Sonic crystal noise barrier using locally resonant scatterers

Transcription

1 PROCEEDINGS of the 22 nd International Congress on Acoustics Road Traffic Noise Modeling and Noise Barrier: Paper ICA Sonic crystal noise barrier using locally resonant scatterers Nicole Kessissoglou (a), Samaneh M.B. Fard (b) (a) UNSW Australia, Sydney, Australia, (b) UNSW Australia, Sydney, Australia, Abstract Sonic crystal barriers have been receiving recent interest as potential noise barriers to reduce traffic noise in certain frequency bands. Sonic crystals comprise of periodic arrangements of sound scatterers for which the simplest scatterer topology is a solid cylinder. This paper investigates the acoustic performance of a sonic crystal noise barrier using vertical cylindrical shells of finite height. Locally resonant scatterers comprising of perforated or C-shaped cylindrical shells are examined. Results for the barrier insertion loss show that attenuation in a broad band gap is generated due to destructive interference between the scattered sound waves within the periodic structure. The local resonance of the scatterers creates an additional peak in insertion loss, approximately predicted by the Helmholtz resonator frequency. For the case of the perforated cylindrical shells, the location of the resonant frequency is shown to be dependent on the number and size of the holes. When the resonant frequency due to the perforations occurs within the Bragg band gap, a significant increase in insertion loss across the band gap is shown to occur. For the case of the C-shaped cylindrical shells, the size of the opening is shown to have a significant effect on both the local resonant frequency and the band gap due to Bragg scattering. Keywords: sonic crystal barrier, perforated cylindrical shell, Helmholtz resonator

2 1 Introduction Sonic crystal noise barrier using locally resonant scatterers Sonic crystals are periodic arrangements of sound scatterers in a homogeneous fluid medium, where there exists a large impedance mismatch between the scatterers and fluid. A luminous and attractive introduction to the concept of sonic crystals for noise control was presented to the general public when a sculpture comprising of a large periodic array of rigid cylinders was displayed in Madrid. An experimental study on this sonic crystal sculpture showed that the unit cell topology (a single rigid hollow cylinder), lattice symmetry (spacing between adjacent cylinder centres) and filling fraction (amount of material used per unit cell) contributed to the presence of Bragg band gaps in which propagation of sound is prohibited [1, 2]. A locally resonant sonic crystal was introduced by Liu et al. [3]. In their seminal paper, a 3D array of lead spheres coated with a thin layer of silicon rubber was stacked in a simple cubic arrangement and excited by sound waves. In addition to the Bragg band gap generated by the spatial periodicity of the cubic arrangement, a second band gap at wavelengths below the first Bragg band gap was generated, attributed to the local resonance of the individual sphere/rubber unit cells. The concept of locally resonant scatterers opened up exciting new possibilities for control of sound. Elford et al. [4] investigated sonic crystal configurations consisting of locally multi-resonant scatterers. By replacing the solid cylinders with C-shaped hollow cylinders, each scatterer essentially became a Helmholtz resonator. C-shaped resonators of increasing size were arranged concentrically around each other in a Russian doll, or Matryoshka, format. The local resonance of each slotted tube created a new band gap below the frequency of the Bragg band gap, dependent only on the dimensions of the resonance cavity and unrelated to the periodicity of the sonic crystal. Since the pioneering work by Martínez-Sala et al. [1], sonic crystals have been designed as noise barriers for reduction of road traffic noise. Krynkin et al. [5] studied the acoustic performance of periodic arrays of cylinders aligned parallel to the ground plane as a potential noise barrier. Koussa et al. [6] combined two geometries of sonic crystal scatterers also aligned parallel to the ground with a rigid straight noise barrier. To further extend the barrier insertion loss, the scatterers were modelled as either rigid cylinders or resonant cavities, where the cavities were either rigid or lined with an absorbent material. Tailoring of locally resonant sonic crystals to generate band gaps that prevent propagation of sound is ideally suited for difficult-to-address low frequency noise problems. However, the sub Bragg band gaps generated by the locally resonant scatterers may be very narrow in bandwidth, particularly at higher order harmonics [5]. This is useful for filtering but impractical for broadband noise control. This work investigates the acoustic performance of a sonic crystal noise barrier using vertical cylindrical shells of finite height. Locally resonant scatterers comprising of perforated or C-shaped cylindrical shells in the same periodic arrangement are examined. In an attempt to broaden the insertion loss, the number and size of the holes of the perforated cylindrical shells, as well as the size of the opening of the C-shaped cylindrical shells, are examined. 2

3 2 Sonic crystal parameters Parameters that affect the acoustic performance of a sonic crystal include the topology of the scatterers. The simplest unit cell comprises a sound hard circular scatterer. Other parameters that play a direct role on the creation of band gaps in sonic crystals are the lattice constant and filling fraction. The lattice constant, a, is defined as the distance between the centres of adjacent scatterers. Figure 1 shows a sonic crystal comprising of sound hard circular scatterers in a square lattice arrangement, showing the lattice constant. Figure 1: Sonic crystal comprising of sound hard circular scatterers in a square lattice The filling fraction is defined as the ratio of the volume occupied by the scattering material with respect to the total volume of the sonic crystal. For a square lattice, the filling fraction is [7] f f = πd2 4a 2 (1) where d is the diameter of the scatterers and a is the lattice constant. A band gap is represented by the centre frequency and its bandwidth. The centre frequency of the periodic structure can be approximately predicted by Bragg s law and is given by f c = nc, n = 1, 2, 3, (2) 2a where c is the speed of sound in the fluid host medium, which in this case is air. 3 Numerical model Three dimensional finite element models of sonic crystal barriers using rigid cylindrical shell scatterers, perforated cylindrical shells and C-shaped cylindrical shells were developed using COMSOL Multiphysics (4.3b). Three rows of cylindrical shell scatterers in the y-direction were considered, as shown in Figure 2. The uniform cylindrical shells were then replaced with the locally resonant rigid perforated cylindrical shells and the C-shaped cylindrical shells shown in Figure 3. Due to the periodic boundary conditions, the number of scatterers in the x-direction was extended to infinity. For all proceeding results, the cylindrical shell diameter, lattice constant 3

4 and filling fraction are respectively d=0.4m, a=0.6m and f f =0.35. The maximum element size in the mesh was dictated by the requirement for a minimum of 6 elements per wavelength, resulting in a finer mesh in the vicinity of the perforated holes. Figure 2: Numerical model of the sonic crystal barrier showing the boundary conditions Figure 3: Sound hard perforated cylindrical shells (left) and C-shaped cylindrical shells (right) 4

5 4 Results and discussion 4.1 Dispersion relation Figures 4 and 5 present dispersion curves for rigid uniform cylindrical shells and C-shaped cylindrical shell scatterers, respectively. For these results, a square lattice sonic crystal array with periodic boundary conditions in both the x- and y-directions was implemented. In Figure 4, the shaded region represents the Bragg band gap in which there is no solution of the frequency for the given wavenumber. As such, this band gap represents the frequencies at which waves cannot propagate in the sonic crystal structure. In Figure 5, two shaded regions can be observed corresponding to the Bragg band gap associated with the overall periodicity of the sonic crystal array and a narrower band gap below the Bragg band gap encompassing the local resonant frequency of the C-shaped scatterers. Figure 4: Dispersion relation for a sonic crystal comprising uniform sound hard cylindrical shells Figure 5: Dispersion relation for a sonic crystal comprising C-shaped cylindrical shells 5

6 4.2 Insertion loss Comparison of the insertion loss for a sonic crystal barrier comprising of uniform, perforated and C-shaped cylindrical shells is presented in Figure 6. All cylindrical shells have a height of 3m and a shell thickness of 20mm. The perforated cylindrical shell scatterers have 4 holes around the circumference and 16 holes along the length, with a radius of 20mm. The C-shaped cylindrical shell scatterers have an opening length of 0.1m. For the parameters chosen, the local resonance of both the perforated and C-shaped cylindrical shells are tuned to the same frequency. The insertion loss was obtained as the sound pressure level at the same receiver position (in the barrier shadow zone) without and with the presence of the barrier. Using uniform cylindrical shells, a broad band gap is generated due to Bragg scattering. Using locally resonant scatterers, a large peak in insertion loss occurs below the Bragg band gap, which corresponds to Helmholtz resonance. It can be observed that the C-shaped cylindrical shell provides a wider insertion loss compared to the perforated cylindrical shell around the Helmholtz resonant frequency. However, the lower frequency of the band gap due to Bragg scattering is shifted to a higher frequency using C-shaped scatterers. Furthermore, the overall attenuation due to Bragg scattering using C-shaped scatterers is reduced. For the case of the perforated cylindrical shells, the Bragg band gap is mostly unaffected by the sub Bragg band gap associated with the local Helmholtz resonant frequency. Figure 6: Insertion loss for a sonic crystal barrier using uniform, C-shaped or perforated cylindrical shell The effect of the size of the hole radius of the perforated cylindrical shells on the barrier insertion loss is presented in Figure 7. Each perforated cylindrical shell scatterer has 8 holes around its circumference and 16 holes along its length. Increasing the hole radius from 10mm to 20mm results in an increase in the frequency at which the narrow band peak insertion loss occurs. When the hole radius is significantly increased to 40mm, the perforated cylindrical shells no longer act as a sonic crystal, attributed to the fact that air can easily pass through the holes. An interesting phenomenon occurs when the resonant frequency due to the perforations occurs 6

7 within the Bragg band gap, as in the case of the hole radius of 20mm. A global increase in insertion loss within the Bragg band gap using perforated cylindrical shell scatterers compared with uniform cylindrical shells can be observed, attributed to Fano resonance [8]. Figure 8 presents the effect of increasing the size of the opening of the C-shaped scatterers on the barrier insertion loss. The resonant frequency increases with an increase in the size of the opening. Furthermore, the lower frequency of the Bragg band gap is significantly affected by the size of the opening of the C-shaped cylindrical shells. Figure 7: Insertion loss for a sonic crystal barrier using uniform cylindrical shells or perforated cylindrical shells with varying size of hole radius Figure 8: Insertion loss for a sonic crystal barrier using uniform cylindrical shells or C-shaped cylindrical shells with varying size of opening 7

8 Figure 9 compares the insertion loss for a sonic crystal barrier using uniform cylindrical shells and perforated cylindrical shells with varying number of holes around the circumference in each row of scatterers. The hole radius is kept constant at 20mm. Two different cases associated with varying the number of holes is examined. In the first case, the first row of scatterers has 8 holes around the circumference, then 6 holes for the second row and 4 holes for the third row. In the second case, the first row of scatterers has 6 holes around the circumference, then 4 and 2 holes for the second and third rows, respectively. Three narrow insertion loss peaks can be observed, where each peak corresponds to the Helmholtz resonant frequency associated with the perforated cylindrical shell in each row. Decreasing the number of holes results in a shift of peak insertion loss to a lower resonant frequency. This is attributed to the fact that the total surface area occupied by the holes has decreased. The peak in insertion loss associated with each locally resonant scatterer only provides narrow band attenuation, with the exception when the local resonance occurs within the Bragg band gap, attributed to Fano resonance. However, since this local resonance is associated with one row of scatterers, only a very slight increase in insertion loss within the Bragg band gap occurs. Figure 9: Insertion loss for a sonic crystal barrier using uniform cylindrical shells or perforated cylindrical shells with varying number of holes around the circumference per row of cylindrical shell scatterers 5 Summary The acoustic performance of sonic crystal barriers comprising of sound hard scatterers in a square lattice periodic arrangement have been examined. Results for insertion loss revealed that a Bragg band gap was generated due to the periodic arrangement of the cylindrical shell scatterers. The band gap is dependent on the distance between the scatterers and the volume occupied by the scatterers. Locally resonant scatterers comprising of perforated and C-shaped cylindrical shells were also considered. The local resonance of the scatterers created an additional peak in insertion loss, approximately predicted by the frequency of a Helmholtz 8

9 resonator. For the case of the perforated cylindrical shells, the location of the resonant frequency was shown to be dependent on the number and size of the holes. When the resonant frequency due to the perforations occurred within the Bragg band gap, a significant increase in insertion loss within the band gap was found to occur. For the case of the C-shaped cylindrical shells, the size of the opening was found to have a significant effect on both the location of the resonant frequency and the band gap due to Bragg scattering. Whilst the use of locally resonant scatterers has been shown to provide greater insertion loss compared with the use of uniform cylindrical shell scatterers, further work is required to design a sonic crystal barrier for broadband noise attenuation. References [1] Martínez-Sala, R.; Sancho, J.; Sánchez, J.V.; Gómez, V.; Llinares J.; Meseguer. F. Sound attenuation by sculpture, Nature, Vol 378, 1995, pp 241. [2] Kushwaha, M.S. Stop-bands for periodic metallic rods: Sculptures that can filter the noise, Applied Physics Letters, Vol 70, 1997, pp [3] Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y.Y.; Yang, Z.; Chan C.T.; Sheng. P. Locally resonant sonic materials, Science, Vol 289 (5485), 2000, pp [4] Elford, D.P.; Chalmers, L.; Kusmartsev, F.V.; Swallowe, G.M. Matryoshka locally resonant sonic crystal. Journal of the Acoustical Society of America, Vol 130 (5), 2011, pp [5] Krynkin, A.; Umnova, O.; Taherzadeh, S.; Attenborough, K. Analytical approximations for low frequency band gaps in periodic arrays of elastic shells, Journal of the Acoustical Society of America, Vol 133, 2013, pp [6] Koussa, F.; Defrance, J.; Jean P.; Blanc-Benon. P. Acoustical efficiency of a sonic crystal assisted noise barrier, Acta Acustica united with Acustica, Vol 99, 2013, pp [7] Gupta, A.; Lim, K.; Chew, C. A quasi two-dimensional model for sound attenuation by the sonic crystals. Journal of the Acoustical Society of America, Vol 132 (4), 2012, pp [8] Xiao, X.; Wu, J.; Miyamaru, F.; Zhang M.; Li, S.; Takeda, M.W.; Wen, W.; Sheng, P. Fano effect of metamaterial resonance in terahertz extraordinary transmission, Applied Physics Letters, Vol 98 (1), 2011, pp