov [16]. In this study, the glass fiber composite honeycomb sandwich panels with two 2D ABHs with various configurations (shown in Figure 48a) are investigated and panels with two 2D ABHs with various configurations (shown in Figure 48a) are investigated and the experimental results of acceleration are compared with a reference sandwich panel without ABH the experimental results of acceleration are compared with a reference sandwich panel without ABH plates. The results show that theoretically the structure in Figure 48b should be the optimum layout to plates. The results show that theoretically the structure in Figure 48b should be the optimum layout perform with the best vibration damping. to perform with the best vibration damping. (a) (b) Figure 48. Cross-section of (a) various configurations of ABHs and (b) optimum configurations of Figure 48. Cross-section of (a) various configurations of ABHs and (b) optimum configurations of ABHs. Reprinted from [16] under a CC BY 3.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2014. ABHs. Reprinted from [16] under a CC BY 3.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2014. Due to the limitation of manufacturing, the ABH may not be perfect. A study on the wave energy Due to the limitation of manufacturing, the ABH may not be perfect. A study on the wave energy focalization in a plate with imperfect 2D ABH is developed by Huang et al. [64]. The imperfect 2D focalization in a plate with imperfect 2D ABH is developed by Huang et al. [64]. The imperfect 2D ABH uses a polynomial profile instead of a power-law profile. The results still show drastic increases ABH uses a polynomial profile instead of a power-law profile. The results still show drastic increases in energy density around the tapered area. Chong also indicates that the smooth and stepped ABH in energy density around the tapered area. Chong also indicates that the smooth and stepped ABH beams are slightly different [22]. Denis et al. also find that the controlled imperfection of the tip of the 1D ABH causes a decrease of the reflection coefficient, which indicates that imperfect extremities are not detrimental to the performance of the ABH effect [65]. These two studies lower the stringent requirement of the ideal power-law thickness variation of ABH. A numerical study on the influence of number of ABHs on a plate in vibration is carried out by Conlon et al. [66,67]. The space averaged acceleration and total radiated sound power results for the 5 × 5 ABH (shown in Figure 49a) and uniform plates are compared in Figure 50. This shows that the plate with the ABH array has very effective results on vibration and noise reduction. Another comparison of total radiated sound power response of ABH panel between the panel with a 5 × 5 ABH array (Figure 49a) and the panel with 13 ABHs (Figure 49b) is shown in Figure 51. It can be seen that for higher frequencies (over 5 kHz) increasing of the number of ABHs results in more sound reduction, but in lower frequency this effect can only be observed in a few areas. This type of ABH array was also applied on the phononic crystals (PC) by Zhu and Semperlotti [68] to investigate peculiar dispersion characteristics. (a) (b) Figure 49. (a) Schematic of a panel with 5 × 5 ABH and (b) a panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 24 of 29 A study on various layout of ABH on glass fiber composite plates with 1D and 2D ABH was studied by Bowyer and Krylov [16]. In this study, the glass fiber composite honeycomb sandwich panels with two 2D ABHs with various configurations (shown in Figure 48a) are investigated and the experimental results of acceleration are compared with a reference sandwich panel without ABH plates. The results show that theoretically the structure in Figure 48b should be the optimum layout to perform with the best vibration damping. (a) (b) Acoustics 2019, 1 246 Figure 48. Cross-section of (a) various configurations of ABHs and (b) optimum configurations of ABHs. Reprinted from [16] under a CC BY 3.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2014. beams are slightly Due to the limitation o different [22]. f mDenis anufactet uring al. , the ABH m also find ay that not be perfect the contr . A st olled udy on t imperfection he wave energy of the tip of focalization in a plate with imperfect 2D ABH is developed by Huang et al. [64]. The imperfect 2D the 1D ABH causes a decrease of the reflection coefficient, which indicates that imperfect extremities ABH uses a polynomial profile instead of a power-law profile. The results still show drastic increases are not detrimental to the performance of the ABH effect [65]. These two studies lower the stringent in energy density around the tapered area. Chong also indicates that the smooth and stepped ABH requirement of the ideal power-law thickness variation of ABH. beams are slightly different [22]. Denis et al. also find that the controlled imperfection of the tip of the A numerical 1D ABH causes study a d on ecre the aseinfluence of the reflection coeffic of number ient of , which ABHs indic on ates th a plate at imp inevibration rfect extremis ities carried are out by not detrimental to the performance of the ABH effect [65]. These two studies lower the stringent Conlon et al. [66,67]. The space averaged acceleration and total radiated sound power results for the requirement of the ideal power-law thickness variation of ABH. 5  5 ABH (shown in Figure 49a) and uniform plates are compared in Figure 50. This shows that A numerical study on the influence of number of ABHs on a plate in vibration is carried out by the plate with the ABH array has very effective results on vibration and noise reduction. Another Conlon et al. [66,67]. The space averaged acceleration and total radiated sound power results for the 5 × 5 ABH (shown in Figure 49a) and uniform plates are compared in Figure 50. This shows that the comparison of total radiated sound power response of ABH panel between the panel with a 5  5 ABH plate with the ABH array has very effective results on vibration and noise reduction. Another array (Figure 49a) and the panel with 13 ABHs (Figure 49b) is shown in Figure 51. It can be seen comparison of total radiated sound power response of ABH panel between the panel with a 5 × 5 that for higher frequencies (over 5 kHz) increasing of the number of ABHs results in more sound ABH array (Figure 49a) and the panel with 13 ABHs (Figure 49b) is shown in Figure 51. It can be seen reduction, but in lower frequency this effect can only be observed in a few areas. This type of ABH that for higher frequencies (over 5 kHz) increasing of the number of ABHs results in more sound reduction, but in lower frequency this effect can only be observed in a few areas. This type of ABH array was also applied on the phononic crystals (PC) by Zhu and Semperlotti [68] to investigate array was also applied on the phononic crystals (PC) by Zhu and Semperlotti [68] to investigate peculiar dispersion characteristics. peculiar dispersion characteristics. (a) (b) Figure 49. (a) Schematic of a panel with 5 × 5 ABH and (b) a panel with 13 ABHs. Reprinted with Figure 49. (a) Schematic of a panel with 5  5 ABH and (b) a panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 25 of 29 Figure 50. The space averaged acceleration (bottom) and total radiated sound power (top) results for Figure 50. The space averaged acceleration (bottom) and total radiated sound power (top) results the 5 × 5 ABHs (read line) and uniform plates (black line). Reprinted with permission from [66]. for the 5  5 ABHs (read line) and uniform plates (black line). Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Figure 51. Total radiated sound power results for the panel with 5 × 5 ABHs and panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. 4.3. Design of Damping Layer The study Feurtado and Conlon [48] mentioned in Section 3.2 indicates that the radius of the damping layer has an optimum value. When the diameter is larger than the optimum value, the damping effect increases slightly. A time domain experimental study based on a laser visualization system is developed by Ji et al. [31]. The wave propagation and attenuation in a plate-embedded 1D ABH is investigated. It is also noted that there exists an optimal thickness for the damping layer. Krylov indicate that the damping material with higher material loss factor makes the damping effect of the ABH have better performance [29]. Denis et al. indicate that a good compromise has been found by using a long additional termination with a moderate length of the damping layer in the framework of nonlinear vibration, which can develop into a turbulent regime [61]. An optimization study was developed by Lee et al. [69] and indicates that the damping performance can be enhanced by treating the tip with an appropriate size of damping layer. Another study developed by Tang et al. [34] shows that the stiffness of the damping layer plays is more critical than mass for a better damping effect but, meanwhile, the effect of the added mass still needs to be accurately designed especially when the thickness of damping is considerable to the tip of the ABH wedge. 4.4. Studies on 3D-Printed Structures with ABH Due to the limitation of traditional manufacturing methods, a power-law curve is difficult to manufacture. Bowyer and Krylov indicate that milling is not a practical way to machine structures with 2D ABH and suggest that 3D printing technology can be applied to produce these structures [23]. Due to the well-known advantages of 3D printing [70], vibration and sound radiation of 3D- Acoustics 2018, 1, x FOR PEER REVIEW 25 of 29 Figure 50. The space averaged acceleration (bottom) and total radiated sound power (top) results for the 5 × 5 ABHs (read line) and uniform plates (black line). Reprinted with permission from [66]. Acoustics 2019, 1 247 Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Figure 51. Total radiated sound power results for the panel with 5 × 5 ABHs and panel with 13 ABHs. Figure 51. Total radiated sound power results for the panel with 5  5 ABHs and panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, Feurtado, P.A., 2015. P.A., 2015. 4.3. Design of Damping Layer 4.3. Design of Damping Layer The study Feurtado and Conlon [48] mentioned in Section 3.2 indicates that the radius of the The study Feurtado and Conlon [48] mentioned in Section 3.2 indicates that the radius of damping layer has an optimum value. When the diameter is larger than the optimum value, the the damping layer has an optimum value. When the diameter is larger than the optimum value, damping effect increases slightly. A time domain experimental study based on a laser visualization the damping effect increases slightly. A time domain experimental study based on a laser visualization system is developed by Ji et al. [31]. The wave propagation and attenuation in a plate-embedded 1D system is developed by Ji et al. [31]. The wave propagation and attenuation in a plate-embedded ABH is investigated. It is also noted that there exists an optimal thickness for the damping layer. 1D ABH is investigated. It is also noted that there exists an optimal thickness for the damping layer. Krylov indicate that the damping material with higher material loss factor makes the damping effect Krylov indicate that the damping material with higher material loss factor makes the damping effect of the ABH have better performance [29]. Denis et al. indicate that a good compromise has been found of the ABH have better performance [29]. Denis et al. indicate that a good compromise has been found by using a long additional termination with a moderate length of the damping layer in the framework by using a long additional termination with a moderate length of the damping layer in the framework of nonlinear vibration, which can develop into a turbulent regime [61]. An optimization study was of nonlinear vibration, which can develop into a turbulent regime [61]. An optimization study was developed by Lee et al. [69] and indicates that the damping performance can be enhanced by treating developed by Lee et al. [69] and indicates that the damping performance can be enhanced by treating the tip with an appropriate size of damping layer. Another study developed by Tang et al. [34] shows the tip with an appropriate size of damping layer. Another study developed by Tang et al. [34] shows that the stiffness of the damping layer plays is more critical than mass for a better damping effect but, that the stiffness of the damping layer plays is more critical than mass for a better damping effect but, meanwhile, the effect of the added mass still needs to be accurately designed especially when the meanwhile, the effect of the added mass still needs to be accurately designed especially when the thickness of damping is considerable to the tip of the ABH wedge. thickness of damping is considerable to the tip of the ABH wedge. 4.4. Studies on 3D-Printed Structures with ABH 4.4. Studies on 3D-Printed Structures with ABH Due to the limitation of traditional manufacturing methods, a power-law curve is difficult to Due to the limitation of traditional manufacturing methods, a power-law curve is difficult to manufacture. Bowyer and Krylov indicate that milling is not a practical way to machine structures manufacture. Bowyer and Krylov indicate that milling is not a practical way to machine structures with 2D ABH and suggest that 3D printing technology can be applied to produce these structures with 2D ABH and suggest that 3D printing technology can be applied to produce these structures [23]. [23]. Due to the well-known advantages of 3D printing [70], vibration and sound radiation of 3D- Due to the well-known advantages of 3D printing [70], vibration and sound radiation of 3D-printed structures are investigated by researchers. Zhou et al. developed a numerical study to investigate the dynamic and static properties of 3D-printed double-layered compound structures with ABHs [24]. Due to small residual thickness of the ABH profile, the structure with ABHs has low local stiffness and high stress concentration. The double-layered compound structure shows good damping performance and also significantly improves the static properties in structural stiffness and strength. Zhao and Prasad developed an experimental case study of vibration energy harvesting of a 3D-printed beam with a modified ABH cavity [8]. Liang et al. developed a numerical simulation for vibration energy harvesting of a 3D-printed beam with multiple ABH cavities [26]. Chong et al. numerically and experimentally studied the dynamic responses of 3D-printed beam with damped ABH grooves [22]. Rothe et al. investigated the dynamic behavior of the 3D-printed beams using a 3D- hexahedron finite element and an isotropic linear elastic homogenized material model [25]. A cantilever beam is embedded with the ABH at its free ends and fully fills the ABH area with flexible thermoplastic Acoustics 2019, 1 248 polyurethane (TPU) to make the overall thickness uniform. In comparison with the uniform beam, the beam-embedded ABH with TPU shows good damping performance. 5. Concluding Remarks This review has presented the recent theoretical and numerical studies on ABH. Applications of ABH on beam- and plate-type structures have been demonstrated. It is shown that the use of ABH in structural design is effective in controlling vibration and noise without adding additional mass. This is particularly important for the design of lightweight structures, such as aircraft panels. ABH has also shown good promise in vibration energy harvesting, however, its practical application is still under research. The current studies show that for 1D ABH, the geometric parameters are critical for the ABH effect. For 2D ABH, besides geometrical parameters, additional variables, such as the diameter of the center hole, the spatial layout of ABH array, and the number of ABHs, influence the performance of the ABH effect. Additionally, the material of the host structure and damping material are important for both 1D and 2D ABH. Optimization of the geometric parameters can significantly improve the damping effect of ABH. The number of ABHs and their spatial layout can expand the effective frequency range of ABH. A higher loss factor for both the host structure and damping layer can further improve the damping effect of ABH using less additive damping. However, there is a need for research efforts to further understand the interdependence of various geometrical parameters so that structural optimization studies can be carried out in designing structures with ABHs for better performance in vibration and noise control. 3D printing technology makes the manufacturing of more complex structures possible, and it is being applied in structures with ABH features. Further studies are required to apply ABH widely to real-life structures and carry out the application for more complex structures. Thus, it is observed from this review of various studies that the use of ABH in structural design for vibration and noise control is significantly effective and has great potential for research and industrial applications. Funding: This research received no external funding. Acknowledgments: The first author thanks the support from Department of Mechanical Engineering of Stevens Institute of Technology. Conflicts of Interest: The authors declare no conflict of interest. References 1. Rao, M.D. Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. J. Sound Vib. 2003, 262, 457–474. [CrossRef] 2. Denis, V. Vibration Damping in Beams Using the Acoustic Black Hole Effect. Ph.D. Thesis, Universit´e du Maine, Le Mans, France, 2014. 3. Bowyer, E.P.; O’Boy, D.J.; Krylov, V.V.; Gautier, F. Experimental investigation of damping flexural vibrations using two-dimensional Acoustic ‘Black Holes’. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acoustics Multidisciplinary Digital Publishing Institute

Acoustic Black Holes in Structural Design for Vibration and Noise Control

Acoustics , Volume 1 (1) – Feb 25, 2019

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acoustics Review Acoustic Black Holes in Structural Design for Vibration and Noise Control Chenhui Zhao and Marehalli G. Prasad * Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, USA; czhao1@stevens.edu * Correspondence: mprasad@stevens.edu; Tel.: +1-201-216-5571 Received: 2 October 2018; Accepted: 28 January 2019; Published: 25 February 2019 Abstract: It is known that in the design of quieter mechanical systems, vibration and noise control play important roles. Recently, acoustic black holes have been effectively used for structural design in controlling vibration and noise. An acoustic black hole is a power-law tapered profile to reduce phase and group velocities of wave propagation to zero. Additionally, the vibration energy at the location of acoustic black hole increases due to the gradual reduction of its thickness. The vibration damping, sound reduction, and vibration energy harvesting are the major applications in structural design with acoustic black holes. In this paper, a review of basic theoretical, numerical, and experimental studies on the applications of acoustic black holes is presented. In addition, the influences of the various geometrical parameters and the configuration of acoustic black holes are presented. The studies show that the use of acoustic black holes results in an effective control of vibration and noise. It is seen that the acoustic black holes have a great potential for quiet design of complex structures. Keywords: acoustic black hole; structure design; noise and vibration control 1. Introduction It is known that with the development of high-speed machinery, the control of unwanted vibration and noise are very important for their stability and reliability, as well as the environmental noise impact [1]. The two well-known methods for passive control of structural vibrations which also results in a reduction of noise are constrained layer damping and tuned dynamic absorbers [2]. The first method is based on using a viscoelastic layer attached to the structure and the second method needs an attachment of additional weight [3] to the target structure. Additionally, the active vibration control devices are also used for vibration damping [4]. However, these active methods require consistent input energy and more complex electro-mechanical design. Thus, for the reasons of limitation of size, budget, or weight, sometimes it is not possible, and also is not desirable, to use these above methods. There is always a need for an effective design of structures for vibration and noise control [5]. In recent times micro-devices, such as portable electronics and wireless remote sensors, are developed and widely used [6]. Most of these low-power electronics are powered by battery. However, even for the long-lasting batteries, they still need to be replaced because of their limited lifecycle. For some applications, such as sensors deployed in remote locations or inside the human body [7], it is challenging and costly, or even impractical. Energy harvesting is the process of capturing and converting ambient energy in the environment into usable electrical energy to extend the life of batteries, which makes the devices self-sustainable and environmental-friendly. Piezoelectric vibration energy harvesting (PVEH) is one of the typical energy harvesting methods. In the design of portable micro devices, the challenge is to reduce the weight and size of the host structure. Thus, approaches to increase the energy harvested from the vibrations of the host structures are desirable [8]. Acoustics 2019, 1, 220–251; doi:10.3390/acoustics1010014 www.mdpi.com/journal/acoustics Acoustics 2018, 1, x FOR PEER REVIEW 2 of 29 structure. Thus, approaches to increase the energy harvested from the vibrations of the host structures are desirable [8]. Acoustics 2019, 1 221 Recently, an approach for passive vibration control, acoustic black holes (ABH), has been developed. An ABH is usually a power-law tapered profile built on structures, such as beams, plates, Recently, an approach for passive vibration control, acoustic black holes (ABH), has been and shells, where the vibration energy is concentrated due to the reduction of wave speed [9] (as developed. An ABH is usually a power-law tapered profile built on structures, such as beams, shown in Figure 1). Therefore, due to this concentration effect of the ABH, the vibration energy can plates, and shells, where the vibration energy is concentrated due to the reduction of wave speed [9] be absorbed by attaching small amount of damping material at the ABH location, which also results (as shown in Figure 1). Therefore, due to this concentration effect of the ABH, the vibration energy can in reduced sound radiation. Additionally, the performance of energy harvesting is enhanced by be absorbed by attaching small amount of damping material at the ABH location, which also results in attaching piezoelectric material at the ABH location [10]. An ABH is a tailing method which cuts reduced sound radiation. Additionally, the performance of energy harvesting is enhanced by attaching material away from the host structure, and it also decreases the usage of damping layer, so it piezoelectric material at the ABH location [10]. An ABH is a tailing method which cuts material away decreases the weight of host structures. Therefore, it is a good option for vibration and noise control from the host structure, and it also decreases the usage of damping layer, so it decreases the weight of of lightweight structures. host structures. Therefore, it is a good option for vibration and noise control of lightweight structures. The ABH effect was first discovered by Pekeris in 1946 [11]. He exploited the central physical The ABH effect was first discovered by Pekeris in 1946 [11]. He exploited the central physical principle of ABH, namely the phase velocity of sound waves that propagate in a stratified fluid are principle of ABH, namely the phase velocity of sound waves that propagate in a stratified fluid are progressively decreased to zero with increasing depth. In 1988 Mironov determined that a flexural progressively decreased to zero with increasing depth. In 1988 Mironov determined that a flexural wave propagates in a thin plate slows down and needs infinite time to reach a tapered edge [12]. wave propagates in a thin plate slows down and needs infinite time to reach a tapered edge [12]. Later, Krylov first used the name “acoustic black hole” to this effect [13], and applied ABH on beams Later, Krylov first used the name “acoustic black hole” to this effect [13], and applied ABH on and plates, also indicating that the ABH approach results in an increased amount of energy to be beams and plates, also indicating that the ABH approach results in an increased amount of energy absorbed by adding a small amount of material attenuation near the ABH locations [14–17]. Then to be absorbed by adding a small amount of material attenuation near the ABH locations [14–17]. Conlon developed further numerical and experimental work to analyze the ABH effect on vibration Then Conlon developed further numerical and experimental work to analyze the ABH effect on and sound radiation of thin plates [18–20]. Later, researchers from different countries around the vibration and sound radiation of thin plates [18–20]. Later, researchers from different countries around world worked on the ABH effect on structural vibration control, sound radiation, and vibration the world worked on the ABH effect on structural vibration control, sound radiation, and vibration energy harvesting. Recently, a review on mechanics problem of the ABH structure was presented by energy harvesting. Recently, a review on mechanics problem of the ABH structure was presented by Ji et al. [21]. This study systematically introduced the theoretical study on mechanics for 1D and 2D Ji et al. [21]. This study systematically introduced the theoretical study on mechanics for 1D and 2D structures and a summary of applications of ABH. Another review on the applications of ABH on structures and a summary of applications of ABH. Another review on the applications of ABH on vibration damping and sound radiation was conducted by Chong et al. [22]. Due to the increase of vibration damping and sound radiation was conducted by Chong et al. [22]. Due to the increase of complexity of structure design with ABH and the limitation of traditional manufacturing methods, complexity of structure design with ABH and the limitation of traditional manufacturing methods, such as milling [23], 3D printing technology is applied. In the study of Chong et al. [22], a numerical such as milling [23], 3D printing technology is applied. In the study of Chong et al. [22], a numerical and experimental study on vibration response of the 3D-printed ABH beams was also developed. and experimental study on vibration response of the 3D-printed ABH beams was also developed. Furthermore, a series of studies on dynamic and static properties [24] and applications in vibration Furthermore, a series of studies on dynamic and static properties [24] and applications in vibration damping [25] and energy harvesting [8,26] of 3D-printed structures embedded with ABH was also damping [25] and energy harvesting [8,26] of 3D-printed structures embedded with ABH was also investigated by other researchers. investigated by other researchers. This paper presents a review of recent studies on the use of ABH in structural design. The paper This paper presents a review of recent studies on the use of ABH in structural design. The paper presents studies on the applications of ABH in structural vibration control, noise reduction, and presents studies on the applications of ABH in structural vibration control, noise reduction, and vibration energy harvesting. In addition, the review particularly focusses on the influence of vibration energy harvesting. In addition, the review particularly focusses on the influence of geometrical parameters of 1D ABH and the layout of the 2D ABH on the structural response in order geometrical parameters of 1D ABH and the layout of the 2D ABH on the structural response in to make the ABH features more efficient in structural design. order to make the ABH features more efficient in structural design. Figure 1. ABH concentration effect, the amplitude of the incident wave increases significantly when Figure 1. ABH concentration effect, the amplitude of the incident wave increases significantly when propagating to the end of the ABH wedge. propagating to the end of the ABH wedge. 2. Theoretical Analysis 2. Theoretical Analysis Mironov [12] showed that the bending wave speed goes to zero for beams and plates whose Mironov [12] showed that the bending wave speed goes to zero for beams and plates whose thickness decreases according to: thickness decreases according to: h(X) = aX , (1) ( ) (1) ℎ 𝑋 =𝑎𝑋 , where h is the thickness, a is constant, m is exponent of the power-law curve, and X is the distance from the tip of the ideal power-law curve. Acoustics 2018, 1, x FOR PEER REVIEW 3 of 29 Acoustics 2019, 1 222 where h is the thickness, 𝑎 is constant, m is exponent of the power-law curve, and X is the distance from the tip of the ideal power-law curve. However, in reality, due to the limitation of manufacturing technics, it is impossible to build a However, in reality, due to the limitation of manufacturing technics, it is impossible to build a zero zero thickness, so there will be a residual thickness ℎ at the free end. Then the equation of the power- thickness, so there will be a residual thickness h at the free end. Then the equation of the power-law law curve (1D ABH) becomes: curve (1D ABH) becomes: h(x) = #x + h , (2) ( ) ℎ 𝑥 =𝜀𝑥 +ℎ , (2) where the exponent m is a positive rational number and m  2, parameter # is a constant, x is the where the exponent m is a positive rational number and 𝑚 2 , parameter 𝜀 is a constant, x is the distance from the tip of the power-law curve with residual thickness, and the scheme is shown in distance from the tip of the power-law curve with residual thickness, and the scheme is shown in Figure 2. Figure 2. Figure 2. Schematic of a 1D ABH [27]. Figure 2. Schematic of a 1D ABH [27]. For the specific thickness of the beam T, Parameter 𝜀 affect the length of ABH: For the specific thickness of the beam T, Parameter # affect the length of ABH: 𝑇− ℎ T h (3) 𝐿 = 1 , L = , (3) ABH 𝜀 thus: thus: T h 𝑇− ℎ # = , (4) 𝜀= L , (4) ABH then the ABH power-law curve is: then the ABH power-law curve is: T h 𝑇− ℎ m h(x) = x + h , (5) ( ) ℎ 𝑥 = 𝑥 +ℎ , (5) ABH where T is the thickness of the beam, h is the residual thickness of the ABH part, L is the length of where T is the thickness of the beam, ℎ is the residual thickness of the ABH part, 𝐿 is the length 1 ABH the ABH, and m is the exponent of the power-law profile [27]. of the ABH, and m is the exponent of the power-law profile [27]. The phase velocity C and group velocity C are given by: The phase velocity 𝐶 p and group velocity 𝐶 g are given by: (6) 𝐶 = 𝜔ℎ(𝑥) , C = wh(x), (6) 3𝜌(1 − 𝜈 ) 3r(1 n ) 4E 4𝐸 C = wh(x), (7) (7) 𝐶 = 𝜔ℎ(𝑥) , 3r 1 n ( ) 3𝜌(1 − 𝜈 ) where E is the Young’s modulus, n is the Poisson’s ratio, r is the mass density, h(x) is the varying where E is the Young’s modulus, ν is the Poisson's ratio, ρ is the mass density, h(x) is the varying thickness at ABH of the beams, and ! is the angular frequency of the flexural wave. When h = 0 and thickness at ABH of the beams, and ω is the angular frequency of the flexural wave. When ℎ =0 x ! 0 , the phase velocity and group velocity tend to zero [9,15]. and 𝑥→ 0 , the phase velocity and group velocity tend to zero [9,15]. Propagation time T from x to x is shown in Figure 2: 0 ABH 0 Propagation time 𝑇 from 𝑥 to 𝑥 is shown in Figure 2: x 2 1 4 12r(1 n ) 1 1m/2 1m/2 12𝜌(1 − 𝜈 ) 1 T = dx = x L , (8) 0 0 ⁄ 1 2 ABH (8) 𝑇 = c 𝑑𝑥 = Ew 2 m 𝑥 −𝐿 , g 𝑐 ABH 𝐸𝜔 2−𝑚 when x tends to 0, h = 0 then T tends to infinity, only if m  2 [12]. 0 1 0 when 𝑥 tends to 0, ℎ =0 then 𝑇 tends to infinity, only if 𝑚 2 [12]. Equations (6)–(8) indicate that the ABH can alter wave speed to decrease and that also result in Equations (6)–(8) indicate that the ABH can alter wave speed to decrease and that also result in the concentration of vibration energy at the ABH location when m  2. the concentration of vibration energy at the ABH location when m ≥ 2. Acoustics 2019, 1 223 If the host structure has a non-zero-loss factor, the reflection coefficient and wave number can be presented as below: (2 Im k(x)dx) ABH R = e , (9) k(x) = 12 , (10) h(x) 2 2 r 1 n w k = , (11) where E is the Young’s modulus, n is the Poisson’s ratio, r is the mass density, h is the thickness of the plate, and w is the angular frequency of flexural wave [28,29]. Equations (9)–(11) indicate that when x tends to zero, if the residual thickness h equal to zero, wave numbers k(x) tend to infinity at ABH locations, and the reflection coefficient tends to zero, which indicates that no wave can escape from the ABH location [30]. When the ABH is partially covered with a damping layer, the reflection coefficient at any point in the damping area can be expressed as: 8 9 r r #x < = m m #x #x h R (x) = + + 1  exp K q (12) 0 2 : ; h h #x 1 1 + 1 1/4 1/2 3 12 k u E d K = (13) 1/2 4# E h 1/4 1/2 12 k h K = , (14) 1/2 2# where u is the loss factor of the material of the absorbing layer, h is the loss factor of the wedge material. d is the layer thickness and E /E is the ratio of Young’s moduli of the absorbing layer and the plate, 2 1 respectively [31,32]. A 2D ABH can be seen as a rotation of the 1D ABH by 360 degrees. When a wave propagates in the 2D ABH, it deviates into the center of the ABH [17,32]. O’Boy et al. developed the theoretical analysis for the thin plate [33]. A systematic summary of theoretical analysis containing a 2D ABH is studied by Ji et al. [21]. When attaching the damping material on the plate, the structure can be seen as a composite structure. The loss factor can be expressed as: E h h h D D D D h 4 + 6 + 3 E h(r) h(r) h(r) h (r) = (15) comp E h h h D D D D 1 + 4 + 6 + 3 E h(r) h(r) h(r) where E is the Young’s modulus of damping, h is the loss factor of the damping material. E is the D D w elasticity modulus of the plate in plural form. h(r) is the thickness of the plate and h is the thickness of damping layer. r is the distance to the center of ABH [33]. When r decreases, the loss factor increases which means more vibration energy is absorbed. The reflection coefficient of ABH becomes: ( ) 1/4 1/4 1/2 2 3 12 w r 1 n R = exp 2 h (r) dr (16) 0 comp 1/2 1/4 4# E R where R is the radius of whole ABH and R is the truncation length which means the radius inner i t hole [33]. The theoretical analysis of the flexural wave propagation in ABH with damping being developed and more new mathematical models are currently under investigation. For example, a Acoustics 2018, 1, x FOR PEER REVIEW 5 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 5 of 29 semi- Acoustics analyt2019 ical mode , 1 l to analyse an Euler-Bernoulli beam with ABH and its full coupling wit 224h the semi-analytical model to analyse an Euler-Bernoulli beam with ABH and its full coupling with the damping layers coated over its surface is presented by Tang et al. [34]. damping layers coated over its surface is presented by Tang et al. [34]. semi-analytical model to analyse an Euler-Bernoulli beam with ABH and its full coupling with the 3. Applications to Structural Design with Acoustic Black Holes damping layers coated over its surface is presented by Tang et al. [34]. 3. Applications to Structural Design with Acoustic Black Holes 3.1. Ap 3. Applications plication of ABHs to Vibration to Structural Design Control with Acoustic Black Holes 3.1. Application of ABHs to Vibration Control 3.1. Application of ABHs to Vibration Control 3.1.1. Vibration Control of Beams with ABHs 3.1.1. Vibration Control of Beams with ABHs 3.1.1. Vibration Control of Beams with ABHs In 2003, Krylov first used the name “acoustic black hole” [13]. The theoretical and numerical In 2003, Krylov first used the name “acoustic black hole” [13]. The theoretical and numerical work [29] on beams with a one-dimensional acoustic black hole (1D ABH), as shown in Figure 3, was In 2003, Krylov first used the name “acoustic black hole” [13]. The theoretical and numerical work [29] on beams with a one-dimensional acoustic black hole (1D ABH), as shown in Figure 3, was developed. Figures 4 and 5 show the effect of various truncation length x0 and thickness of absorbing work [29] on beams with a one-dimensional acoustic black hole (1D ABH), as shown in Figure 3, developed. Figures 4 and 5 show the effect of various truncation length x0 and thickness of absorbing was developed. Figures 4 and 5 show the effect of various truncation length x and thickness of film δ on the reflection coefficient of the ABH. First, it can be observed from the behavior of the film δ on the reflection coefficient of the ABH. First, it can be observed from the behavior of the absorbing film d on the reflection coefficient of the ABH. First, it can be observed from the behavior of reflection coefficient for the uncovered wedge (solid curve) that a small truncation can result in a reflection coefficient for the uncovered wedge (solid curve) that a small truncation can result in a the reflection coefficient for the uncovered wedge (solid curve) that a small truncation can result in large increase of the refection coefficient. Second, it can be observed that the reflection coefficient large increase of the refection coefficient. Second, it can be observed that the reflection coefficient a large increase of the refection coefficient. Second, it can be observed that the reflection coefficient increases with the increase of truncation length. Third, damping layers with higher relative stiffness increases with the increase of truncation length. Third, damping layers with higher relative stiffness increases with the increase of truncation length. Third, damping layers with higher relative stiffness can decrease the reflection coefficient further. Finally, the reflection coefficient decreases with the can decrease the reflection coefficient further. Finally, the reflection coefficient decreases with the can decrease the reflection coefficient further. Finally, the reflection coefficient decreases with the increase of excitation frequency. These two figures indicate that the presence of thin absorbing layers incre incr ase o ease f ex of citation excitation frefr qequency uency. The . These se tw two o figure figures s indicate indicate tha that the t the presence of presence of thin thi absorbing n absorblayers ing layers on the surfaces of ABH wedges can result in very low reflection coefficients of flexural waves from on the surfaces of ABH wedges can result in very low reflection coefficients of flexural waves from on the surfaces of ABH wedges can result in very low reflection coefficients of flexural waves from their edges. their edges. their edges. Figure 3. Truncated quadratic wedges covered by thin damping layers. x0 is the truncation length. Figure 3. Truncated quadratic wedges covered by thin damping layers. x0 is the truncation length. Figure 3. Truncated quadratic wedges covered by thin damping layers. x is the truncation length. Reprinted with permission from [29]. Copyright Elsevier, 2004. Reprinted with permission from [29]. Copyright Elsevier, 2004. Reprinted with permission from [29]. Copyright Elsevier, 2004. Figure 4. Reflection coefficient for the wedge covered by a thick absorbing film. The solid curve Figure 4. Reflection coefficient for the wedge covered by a thick absorbing film. The solid curve Figure 4. Reflection coefficient for the wedge covered by a thick absorbing film. The solid curve corresponds to an uncovered wedge, and dotted and dashed curves correspond to wedges covered corresponds to an uncovered wedge, and dotted and dashed curves correspond to wedges covered by corresponds to an uncovered wedge, and dotted and dashed curves correspond to wedges covered by thin absorbing thin absorbing film films s with t with the he valu valueses of rel of relative ative s stiffness tiffnes E s /E E2/= E12/30 =2/30 and and E E / 2E /E1= =2/3 2/3, , resp respectively; ectively; the 2 1 2 1 by thin absorbing films with the values of relative stiffness E2/E1=2/30 and E2/E1=2/3, respectively; the the film material loss factor n is 0.2, and the film thickness d is 5 m. Reprinted with permission film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. from [29]. Copyright Elsevier, 2004. Copyright Elsevier, 2004. Copyright Elsevier, 2004. Acoustics Acoustics 2019 2018,, 1 1, x FOR PEER REVIEW 6 of 225 29 Acoustics 2018, 1, x FOR PEER REVIEW 6 of 29 Figure 5. Frequency dependence of the reflection coefficient R: x0 = 1.5 and 2.5 cm (thicker and thinner Figure 5. Frequency dependence of the reflection coefficient R: x = 1.5 and 2.5 cm (thicker and thinner Figure 5. Frequency dependence of the reflection coefficient R: x0 = 1.5 and 2.5 cm (thicker and thinner curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to uncovered wedges; the film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted uncovered wedges; the film material loss factor n is 0.2, and the film thickness d is 5 m. Reprinted uncovered wedges; the film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. Copyright Elsevier, 2004. with permission from [29]. Copyright Elsevier, 2004. with permission from [29]. Copyright Elsevier, 2004. A numerical study of beams with a spiral ABH was developed by Jeon et al. [35,36]. The spiral A num A numerical erical st study udy of b of beams eams wit with h a sp a spiral iral ABH w ABH was as dev developed eloped b by y Jeon et Jeon et al al. . [[3 35 5,,36 36]. The sp ]. The spiral iral ABH is a compact and curvilinear shape by using an Archimedean spiral with a uniform gap-distance ABH ABH is is a com a compact pact and and curv curvili ilin near ear sh shape ape by us by using ing an anA Ar rchimede chimedean an sspiral piral w with ith a au uniform niform gap gap-distance -distance between adjacent baselines of the spiral as shown in Figure 6. Figure 7 shows the driving point bet between ween adj adjacent acent base baselines lines of t of the he spir spiral al a as s shown shown iin n Fig Figur ure e 6 6.. F Figur iguree 7 show 7 shows s tthe he driv driving ing point point mobility of the beam with a 720 mm spiral ABH compared with reference uniform beam (black line). mobilit mobility y of of tthe he be beam am w with ith a a 7 720 20 mm mm sp spiral iral AB ABH H compared compared wit with h re re ffer erence enceuni uniform form beam beam (b (black lack line line). ). The beam with the ABH is 10% lighter than the reference beam, but it reduces the resonant peak The bea The beam m with the ABH is 10 with the ABH is 10% % li lighter ghter tha than n the re the refer ference beam, ence beam, but it reduce but it reduces s the reson the resonant ant peak peak levels to 90% without additional damping. This indicates that the spiral ABH has great potential for llevels evels to 9 to 90% 0% wi without thout a additional dditional da damping. mping. Thi This s iindicates ndicates tha that t the spi the spiral ral ABH has grea ABH has great t potent potential ial ffor or vibration damping. vibrat vibration ion da damping. mping. Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with permission from [36]. Copyright Acoustic Society of America, 2017. permission from [36]. Copyright Acoustic Society of America, 2017. permission from [36]. Copyright Acoustic Society of America, 2017. Figure 7. Driving point mobility of the beam with a 720 mm spiral ABH (grey line) and the reference Figure 7. Driving point mobility of the beam with a 720 mm spiral ABH (grey line) and the reference uniform beam (black line). Reprinted with permission from [36]. Copyright Acoustic Society of uniform beam (black line). Reprinted with permission from [36]. Copyright Acoustic Society of America, 2017. America, 2017. Acoustics 2018, 1, x FOR PEER REVIEW 6 of 29 Figure 5. Frequency dependence of the reflection coefficient R: x0 = 1.5 and 2.5 cm (thicker and thinner curves, respectively); solid and dotted curves correspond to wedges with absorbing films and to uncovered wedges; the film material loss factor ν is 0.2, and the film thickness δ is 5 μm. Reprinted with permission from [29]. Copyright Elsevier, 2004. A numerical study of beams with a spiral ABH was developed by Jeon et al. [35,36]. The spiral ABH is a compact and curvilinear shape by using an Archimedean spiral with a uniform gap-distance between adjacent baselines of the spiral as shown in Figure 6. Figure 7 shows the driving point mobility of the beam with a 720 mm spiral ABH compared with reference uniform beam (black line). The beam with the ABH is 10% lighter than the reference beam, but it reduces the resonant peak levels to 90% without additional damping. This indicates that the spiral ABH has great potential for vibration damping. Figure 6. Shape of the beam with a spiral ABH. The length of the ABH is 720 mm. Reprinted with Acoustics 2019, 1 226 permission from [36]. Copyright Acoustic Society of America, 2017. Figure Figure 7. 7. Driving Driving point mobility of the beam point mobility of the beam with a 720 mm spiral A with a 720 mm spiral ABH BH (gr (grey ey line) and th line) and the e reference reference uniform uniform beam beam (bla (black ck l line). ine). Repr Reprinted wit inted with h perm permission ission f frrom om [36]. Copyright Acousti [36]. Copyright Acoustic c S Society ociety of of America, America, 2017. 2017. Acoustics 2018, 1, x FOR PEER REVIEW 7 of 29 Recently, Zhou and Cheng proposed a numerical and experimental work to develop an ABH that Recently, Zhou and Cheng proposed a numerical and experimental work to develop an ABH featured a resonant beam damper, as shown in Figure 8 (ABH-RBD) [37]. Figure 9 shows the measured that featured a resonant beam damper, as shown in Figure 8 (ABH-RBD) [37]. Figure 9 shows the driving and cross point mobility of the host beam with and without ABH-RBD. It can be observed that measured driving and cross point mobility of the host beam with and without ABH-RBD. It can be a significant vibration reduction is obtained with mounting the ABH-RBD. This study shows a great observed that a significant vibration reduction is obtained with mounting the ABH-RBD. This study damping treatment of ABH-RBD to control the vibration of the host structure. shows a great damping treatment of ABH-RBD to control the vibration of the host structure. Figure 8. Schematics of the host beam with an ABH-RBD. Reprinted with permission from [37]. Figure 8. Schematics of the host beam with an ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. Copyright Elsevier, 2018. (a) (b) Figure 9. Measured (a) driving and (b) cross-point mobility of the primary host beam with and without ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. 3.1.2. Vibration Damping of Plates with ABH The vibration performance of the plates with various 1D ABH slots was studied by Bowyer [38]. One of the samples is shown in Figure 10a. From the experimental result of this sample, it can be observed that a substantial reduction of acceleration is obtained in comparison with the reference plate. The experimental results of other steel plates and composite plates with various slots also show acceptable damping performance. The composite plates have good damping performance even without the damping layer attached due to the large material loss factor. Acoustics 2018, 1, x FOR PEER REVIEW 7 of 29 Recently, Zhou and Cheng proposed a numerical and experimental work to develop an ABH that featured a resonant beam damper, as shown in Figure 8 (ABH-RBD) [37]. Figure 9 shows the measured driving and cross point mobility of the host beam with and without ABH-RBD. It can be observed that a significant vibration reduction is obtained with mounting the ABH-RBD. This study shows a great damping treatment of ABH-RBD to control the vibration of the host structure. Figure 8. Schematics of the host beam with an ABH-RBD. Reprinted with permission from [37]. Acoustics 2019, 1 227 Copyright Elsevier, 2018. (a) (b) Figure 9. Measured (a) driving and (b) cross-point mobility of the primary host beam with and Figure 9. Measured (a) driving and (b) cross-point mobility of the primary host beam with and without without ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. ABH-RBD. Reprinted with permission from [37]. Copyright Elsevier, 2018. 3.1.2. Vibration Damping of Plates with ABH 3.1.2. Vibration Damping of Plates with ABH The vibration performance of the plates with various 1D ABH slots was studied by Bowyer [38]. The vibration performance of the plates with various 1D ABH slots was studied by Bowyer [38]. One of the samples is shown in Figure 10a. From the experimental result of this sample, it can be One of the samples is shown in Figure 10a. From the experimental result of this sample, it can be observed that a substantial reduction of acceleration is obtained in comparison with the reference observed that a substantial reduction of acceleration is obtained in comparison with the reference plate. The experimental results of other steel plates and composite plates with various slots also show plate. The experimental results of other steel plates and composite plates with various slots also acceptable damping performance. The composite plates have good damping performance even show acceptable damping performance. The composite plates have good damping performance even Acoustics 2018, 1, x FOR PEER REVIEW 8 of 29 without the damping layer attached due to the large material loss factor. without the damping layer attached due to the large material loss factor. (a) (b) Figure 10. (a) Carbon fiber composite sample and longitudinal cross-section of the sample; and (b) Figure 10. (a) Carbon fiber composite sample and longitudinal cross-section of the sample; Acceleration for the plate with ABH slot (solid line) and reference plate (dashed line). Reprinted from and (b) Acceleration for the plate with ABH slot (solid line) and reference plate (dashed line). Reprinted [38] under a CC BY 4.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2016. from [38] under a CC BY 4.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2016. Experimental investigation on damping flexural vibrations using two-dimensional acoustic Experimental investigation on damping flexural vibrations using two-dimensional acoustic black holes (2D ABHs) was developed by Bowyer et al. [3]. Figure 11 shows the schematics of the black holes (2D ABHs) was developed by Bowyer et al. [3]. Figure 11 shows the schematics of experimental samples. The forced excitation was applied to the center of the plate via shaker over a the experimental samples. The forced excitation was applied to the center of the plate via shaker over frequency range of 0–9 kHz. The experimental results show that, in comparison with the reference plate, the plate embedded with single 2D ABH with a damping layer provides little damping below 3 kHz. In the region of 3.8–9 kHz, damping varies between 3–8 dB, and maximum damping occurs at 6.6 kHz. Due to the long wave length, the damping performance of the ABH is not evident at low frequency, but it has better damping performance for higher frequencies. It is also indicated by this study that the plates with multiple holes in the current random layout does not significantly improve the damping performance of the 2D ABHs in comparison with the plate with a single 2D ABH. (a) (b) Figure 11. (a) A singular 2D ABH with a center hole, and (b) three 2D ABH with central holes. Adapted from [3] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2010. Later, another experimental investigation of damping flexural vibrations in plates was also developed by Bowyer et al. [39]. Figure 12 shows the steel plate containing an array of six 2D ABH. Figure 13 clearly shows that, compared with the reference plate without ABH, the acceleration of the plate with six ABHs attaching damping layers sharply decreases. Additionally, it shows that the minimum frequency of effective damping performance is about 1.5 kHz. These two experimental investigations indicate that ABHs with damping layers can decrease the vibration of plates and the number and configuration of ABHs affect the performance of vibration damping. The increase in the number of ABHs can expand the effective frequency of damping performance of the plate with a 2D ABH and damping layer. Acoustics 2018, 1, x FOR PEER REVIEW 8 of 29 (a) (b) Figure 10. (a) Carbon fiber composite sample and longitudinal cross-section of the sample; and (b) Acceleration for the plate with ABH slot (solid line) and reference plate (dashed line). Reprinted from [38] under a CC BY 4.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2016. Experimental investigation on damping flexural vibrations using two-dimensional acoustic Acoustics 2019, 1 228 black holes (2D ABHs) was developed by Bowyer et al. [3]. Figure 11 shows the schematics of the experimental samples. The forced excitation was applied to the center of the plate via shaker over a frequency range of 0–9 kHz. The experimental results show that, in comparison with the reference a frequency range of 0–9 kHz. The experimental results show that, in comparison with the reference plate, the plate embedded with single 2D ABH with a damping layer provides little damping below plate, the plate embedded with single 2D ABH with a damping layer provides little damping below 3 kHz. In the region of 3.8–9 kHz, damping varies between 3–8 dB, and maximum damping occurs 3 kHz. In the region of 3.8–9 kHz, damping varies between 3–8 dB, and maximum damping occurs at 6.6 kHz. Due to the long wave length, the damping performance of the ABH is not evident at low at 6.6 kHz. Due to the long wave length, the damping performance of the ABH is not evident at low frequency, but it has better damping performance for higher frequencies. It is also indicated by this frequency, but it has better damping performance for higher frequencies. It is also indicated by this study that the plates with multiple holes in the current random layout does not significantly improve study that the plates with multiple holes in the current random layout does not significantly improve the damping performance of the 2D ABHs in comparison with the plate with a single 2D ABH. the damping performance of the 2D ABHs in comparison with the plate with a single 2D ABH. (a) (b) Figure 11. (a) A singular 2D ABH with a center hole, and (b) three 2D ABH with central holes. Figure 11. (a) A singular 2D ABH with a center hole, and (b) three 2D ABH with central holes. Adapted Adapted from [3] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2010. from [3] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2010. Later, another experimental investigation of damping flexural vibrations in plates was also Later, another experimental investigation of damping flexural vibrations in plates was also developed by Bowyer et al. [39]. Figure 12 shows the steel plate containing an array of six 2D ABH. developed by Bowyer et al. [39]. Figure 12 shows the steel plate containing an array of six 2D ABH. Figure 13 clearly shows that, compared with the reference plate without ABH, the acceleration of the Figure 13 clearly shows that, compared with the reference plate without ABH, the acceleration of plate with six ABHs attaching damping layers sharply decreases. Additionally, it shows that the the plate with six ABHs attaching damping layers sharply decreases. Additionally, it shows that the minimum frequency of effective damping performance is about 1.5 kHz. These two experimental minimum frequency of effective damping performance is about 1.5 kHz. These two experimental investigations indicate that ABHs with damping layers can decrease the vibration of plates and the investigations indicate that ABHs with damping layers can decrease the vibration of plates and the number and configuration of ABHs affect the performance of vibration damping. The increase in the number and configuration of ABHs affect the performance of vibration damping. The increase in the number of ABHs can expand the effective frequency of damping performance of the plate with a 2D number of ABHs can expand the effective frequency of damping performance of the plate with a 2D ABH and damping layer. ABH and damping layer. Acoustics 2018, 1, x FOR PEER REVIEW 9 of 29 Figure 12. Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission Figure 12. Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission from [39]. Copyright Elsevier, 2013. Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central ABH with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The point- and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared with the reference plate. A numerical and experimental study of the acoustic black hole effect for vibration damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with ABH and without ABH were tested. An elliptical plate with disks of resin placed at the location of the ABHs and an elliptical plate completely covered by resin were also tested. Figure 14 shows velocity fields of plates with ABH (b) and without ABH (a). The excitation force was applied to the left focus whereas the ABH is in the right one. It can be observed that the plate with ABH (b) has a lower amplitude of vibration. Figure 15 shows the point mobility measured at the left focus of the plate. It can be observed that the point mobility of the plate with ABH with the damping material at its location was reduced over 2 kHz. (a) (b) Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Hz. Reprinted with permission from [41]. Copyright Elsevier, 2011. Acoustics 2018, 1, x FOR PEER REVIEW 9 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 9 of 29 Acoustics Figure 12. 2019, 1 Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission 229 Figure 12. Manufactured steel plate containing an array of six 2D ABHs. Reprinted with permission from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and Figure 13. Measured acceleration for a plate containing six 2D ABHs with 14 mm central holes and additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission additional damping layers (solid line) and a reference plate (dashed line). Reprinted with permission from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. from [39]. Copyright Elsevier, 2013. Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central Other shapes of plates with ABH are also studied. Mobilities for a circular plate with a central ABH ABH with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The ABH with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The with a thin damping layer and a constrained layer are studied by O’Boy and Krylov [40]. The point- point- and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared point- and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared and cross-mobilities show a suppression of resonant peaks which is up to 17 dB compared with the with the reference plate. A numerical and experimental study of the acoustic black hole effect for with the reference plate. A numerical and experimental study of the acoustic black hole effect for reference plate. A numerical and experimental study of the acoustic black hole effect for vibration vibration damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with vibration damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with damping in elliptical plates has been studied by Georgiev et al. [41]. Elliptical plates with ABH and ABH and without ABH were tested. An elliptical plate with disks of resin placed at the location of ABH and without ABH were tested. An elliptical plate with disks of resin placed at the location of without ABH were tested. An elliptical plate with disks of resin placed at the location of the ABHs and the ABHs and an elliptical plate completely covered by resin were also tested. Figure 14 shows the ABHs and an elliptical plate completely covered by resin were also tested. Figure 14 shows an elliptical plate completely covered by resin were also tested. Figure 14 shows velocity fields of plates velocity fields of plates with ABH (b) and without ABH (a). The excitation force was applied to the velocity fields of plates with ABH (b) and without ABH (a). The excitation force was applied to the with ABH (b) and without ABH (a). The excitation force was applied to the left focus whereas the ABH left focus whereas the ABH is in the right one. It can be observed that the plate with ABH (b) has a left focus whereas the ABH is in the right one. It can be observed that the plate with ABH (b) has a is in the right one. It can be observed that the plate with ABH (b) has a lower amplitude of vibration. lower amplitude of vibration. Figure 15 shows the point mobility measured at the left focus of the lower amplitude of vibration. Figure 15 shows the point mobility measured at the left focus of the Figure 15 shows the point mobility measured at the left focus of the plate. It can be observed that the plate. It can be observed that the point mobility of the plate with ABH with the damping material at plate. It can be observed that the point mobility of the plate with ABH with the damping material at point mobility of the plate with ABH with the damping material at its location was reduced over 2 kHz. its location was reduced over 2 kHz. its location was reduced over 2 kHz. (a) (a) (b) (b) Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Figure 14. Velocity fields of an elliptical plate (a) without ABH at 8671 Hz, and (b) with ABH at 8117 Hz. Hz. Reprinted with permission from [41]. Copyright Elsevier, 2011. Hz. Reprinted with permission from [41]. Copyright Elsevier, 2011. Reprinted with permission from [41]. Copyright Elsevier, 2011. Acoustics 2018, 1, x FOR PEER REVIEW 10 of 29 Acoustics 2019, 1 230 Acoustics 2018, 1, x FOR PEER REVIEW 10 of 29 Figure 15. Measured point mobilities of the elliptical plate. The black line shows the plate with ABH Figure 15. Measured point mobilities of the elliptical plate. The black line shows the plate with ABH by Figure 15. by attaching damping materi Measured point mobiliti al at its es of the e locatilon, liptica the dashe l plate. T dh line e black l shows t ine sh he plate witho ows the plate u with ABH t ABH by attaching damping material at its location, the dashed line shows the plate without ABH by attaching by attaching damping materi attaching damping at the same locat al at its ion, and location, the gr the dashe een line shows the plate d line shows the plate witho without ABH b ut ABH by y covering damping at the same location, and the green line shows the plate without ABH by covering the attaching damping at the same locat the damping material over the whole p ion, and late. R the gr eprinte een line shows the plate d with permission from without ABH b [41]. Copyrigy h covering t Elsevier, damping material over the whole plate. Reprinted with permission from [41]. Copyright Elsevier, 2011. the damping material o 2011. ver the whole plate. Reprinted with permission from [41]. Copyright Elsevier, Not only were plates with ABH with damping material, but also ABH with dynamic vibration Not only were plates with ABH with damping material, but also ABH with dynamic vibration Not only were plates with ABH with damping material, but also ABH with dynamic vibration absorbers (DVA) were studied by Jia et al., as shown in Figure 16 [42]. It can be observed from the absorbers (DVA) were studied by Jia et al., as shown in Figure 16 [42]. It can be observed from the absorber simulatis (D on results that VA) were st there is udied b a reduction of ov y Jia et al., as shown er 10 in dB Fig at maj ure 1 or r 6 [4 esponse 2]. It ca peak n be observed s over 1 kH fro z for the m the simulation results that there is a reduction of over 10 dB at major response peaks over 1 kHz for simulation results that there is a reduction of over 10 dB at major response peaks over 1 kHz for the plate with ABH with DVA. This result shows great potential of combining ABHs and DVAs for the plate with ABH with DVA. This result shows great potential of combining ABHs and DVAs for plat vibrat e wit ion con h ABtH wit rol. h DVA. This result shows great potential of combining ABHs and DVAs for vibration control. vibration control. (b) (b) (c) (a) (a) (c) Figure 16. Cross-section of the plate structure considered in the FE model and the experiments: (a) a Figure 16. Cross-section of the plate structure considered in the FE model and the experiments: (a) a Figure 16. top-view of p Cros la s-sec te structure wit tion of the plate h two ABH structu s, re con (b) a plate sidere embe d in the F dded with ABH and dampin E model and the experim g layer, and ents: (a) a top-view of plate structure with two ABHs, (b) a plate embedded with ABH and damping layer, and (c) top-view of p (c) a plate em la bedded te structure wit with ABh H and DVA two ABHs, . ( A bd ) a plate apted w embe ith pe dded with ABH and dampin rmission from [42]. Copyright American g layer, and a plate embedded with ABH and DVA. Adapted with permission from [42]. Copyright American (c Society ) a plate of Me embedded chanical Engine with ABH and DVA ers, 2015. . Adapted with permission from [42]. Copyright American Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Since the structures with multiple ABHs have great damping effect for vibration and sound, Since the structures with multiple ABHs have great damping effect for vibration and sound, Since the structures with multiple ABHs have great damping effect for vibration and sound, such structures can be considered as metamaterials [43]. Semperlotti and Zhu developed a meta- such structures can be considered as metamaterials [43]. Semperlotti and Zhu developed a such structures can be co structure based on the consider ncept of ed as ABmeta H [44] ma . This terials [ load-be 43]. Semper aring thlotti and Zh in-wall struu develope cture element d a meta- enables meta-structure based on the concept of ABH [44]. This load-bearing thin-wall structure element structure based on the concept of ABH [44]. This load-bearing thin-wall structure element enables propagation characteristics comparable with resonant metamaterials without the fabrication enables propagation characteristics comparable with resonant metamaterials without the fabrication pcompl ropagat exiion ty. The experi characterist menta ics com l work shows tha parable with reso t the ABH trea nant meta tm ments ca aterials wit n signi hout fica t ntlhy improve the e fabrication complexity. The experimental work shows that the ABH treatments can significantly improve the complexity. The experimental work shows that the ABH treatments can significantly improve the damping effect. damping effect. damping A waveg effect. uide is designed to observe travelling waves by Foucaud et al. [45], which is inspired A waveguide is designed to observe travelling waves by Foucaud et al. [45], which is inspired by A waveguide is designed to observe travelling waves by Foucaud et al. [45], which is inspired by artificial cochlea. The experimental study (shown in Figure 17) uses a varying width plate artificial cochlea. The experimental study (shown in Figure 17) uses a varying width plate immersed by art immeirsed ficia in flu l cochlea id and . The te experiment rminated wit alh st an ud ABH. y (sho It wn in F shows itgu hat re a 17 n ) AB use H use s a d varyin as ang wi anecho dth plat ic ened in fluid and terminated with an ABH. It shows that an ABH used as an anechoic end improves the immersed in fluid and terminated with an ABH. It shows that an ABH used as an anechoic end improves the quality of measurements and the accuracy of tonotopic maps due to the attenuation of quality of measurements and the accuracy of tonotopic maps due to the attenuation of reflected waves. improves the reflected waves. quality of measurements and the accuracy of tonotopic maps due to the attenuation of reflected waves. Acoustics 2019, 1 231 Acoustics 2018, 1, x FOR PEER REVIEW 11 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 11 of 29 Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Copyright Elsevier, 2014. Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Copyright Elsevier, 2014. Copyright Elsevier, 2014. 3.1.3. Vibration Damping of Turbofan Blades with ABH 3.1.3. Vibration Damping of Turbofan Blades with ABH 3.1.3. Vibration Damping of Turbofan Blades with ABH An experimental work of damping of flexural vibrations in turbofan blades using ABH was An experimental work of damping of flexural vibrations in turbofan blades using ABH was An experimental work of damping of flexural vibrations in turbofan blades using ABH was studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a blade blade with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has blade with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has about 50% about 50% reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with about 50% reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with appropriate appropriate damping layers are efficient in the reduction of airflow-excited vibrations of the fan appropriate damping layers are efficient in the reduction of airflow-excited vibrations of the fan damping layers are efficient in the reduction of airflow-excited vibrations of the fan blades. This study blades. This study shows the great potential of the ABH in jet engine design to reduce flexural blades. This study shows the great potential of the ABH in jet engine design to reduce flexural shows the great potential of the ABH in jet engine design to reduce flexural vibration in the blades, vibration in the blades, thus reducing internal stresses in the blades and increasing their fatigue life vibration in the blades, thus reducing internal stresses in the blades and increasing their fatigue life thus reducing internal stresses in the blades and increasing their fatigue life cycle. cycle. cycle. (a) (a) (b) (b) Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted from Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Acoustics 2018, 1, x FOR PEER REVIEW 11 of 29 Figure 17. The photo for the assembled experimental setup. Reprinted with permission from [45]. Copyright Elsevier, 2014. 3.1.3. Vibration Damping of Turbofan Blades with ABH An experimental work of damping of flexural vibrations in turbofan blades using ABH was studied by Bowyer and Krylov [46]. Figure 18a shows the fan blade profile with a tapered ABH geometry. The experimental setup with four experimental samples are shown in Figure 19. Figure 20 shows the measurement of acceleration for a twisted reference blade (dashed line) compared to a blade with a 1D ABH and damping layer (solid line). It can be observed that the acceleration has about 50% reduction at 60 Hz and 360 Hz. This indicates that the trailing edges of the 1D ABH with appropriate damping layers are efficient in the reduction of airflow-excited vibrations of the fan blades. This study shows the great potential of the ABH in jet engine design to reduce flexural vibration in the blades, thus reducing internal stresses in the blades and increasing their fatigue life cycle. (a) (b) Acoustics 2019, 1 232 Figure 18. (a) Fan blade profile with tapered ABH geometry. (b) Experimental setup. Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Acoustics 2018, 1, x FOR PEER REVIEW 12 of 29 Figure 19. Experimental setup of closed-circuit wind tunnel and samples of the test. Experimental Figure 19. Experimental setup of closed-circuit wind tunnel and samples of the test. Experimental samples: Flow visualization diagram for: (a) reference fan blade, (b) fan blade with a power-law samples: Flow visualization diagram for: (a) reference fan blade, (b) fan blade with a power-law wedge, ( wedge, (c c) fan ) fan blade having a blade having a power-law wedge power-law wedge with a with a sing single le dam damping ping lay layer er, and ( , and (d d) fan ) fan blade hav blade having ing a power-law w a power-law wedge edge with a with a shaped damping layer. shaped damping layer. Reprint Reprinted ed from [46] u from [46] under nder a CC BY a CC BY3.0 license 3.0 license. . Copy Copyright right Bowy Bowyer er, E. , E.P P. . and and K Krylov rylov, V.V., 201 , V.V., 2014.4. Figure 20. Acceleration for a twisted reference blade (dashed line) compared to a twisted blade with Figure 20. Acceleration for a twisted reference blade (dashed line) compared to a twisted blade with a a 1D ABH and damping layer (solid line). Reprinted from [46] under a CC BY 3.0 license. Copyright 1D ABH and damping layer (solid line). Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Bowyer, E.P. and Krylov, V.V., 2014. 3.2. Application of ABH to Sound Reduction 3.2. Application of ABH to Sound Reduction An experimental work was developed by Bowyer and Krylov [47]. A 300 × 400 mm × 5-mm thick An experimental work was developed by Bowyer and Krylov [47]. A 300 400 mm 5-mm thick plate embedded with six 2D ABHs (Figure 21) with damping at the center of each ABH was plate embedded with six 2D ABHs (Figure 21) with damping at the center of each ABH was suspended suspended vertically. The excitation force was applied centrally on the plate. The results compare the vertically. The excitation force was applied centrally on the plate. The results compare the sound sound radiation power level of a plate containing six 2D ABHs with a damping layer with the sound radiation power level of a plate containing six 2D ABHs with a damping layer with the sound radiation radiation power level of the reference plate, which are shown in Figure 22. Below 1 kHz there is little power level of the reference plate, which are shown in Figure 22. Below 1 kHz there is little reduction reduction in the sound power level. Between 1 and 3 kHz, the sound power level is reduced by 10– in the sound power level. Between 1 and 3 kHz, the sound power level is reduced by 10–18 dB in 18 dB in comparison with the reference plate, and the maximum reduction in the sound radiation comparison with the reference plate, and the maximum reduction in the sound radiation occurs at 1.6 occurs at 1.6 kHz. Above 3 kHz, almost all responses in sound radiation are absorbed. This indicates kHz. Above 3 kHz, almost all responses in sound radiation are absorbed. This indicates that the plate that the plate with ABHs having damping layers effectively reduce the sound radiation of the steel with ABHs having damping layers effectively reduce the sound radiation of the steel plate. plate. Figure 21. Sample of the plate with six 2D ABHs. Reprinted from [47] under a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. Acoustics 2018, 1, x FOR PEER REVIEW 12 of 29 Figure 19. Experimental setup of closed-circuit wind tunnel and samples of the test. Experimental samples: Flow visualization diagram for: (a) reference fan blade, (b) fan blade with a power-law wedge, (c) fan blade having a power-law wedge with a single damping layer, and (d) fan blade having a power-law wedge with a shaped damping layer. Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. Figure 20. Acceleration for a twisted reference blade (dashed line) compared to a twisted blade with a 1D ABH and damping layer (solid line). Reprinted from [46] under a CC BY 3.0 license. Copyright Bowyer, E.P. and Krylov, V.V., 2014. 3.2. Application of ABH to Sound Reduction An experimental work was developed by Bowyer and Krylov [47]. A 300 × 400 mm × 5-mm thick plate embedded with six 2D ABHs (Figure 21) with damping at the center of each ABH was suspended vertically. The excitation force was applied centrally on the plate. The results compare the sound radiation power level of a plate containing six 2D ABHs with a damping layer with the sound radiation power level of the reference plate, which are shown in Figure 22. Below 1 kHz there is little reduction in the sound power level. Between 1 and 3 kHz, the sound power level is reduced by 10– 18 dB in comparison with the reference plate, and the maximum reduction in the sound radiation occurs at 1.6 kHz. Above 3 kHz, almost all responses in sound radiation are absorbed. This indicates Acoustics 2019, 1 233 that the plate with ABHs having damping layers effectively reduce the sound radiation of the steel plate. Figure 21. Sample of the plate with six 2D ABHs. Reprinted from [47] under a CC BY license. Figure 21. Sample of the plate with six 2D ABHs. Reprinted from [47] under a CC BY license. Copyright Copyright Bowyer E.P. and Krylov, V.V., 2012. Bowyer E.P. and Krylov, V.V., 2012. Acoustics 2018, 1, x FOR PEER REVIEW 13 of 29 Figure 22. Sound power level comparison for a plate containing six indentations of power-law profile Figure 22. Sound power level comparison for a plate containing six indentations of power-law profile with a damping layer (black line) compared to a reference plate (grey line). Reprinted from [47] under with a damping layer (black line) compared to a reference plate (grey line). Reprinted from [47] under a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. Feurtado and Conlon developed a numerical and experimental investigation of sound power of Feurtado and Conlon developed a numerical and experimental investigation of sound power a plate with an array of ABH [48]. A 4 × 5 array of 10-cm diameter 2D ABHs, which is shown in Figure of a plate with an array of ABH [48]. A 4  5 array of 10-cm diameter 2D ABHs, which is shown in 23, was machined into a 6.35-mm thick, 61 cm × 91 cm aluminum plate. The ABH with various Figure 23, was machined into a 6.35-mm thick, 61 cm  91 cm aluminum plate. The ABH with various diameters of damping layers was tested to assess the effects of the amount of damping layer on ABH diameters of damping layers was tested to assess the effects of the amount of damping layer on ABH performance. The experimental setup is shown in Figure 24. The plate was mounted on a frame and performance. The experimental setup is shown in Figure 24. The plate was mounted on a frame and excited with band-limited white noise. Figure 25 shows one-third octave band radiated sound power excited with band-limited white noise. Figure 25 shows one-third octave band radiated sound power tested by an intensity probe for a uniform plate and ABH plate with various diameters of damping tested by an intensity probe for a uniform plate and ABH plate with various diameters of damping material. It can be observed that an ABH with a damping layer effectively reduced the radiated sound material. It can be observed that an ABH with a damping layer effectively reduced the radiated sound power over 1.5 kHz compared with the reference plate (blue line). It can also be observed that sound power over 1.5 kHz compared with the reference plate (blue line). It can also be observed that sound radiation of the plate with the diameters of the damping layer of 6.75 cm and 10 cm have almost the radiation of the plate with the diameters of the damping layer of 6.75 cm and 10 cm have almost the same performance, namely higher radiated sound power. The 3.5 cm diameter damping layer shows same performance, namely higher radiated sound power. The 3.5 cm diameter damping layer shows comparable performance to the larger damping diameters. comparable performance to the larger damping diameters. (a) (b) Figure 23. (a) FEA model and (b) cross-section of an aluminum plate (green) with a 2D ABH attached damping layer (yellow). Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Acoustics 2018, 1, x FOR PEER REVIEW 13 of 29 Figure 22. Sound power level comparison for a plate containing six indentations of power-law profile with a damping layer (black line) compared to a reference plate (grey line). Reprinted from [47] under a CC BY license. Copyright Bowyer E.P. and Krylov, V.V., 2012. Feurtado and Conlon developed a numerical and experimental investigation of sound power of a plate with an array of ABH [48]. A 4 × 5 array of 10-cm diameter 2D ABHs, which is shown in Figure 23, was machined into a 6.35-mm thick, 61 cm × 91 cm aluminum plate. The ABH with various diameters of damping layers was tested to assess the effects of the amount of damping layer on ABH performance. The experimental setup is shown in Figure 24. The plate was mounted on a frame and excited with band-limited white noise. Figure 25 shows one-third octave band radiated sound power tested by an intensity probe for a uniform plate and ABH plate with various diameters of damping material. It can be observed that an ABH with a damping layer effectively reduced the radiated sound power over 1.5 kHz compared with the reference plate (blue line). It can also be observed that sound radiation of the plate with the diameters of the damping layer of 6.75 cm and 10 cm have almost the same performance, namely higher radiated sound power. The 3.5 cm diameter damping layer shows Acoustics 2019, 1 234 comparable performance to the larger damping diameters. (a) (b) Figure 23. (a) FEA model and (b) cross-section of an aluminum plate (green) with a 2D ABH attached Figure 23. (a) FEA model and (b) cross-section of an aluminum plate (green) with a 2D ABH attached damping layer (yellow). Reprinted with permission from [48]. Copyright American Society of damping layer (yellow). Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Mechanical Engineers, 2015. Acoustics 2018, 1, x FOR PEER REVIEW 14 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 14 of 29 Figure 24. Aluminum plate with a 4 × 5 array of embedded ABHs with full-diameter damping layers Figure 24. Aluminum plate with a 4  5 array of embedded ABHs with full-diameter damping layers Figure 24. Aluminum plate with a 4 × 5 array of embedded ABHs with full-diameter damping layers in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with varying diameters of damping material. Reprinted with permission from [48]. Copyright American varying diameters of damping material. Reprinted with permission from [48]. Copyright American varying diameters of damping material. Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. Society of Mechanical Engineers, 2015. A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on sound radiation of engine cover [49].The ABH were machined on two plates and then bonded into sound radiation of engine cover [49].The ABH were machined on two plates and then bonded into the engine cover with glue as shown in Figure 26. Figure 27 shows the sound radiation from a the engine cover with glue as shown in Figure 26. Figure 27 shows the sound radiation from a reference engine cover (dashed line) in comparison with the engine cover attaching plates with ABH reference engine cover (dashed line) in comparison with the engine cover attaching plates with ABH (black line) at 2100 rpm with the bonnet closed. A total average reduction from the reference specimen (black line) at 2100 rpm with the bonnet closed. A total average reduction from the reference specimen of 6.5 dB was recorded. This indicates that engine covers with ABHs can decrease the sound radiation of 6.5 dB was recorded. This indicates that engine covers with ABHs can decrease the sound radiation from the vehicle engine. from the vehicle engine. (b) (a) (b) (a) Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted from Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 14 of 29 Figure 24. Aluminum plate with a 4 × 5 array of embedded ABHs with full-diameter damping layers in a frame with a mechanical point drive. Reprinted with permission from [48]. Copyright American Society of Mechanical Engineers, 2015. Figure 25. One-third octave band radiated sound power for a uniform plate and ABH plate with varying diameters of damping material. Reprinted with permission from [48]. Copyright American Acoustics 2019, 1 235 Society of Mechanical Engineers, 2015. A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on sound A practical experiment is studied by Bowyer and Krylov to investigate the effect of ABH on radiation sound rof adi engine ation of cover engine [49 cov ].The er [ ABH 49].Th wer e AB e machined H were mon achined two plates on twand o plat then es an bonded d then b into onde the d int engine o cover the engine c with glue over with glue as shown as shown in Figure 26. in Figur Figure 27e shows 26. Figure the sound 27 shows the sound ra radiation from adia refer tioence n from a engine reference engine cover (dashed line) in comparison with the engine cover attaching plates with ABH cover (dashed line) in comparison with the engine cover attaching plates with ABH (black line) at (black line) at 2100 rpm with the bonnet closed. A total average reduction from the reference specimen 2100 rpm with the bonnet closed. A total average reduction from the reference specimen of 6.5 dB of 6.5 dB was recorded. This indicates that engine covers with ABHs can decrease the sound radiation was recorded. This indicates that engine covers with ABHs can decrease the sound radiation from the from the vehicle engine. vehicle engine. (a) (b) Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted from Figure 26. Engine cover attaching two plates with ABHs (a) and reference cover (b). Reprinted [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 15 of 29 Figure 27. Sound radiation from a reference engine cover (dashed line) compared with the engine Figure 27. Sound radiation from a reference engine cover (dashed line) compared with the engine cover attaching plates with ABH (black line) at 2100 rpm with the bonnet closed. Reprinted from [49] cover attaching plates with ABH (black line) at 2100 rpm with the bonnet closed. Reprinted from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. An experimental study on sound absorption in air of ABH based inhomogeneous acoustic An experimental study on sound absorption in air of ABH based inhomogeneous acoustic waveguides [50] was developed by Azbaid et al. [51,52]. The inner radius of the structures as shown waveguides [50] was developed by Azbaid et al. [51,52]. The inner radius of the structures as shown in in Figure 28 are built following the linear function (exponent m = 1) and power-law function Figure 28 are built following the linear function (exponent m = 1) and power-law function (exponent m (exponent m = 2), respectively. Using two microphone transfer function methods, the experiment = 2), respectively. Using two microphone transfer function methods, the experiment results show a results show a substantial reduction in the reflection coefficient. The adding of absorbing porous substantial reduction in the reflection coefficient. The adding of absorbing porous materials results in materials results in a further reduction of the reflection coefficient. a further reduction of the reflection coefficient. (b) (a) Figure 28. Photo of a linear ABH (exponent m = 1) and a quadratic ABH (exponent m = 2). Reprinted from [51] under a CC BY-NC-ND 4.0 license. Copyright Azbaid El Ouahabi, A., Krylov, V.V. and O'Boy, D.J., 2015. A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption performance was developed by Zhou and Semperlotti [53]. The HR cavity shown in Figure 29 is partitioned by a flexible plate-embedded ABH and the numerical results show a great increase in the energy absorption of the HR system. Figure 29. Schematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Foundation, 2016. Acoustics 2018, 1, x FOR PEER REVIEW 15 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 15 of 29 Figure 27. Sound radiation from a reference engine cover (dashed line) compared with the engine cover Figure 27. attaching plate Sound radiation s with ABH (bla from a reference engine co ck line) at 2100 rpm wit ver h the b (dashed line) compared onnet closed. Reprinte with the d from [49] engine under a CC BY cover attaching plate -NC-ND 4.0 s with license. Copyri ABH (black line) ght Bowyer E.P. a at 2100 rpm wit nd Krylov, V.V h the bonnet closed. Reprinte ., 2015. d from [49] under a CC BY-NC-ND 4.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2015. An experimental study on sound absorption in air of ABH based inhomogeneous acoustic An experimental study on sound absorption in air of ABH based inhomogeneous acoustic waveguides [50] was developed by Azbaid et al. [51,52]. The inner radius of the structures as shown in Fi waveg gure uide 28 a s [r5e buil 0] was deve t followi loped by Azb ng the linea aid r et al functi . [on ( 51,52]. The inner exponent m rad = 1) ius o and power- f the structure law functi s as shown on in Figure 28 are built following the linear function (exponent m = 1) and power-law function (exponent m = 2), respectively. Using two microphone transfer function methods, the experiment resu (exponent lts show a subst m = 2), a resp ntial reduction ectively. Us in the ing two m reflecti icrop on ho coe ne t fficient. The addin ransfer function methods, the experi g of absorbing porous ment Acoustics 2019, 1 236 results show a substantial reduction in the reflection coefficient. The adding of absorbing porous materials results in a further reduction of the reflection coefficient. materials results in a further reduction of the reflection coefficient. (b) (a) (b) (a) Figure 28. Photo of a linear ABH (exponent m = 1) and a quadratic ABH (exponent m = 2). Reprinted Figure 28. Photo of (a) a linear ABH (exponent m = 1) and (b) a quadratic ABH (exponent m = 2). from [51] unde Figure 28. Phot r a CC o of a l BY-NC inear A -ND 4 BH (exponent .0 license. Copy m = 1) and a right Azbaid El Ouah quadratic ABH ( abi, exponent A., Kryl mo = 2). Reprinted v, V.V. and Reprinted from [51] under a CC BY-NC-ND 4.0 license. Copyright Azbaid El Ouahabi, A., Krylov, V.V. O'Boy, D.J from [51] unde ., 2015. r a CC BY-NC-ND 4.0 license. Copyright Azbaid El Ouahabi, A., Krylov, V.V. and and O’Boy, D.J., 2015. O'Boy, D.J., 2015. A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption A numerical study on Helmholtz resonators (HR) using ABH to enhance the energy absorption performance was developed by Zhou and Semperlotti [53]. The HR cavity shown in Figure 29 is performance was developed by Zhou and Semperlotti [53]. The HR cavity shown in Figure 29 is part performa itioned by a nce wa flexibl s developed by Zhou e plate-embedded ABH and and Semperlotti the numer [53] ica . The HR l results show cavity shown i a great incre n Fi ase gure in t29 he is partitioned by a flexible plate-embedded ABH and the numerical results show a great increase in the partitioned by a flexible plate-embedded ABH and the numerical results show a great increase in the energy absorption of the HR system. energy absorption of the HR system. energy absorption of the HR system. Figure 29. Schematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Foundation, 20 Figure 29. Sch 16. ematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Figure 29. Schematic of the HR system with ABH. Adapted from [53]. Copyright Noise Control Foundation, 2016. Foundation, 2016. 3.3. Application of ABH to Vibration Energy Harvesting Zhao and Conlon developed a numerical study to investigate the structures tailored with ABHs to enhance vibration energy harvesting under both steady state and transient excitation [10]. Figure 30 shows the schematic of the plate with five equally spaced 1D ABH grooves attached with transducers. The plate is excited on the right side by a 400 N force sweeping from 0 to 10 kHz. The external resistance is 1 W. The numerical results of this study show the performance of the energy harvesting under steady state excitation and transient excitation. It can be observed that, by comparing with the flat plate, the normalized energy ratio of the plate with ABH increases drastically up to 80% in the 5–10 kHz frequency band at all the five ABH locations, and it increases most at Location 2. This study shows that the structure with ABH can drastically increase the efficiency of the energy harvesting. Later, another numerical and experimental investigation of a plate with three 2D ABHs (as shown in Figure 31) was also studied by Zhao and Conlon [54]. The ABHs effectively focus broadband energy to the center of the ABH. This also indicates that the focusing ability of the ABH is independent of the spectral and spatial characteristics of the external mechanical load. Acoustics 2018, 1, x FOR PEER REVIEW 16 of 29 3.3. Application of ABH to Vibration Energy Harvesting Zhao and Conlon developed a numerical study to investigate the structures tailored with ABHs Acoustics 2018, 1, x FOR PEER REVIEW 16 of 29 to enhance vibration energy harvesting under both steady state and transient excitation [10]. Figure 30 shows the schematic of the plate with five equally spaced 1D ABH grooves attached with 3.3. Application of ABH to Vibration Energy Harvesting transducers. The plate is excited on the right side by a 400 N force sweeping from 0 to 10 kHz. The Zhao and Conlon developed a numerical study to investigate the structures tailored with ABHs external resistance is 1 Ω. The numerical results of this study show the performance of the energy to enhance vibration energy harvesting under both steady state and transient excitation [10]. Figure harvesting under steady state excitation and transient excitation. It can be observed that, by 30 shows the schematic of the plate with five equally spaced 1D ABH grooves attached with comparing with the flat plate, the normalized energy ratio of the plate with ABH increases drastically transducers. The plate is excited on the right side by a 400 N force sweeping from 0 to 10 kHz. The up to 80 exte% i rnal r n e the 5– sistance is 10 kHz 1 Ω f . The numeric requency ba al re nd a sults of t t all th he fi is study sho ve ABH wloca the perfor tions, mance of the and it increa energy ses most at harvesting under steady state excitation and transient excitation. It can be observed that, by Location 2. This study shows that the structure with ABH can drastically increase the efficiency of comparing with the flat plate, the normalized energy ratio of the plate with ABH increases drastically the energy harvesting. Later, another numerical and experimental investigation of a plate with three up to 80% in the 5–10 kHz frequency band at all the five ABH locations, and it increases most at 2D ABHs (as shown in Figure 31) was also studied by Zhao and Conlon [54]. The ABHs effectively Location 2. This study shows that the structure with ABH can drastically increase the efficiency of Acoustics 2019, 1 237 focus broadband energy to the center of the ABH. This also indicates that the focusing ability of the the energy harvesting. Later, another numerical and experimental investigation of a plate with three ABH is independent of the spectral and spatial characteristics of the external mechanical load. 2D ABHs (as shown in Figure 31) was also studied by Zhao and Conlon [54]. The ABHs effectively focus broadband energy to the center of the ABH. This also indicates that the focusing ability of the ABH is independent of the spectral and spatial characteristics of the external mechanical load. Figure 30. A schematic of the plate with five equally spaced 1D ABH grooves, and one of the ABH Figure 30. A schematic of the plate with five equally spaced 1D ABH grooves, and one of the ABH Figure 30. A schematic of the plate with five equally spaced 1D ABH grooves, and one of the ABH with the surface mounted piezo-transducer. Adapted from [10]. Copyright IOP Publishing, 2014. with the surface mounted piezo-transducer. Adapted from [10]. Copyright IOP Publishing, 2014. with the surface mounted piezo-transducer. Adapted from [10]. Copyright IOP Publishing, 2014. Figure 31. A schematic of the plate with three equally-spaced 2D ABHs. Adapted from [54]. Copyright FigureFigure 31. 31. A schematic A schematic of of the the plate plate with thre with threee equally-s equally-spaced paced 2D 2D ABHs. ABHs. Adapted from Adapted fr[54] om. Copyright [54]. Copyright IOP Publishing, 2015. IOP Publishing, 2015. IOP Publishing, 2015. An experimental study on vibration energy harvesting using a cantilever beam with a modified An exper ABH caviitme y was ntal deve study lope on d by vibration Zhao an ene d Pras rgy h ad [8 a]. rvest The cant ing using ilever be a cam antilever is designed beam wit with a modified h an ABH An experimental study on vibration energy harvesting using a cantilever beam with a modified cavity near the fixed end due to the presence of higher strain energy. The experimental setup of a ABH cavity was developed by Zhao and Prasad [8]. The cantilever beam is designed with an ABH cantilever beam with modified ABH cavity attaching a piezo sensor is shown in Figure 32. Figure 33 ABH cavity was developed by Zhao and Prasad [8]. The cantilever beam is designed with an ABH cavity near the fixed end due to the presence of higher strain energy. The experimental setup of a shows the energy concentration effect of the ABH cavity at 2000 Hz. Figure 34 shows the experimental cavity near the fixed end due to the presence of higher strain energy. The experimental setup of a cantilever beam with modified ABH cavity attaching a piezo sensor is shown in Figure 32. Figure 33 results of the voltage power spectra. The vertical axis presents the decibel value referring to 1 volt. It shows the energy concentration effect of the ABH cavity at 2000 Hz. Figure 34 shows the experimental cantilever beam with modified ABH cavity attaching a piezo sensor is shown in Figure 32. Figure 33 can be observed that the beam with ABH cavity (red line) has a higher voltage output level within results of the voltage power spectra. The vertical axis presents the decibel value referring to 1 volt. It shows the energy concentration effect of the ABH cavity at 2000 Hz. Figure 34 shows the experimental the frequency range from 900–500 Hz, the voltage output is 10 dB higher than the beam without the can be observed that the beam with ABH cavity (red line) has a higher voltage output level within ABH (blue line). The increases due to the ABH cavity are substantial, even without considering the results of the voltage power spectra. The vertical axis presents the decibel value referring to 1 volt. the frequency range from 900–500 Hz, the voltage output is 10 dB higher than the beam without the neutralization of the piezo material. Embedding a modified ABH cavity uses less piezoelectrical It can be observed that the beam with ABH cavity (red line) has a higher voltage output level within ABH (b material lue line). The and decreases the increase weight of th s due to the host stru e ABH cacture. Tuna vity are subst bility ca antial, even wit n be achieved by a hout conside djusting the ring the the frequency range from 900–500 Hz, the voltage output is 10 dB higher than the beam without length of the ABH cavity. Thus, ABH cavity design into structures has good potential in increasing neutralization of the piezo material. Embedding a modified ABH cavity uses less piezoelectrical the ABH (blue the energy line). harvested. The increases due to the ABH cavity are substantial, even without considering material and decreases the weight of the host structure. Tunability can be achieved by adjusting the the neutralization of the piezo material. Embedding a modified ABH cavity uses less piezoelectrical length of the ABH cavity. Thus, ABH cavity design into structures has good potential in increasing material the energy and decr harvested. eases the weight of the host structure. Tunability can be achieved by adjusting the length of the ABH cavity. Thus, ABH cavity design into structures has good potential in increasing the Acoustics 2018, 1, x FOR PEER REVIEW 17 of 29 energy harvested. (a) (b) Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached piezo sensor. (b) Schematic of a 3D-printed beam with an ABH cavity [8]. piezo sensor. (b) Schematic of a 3D-printed beam with an ABH cavity [8]. (a) (b) Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity and (b) with an ABH cavity at 2000 Hz [8]. Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity [8]. A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the numerical result of power harvested. The results show that the beam with ABH cavities are more effective for vibration energy harvesting than uniform structures. Acoustics 2018, 1, x FOR PEER REVIEW 17 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 17 of 29 (a) (b) (a) (b) Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached Figure 32. (a) The experimental setup of a cantilever beam with modified ABH cavity with an attached Acoustics piezo sensor. ( 2019, 1 b) Schematic of a 3D-printed beam with an ABH cavity [8]. 238 piezo sensor. (b) Schematic of a 3D-printed beam with an ABH cavity [8]. (a) (b) (a) (b) Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity and (b) Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity Figure 33. Numerical results for the energy of (a) the cantilever beam without an ABH cavity and (b) with an ABH cavity at 2000 Hz [8]. and (b) with an ABH cavity at 2000 Hz [8]. with an ABH cavity at 2000 Hz [8]. Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 34. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity [8]. the ABH cavity and red line is for the beam with the ABH cavity [8]. the ABH cavity and red line is for the beam with the ABH cavity [8]. A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than A numerical study on energy harvesting using multiple ABH cavities as shown in Figure 35 was studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the studied by Liang et al. [26]. The size of piezoelectric patches is designed to be relatively smaller than the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the numerical result of power harvested. The results show that the beam with ABH cavities are more the wavelengths of ABH features, avoiding neutralization of the electric charge. Figure 36 shows the numerical result of power harvested. The results show that the beam with ABH cavities are more effective for vibration energy harvesting than uniform structures. numerical result of power harvested. The results show that the beam with ABH cavities are more effective for vibration energy harvesting than uniform structures. Acoustics 2018, 1, x FOR PEER REVIEW 18 of 29 effective for vibration energy harvesting than uniform structures. Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from [26]. the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from Copyright IEEE, 2018. [26]. Copyright IEEE, 2018. Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 Kω. Reprinted with permission from [26]. Copyright IEEE, 2018. 3.4. Discussion Applications of ABH for vibration control, sound radiation and vibration energy harvesting are presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with damping materials or piezo transducers are the major methods for the design of structures to enhance the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. 4. Design of Structures Using Acoustic Black Holes 4.1. Design of 1D ABH The length of the ABH (LABH), the exponent m, and the residual thickness h1 are the three geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform thickness (no ABH). LABH affects the length of ABH part when m is fixed as shown in Figure 37b. For 2D ABH, LABH represents the radius of outer circle abstracting the radius of inner hole. (a) (b) Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with various LABH(in). Acoustics 2018, 1, x FOR PEER REVIEW 18 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 18 of 29 Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without Figure 35. Experimental result for the voltage power spectrum. The blue line is for the beam without the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from Acoustics 2019, 1 239 the ABH cavity and red line is for the beam with the ABH cavity. Reprinted with permission from [26]. Copyright IEEE, 2018. [26]. Copyright IEEE, 2018. Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 Kω. Reprinted Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 K!. Reprinted Figure 36. Harvested power spectra from the beam with and without ABHs, when R = 1 Kω. Reprinted with permission from [26]. Copyright IEEE, 2018. with permission from [26]. Copyright IEEE, 2018. with permission from [26]. Copyright IEEE, 2018. 3.4. Discussion 3.4. Discussion 3.4. Discussion Applications of ABH for vibration control, sound radiation and vibration energy harvesting are Applications of ABH for vibration control, sound radiation and vibration energy harvesting are Applications of ABH for vibration control, sound radiation and vibration energy harvesting are presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with presented. Generally, 1D ABH edges or grooves on beams and multiple 2D ABHs on plates with damping materials or piezo transducers are the major methods for the design of structures to enhance damping materials or piezo transducers are the major methods for the design of structures to enhance damping materials or piezo transducers are the major methods for the design of structures to enhance the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding the vibration damping, sound reduction, and energy harvesting. Some specific structures embedding ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. ABHs for practical applications, such as turbo fan blades, engine covers, etc., are also reviewed. 4. Design of Structures Using Acoustic Black Holes 4. Design of Structures Using Acoustic Black Holes 4. Design of Structures Using Acoustic Black Holes 4.1. Design of 1D ABH 4.1. Design of 1D ABH 4.1. Design of 1D ABH The length of the ABH (LABH), the exponent m, and the residual thickness h1 are the three The length of the ABH (L ), the exponent m, and the residual thickness h are the three ABH 1 The length of the ABH (LABH), the exponent m, and the residual thickness h1 are the three geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH geometrical parameters that affect the shape of 1D ABHs. Exponent m affects the depth of the ABH curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various curve when the length of the ABH and residual thickness are fixed. Figure 37a shows the various shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform shapes of power-law profiles with different values of exponent m. m = 0 means the beam is of uniform thickness (no ABH). LABH affects the length of ABH part when m is fixed as shown in Figure 37b. For thickness (no ABH). L affects the length of ABH part when m is fixed as shown in Figure 37b. ABH thickness (no ABH). LABH affects the length of ABH part when m is fixed as shown in Figure 37b. For 2D ABH, LABH represents the radius of outer circle abstracting the radius of inner hole. For 2D ABH, L represents the radius of outer circle abstracting the radius of inner hole. ABH 2D ABH, LABH represents the radius of outer circle abstracting the radius of inner hole. (a) (a) (b) (b) Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with Figure 37. (a) Shapes of power law profiles with various m, and (b) shapes of power law profiles with various LABH(in). various LABH(in). various L (in). ABH In the study by Krylov, which is mentioned in Section 3.1.1 [29], the concept of “truncation” is applied. For a given value of exponent m, the truncation length is controlled by two variables, the length of the ABH L and the residual thickness h . In this numerical study, it can be observed ABH 1 that when the truncation length is larger than about 0.01 m the reflection coefficient of the beam with ABH (the solid line in Figure 4) increases sharply. This indicates that even a small truncation length results in a large increase in the reflection coefficient, which weakens the ABH effect. Thus, a sharp Acoustics 2018, 1, x FOR PEER REVIEW 19 of 29 In the study by Krylov, which is mentioned in Section 3.1.1 [29], the concept of “truncation” is applied. For a given value of exponent m, the truncation length is controlled by two variables, the length of the ABH LABH and the residual thickness h1. In this numerical study, it can be observed that when the truncation length is larger than about 0.01 m the reflection coefficient of the beam with ABH Acoustics 2019, 1 240 (the solid line in Figure 4) increases sharply. This indicates that even a small truncation length results in a large increase in the reflection coefficient, which weakens the ABH effect. Thus, a sharp end of end of an ABH is critical for structural design. It is noted that a small amount of damping material can an ABH is critical for structural design. It is noted that a small amount of damping material can effectively restrain the increase of reflection coefficient. effectively restrain the increase of reflection coefficient. A numerical and experimental study on sound radiation of a beam with a 1D ABH is developed A numerical and experimental study on sound radiation of a beam with a 1D ABH is developed by Li and Ding [55]. In Figure 38, it can be seen that the increase of the truncation thickness results by Li and Ding [55]. In Figure 38, it can be seen that the increase of the truncation thickness results in in decrease of radiated sound power in the frequency from 43 to 160 Hz and increase from 626 Hz decrease of radiated sound power in the frequency from 43 to 160 Hz and increase from 626 Hz to 6 to 6 kHz. The reason for this phenomenon was discussed and explained. The increase of truncation kHz. The reason for this phenomenon was discussed and explained. The increase of truncation thickness causes added mass and equivalent stiffness effects, which makes an enhancement of the thickness causes added mass and equivalent stiffness effects, which makes an enhancement of the low frequency bandwidth. On the other hand, an increase of truncation thickness causes an increase low frequency bandwidth. On the other hand, an increase of truncation thickness causes an increase of the reflection coefficient and a decrease of the degree of wave concentration in the ABH, therefore of the reflection coefficient and a decrease of the degree of wave concentration in the ABH, therefore weakening noise suppression at high frequencies. This indicates that an optimization of the truncation weakening noise suppression at high frequencies. This indicates that an optimization of the thickness is required for the best performance of noise reduction. truncation thickness is required for the best performance of noise reduction. Figure 38. Sound radiation from the beams with various truncation thicknesses. Reprinted with Figure 38. Sound radiation from the beams with various truncation thicknesses. Reprinted with permission from [55]. Copyright Elsevier, 2019. permission from [55]. Copyright Elsevier, 2019. A wavelet-decomposed energy model is developed and investigated by Tang and Cheng [56]. A wavelet-decomposed energy model is developed and investigated by Tang and Cheng [56]. A new type of beam structure to achieve broad attenuation bands in relatively low frequencies is A new type of beam structure to achieve broad attenuation bands in relatively low frequencies is proposed. The model can be used to predict the frequency bounds of the band gaps. A parametric proposed. The model can be used to predict the frequency bounds of the band gaps. A parametric analysis is also illustrated. The increase of m and decrease of the truncation thickness h0 would analysis is also illustrated. The increase of m and decrease of the truncation thickness h would decrease the lower band gaps so as to enhance the ABH effect. decrease the lower band gaps so as to enhance the ABH effect. A numerical study on the influence of geometrical parameters of ABHs on vibration of cantilever A numerical study on the influence of geometrical parameters of ABHs on vibration of cantilever beams was studied by Zhao and Prasad [57]. Figure 39 shows the 3rd mode shape simulation results beams was studied by Zhao and Prasad [57]. Figure 39 shows the 3rd mode shape simulation results of the beam without ABH [58] and beams with 1D ABH with various m values. It can be observed of the beam without ABH [58] and beams with 1D ABH with various m values. It can be observed that the amplitude of displacement increases at the ABH location (with the increase of m value) and that the amplitude of displacement increases at the ABH location (with the increase of m value) and decreases at other locations of nodes. decreases at other locations of nodes. The numerical study on the sound radiation from vibrating cantilever aluminum beams with The numerical study on the sound radiation from vibrating cantilever aluminum beams with various exponent m, which are 10 inches long, 1 inch wide, and 1/8 inch thick, with various shapes various exponent m, which are 10 inches long, 1 inch wide, and 1/8 inch thick, with various shapes of ABHs at the free end, and the residual thickness h1 equal to 1/64 inch is developed by Zhao and of ABHs at the free end, and the residual thickness h equal to 1/64 inch is developed by Zhao and Prasad [27]. The results show that, in this group of beams with given residual thickness and length Prasad [27]. The results show that, in this group of beams with given residual thickness and length of of the ABH, the concentration effect performs better when m is less than 5. However, when m is larger the ABH, the concentration effect performs better when m is less than 5. However, when m is larger than 5, the ABH loses its concentration effect. Figure 40b shows how the logarithmic ratio of near than 5, the ABH loses its concentration effect. Figure 40b shows how the logarithmic ratio of near field sound radiation at the free end and the other three anti-node locations changes with various m field sound radiation at the free end and the other three anti-node locations changes with various m values, which essentially shows how the m values affect the concentration effect. This indicates that values, which essentially shows how the m values affect the concentration effect. This indicates that the m value has a maximum limitation. Another theoretical and numerical study on vibration energy the m value has a maximum limitation. Another theoretical and numerical study on vibration energy concentration of the 1D ABH was developed by Li and Ding [59]. The length of the edge part with concentration of the 1D ABH was developed by Li and Ding [59]. The length of the edge part with comparatively large deflection deformation Le, as show in Figure 41, was investigated. With the given comparatively large deflection deformation L , as show in Figure 41, was investigated. With the given parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy are trapped within a small area (L ) from the wedge tip, which shows the energy concentration effect. For a e Acoustics 2018, 1, x FOR PEER REVIEW 20 of 29 parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy are trapped within a small area (Le) from the wedge tip, which shows the energy concentration effect. Acoustics 2018, 1, x FOR PEER REVIEW 20 of 29 For a fixed value of Le, the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. This study also indicates that the exponent m has an optimum value to provide the best concentration parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy Acoustics 2019, 1 241 effect. are trapped within a small area (Le) from the wedge tip, which shows the energy concentration effect. Acoustics 2018, 1, x FOR PEER REVIEW 20 of 29 For a fixed value of Le, the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. fixed value of L , the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. This study This study also indicates that the exponent m has an optimum value to provide the best concentration parameters of the ABH, the results show that more than 80% strain energy and 96% kinetic energy also indicates that the exponent m has an optimum value to provide the best concentration effect. effect. are trapped within a small area (Le) from the wedge tip, which shows the energy concentration effect. For a fixed value of Le, the peak ratio of energy occurs in the range of m from 2.5–3.0 for this case. This study also indicates that the exponent m has an optimum value to provide the best concentration effect. Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. Figure 39. Third model shape simulation results. The bottom is the fixed end, and the top is the free (a) (a) end. Blue color shows the locations of nodes and red color shows the peak value locations [57]. (b) (a) (b) (b) Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at free end and other three anti-nodes along with the various m values in Group 1 with excitation of 1600 Hz [57]. Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at Figure 40. (a) The fourth mode of cantilever beam [58]. (b) The logarithmic ratio of displacement at free end and o free endther three anti-nodes and other three anti-nodes along w along with ith the various m values the various m values in Gr in G oupr1oup 1 with excitation o with excitation of f free end and other three anti-nodes along with the various m values in Group 1 with excitation of 1600 Hz [57]. 1600 Hz [57]. 1600 Hz [57]. Figure 41. Schematic in 1D shows the ABH length of the edge part with a comparatively large deflection deformation Le. Adapted from [59]. Copyright SAGE Publications, 2018. One possible explanation of the phenomenon is that when the m value is larger, the ABH part Figure Figure 41. 41. Schematic Schematic in 1D in 1D shows shows the ABH length of the ABH length of the edge the edge part w part with a comparatively ith a comparatively large large deflection becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration deformation deflection defo L . rm Adapted ation Le. Adapted from [59 from [59]. Copyright ]. Copyright SAGE Publications, 2018 SAGE Publications, 2018. . Figure 41. Schematic in 1D shows the ABH length of the edge part with a comparatively large systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a deflection deformation Le. Adapted from [59]. Copyright SAGE Publications, 2018. untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 One possible explanation of the phenomenon is that when the m value is larger, the ABH part One possible explanation of the phenomenon is that when the m value is larger, the ABH part generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration One possible explanation of the phenomenon is that when the m value is larger, the ABH part [9] applied another assumption. In this study, it is shown that the violation of smoothness criterion systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a is a significant design problem for ABHs. Smoothness criterion is an assumption that the change of becomes so thin that the uniform part of the beam and ABH part becomes two isolated vibration untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon systems and it loses the pattern of the response of the beam vibration, the ABH part performs as a generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon [9] [9] applied another assumption. In this study, it is shown that the violation of smoothness criterion untuned dynamic absorber of the uniform part, which can be seen in Figure 39, the beam with m = 10 is a significant design problem for ABHs. Smoothness criterion is an assumption that the change of generated an extra node at the ABH location. A previous study developed by Feurtado and Conlon [9] applied another assumption. In this study, it is shown that the violation of smoothness criterion is a significant design problem for ABHs. Smoothness criterion is an assumption that the change of Acoustics 2019, 1 242 Acoustics 2018, 1, x FOR PEER REVIEW 21 of 29 Acoustics 2018, 1, x FOR PEER REVIEW 21 of 29 applied another assumption. In this study, it is shown that the violation of smoothness criterion is a the local wave number is small enough over distances to make the ABH perform better, which is significant design problem for ABHs. Smoothness criterion is an assumption that the change of the the local wave number is small enough over distances to make the ABH perform better, which is stated by Mironov [12]. Wave number variation (NWV) is investigated to assess the validity of the local wave number is small enough over distances to make the ABH perform better, which is stated by stated by Mironov [12]. Wave number variation (NWV) is investigated to assess the validity of the smoothness criterion. Equation (17) shows that NWV is a function of frequency and position and Mironov [12]. Wave number variation (NWV) is investigated to assess the validity of the smoothness smoothness criterion. Equation (17) shows that NWV is a function of frequency and position and need to be much less than 1. criterion. Equation (17) shows that NWV is a function of frequency and position and need to be much need to be much less than 1. less than 1. / (𝑥) 1   1 𝐸  1 𝑑ℎ 1/4 (𝑥) 1 1 𝐸 1 𝑑ℎ (17) dk(x) 1 = 1 E 1 dh ≪1 , (17) = = ( ) ≪1  1, , (17) 𝑑𝑥 𝑘 2 𝜌𝜔 12 1−𝜐 ℎ 𝑑𝑥 2 2 2 1/2 𝑑𝑥 𝑘 2 𝜌𝜔 12(1−𝜐 ) ℎ 𝑑𝑥 dx k 2 rw 12(1 u ) dx where E is the Young’s modulus, ν is the Poisson’s ratio, ρ is the density, h is the varying thickness at where where EE is th is the e Young’s mo Young’s modulus, dulus, νn is the is the Poisson’ Poisson’s ratio, s ratio,ρ is the den is the density sity, , hh is t is the he v varyin arying t g thickness hickness at at the ABH of the beams, and ω is the angular frequency of the flexural wave. According to this theory, the ABH of the beams, and w is the angular frequency of the flexural wave. According to this theory, the ABH of the beams, and ω is the angular frequency of the flexural wave. According to this theory, an increased m value increases the NWV and the smoothness criterion is violated for a greater range a an n iincr ncrea eased sed m val m value ue iincr ncrea eases ses the NWV the NWV a and nd the the ssmoothness moothness ccriterion riterion iis s vi violated olated ffor or a g a gr rea eater ter ra range nge of frequencies. For m = 10, the NWV is above one for the frequency range of study, which is shown of frequencies. For m = 10, the NWV is above one for the frequency range of study, which is shown in of frequencies. For m = 10, the NWV is above one for the frequency range of study, which is shown in Figure 42. Further work investigates that NWV < 0.4 is a good rule for designing ABHs [60]. in Figur Figu ere 42 42. Further . Further work worinvestigates k investigate that s that NWV NWV < 0. < 0.4 is 4 is a good a good rule rule for for designing designinABHs g ABHs [6 [60].0]. (a) (b) (a) (b) Figure 42. (a) Reflection coefficients and NWV for 5-cm long ABHs with m = 2, 4, 10. (b) Increased m Figure 42. (a) Reflection coefficients and NWV for 5-cm long ABHs with m = 2, 4, 10. (b) Increased m Figure 42. (a) Reflection coefficients and NWV for 5-cm long ABHs with m = 2, 4, 10. (b) Increased increases the NWV and violates the smoothness criterion. Reprinted with permission from [9]. increases the NWV and violates the smoothness criterion. Reprinted with permission from [9]. m increases the NWV and violates the smoothness criterion. Reprinted with permission from [9]. Copyright Acoustic Society of America, 2014. Copyright Acoustic Society of America, 2014. Copyright Acoustic Society of America, 2014. In the study of Denis et al. which investigated ABH as a vector for energy transfer from the low In the study of Denis et al. which investigated ABH as a vector for energy transfer from the low In the study of Denis et al. which investigated ABH as a vector for energy transfer from the frequencies to high frequencies in the framework of nonlinear vibration, a parametric study on a frequencies to high frequencies in the framework of nonlinear vibration, a parametric study on a low frequencies to high frequencies in the framework of nonlinear vibration, a parametric study on a number of beams shown in Figure 43 was developed [61]. Longer additional termination, which number of beams shown in Figure 43 was developed [61]. Longer additional termination, which number of beams shown in Figure 43 was developed [61]. Longer additional termination, which means means longer LABH, can improve the efficiency of energy transfer. It is also noted that a 2D ABH can means longer LABH, can improve the efficiency of energy transfer. It is also noted that a 2D ABH can longer L , can improve the efficiency of energy transfer. It is also noted that a 2D ABH can present present larger LABH with minimal thickness and obtains larger modal density in the low-frequency ABH present larger LABH with minimal thickness and obtains larger modal density in the low-frequency larger L with minimal thickness and obtains larger modal density in the low-frequency range. range. ABH range. Figure 43. Beams with various lengths of ABHs and various lengths of damping. Reprinted with Figure 43. Beams with various lengths of ABHs and various lengths of damping. Reprinted with Figure 43. Beams with various lengths of ABHs and various lengths of damping. Reprinted with permission from [61]. Copyright Elsevier, 2017. permission from [61]. Copyright Elsevier, 2017. permission from [61]. Copyright Elsevier, 2017. The effect of the length of ABH is also investigated by Zhao and Prasad [8,27]. For a given beam The ef The effect fect o of f tthe he len length gth of of ABH ABHis is also also iinvestigated nvestigated b by y Z Zhao hao and andP Prasad rasad [[8 8,,2 27 7]]. . F For or a a g given iven b beam eam at a given excitation frequency there are specific optimum values of LABH. The simulation results of a at t a a gi given ven exci excitation tation ffr req equency uency ther there e ar are specific e specific optimum values optimum values of of L LABH. The si . The simulation mulation resu results lts of of ABH total energy at the ABH location of beams with the ABH cavity (shown in Figure 32b) is shown in total energy at the ABH location of beams with the ABH cavity (shown in Figure 32b) is shown in Figure 44. It is seen that the beam with LABH of 19 mm has maximum energy output. This indicates Figure 44. It is seen that the beam with LABH of 19 mm has maximum energy output. This indicates that the ABH is tunable by changing LABH to obtain a higher concentration of vibration energy [8]. that the ABH is tunable by changing LABH to obtain a higher concentration of vibration energy [8]. 𝑑𝑘 𝑑𝑘 Acoustics 2019, 1 243 total energy at the ABH location of beams with the ABH cavity (shown in Figure 32b) is shown in Figure 44. It is seen that the beam with L of 19 mm has maximum energy output. This indicates ABH that the ABH is tunable by changing L to obtain a higher concentration of vibration energy [8]. ABH Acoustics 2018, 1, x FOR PEER REVIEW 22 of 29 Figure 44. Simulation results of total energy output of 3D-printed beams with ABH cavities with Figure 44. Simulation results of total energy output of 3D-printed beams with ABH cavities with various LABH [8]. various L [8]. ABH From previous studies in this section, it can be seen that, all the three geometrical parameters From previous studies in this section, it can be seen that, all the three geometrical parameters need an optimization design to get a best performance on vibration damping and sound reduction. need an optimization design to get a best performance on vibration damping and sound reduction. An investigation of the optimization design and position of an embedded 1D ABH is studied by An investigation of the optimization design and position of an embedded 1D ABH is studied by McCormick [62]. This study is on a thin simply-supported beam with a 1D ABH at some position McCormick [62]. This study is on a thin simply-supported beam with a 1D ABH at some position along along the beam. The relevant design variables are shown in Figure 45, which are the length of the the beam. The relevant design variables are shown in Figure 45, which are the length of the ABH L , ABH ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point the portion of the damped taper L , the thickness of the damping layer h , the drive point location D, d d location D, and the offset between the center of the beam and the center of the ABH B. The and the offset between the center of the beam and the center of the ABH B. The optimization result optimization result is shown in Figure 46. Comparing with the uniform beam, the simple damped is shown in Figure 46. Comparing with the uniform beam, the simple damped beam can decrease beam can decrease the surface-averaged velocity, but increases the mass by 5%. The optimized design the surface-averaged velocity, but increases the mass by 5%. The optimized design can decrease the can decrease the total mass of the beam by 15% and decreases the total surface-averaged velocity total mass of the beam by 15% and decreases the total surface-averaged velocity response by 12 dB, response by 12 dB, which has more reduction than a simple damped beam. The aim of the study is which has more reduction than a simple damped beam. The aim of the study is to figure out the to figure out the ABH design that minimizes the total mass of the beam and, at the same time, ABH design that minimizes the total mass of the beam and, at the same time, minimizes the total minimizes the total surface-averaged velocity response. This benefits the design of ABHs for surface-averaged velocity response. This benefits the design of ABHs for vibration reduction without vibration reduction without adding mass. adding mass. As shown in the studies on vibration of beams and plates with ABH in Section 3.1, there is a As shown in the studies on vibration of beams and plates with ABH in Section 3.1, there is a minimum effective frequency, below which the ABH damping performance is not evident. A minimum effective frequency, below which the ABH damping performance is not evident. A numerical numerical and experimental study on the investigation of disappearance of the ABH effect is and experimental study on the investigation of disappearance of the ABH effect is presented by Tang presented by Tang and Cheng [63]. It shows that cut-on frequency bands close to the low-order local and Cheng [63]. It shows that cut-on frequency bands close to the low-order local resonant frequencies resonant frequencies of the beam exist and the ABH effect failure in these frequency bands. The of the beam exist and the ABH effect failure in these frequency bands. The failure frequencies of the failure frequencies of the beam are delimited by the excitation point in order to avoid the beam are delimited by the excitation point in order to avoid the phenomenon in the structural design. phenomenon in the structural design. Figure 45. Schematic of the simple supported beams with the relevant design variables: Length of ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point location D, and the offset between the center of the beam and the center of the ABH B. Reprinted with permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. Acoustics 2018, 1, x FOR PEER REVIEW 22 of 29 Figure 44. Simulation results of total energy output of 3D-printed beams with ABH cavities with various LABH [8]. From previous studies in this section, it can be seen that, all the three geometrical parameters need an optimization design to get a best performance on vibration damping and sound reduction. An investigation of the optimization design and position of an embedded 1D ABH is studied by McCormick [62]. This study is on a thin simply-supported beam with a 1D ABH at some position along the beam. The relevant design variables are shown in Figure 45, which are the length of the ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point location D, and the offset between the center of the beam and the center of the ABH B. The optimization result is shown in Figure 46. Comparing with the uniform beam, the simple damped beam can decrease the surface-averaged velocity, but increases the mass by 5%. The optimized design can decrease the total mass of the beam by 15% and decreases the total surface-averaged velocity response by 12 dB, which has more reduction than a simple damped beam. The aim of the study is to figure out the ABH design that minimizes the total mass of the beam and, at the same time, minimizes the total surface-averaged velocity response. This benefits the design of ABHs for vibration reduction without adding mass. As shown in the studies on vibration of beams and plates with ABH in Section 3.1, there is a minimum effective frequency, below which the ABH damping performance is not evident. A numerical and experimental study on the investigation of disappearance of the ABH effect is presented by Tang and Cheng [63]. It shows that cut-on frequency bands close to the low-order local resonant frequencies of the beam exist and the ABH effect failure in these frequency bands. The failure frequencies of the beam are delimited by the excitation point in order to avoid the Acoustics 2019, 1 244 phenomenon in the structural design. Figure 45. Schematic of the simple supported beams with the relevant design variables: Length of Figure 45. Schematic of the simple supported beams with the relevant design variables: Length of ABH LABH, the portion of the damped taper Ld, the thickness of the damping layer hd, the drive point ABH L , the portion of the damped taper L , the thickness of the damping layer h , the drive point ABH d d location D, and the offset between the center of the beam and the center of the ABH B. Reprinted with location D, and the offset between the center of the beam and the center of the ABH B. Reprinted with Acoustics 2018, 1, x FOR PEER REVIEW 23 of 29 permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. Figure 46. Approximate Pareto front (black curve) after 10,000 evaluations. Objective function Figure 46. Approximate Pareto front (black curve) after 10,000 evaluations. Objective function evaluations for a sample of designs are denoted by red dots. The black × shows the function evaluation evaluations for a sample of designs are denoted by red dots. The black shows the function evaluation for an unmodified beam. The black circle shows the optimization. The black + shows the evaluation for an unmodified beam. The black circle shows the optimization. The black + shows the evaluation of of a uniform beam with the same damping. Reprinted with permission from [62]. Copyright a uniform beam with the same damping. Reprinted with permission from [62]. Copyright McCormick, McCormick, C.A.; Shepherd, M.R., 2018. C.A.; Shepherd, M.R., 2018. 4.2. Design of a 2D ABH 4.2. Design of a 2D ABH For a 2D ABH, it is to be noted that, besides geometrical parameters, the spatial layout of the For a 2D ABH, it is to be noted that, besides geometrical parameters, the spatial layout of the ABH, the material of the host structure and damping material, the diameter of center hole, and the ABH, the material of the host structure and damping material, the diameter of center hole, and the number of ABH also affect the performance of the ABH. number of ABH also affect the performance of the ABH. A numerical analysis of flexural ray trajectories in 2D ABH is developed by Huang et al. [32]. A numerical analysis of flexural ray trajectories in 2D ABH is developed by Huang et al. [32]. Figure 47 shows that for a given value of exponent m, the 2D ABH with thinner residual thickness Figure 47 shows that for a given value of exponent m, the 2D ABH with thinner residual thickness (h1 = 0.0002 m) absorbs more propagation energy than the one with residual thickness of h1 = 0.001 m. (h = 0.0002 m) absorbs more propagation energy than the one with residual thickness of h = 0.001 m. 1 1 In this study, the results also show the ratio of the rays converging to the center is 99.69% in the ideal In this study, the results also show the ratio of the rays converging to the center is 99.69% in the ideal ABH. The wave focalization can be enhanced with a larger m and a small h1. ABH. The wave focalization can be enhanced with a larger m and a small h . (a) (b) Figure 47. Numerical analysis of flexural ray trajectories in 2D ABH with various geometrical analysis (a) m = 2; and (b) m = 3. Reprinted with permission from [32]. Copyright Elsevier, 2018. Acoustics 2018, 1, x FOR PEER REVIEW 23 of 29 Figure 46. Approximate Pareto front (black curve) after 10,000 evaluations. Objective function evaluations for a sample of designs are denoted by red dots. The black × shows the function evaluation for an unmodified beam. The black circle shows the optimization. The black + shows the evaluation of a uniform beam with the same damping. Reprinted with permission from [62]. Copyright McCormick, C.A.; Shepherd, M.R., 2018. 4.2. Design of a 2D ABH For a 2D ABH, it is to be noted that, besides geometrical parameters, the spatial layout of the ABH, the material of the host structure and damping material, the diameter of center hole, and the number of ABH also affect the performance of the ABH. A numerical analysis of flexural ray trajectories in 2D ABH is developed by Huang et al. [32]. Figure 47 shows that for a given value of exponent m, the 2D ABH with thinner residual thickness (h1 = 0.0002 m) absorbs more propagation energy than the one with residual thickness of h1 = 0.001 m. Acoustics 2019, 1 245 In this study, the results also show the ratio of the rays converging to the center is 99.69% in the ideal ABH. The wave focalization can be enhanced with a larger m and a small h1. (a) (b) Figure 47. Numerical analysis of flexural ray trajectories in 2D ABH with various geometrical analysis Figure 47. Numerical analysis of flexural ray trajectories in 2D ABH with various geometrical analysis (a) m = 2; and (b) m = 3. Reprinted with permission from [32]. Copyright Elsevier, 2018. (a) m = 2; and (b) m = 3. Reprinted with permission from [32]. Copyright Elsevier, 2018. Acoustics 2018, 1, x FOR PEER REVIEW 24 of 29 A study on various layout of ABH on glass fiber composite plates with 1D and 2D ABH was A study on various layout of ABH on glass fiber composite plates with 1D and 2D ABH was studied by Bowyer and Krylov [16]. In this study, the glass fiber composite honeycomb sandwich studied by Bowyer and Krylov [16]. In this study, the glass fiber composite honeycomb sandwich panels with two 2D ABHs with various configurations (shown in Figure 48a) are investigated and panels with two 2D ABHs with various configurations (shown in Figure 48a) are investigated and the experimental results of acceleration are compared with a reference sandwich panel without ABH the experimental results of acceleration are compared with a reference sandwich panel without ABH plates. The results show that theoretically the structure in Figure 48b should be the optimum layout to plates. The results show that theoretically the structure in Figure 48b should be the optimum layout perform with the best vibration damping. to perform with the best vibration damping. (a) (b) Figure 48. Cross-section of (a) various configurations of ABHs and (b) optimum configurations of Figure 48. Cross-section of (a) various configurations of ABHs and (b) optimum configurations of ABHs. Reprinted from [16] under a CC BY 3.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2014. ABHs. Reprinted from [16] under a CC BY 3.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2014. Due to the limitation of manufacturing, the ABH may not be perfect. A study on the wave energy Due to the limitation of manufacturing, the ABH may not be perfect. A study on the wave energy focalization in a plate with imperfect 2D ABH is developed by Huang et al. [64]. The imperfect 2D focalization in a plate with imperfect 2D ABH is developed by Huang et al. [64]. The imperfect 2D ABH uses a polynomial profile instead of a power-law profile. The results still show drastic increases ABH uses a polynomial profile instead of a power-law profile. The results still show drastic increases in energy density around the tapered area. Chong also indicates that the smooth and stepped ABH in energy density around the tapered area. Chong also indicates that the smooth and stepped ABH beams are slightly different [22]. Denis et al. also find that the controlled imperfection of the tip of the 1D ABH causes a decrease of the reflection coefficient, which indicates that imperfect extremities are not detrimental to the performance of the ABH effect [65]. These two studies lower the stringent requirement of the ideal power-law thickness variation of ABH. A numerical study on the influence of number of ABHs on a plate in vibration is carried out by Conlon et al. [66,67]. The space averaged acceleration and total radiated sound power results for the 5 × 5 ABH (shown in Figure 49a) and uniform plates are compared in Figure 50. This shows that the plate with the ABH array has very effective results on vibration and noise reduction. Another comparison of total radiated sound power response of ABH panel between the panel with a 5 × 5 ABH array (Figure 49a) and the panel with 13 ABHs (Figure 49b) is shown in Figure 51. It can be seen that for higher frequencies (over 5 kHz) increasing of the number of ABHs results in more sound reduction, but in lower frequency this effect can only be observed in a few areas. This type of ABH array was also applied on the phononic crystals (PC) by Zhu and Semperlotti [68] to investigate peculiar dispersion characteristics. (a) (b) Figure 49. (a) Schematic of a panel with 5 × 5 ABH and (b) a panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 24 of 29 A study on various layout of ABH on glass fiber composite plates with 1D and 2D ABH was studied by Bowyer and Krylov [16]. In this study, the glass fiber composite honeycomb sandwich panels with two 2D ABHs with various configurations (shown in Figure 48a) are investigated and the experimental results of acceleration are compared with a reference sandwich panel without ABH plates. The results show that theoretically the structure in Figure 48b should be the optimum layout to perform with the best vibration damping. (a) (b) Acoustics 2019, 1 246 Figure 48. Cross-section of (a) various configurations of ABHs and (b) optimum configurations of ABHs. Reprinted from [16] under a CC BY 3.0 license. Copyright Bowyer E.P. and Krylov, V.V., 2014. beams are slightly Due to the limitation o different [22]. f mDenis anufactet uring al. , the ABH m also find ay that not be perfect the contr . A st olled udy on t imperfection he wave energy of the tip of focalization in a plate with imperfect 2D ABH is developed by Huang et al. [64]. The imperfect 2D the 1D ABH causes a decrease of the reflection coefficient, which indicates that imperfect extremities ABH uses a polynomial profile instead of a power-law profile. The results still show drastic increases are not detrimental to the performance of the ABH effect [65]. These two studies lower the stringent in energy density around the tapered area. Chong also indicates that the smooth and stepped ABH requirement of the ideal power-law thickness variation of ABH. beams are slightly different [22]. Denis et al. also find that the controlled imperfection of the tip of the A numerical 1D ABH causes study a d on ecre the aseinfluence of the reflection coeffic of number ient of , which ABHs indic on ates th a plate at imp inevibration rfect extremis ities carried are out by not detrimental to the performance of the ABH effect [65]. These two studies lower the stringent Conlon et al. [66,67]. The space averaged acceleration and total radiated sound power results for the requirement of the ideal power-law thickness variation of ABH. 5  5 ABH (shown in Figure 49a) and uniform plates are compared in Figure 50. This shows that A numerical study on the influence of number of ABHs on a plate in vibration is carried out by the plate with the ABH array has very effective results on vibration and noise reduction. Another Conlon et al. [66,67]. The space averaged acceleration and total radiated sound power results for the 5 × 5 ABH (shown in Figure 49a) and uniform plates are compared in Figure 50. This shows that the comparison of total radiated sound power response of ABH panel between the panel with a 5  5 ABH plate with the ABH array has very effective results on vibration and noise reduction. Another array (Figure 49a) and the panel with 13 ABHs (Figure 49b) is shown in Figure 51. It can be seen comparison of total radiated sound power response of ABH panel between the panel with a 5 × 5 that for higher frequencies (over 5 kHz) increasing of the number of ABHs results in more sound ABH array (Figure 49a) and the panel with 13 ABHs (Figure 49b) is shown in Figure 51. It can be seen reduction, but in lower frequency this effect can only be observed in a few areas. This type of ABH that for higher frequencies (over 5 kHz) increasing of the number of ABHs results in more sound reduction, but in lower frequency this effect can only be observed in a few areas. This type of ABH array was also applied on the phononic crystals (PC) by Zhu and Semperlotti [68] to investigate array was also applied on the phononic crystals (PC) by Zhu and Semperlotti [68] to investigate peculiar dispersion characteristics. peculiar dispersion characteristics. (a) (b) Figure 49. (a) Schematic of a panel with 5 × 5 ABH and (b) a panel with 13 ABHs. Reprinted with Figure 49. (a) Schematic of a panel with 5  5 ABH and (b) a panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Acoustics 2018, 1, x FOR PEER REVIEW 25 of 29 Figure 50. The space averaged acceleration (bottom) and total radiated sound power (top) results for Figure 50. The space averaged acceleration (bottom) and total radiated sound power (top) results the 5 × 5 ABHs (read line) and uniform plates (black line). Reprinted with permission from [66]. for the 5  5 ABHs (read line) and uniform plates (black line). Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Figure 51. Total radiated sound power results for the panel with 5 × 5 ABHs and panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. 4.3. Design of Damping Layer The study Feurtado and Conlon [48] mentioned in Section 3.2 indicates that the radius of the damping layer has an optimum value. When the diameter is larger than the optimum value, the damping effect increases slightly. A time domain experimental study based on a laser visualization system is developed by Ji et al. [31]. The wave propagation and attenuation in a plate-embedded 1D ABH is investigated. It is also noted that there exists an optimal thickness for the damping layer. Krylov indicate that the damping material with higher material loss factor makes the damping effect of the ABH have better performance [29]. Denis et al. indicate that a good compromise has been found by using a long additional termination with a moderate length of the damping layer in the framework of nonlinear vibration, which can develop into a turbulent regime [61]. An optimization study was developed by Lee et al. [69] and indicates that the damping performance can be enhanced by treating the tip with an appropriate size of damping layer. Another study developed by Tang et al. [34] shows that the stiffness of the damping layer plays is more critical than mass for a better damping effect but, meanwhile, the effect of the added mass still needs to be accurately designed especially when the thickness of damping is considerable to the tip of the ABH wedge. 4.4. Studies on 3D-Printed Structures with ABH Due to the limitation of traditional manufacturing methods, a power-law curve is difficult to manufacture. Bowyer and Krylov indicate that milling is not a practical way to machine structures with 2D ABH and suggest that 3D printing technology can be applied to produce these structures [23]. Due to the well-known advantages of 3D printing [70], vibration and sound radiation of 3D- Acoustics 2018, 1, x FOR PEER REVIEW 25 of 29 Figure 50. The space averaged acceleration (bottom) and total radiated sound power (top) results for the 5 × 5 ABHs (read line) and uniform plates (black line). Reprinted with permission from [66]. Acoustics 2019, 1 247 Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, P.A., 2015. Figure 51. Total radiated sound power results for the panel with 5 × 5 ABHs and panel with 13 ABHs. Figure 51. Total radiated sound power results for the panel with 5  5 ABHs and panel with 13 ABHs. Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Reprinted with permission from [66]. Copyright Conlon, S.C.; Fahnline, J.B.; Shepherd, M.R.; Feurtado, Feurtado, P.A., 2015. P.A., 2015. 4.3. Design of Damping Layer 4.3. Design of Damping Layer The study Feurtado and Conlon [48] mentioned in Section 3.2 indicates that the radius of the The study Feurtado and Conlon [48] mentioned in Section 3.2 indicates that the radius of damping layer has an optimum value. When the diameter is larger than the optimum value, the the damping layer has an optimum value. When the diameter is larger than the optimum value, damping effect increases slightly. A time domain experimental study based on a laser visualization the damping effect increases slightly. A time domain experimental study based on a laser visualization system is developed by Ji et al. [31]. The wave propagation and attenuation in a plate-embedded 1D system is developed by Ji et al. [31]. The wave propagation and attenuation in a plate-embedded ABH is investigated. It is also noted that there exists an optimal thickness for the damping layer. 1D ABH is investigated. It is also noted that there exists an optimal thickness for the damping layer. Krylov indicate that the damping material with higher material loss factor makes the damping effect Krylov indicate that the damping material with higher material loss factor makes the damping effect of the ABH have better performance [29]. Denis et al. indicate that a good compromise has been found of the ABH have better performance [29]. Denis et al. indicate that a good compromise has been found by using a long additional termination with a moderate length of the damping layer in the framework by using a long additional termination with a moderate length of the damping layer in the framework of nonlinear vibration, which can develop into a turbulent regime [61]. An optimization study was of nonlinear vibration, which can develop into a turbulent regime [61]. An optimization study was developed by Lee et al. [69] and indicates that the damping performance can be enhanced by treating developed by Lee et al. [69] and indicates that the damping performance can be enhanced by treating the tip with an appropriate size of damping layer. Another study developed by Tang et al. [34] shows the tip with an appropriate size of damping layer. Another study developed by Tang et al. [34] shows that the stiffness of the damping layer plays is more critical than mass for a better damping effect but, that the stiffness of the damping layer plays is more critical than mass for a better damping effect but, meanwhile, the effect of the added mass still needs to be accurately designed especially when the meanwhile, the effect of the added mass still needs to be accurately designed especially when the thickness of damping is considerable to the tip of the ABH wedge. thickness of damping is considerable to the tip of the ABH wedge. 4.4. Studies on 3D-Printed Structures with ABH 4.4. Studies on 3D-Printed Structures with ABH Due to the limitation of traditional manufacturing methods, a power-law curve is difficult to Due to the limitation of traditional manufacturing methods, a power-law curve is difficult to manufacture. Bowyer and Krylov indicate that milling is not a practical way to machine structures manufacture. Bowyer and Krylov indicate that milling is not a practical way to machine structures with 2D ABH and suggest that 3D printing technology can be applied to produce these structures with 2D ABH and suggest that 3D printing technology can be applied to produce these structures [23]. [23]. Due to the well-known advantages of 3D printing [70], vibration and sound radiation of 3D- Due to the well-known advantages of 3D printing [70], vibration and sound radiation of 3D-printed structures are investigated by researchers. Zhou et al. developed a numerical study to investigate the dynamic and static properties of 3D-printed double-layered compound structures with ABHs [24]. Due to small residual thickness of the ABH profile, the structure with ABHs has low local stiffness and high stress concentration. The double-layered compound structure shows good damping performance and also significantly improves the static properties in structural stiffness and strength. Zhao and Prasad developed an experimental case study of vibration energy harvesting of a 3D-printed beam with a modified ABH cavity [8]. Liang et al. developed a numerical simulation for vibration energy harvesting of a 3D-printed beam with multiple ABH cavities [26]. Chong et al. numerically and experimentally studied the dynamic responses of 3D-printed beam with damped ABH grooves [22]. Rothe et al. investigated the dynamic behavior of the 3D-printed beams using a 3D- hexahedron finite element and an isotropic linear elastic homogenized material model [25]. A cantilever beam is embedded with the ABH at its free ends and fully fills the ABH area with flexible thermoplastic Acoustics 2019, 1 248 polyurethane (TPU) to make the overall thickness uniform. In comparison with the uniform beam, the beam-embedded ABH with TPU shows good damping performance. 5. Concluding Remarks This review has presented the recent theoretical and numerical studies on ABH. Applications of ABH on beam- and plate-type structures have been demonstrated. It is shown that the use of ABH in structural design is effective in controlling vibration and noise without adding additional mass. This is particularly important for the design of lightweight structures, such as aircraft panels. ABH has also shown good promise in vibration energy harvesting, however, its practical application is still under research. The current studies show that for 1D ABH, the geometric parameters are critical for the ABH effect. For 2D ABH, besides geometrical parameters, additional variables, such as the diameter of the center hole, the spatial layout of ABH array, and the number of ABHs, influence the performance of the ABH effect. Additionally, the material of the host structure and damping material are important for both 1D and 2D ABH. Optimization of the geometric parameters can significantly improve the damping effect of ABH. The number of ABHs and their spatial layout can expand the effective frequency range of ABH. A higher loss factor for both the host structure and damping layer can further improve the damping effect of ABH using less additive damping. However, there is a need for research efforts to further understand the interdependence of various geometrical parameters so that structural optimization studies can be carried out in designing structures with ABHs for better performance in vibration and noise control. 3D printing technology makes the manufacturing of more complex structures possible, and it is being applied in structures with ABH features. Further studies are required to apply ABH widely to real-life structures and carry out the application for more complex structures. Thus, it is observed from this review of various studies that the use of ABH in structural design for vibration and noise control is significantly effective and has great potential for research and industrial applications. Funding: This research received no external funding. Acknowledgments: The first author thanks the support from Department of Mechanical Engineering of Stevens Institute of Technology. Conflicts of Interest: The authors declare no conflict of interest. References 1. Rao, M.D. Recent applications of viscoelastic damping for noise control in automobiles and commercial airplanes. J. Sound Vib. 2003, 262, 457–474. [CrossRef] 2. Denis, V. Vibration Damping in Beams Using the Acoustic Black Hole Effect. Ph.D. Thesis, Universit´e du Maine, Le Mans, France, 2014. 3. Bowyer, E.P.; O’Boy, D.J.; Krylov, V.V.; Gautier, F. Experimental investigation of damping flexural vibrations using two-dimensional Acoustic ‘Black Holes’. 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Journal

AcousticsMultidisciplinary Digital Publishing Institute

Published: Feb 25, 2019

Keywords: acoustic black hole; structure design; noise and vibration control

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