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Wave-based method for curvilinear acoustic black hole and study on cut-on frequency
Jae Yeon Lee, Wonju Jeon
To cite this version:
Jae Yeon Lee, Wonju Jeon. Wave-based method for curvilinear acoustic black hole and study on cut-on frequency. Forum Acusticum, Dec 2020, Lyon, France. pp.3301-3303, �10.48465/fa.2020.1118�.
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Wave-based method for curvilinear acoustic black hole and study on cut-on frequency
Jae Yeon Lee
1Wonju Jeon
11 Department of Mechanical Engineering, KAIST, Korea [email protected]
ABSTRACT
Recently, the curvilinear acoustic black hole (ABH), a thin structure whose thickness is tapered according to the power-law while the baseline is curved, was proposed and investigated numerically and experimentally. The curvilinear ABH was shown to be an effective and compact solution to absorption of elastic waves in thin structures. Despite the fine features of the curvilinear ABH, further researches on the fundamentals such as the effect of baseline curvature profile or the cut-on frequency are required for systematical design of the curvilinear ABH for various applications. In this study, we develop a semi- analytical method that gives the reflection coefficient of the curvilinear ABH, and use the method to investigate the reflection coefficients of various types of curvilinear ABHs. Specifically, two types of curvilinear ABHs: the arc ABH whose baseline curvature is constant and the spiral ABH whose baseline curvature is varying are investigated for various geometrical parameters and frequencies. Based on the investigations, we generalize the cut-on frequency for the standard ABHs to the cut-on frequency for the curvilinear ABHs, and show that the cut- on frequency of the curvilinear ABH is dependent on the rate of change of baseline curvature as well as the rate of change of thickness.
INTRODUCTION
An acoustic black hole (ABH) is a thin structure whose thickness is reduced according to a power law with power equal to or greater than two, and the tip region covered with a viscoelastic layer. Due to the combined effects of the power-law thickness and the viscoelastic layer, flexural waves propagating to the tip are slowed down, and the wave energy carried by the waves are dissipated into heat.
Ever since the discovery of the ABH effect [1] and the viscoelastic-layer treatment solution [2], abundant researches have been conducted on the ABHs. Some of the researches are listed in the following: the energy harvesting from the tip of an ABH [3], semi-analytical methods for evaluating the reflection coefficient of an ABH and frequency response function of a beam attached to an ABH [4–6], shape optimizations of ABHs and viscoelastic layers [7,8], exact solutions of the wave motions in ABHs [9,10], and the use of a functionally graded material to an ABH [11].
In 2017, Lee and Jeon [12] proposed the concept of the
“curvilinear” ABH, which has a curved baseline in contrast to the straight baseline of a standard ABH, to dampen the vibration of thin structures in a compact space. Lee and Jeon investigated numerically the damping performance of
the curvilinear ABHs such as the arc ABH and the Archimedean spiral ABHs, and showed that the damping performances of the curvilinear ABHs were almost the same as that of a standard ABH with the same arc length.
Recently, their numerical investigations were validated experimentally by Park, Kim, and Jeon [13] with further investigations of the viscoelastic-layer treatment on the tip of a curvilinear ABH.
In this paper, one observation from the previous researches [12,13] is investigated and discussed using the impedance method for a curvilinear ABH [14], which allows efficient evaluation of the reflection coefficient of a curvilinear ABH. For curvilinear ABHs with baseline shapes that are an arc of a circle or an Archimedean spiral, the absorption performances above a certain frequency were almost the same as those of a standard ABH of the same arc length. This critical frequency, which is named
“cut-on” frequency, is investigated for arc ABHs for various curvatures in this paper. From investigations, we show that the cut-on frequency of an arc ABH increases linearly proportional to its curvature.
The paper is organized as follows. In Section 1, the effect of curvature on the reflection coefficient of curvilinear ABHs is discussed. In Section 2, the effect of curvature on the cut-on frequency of arc ABHs is investigated. Finally, conclusions are presented in Section 4.
1. EFFECT OF BASELINE CURVATURE ON REFLECTION COEFFICIENT
In this section, we investigate the effect of curvature on the reflection coefficient R of an arc ABH.
Figure 1 shows R of an arc ABH whose thickness profiles and material properties are listed in Tables 1 and 2, respectively. The x axis indicates kf L0, where kf is the flexural wavenumber and L0 is the arc length of the ABH without tip truncation. The y axis indicates κL, where κ is the baseline curvature and L is the arc length of the ABH, and the color at each point in Fig. 3 indicates R. The upper limit of the y axis κL is chosen as 2π because the radius of curvature of an arc ABH cannot be smaller than L/2π for a fixed L.
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Figure 1. The reflection coefficient of an arc ABH with power 𝑚 = 2, arc length L = 360 mm, and 100- mm-long and 1-mm-thick viscoelastic layers attached at the tips. The x axis indicates normalized frequency, the y axis indicates normalized baseline curvature, and the colors indicate R.
Thickness profile [m]
Arc ABH (ℎ) ℎ(𝑠) = 0.0312𝑠2, 𝑠 ∈ [0.04, 0.4) viscoelastic layer (ℎd) ℎd(𝑠) = 0.001,
𝑠 ∈ [0.04, 0.14) Table 1. Thickness profile of the arc ABHs and the viscoelastic layers.
Material Arc ABHs Viscoelastic
layer Young’s
modulus 190 GPa 0.5 GPa
Mass density 7,850 kg/m3 950 kg/m3
Poisson ratio 0.33 0.45
Loss factor 0 0.15
Table 2. Material properties of the arc ABHs and the viscoelastic layers.
At low frequencies, flexural waves are almost perfectly reflected from the interface between the arc ABH and the base beam because the wavelength of the incident waves are longer than the length of the ABH, and so the waves cannot be smoothly guided to the tips. In other words, the low frequency range is below the cut-on frequency.
Conversely, at higher frequencies, the reflection coefficient decreases as the frequency increases. This implies that the curvilinear ABH smoothly guides incoming elastic waves to the viscoelastic layer located at the tip. Because the material for the arc ABHs was assumed to be lossless as shown in Table 2, only the viscoelastic layers at the tips absorb elastic waves. Similar to a standard ABH, it is concluded here that an arc ABH is an effective absorber of flexural wave energy at high frequencies and the absorption performance improves as the frequency increases.
2. CUT-ON FREQUENCY OF ARC ABH In this section, we investigate the reflection coefficients of the same arc ABHs with non-reflecting boundary condition imposed at the tips instead of the viscoelastic layers to investigate the effect of the curvature on their cut-on frequencies. The non-reflecting boundary condition is imposed at the tip of each ABH to obtain the reflection coefficient without influence of reflection from the tip.
For a standard ABH with m=2, the cut-on frequency is obtained theoretically as kfL0=1.94. [9] At the cut-on frequency, the reflection coefficient of the standard ABH with non-reflecting boundary condition at the tip is obtained numerically as 0.834. In analogy with the standard ABH, we define the frequency at which R becomes less than 0.834 as the “generalized cut-on frequency” of a curvilinear ABH with m=2 and investigate it throughout the section.
The reflection coefficient of an arc ABH with respect to the non-dimensional frequency and its normalized curvature is shown in Fig. 2. The x axis is kf L0, the y axis is κL and the colors at each point indicate R. The baseline curvature κ was varied from 0 to 2π/L, which is the maximum curvature, while the power m, tip thickness ht, and arc length L were fixed as 2, 0.5 mm, 0.0312 m−1 and 360 mm, respectively. The generalized cut-on frequency of the arc ABHs are marked with black markers in Fig. 2.
Figure 2. The reflection coefficient of an arc ABH with power m = 2, arc length L = 360 mm, and non- reflecting boundary condition imposed at the tips. The x axis indicates normalized frequency, the y axis indicates normalized baseline curvature, and the colors indicate R. The black markers indicate the generalized cut-on frequency for each curvature. The yellow dashed line indicates the regressed line for the generalized cut-on frequencies.
As shown in Fig. 2, the reflection coefficients of the arc ABHs are close to 1 at low frequencies and close to 0 at higher frequencies, and the cut-on frequencies are located
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in between the low and high frequency ranges depending on the curvature. For an arc ABH with normalized curvature up to 0.12, the generalized cut-on frequency is close to the cut-on frequency of the standard ABH, kf
L0=1.94. Above κL=0.12, the generalized curvature is low for κL<2.58, and increases as the curvature increases for κL>2.58.
For arc ABHs with normalized curvatures greater than 2.58, it seems that the cut-on frequency increases linearly proportional to the curvature, and the slope of increase can be observed more clearly by least-squares fitting. A regressed line for the generalized cut-on frequency for an arc ABH with ht=0.05 mm are shown in yellow dashed line in Fig. 2. The coefficient of determination is 0.97, which shows that the cut-on frequency indeed increases linearly with respect to the curvature.
The increase of the generalized cut-on frequency is in accordance with the numerical and experimental results obtained in previous researches [12,13], which showed that the mobilities of arc ABHs with various curvatures are different from the mobility of a standard ABH at low frequencies, in contrast to the behavior at higher frequencies. At low frequency range, the mobilities and the reflection coefficients are notably different from those found in a standard ABH because the cut-on frequency of an arc ABH increases with curvature. Conversely, at frequency sufficiently above the cut-on frequencies, the mobility of an arc ABH is similar to that of a standard ABH because reflections from the power-law taper and the curvature are negligible and reflection from the tip of an ABH is the most significant factor affecting its mobility.
3. CONCLUSION
Using the impedance method for the curvilinear ABH, we calculated the reflection coefficients of arc ABHs with viscoelastic layers at the tip and with non-reflecting boundary condition to investigate the effect of curvature on the reflection coefficient and the cut-on frequency. We showed that the cut-on frequency increases with the curvature for an arc ABH, and the absorption performance is improved at frequency range higher than the cut-on frequency.
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