Sound absorption of Helmholtz resonator included a winding built-in neck extension Shinsuke NAKANISHI 1 1 Hiroshima International University, Japan ABSTRACT Acoustic resonant absorber like a perforated panel or a Helmholtz resonator can be tuned at a low frequency by extending its neck or enlarging cavity volume. However, a total size of the resonators is often quite large when the neck or the cavity is simply extended for tuning at a low frequency. Previous researchers have studied Helmholtz resonator shortened in its size by subsided neck into back air cavity, and confirm that this resonator is tuned at low frequency without a deep cavity. The author has studied effects of a winding built-in neck extension to sound absorption of perforated panels, which shows same effects as the subsided neck into back cavity even if section of the winding extension is 1.8 mm square and 40 mm length. This study obtains sound absorption coefficient by measuring surface acoustic impedance at Helmholtz resonator, and discusses sound absorption of the resonator included the various winding neck extension built in a surface panel. Discussions in this paper focus effects of path length, patterns or number of turns of the winding neck extension and cavity volume to the sound absorption of the Helmholtz resonator. Keywords: Sound absorption, Helmholtz resonator I-INCE Classification of Subjects Number(s): 35.7 1. INTRODUCTION The previous studies have studied effects of sound absorption by a small size resonator tuning at a low and middle frequency, and it is typical method to enlarge cavity volume or neck length of the resonator tuned at low frequency. To shorten the size of resonator, Iwase et al. 1 have studied the effects on acoustic resonance by the subsiding changes of neck from original type Helmholtz resonator and by the adapting of extension parts to opening hole of perforated panel. From their experimental research to make shortening size of Helmholtz resonator and lowering resonant frequency of sound absorption of perforated panel, they conclude that use of extension parts to the opening hole on the perforated plate is effective to lower the resonance frequency simply. They have also obtained the same effects of extension parts with turn back shape in the back air cavity as with straight shape. And, Iwase et al. 2 have verified the high sound absorption at low frequency by the method adding extension part behind opening holes. They could also have the empiric model to calculate the acoustic impedance and the sound absorption coefficient for their new type resonator. As applications from the previous study by Iwase et al., the author 3 considered that the extension parts are built in perforated panel with multi-turn back shape. Measurements of its sound absorption coefficient and considerations on effects of the winding extension to the resonant sound absorption show that the extension shifts its resonant peak to lower frequency without change of depth of air cavity which affects total thickness of the sound resonator. The shift amount of the resonant frequency is quite close between the straight extension and the winding extension when its section area is 3 mm side of square and 2.4 mm side of square, although the shift amount increases when the section area is 1.8 mm side of square as the winding extension including twice of rectangle corners to same length of the above cases. This study discusses sound absorption of the resonator included the various winding neck extension built in a surface panel. Its sound absorption coefficient is calculated by measuring surface 1 s-nakani@it.hirokoku-u.ac.jp 4079
acoustic impedance at Helmholtz resonator which has single winding neck extension. And, discussions focus effects of path length, patterns or number of turns of the winding neck extension and cavity volume to the sound absorption of the Helmholtz resonator. 2. MEASUREMENT 2.1 Procedure The sound absorption coefficient of test Helmholtz resonator including a winding neck extension in a surface panel is obtained by surface normal impedance at the opening hole, which is measured by in-situ measurement method 4. Figure 1 and Figure 2 indicate procedure of measuring their surface normal impedance in workshop of Hiroshima International University. The test Helmholtz resonator is built in a box of 120 mm side of upper square surface and 60 mm height which is placed on the rigid floor. Two microphones are placed at the opening hole of test Helmholtz resonator for measuring transmission function between them, is needed to calculate normal surface impedance by in-situ measurement method. Incoherent pink noise from five loudspeakers in the workshop makes random sound incidence to the surface of test Helmholtz resonator. The normal surface impedance of test perforated panel is obtained by following equation: Here, d is distance between the surface of test piece and lower microphone, l is 14 mm distance between centers of two microphones, is air density, c is sound speed in the air, k is wave number and j is the imaginary unit. Figure 1 Procedure of in-situ measurement method for measuring normal surface impedance at an opening hole of test Helmholtz resonator. Figure 2 Positions of the test Helmholtz resonator on a floor in the workshop and two microphones at an opening hole of the resonator. 4080
2.2 Structures of test Helmholtz resonators Figure 3 shows a test Helmholtz resonator built in the box like double shells filled clay between them. The clay is used to fix vibration of shells by its large mass and large viscosity. Each part is modeled by Google SketchUp and is made by 3D printing (MakerBot, Replicator2) with polylactic acid plastics. The resonator has a square opening hole, a winding neck extension tube with a square section and a rectangular cavity behind it. Each resonator is designed to be tuned at 200 Hz by the following formula: Here, c is sound speed in the air [m/s], S is area of opening hole [m 2 ], V is volume of cavity [m 3 ], l is length of neck and is diameter of equivalent circle with same area of the square opening hole. At the first, this study considers effects of a turn back structure in the winding neck extension to sound absorption of the Helmholtz resonator. The extension tube has 40 mm length along a path at center of its cross section, and it has (a) no turn back, (b) 1 turn back, (c) 2 turn backs and (d) 3 turn backs. Figure 4 shows that length of a straight path between turn backs is unified in each test resonator. A side length of square opening hole has 1.8 mm, 2.4 mm, 3.0 mm and 3.6 mm, and its cube cavity has a side of 18 mm, 21.6 mm, 25.6 mm and 28 mm, respectively. Figure 3 A cross section of a test Helmholtz resonator built in a box of 120 mm side of upper square surface and 60 mm height filled clay between them. This example has following specifications: (1) an opening hole is 3.6 mm side of square, (2) a winding extension tube has 40 mm length with three turn backs, and (3) a cavity behind the extension is 28 mm side of cube. (a) No turn back (b) 1 turn back (c) 2 turn backs (d) 3 turn backs Figure 4 Examples of winding extension tube built in an upper plate of the Helmholtz resonator, which has 40 mm length of winding extension behind 3.6 mm side of square opening hole. (a) No turn back, (b) 1 turn backs, (c) 2 turn backs and (d) 3 turn backs. 4081
Secondly, this study considers a low height Helmholtz resonator which has 80 mm length of the neck extension tube. Its cavity behind the neck extension tube has 14.6 mm or 11 mm for 3.6 mm side of square opening hole, and turn back structure in the extension also has 8 variations. Their cavity height is almost half of that for case of 40 mm of neck extension mentioned above. 3. RESULTS AND DISCUSSION 3.1 Effects of winding extension tube to sound absorption This section discusses effects of a turn back structure in the winding neck extension to sound absorption of the Helmholtz resonator. Table 1 indicates resonant frequency of each test Helmholtz resonator which causes a significant peak in sound absorption coefficient, and it is also obtained when imaginary part of normal surface impedance takes zero. In cases of 1.8 mm side of square opening hole, their measured results do not show a significant peak in sound absorption coefficient even if a turn back structure varies. The resonant frequency of test resonators is designed to be tuned at 200 Hz, and their measured results show that they are around 200 Hz with plus or minus 10 % of fluctuations. A two way factorial analysis of variance indicates that the resonant frequencies except case of 1.8 mm side of square opening hole are not different (P=0.05) among variations of side length and turn back. This result suggests that the resonant frequency will be obtained opening area, neck extension length and cavity volume of the resonator even if structure of winding neck extension varies for possible turn back of 40 mm of extension. Figure 5 indicates sound absorption coefficient of the test Helmholtz resonators attached a 40 mm of winding extension tube having (a) No turn back, (b) 1 turn back, (c) 2 turn backs and (d) 3 turn backs. A significant peak of the sound absorption coefficient causes at about 200 Hz in cases except test resonators with 1.8 mm side of square opening hole. For every turn back structures, peak value of sound absorption coefficient becomes large as the opening hole enlarges. It is well known that resonance of the resonator becomes weak and small value due to a large friction loss on an inner wall of a straight cylindrical duct is inversely proportional to its diameter as following formula: Here, is a friction factor of the duct, [m] is its length, [m] is its diameter, [kg/m 3 ] is the density of the air and [m/s] is speed of air flow in the duct. In addition, a winding extension tube also a local resistance at its turn back, and the extension is added the local resistance and reduced the friction loss of straight part as increasing turn back if total length of the extension is constant. Results in this study do not show a significant difference even if varying number of turn back in each size of square opening hole. It suggests that sound absorption coefficient is obtained by length of the extension and size of opening hole, although it is need to discuss total resistance of the winding extension tube in a future study. Table 1 Resonant frequency of the test Helmholtz resonators. side length of square opening hole 1.8 mm 2.4 mm 3.0 mm 3.6 mm No turn back - 202 189 190 1 turn back - 192 200 203 2 turn backs - 200 224 211 3 turn backs - 187 224 221 (Hz) 4082
(a) No turn back (b) 1 turn back (c) 2 turn backs (d) 3 turn backs Figure 5 Sound absorption coefficient of the test Helmholtz resonators attached a 40 mm of winding extension tube having (a) No turn back, (b) 1 turn back, (c) 2 turn backs and (d) 3 turn backs. 4083
3.2 Considerations on low-height Helmholtz resonator This section discusses effects of 80 mm of neck extension tube to the sound absorption of the resonator for low-height Helmholtz resonator. Figure 6 indicates sound absorption coefficient of the resonators with an 80 mm of extension tube, which has 1, 3, 5 or 7 turn backs behind a 3.6 mm side of square opening hole. Actual measured sound absorption coefficients could be divided into three groups as follows (1) 1 or 2 turn backs, (2) 3, 4 or 5 turn backs and (3) 6, 7 or 8 turn backs. The results show that a behavior around resonant peak in sound absorption coefficient is quite similar in each group. (1) 1 or 2 turn backs give a peak with about 0.4 of absorption coefficient around 200 Hz with plus or minus 10 % fluctuations. (2) 3, 4 or 5 turn backs give a narrow peak with about 0.6 to 0.7 of absorption coefficient at 200 Hz. (3) 6, 7 or 8 turn backs give a gently-sloping peak with about 0.4 of absorption coefficient at 215 Hz to 246 Hz. Figure 7 indicates that its resonant frequency causes at higher frequency as increasing turn backs, although it is caused the resonant frequencies around 200 Hz with plus or minus 10 % fluctuations even if the resonators have 1 to 6 turn-backed extensions. On these conditions of the turn-backed extension, it can suggest that the resonant frequency is obtained by size of opening hole, length of the extension and cavity volume even if varying turn back structure. And sound absorption coefficient reduces as longer by length of the winding extension under the constant size of opening hole. However, 7 or 8 turn backed extension tube significant shifts resonant frequency of the resonator to 14 % or 23 % higher frequency than each designed resonant frequency, and reduces peak value of sound absorption coefficient again. These results are different from sound absorption coefficient of the resonator with a 40 mm length of extension tube behind 3.6 side of square opening hole mentioned above. Figure 6 Sound absorption coefficient of the test Helmholtz resonators attached an 80 mm of winding extension tube with 1, 3, 5 or 7 turn backs behind a 3.6 mm side of square opening hole. Figure 7 Resonant frequency of the test Helmholtz resonators to number of turn back of an 80 mm of winding extension tube. 4084
Figure 8 indicates structure of winding extension tube with 6, 7 or 8 turn backs, which is built in upper plate of test Helmholtz resonator. Straight part between turn backs reduces to comparable with a size of 180-dgree turn as increasing up to 8 turn backs. Here, actual path of air flow in the winding extension tube can be assumed to be narrower than physical size of cross section of the tube. Sugiyama et al. numerically analyze turbulent flow separation in a rectangular duct a sharp 180-degree turn, and suggest that air flow after the turn is compressed to outer wall and speed of air flow increases. Moreover, stream function over the separated flow shows that a vortex appears at inner wall near the 180-degree turn and it makes the air flow path narrower than the physical size due to succession of vortex at 180-degree turns with quite short straight part between turns. As results, cross section of the tube virtually becomes small to cause the resonance at lower frequency, and to increase a friction loss as mentioned in Equation 2 which reduces the resonant peak value of sound absorption as mentioned in Section 3.1. (a) 6 turn backs (b) 7 turn backs (c) 8 turn backs Figure 8 Structure of winding extension tube built in upper plate of test Helmholtz resonator. Figure 9 Streamwise velocity and stream function over the separated flow in the X 1-X 2 plain of a rectangular duct with a sharp 180-degree turn numerically analyzed by Sugiyama et al. 5. 4. CONCLUSIONS This study discussed sound absorption of the resonator included the various winding neck extension built in a surface panel. Its sound absorption coefficient was calculated by measuring surface acoustic impedance at Helmholtz resonator which has single winding neck extension. And, discussions focus effects of path length, patterns or number of turns of the winding neck extension and cavity volume to the sound absorption of the Helmholtz resonator. The author suggests following conclusions. (1) A winding neck extension built in upper plate of the Helmholtz resonator is useful method to tune it at lower resonant frequency. (2) Structure of winding extension tube is generally made with any turn-backed tube, although 4085
this study show that the resonant does not appear for 1.8 mm side of square opening with 40 mm of extension tube. (3) This study indicate that 8 turn back of 80 mm of extension tube shifts the resonance to 23 % higher frequency than designed resonance frequency, because a virtual path of air flow in the tube is narrower than the physical cross section. It seemed to be caused by a vortex at inner wall of 180-degree turn due to turbulent flow separation in a rectangular tube. REFERENCES 1. Teruo Iwase, Keiko Shirahata, Akiko Igarashi, Satoshi Sugie, Yasuaki Okada, "Shortening Helmholtz resonator by subsided neck and application to perforated plate structure for low frequency sound resonance," Proceedings in DVD-ROM on internoise2012 (2012, New York, NY), 12p (2012). 2. Teruo Iwase, Satoshi Sugie, Masayuki Abe, Hiroyasu Kurono, Shinya Nishimura, Yasuaki Okada and Koichi Yoshihisa, "Modeling and verification of perforated plate structure for high sound absorption at low frequency with extending parts behind holes into shallow air space," Proceedings in DVD-ROM on internoise2015 (2015, San Francisco, CA), 12p (2015). 3. Shinsuke Nakansihi, "A pilot study of acoustic absorption by a perforated panel with bending tube extension built in the panel," Proceedings in DVD-ROM on internoise2015 (2015, San Francisco, CA), 10p (2015). 4. Yasuo Takahashi, Toru Otsuru and Reiji Tomiku, "In situ measurements of surface impedance and absorption coefficients of porous materials using two microphones and ambient noise," Applied Acoustics, 66, 845-865 (2005) 5. Hitoshi Sugiyama, Tatsuya Tanaka and Hideaki Mukai, "Numerical analysis of turbulent flow separation in a rectangular duct with a sharp 180 degree turn by algebraic Reynolds stress model," International Journal for Numerical Methods in Fluids, 56(12), 2207 2228 4086