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Several explanations on the theoretical formula of Helmholtz resonator
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Research paper
Several explanations on the theoretical formula of Helmholtz resonator Lijun Li∗, Yiran Liu, Fan Zhang, Zhenyong Sun School of Transportation and Vehicle Engineering, Shandong University of Technology NO.12, Zhangzhou Road Zhangdian, Zibo 255049, Shandong, China
a r t i c l e
i n f o
Article history: Received 28 June 2017 Revised 19 July 2017 Accepted 6 August 2017 Available online xxx Keywords: Helmholtz resonator Finite element Acoustic pressure Mode shape Impedance
a b s t r a c t Helmholtz resonator is one of the most basic acoustic models in acoustic theoretical research and engineering applications. It is simple and effective to directly apply the theoretical formula for its resonant frequency calculation, but sometimes the calculation error is too large or even wrong. In this paper, the characteristics of Helmholtz resonators are studied based on the finite element numerical analysis. The influence of structural parameters and boundary conditions of the Helmholtz resonator on the resonant frequency is given, several related problems of the theoretical formula are supplemented and the relevant conclusions are obtained. The explanations employed in this paper could be used as a supplement to the theoretical formula for theoretical study and engineering applications of Helmholtz resonators. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Helmholtz resonator is one of the most basic acoustic resonant structures, which consists a short tube and an acoustic cavity, as shown in Fig. 1. About the Helmholtz resonator, there are many theoretical studies in the field of acoustics, and there are many direct or indirect applications in engineering, such as the muffler of vehicle’s exhaust tube, the intubation expansion chamber, and the porous plate sound absorbing material, etc. [1–5]. The theoretical formula of the Helmholtz resonator is simple, but some pertinent questions on the resonant frequency are poorly described in references, which lead to used blindly. For example, perforated plates are often equivalent to multiple Helmholtz resonators in parallel, the theoretical formula can be used to calculate the resonant sound absorption frequency, but significant errors may exist. When the perforated plate is used, it is sometimes necessary to fill the sound absorbing material on wall surfaces to enhance the sound absorption effect. As a result, the boundary condition no longer satisfies the sound hard boundary of the theoretical formula, in this case the use of theoretical formula to calculate the resonant frequency lead to large error. Based on the acoustic finite element analysis, the Helmholtz resonator is calculated under various parameters and working con-
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (L. Li), [email protected] (Y. Liu), [email protected] (F. Zhang), [email protected] (Z. Sun).
ditions, some relevant conclusions can be obtained as a supplement to the theoretical formula.
2. Theoretical formula As shown in Fig. 1, The Helmholtz resonator can be equivalent to a single-freedom vibration system. In the low frequency, the air in short tube is equivalent to mass in vibration system or inductance in electricity; the air in cavity is equivalent to spring in vibration system or capacitance in electricity. The theoretical formula for the Helmholtz resonator is
c0 f = 2π
S V · la
(1)
where f is the resonant frequency, C0 is the velocity of sound in the air, S is the cross-sectional area of the short tube, V is the volume of the acoustic cavity, and la is the total length of the short tube, the actual length of the short tube is l. la is l plus correction length. The correction length of the short tube depends on the acoustic radiation at the two ends of the short tube. The effective length of the tube is increased by l than the actual length. If it is a separate Helmholtz resonator, the correction length is 0.85a, a is the radius of short tube; if the Helmholtz resonator is a bypass tube to a pipe, the correction length of tube is 1.7a.
http://dx.doi.org/10.1016/j.advengsoft.2017.08.004 0965-9978/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: L. Li et al., Several explanations on the theoretical formula of Helmholtz resonator, Advances in Engineering Software (2017), http://dx.doi.org/10.1016/j.advengsoft.2017.08.004
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L. Li et al. / Advances in Engineering Software 000 (2017) 1–11 Table 1 Natural frequency of Helmholtz resonator of Example 1. Acoustic mode
Natural frequency (Hz)
Acoustic mode
Natural frequency (Hz)
1 2 3 5 7 8 10 12
184.18 1372.2 2013.4 (2) 2424.5 (2) 2704.8 3336.9 (2) 3365.7 (2) 3599 (2)
14 15 16 18 19 21 23 25
4045 4188.1 4290 (2) 4395.1 4518.1 (2) 4590.2 (2) 4784.1 (2) 4955.8
Table 2 Cutoff frequency of Helmholtz resonators. Radius of cylindrical cavity (m)
Theoretical cutoff frequency (Hz)
Numerical cutoff frequency (Hz)
Error
0.04 0.05 0.06 0.07 0.08 0.09 0.1
2513.8 2011.0 1675.9 1436.4 1256.9 1117.2 1005.5
2516.6 2013.4 1677.8 1438.2 1258.4 1118.6 1006.7
0.111% 0.119% 0.113% 0.125% 0.119% 0.125% 0.119%
Table 3 Effect of geometric parameters on resonant frequency. Radius of cylindrical cavity (m)
Numerical resonant frequency (Hz)
Underside length of block (m)
Numerical resonant frequency (Hz)
0.05 0.06 0.07 0.08 0.09 0.10
184.34 186.10 186.49 186.37 185.83 184.92
0.10 0.13 0.16 0.19 0.22 0.25
185.71 186.48 185.81 183.96 181.06 177.09
Fig. 1. Schematic diagram of the Helmholtz resonator.
3. Some research on theoretical formula 3.1. Relation between Helmholtz resonant frequency and modal The resonant frequency of Helmholtz resonator has several orders, but the first order is studied generally. The resonant frequency calculated by theoretical formula corresponds to the first order natural frequency. Under the plane wave excitation, all plane wave modes can be excited by resonance. If using appropriate acoustic excitation, the other modes can also cause resonance, but the application of circumferential and radial modes is less. Fig. 2. Acoustic finite element model.
Example 1. The Helmholtz resonator with a right cylindrical cavity is selected. The length of the short tube is 0.02 m, the radius of the short tube is 0.01 m, the radius of the cylinder cavity is 0.05 m, the volume of the cylinder cavity is 0.001 m3 . In this example, the Helmholtz resonator is separate, and the resonant frequency calculated from the theoretical formula Eq. (1) is then 181.25 Hz. The acoustic finite element model is built, as shown in Fig. 2, the analyzing frequency is selected that the maximum size of the mesh is less than 1/6 of the minimum wave-
length. The acoustic finite element equation is
[KF + iωCF − ω2 MF ]{ p} = {FA }
(2)
In the formula, FA is the acoustic excitation, which is proportional to the normal velocity on the boundary condition. KF, MF, CF are the fluid stiffness, mass and damping matrix respectively, p is
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Table 4 Resonant frequency of Helmholtz resonator with different short tube radius. Radius of short tube (m)
Theoretical resonant frequency (Hz)
Numerical resonant frequency (Hz)
Error
0.006 0.010 0.014 0.018 0.022 0.026
115.88 181.25 239.84 293.14 342.18 387.72
116.99 184.34 245.57 301.85 353.62 401.00
0.96% 1.70% 2.39% 2.97% 3.34% 3.43%
Fig. 3. The mode shape of Example 1.
pressure.
{ p} =
n
αi {φi } = [φ ]{α}
(3)
i=1
In the formula, {α } is modal participation coefficient, [] is eigen vector. The first order natural frequency obtained from simulation is 184.18 Hz, and some modes is shown in Fig. 3. Apply plane wave excitation at the tube inlet of the Helmholtz resonator from 0–50 0 0 Hz, the frequency response is calculated as shown in the Fig. 4. In Table 1, ‘(2)’ means repeated root. Therefore, the mode shapes of plane wave or nearly plane wave could be excited by resonance, such as 1st, 2nd, 7th, 14th, 15th, 18th and 25th. The
application of the theoretical formula emphasizes the incidence of plane waves, and just the 1st resonant frequency. The higher order resonant frequency is out of the theoretical formula, but can be calculated by finite element method or others.
3.2. Premise of theoretical formula The Helmholtz resonator theoretical formula is confined to the plane wave incident vertically. For most engineering problems, the propagation of sound waves in the tube can satisfy the condition of plane wave, but for higher order waves, like oblique incident or nonlinear wave, we cannot use the theoretical formula. In the low frequency range, the resonant cavity geometry satisfies kh < 0.5, where k is wavenumber, and h is length of cavity. If
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the frequency is too high or the tube is too long, the plane wave cannot be satisfied, and the calculation error is not negligible. Example 2. The geometric parameters of the Helmholtz resonator is first selected as in Example 1, the resonant frequency is 181.25 Hz by theoretical formula, and 184.18 Hz by finite element method. So that
kh =
Fig. 4. Helmholtz resonator harmonic response curve of Example 1.
2π f h = 0.43 c0
(4)
Due to meet the prerequisite kh < 0.5, the error using the theoretical formula to calculate the resonant frequency is small. Then the volume of this example is changed to 0.005 m3 , which means the length of the cavity becomes 5 times longer than the original modal. The first order natural frequency is then 73.383 Hz calculated by finite element method, shown in Fig. 5, but 81.055 Hz calculated by theoretical formula. It is apparently that when kh > 0.5, deviating from the precondition of the theoretical formula, the calculation error is great. 3.3. Cutoff frequency of the Helmholtz resonator The cutoff frequency of the Helmholtz resonator corresponds to a mode, which is the mode of the first order that is not a plane wave. The cutoff frequency mode depends on the acoustic cavity and independent of the short tube. For cylindrical cavity, the plane wave cutoff frequency is
fcuto f f =
1.841c0 2π a
(5)
In the formula, a is the radius of cylindrical cavity. Example 3. The volume of the cavity is fixed at 0.001 m3 . The radius of the acoustic cavity is parameterized and the calculated numerical results are shown in Table 2. Modes below the cutoff frequency are plane wave modes. For the cylindrical cavity, because of its structural symmetry, the first mode which is not the plane wave is circumferential mode, and is the repeated root, shown in Fig. 6. Fig. 5. The 1st mode of Example 2 with cavity 0.005 m3 .
Fig. 6. Mode of cutoff frequency various radius.
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Fig. 7. Sound absorbing material added at the bottom of cylindrical cavity. Fig. 9. Sound absorbing material is added at the side of cylindrical cavity.
3.4. Effect of the shape of the cavity The resonant frequency is affected by the shape of the cavity and short tube but is not considered in the theoretical formula. If you want to design a good resonant cavity, only according to the theoretical formula is not enough. Example 4. The volume of the cavity is fixed at 0.001 m3 , Helmholtz resonators with right cylindrical cavities and rectangular cavities are selected. The short tube is fixed, the volume of the cavity is still fixed at 0.001 m3 , but the radius or the Underside length of the cavity is parameterized. Then the first order natural frequency is simulated as shown in Table 3.
is thinner, and the difference of the radius and width of the cavity is greater, the correction length will be more accurate, and the theoretical resonant frequency will be closer to actual. Example 5. The cylindrical cavity volume of the Helmholtz resonator is fixed at 0.001 m3 , the radius is 0.05 m, and the length of the short tube is 0.02 m, the Radius of the short tube is parameterized, and the resonant frequencies obtained from the theoretical formula and finite element simulation are shown in Table 4.
3.6. Effect of acoustic boundary on resonant frequency It is apparent that the resonant frequency of the Helmholtz resonator is related to the shape of the resonator. 3.5. Correction length of short tube The Correction length is obtained by approximating the end of tube as the infinite baffle. For the same cavity, when the short tube
The resonant frequency of the Helmholtz resonator is deduced at the hard boundary. If there are other kind of boundaries in the resonant cavity, the theoretical formula would not be used directly. In engineering practice, many walls are elastic. If the elastic of wall is considered, the resonant frequency is not the same as theory formula, which is researched by many people [6–10].
Fig. 8. Response curves with and without absorptive liner at the bottom of cavity.
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Fig. 10. Response curves with and without absorptive liner at the side of cavity.
When the sound absorbing material is added at the bottom of the cavity shown as the Fig. 7, the resonant frequency is not affected as shown in the Fig. 8. The conclusion is the same when the sound absorbing material is added at the bottom of the cavity, shown as the Figs. 9 and 10. When the sound absorbing material is added to the short tube shown as the Fig. 11, the resonance frequency is greatly affected, as shown in the Fig. 12. It is apparent that when the sound absorbing material is added to the short tube, the resonance will be weakened.
3.7. Effect of position of short tube on resonant frequency The position of the short tube is not considered in the theoretical formula. However, the resonant frequency of the Helmholtz resonator is effected by position of the short tube. More importantly, the premise of the theory formula on the porous plate is that the holes are evenly distributed, if the pore distribution is a lot of difference, it cannot be equivalent to the same parallel Helmholtz resonant cavity, and cannot use the formula to calculate the resonant frequency.
Fig. 11. Sound absorbing material is added at side of short tube.
Example 6. The geometric parameters are the same as Example 1, and the resonant cavity has sound absorbing material. This model uses a rather lightweight glass wool with ρ ap = 12 kg/m3 , dav = 10 μm, Rf = 1424.2 Pa. s/m2 , where ρ ap is the material’s apparent density, dav is the mean fiber diameter, and Rf is the flow resistivity by an empirical correlation [11]. The thickness of absorptive material is 0.002 m. In this example, the porous material is equivalent to Delany– Bazley–Miki model. The Delany–Bazley–Miki model is an empirical model used to describe fibrous materials such as rockwool or glass fiber. When 103 ≤ Rf ≤ 50 × 103 Pa s m−2 , the Delany–Bazley– Miki could be used as approximation for the porous material.
Example 7. The length of short tube is 0.02 m, the radius of short tube is 0.01 m, the radius of cylinder cavity is 0.08 m, and the volume is 0.001 m3 , d is the distance of the center of the short tube to center of the cavity. The numerical resonant frequency is shown as Fig. 13. From Fig. 13, we could see the short tube is farther away from the central axis of the cavity, lower the resonant frequency. If the short tube is far from the central axis of the cavity, the calculation of resonant frequency by theoretical formula will lead to great errors.
3.8. Effect of internal intubation on resonant frequency If the short tube is an internal intubation, the effective length of the short tube is about the total length of the inner tube plus the correction value, that is l + 0.85a.
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Fig. 12. Response curves with and without absorptive liner at the side of short tube.
Fig. 13. The effect of position of short tube on resonant frequency.
Example 8. Helmholtz resonator geometric parameters are set as in Example 1, except the short tube is 0.04 m with internal length 0.02 m. The model is shown as Fig. 14. The numerical resonant frequency is 142.68 Hz, and the theoretical value is calculated as 138.94 Hz. In addition, there are some researches on conical tubes, rotating tubes, etc. [12–15], which shows the influence of short tube shape on resonant frequency.
ently. Second, the flow encounters a sudden change in the section of resonant cavity, resulting in eddy current and generate noise, this noise will offset the transmission loss [16–18]. The mean background flow in the system is calculated with the SST turbulence model for Mach numbers Ma = 0, M = 0.05 and Ma = 0.1. For the consideration of the flow, Helmholtz resonator is set as a side branch, shown as Fig. 16, and the main pipeline transmission loss is calculated, shown as Fig. 17. As the flow rate increases, the resonance decreases, and resonant frequency shifts.
3.9. Effect of flow velocity on resonant frequency 4. Impedance tube test The effect of flow on the performance of the resonator is mainly through two paths: first, the overall motion of the flow affects the propagation of the acoustic waves, and the boundary conditions on the surface of the resonant cavity structure are changed accordingly, so the sound waves propagate in the resonant cavity differ-
The Helmholtz resonator is tested by impedance tube testing. Impedance tube measurement of sound absorption coefficient include two ways, one is the standing wave ratio method, and the other is the transfer function method. Transfer function method is
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Incident sound pressure PI and reflected pressure PR are respectively
PI = Pi e jkx
PR = Pr e− jkx
(6)
If the distance from microphone to material are x1 and x2 respectively, and the pressure of the two microphones are
P1 = Pi e jkx1 + Pr e− jkx1 P2 = Pi e jk2 x + Pr e− jk2 x
(7)
Pr = r Pi
(8)
Where r is the reflection coefficient. The transfer function of sound field H12 is
H12 = r=
P2 e jkx2 + r e− jkx2 = jkx P1 e 1 + r e− jkx1
H12 − HI 2 jkx1 e HR − H12
(9) (10)
The normal incidence sound absorption coefficient α is
α = 1 − |r|2 Fig. 14. Geometric model of internal intubation.
(11)
Fig. 19 is test photo. The acoustic source is white noise, and the data analysis software is Smart Office of Germany M + P International company. The specimen of acrylic plate has a central hole of 10 mm in diameter, whose outside diameter is 100 mm, thickness is 10 mm, shown as Fig. 20. The plate is put into the impedance tube whose location can be adjusted, so the hole and wall of impedance tube forms a Helmholtz resonator, the center hole as a short tube and the back cavity is the resonant cavity, shown as Fig. 21. The absorption coefficient of single hole plate specimens with different volumes of the back cavity is obtained by experiment, as shown in Fig. 22. The frequency of the maximum sound absorption coefficient corresponds to the resonant frequency. The resonant frequencies obtained from the experimental tests are not very different from those obtained by finite element analysis. The validity of the finite element numerical method is proved by experiments. 5. Conclusions In this paper, the theory of Helmholtz resonator is studied deeply. The influence factors of resonant frequency are analyzed by several finite element examples, such as geometric parameters, boundary conditions, and flow velocity.
Fig. 15. The first mode of Helmholtz resonator with internal intubation.
Fig. 16. The Helmholtz resonator as a side branch.
more effective compared to the standing wave ratio method, so the choice of measurement of the absorption coefficient is the transfer function method. Fig. 18 is impedance tube system diagram, and absorption coefficient is computed through Matlab by transfer function method.
(1) The theoretic resonant frequency corresponds to the first order natural frequency. Under the plane wave excitation, all plane wave modes can be excited by resonance. (2) The application of theoretic formula is limited to low frequency or short tube, and the product of the frequency and the length must be less than a certain value (3) The cutoff frequency of Helmholtz resonator depends on the acoustic cavity and independent of the short tube. (4) The resonant frequencies of Helmholtz resonator depend on the modes, and the modes depend on the geometry, so the resonant frequencies depend on shape. (5) The correction length is 0.85a when resonator is separate, and 1.7a when resonator is side branch. The difference of the short tube radius and the cavity radius is greater, the correction length will be more accurate, and the theoretical resonant frequency will be closer to actual. (6) The boundary conditions influence the resonant frequency, such as soft wall, elastic wall, and absorbing material. (7) The position of short tube influences the resonant frequency.
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Fig. 17. The effect of flow velocity on resonant frequency.
Fig. 18. Impedance tube test system diagram.
Fig. 20. Entity model of perforated panel with one hole. Fig. 19. Impedance tube and other experimental instruments.
(8) When the short tube is an internal intubation, the effective length of the short tube is about the total length of the inner tube plus the correction value, that is l + 0.85a. (9) As the flow rate increases, the resonance decreases, and resonant frequency shifts.
There are many restrictions on the application of the theoretical formula, and attention should be paid to applying the theoretical formula. Numerical calculations, such as finite element method yield more accurate results and do not have so many constraints.
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References
Fig. 21. Diagrammatic sketch of impedance tube test.
Fig. 22. Sound absorption coefficient of one hole panel with different back cavity volumes.
[1] Corá R, Martins CA, Lacava PT. Acoustic instabilities control using Helmholtz resonators. Appl Acoust 2014;77:1–10. [2] Sheng M, Zhang L, Zhang H. Research on broadband resonator with extended soft neck for low frequency. J Vib Eng 2007;20(02):145–8. [3] Liu H, Zheng S, Lian X, Dan J. The acoustic performance prediction of the Helmholtz resonator with a conical neck. ACTA Acoust 2014(03):353–9. [4] Unnikrishnan NS, Shete CD, Subramoniam A, Handoo KL, Padmanabhan C. Experimental and computational investigation of coupled resonator–cavity systems. Appl Acoust 2010;71(1):61–7. [5] Klaus J. On the adjustment of Helmholtz resonators. Appl Acoust 2014;77:37–41. [6] Boutin C, Becot FOX. Theory and experiments on poro-acoustics with inner resonators. Wave Motion 2015;54:76–99. [7] Yasuda T, Wu C, Nakagawa N, Nagamura K. Studies on an automobile muffler with the acoustic characteristic of low-pass filter and Helmholtz resonator. Appl Acoust 2013;74(1):49–57. [8] Lan Y. Semi-analytical calculation of resonant modes in axially asymmetric microtube resonators. Opt Commun 2017;386:72–6. [9] Coulon J, Atalla N, Desrochers A. Optimization of concentric array resonators for wide band noise reduction. Appl Acoust 2016;113:109–15. [10] Romadhona IC. On the use of coupled cavity Helmholtz resonator inclusion for improving absorption performance of wooden sound diffuser element. Procedia Eng 2017;170:458–62. [11] Bies DA, Hansen CH. Flow resistance information for acoustical design. Appl Acoust 1980;13(5):357–91. [12] Nemati N, Kumar A, Lafarge D, Fang NX. Nonlocal description of sound propagation through an array of Helmholtz resonators. Comptes Rendus Mécanique 2015;343(12):656–69. [13] Cai C, Mak C-M, Shi X. An extended neck versus a spiral neck of the Helmholtz resonator. Appl Acoust 2017;115:74–80. [14] Ghanadi F, Arjomandi M, Cazzolato B, Zander A. Interaction of a flow-excited Helmholtz resonator with a grazing turbulent boundary layer. Exp Thermal Fluid Sci 2014;58:80–92. [15] Selamet A, Lee L. Helmholtz resonator with extended neck. J Acoust Soc Am 2003;113(4):1975–85. [16] Lan J, Li Y, Yua H, Li B, Liu X. Nonlinear effects in acoustic metamaterial based on a cylindrical pipe with ordered Helmholtz resonators. Phys Lett A 2017;381(13):1111–17. [17] Cai C, Mak CM, Wang X. Noise attenuation performance improvement by adding Helmholtz resonators on the periodic ducted Helmholtz resonator system. Appl Acoust 2017;122:8–15. [18] Richoux O, Lombard B, Mercier JO. Generation of acoustic solitary waves in a lattice of Helmholtz resonators. Wave Motion 2015;56:85–99.
Acknowledgments The authors would like to thank the National Natural Science Foundation of China (Grant No. 51505261), and the Natural Science Foundation of Shandong Province, China (Grant No. ZR2015AM013).
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Lijun Li, born in 1977, is currently an associate professor in Shandong University of Technology, China. She received her PhD degree from Shanghai Jiao tong University, China. Her research interests include the finite element analysis of structural and mechanical vibration and noise control. Yiran Liu, born in 1992, is currently a master candidate in School of Transportation and Vehicle Engineering, Shandong University of Technology, China. Fan Zhang, born in 1989, is currently a lecturer in School of Transportation and Vehicle Engineering, Shandong University of Technology, China. Zhenyong Sun, born in 1991, is currently a master candidate in School of Transportation and Vehicle Engineering, Shandong University of Technology, China.
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Research paper
Several explanations on the theoretical formula of Helmholtz resonator Lijun Li∗, Yiran Liu, Fan Zhang, Zhenyong Sun School of Transportation and Vehicle Engineering, Shandong University of Technology NO.12, Zhangzhou Road Zhangdian, Zibo 255049, Shandong, China
a r t i c l e
i n f o
Article history: Received 28 June 2017 Revised 19 July 2017 Accepted 6 August 2017 Available online xxx Keywords: Helmholtz resonator Finite element Acoustic pressure Mode shape Impedance
a b s t r a c t Helmholtz resonator is one of the most basic acoustic models in acoustic theoretical research and engineering applications. It is simple and effective to directly apply the theoretical formula for its resonant frequency calculation, but sometimes the calculation error is too large or even wrong. In this paper, the characteristics of Helmholtz resonators are studied based on the finite element numerical analysis. The influence of structural parameters and boundary conditions of the Helmholtz resonator on the resonant frequency is given, several related problems of the theoretical formula are supplemented and the relevant conclusions are obtained. The explanations employed in this paper could be used as a supplement to the theoretical formula for theoretical study and engineering applications of Helmholtz resonators. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Helmholtz resonator is one of the most basic acoustic resonant structures, which consists a short tube and an acoustic cavity, as shown in Fig. 1. About the Helmholtz resonator, there are many theoretical studies in the field of acoustics, and there are many direct or indirect applications in engineering, such as the muffler of vehicle’s exhaust tube, the intubation expansion chamber, and the porous plate sound absorbing material, etc. [1–5]. The theoretical formula of the Helmholtz resonator is simple, but some pertinent questions on the resonant frequency are poorly described in references, which lead to used blindly. For example, perforated plates are often equivalent to multiple Helmholtz resonators in parallel, the theoretical formula can be used to calculate the resonant sound absorption frequency, but significant errors may exist. When the perforated plate is used, it is sometimes necessary to fill the sound absorbing material on wall surfaces to enhance the sound absorption effect. As a result, the boundary condition no longer satisfies the sound hard boundary of the theoretical formula, in this case the use of theoretical formula to calculate the resonant frequency lead to large error. Based on the acoustic finite element analysis, the Helmholtz resonator is calculated under various parameters and working con-
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (L. Li), [email protected] (Y. Liu), [email protected] (F. Zhang), [email protected] (Z. Sun).
ditions, some relevant conclusions can be obtained as a supplement to the theoretical formula.
2. Theoretical formula As shown in Fig. 1, The Helmholtz resonator can be equivalent to a single-freedom vibration system. In the low frequency, the air in short tube is equivalent to mass in vibration system or inductance in electricity; the air in cavity is equivalent to spring in vibration system or capacitance in electricity. The theoretical formula for the Helmholtz resonator is
c0 f = 2π
S V · la
(1)
where f is the resonant frequency, C0 is the velocity of sound in the air, S is the cross-sectional area of the short tube, V is the volume of the acoustic cavity, and la is the total length of the short tube, the actual length of the short tube is l. la is l plus correction length. The correction length of the short tube depends on the acoustic radiation at the two ends of the short tube. The effective length of the tube is increased by l than the actual length. If it is a separate Helmholtz resonator, the correction length is 0.85a, a is the radius of short tube; if the Helmholtz resonator is a bypass tube to a pipe, the correction length of tube is 1.7a.
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L. Li et al. / Advances in Engineering Software 000 (2017) 1–11 Table 1 Natural frequency of Helmholtz resonator of Example 1. Acoustic mode
Natural frequency (Hz)
Acoustic mode
Natural frequency (Hz)
1 2 3 5 7 8 10 12
184.18 1372.2 2013.4 (2) 2424.5 (2) 2704.8 3336.9 (2) 3365.7 (2) 3599 (2)
14 15 16 18 19 21 23 25
4045 4188.1 4290 (2) 4395.1 4518.1 (2) 4590.2 (2) 4784.1 (2) 4955.8
Table 2 Cutoff frequency of Helmholtz resonators. Radius of cylindrical cavity (m)
Theoretical cutoff frequency (Hz)
Numerical cutoff frequency (Hz)
Error
0.04 0.05 0.06 0.07 0.08 0.09 0.1
2513.8 2011.0 1675.9 1436.4 1256.9 1117.2 1005.5
2516.6 2013.4 1677.8 1438.2 1258.4 1118.6 1006.7
0.111% 0.119% 0.113% 0.125% 0.119% 0.125% 0.119%
Table 3 Effect of geometric parameters on resonant frequency. Radius of cylindrical cavity (m)
Numerical resonant frequency (Hz)
Underside length of block (m)
Numerical resonant frequency (Hz)
0.05 0.06 0.07 0.08 0.09 0.10
184.34 186.10 186.49 186.37 185.83 184.92
0.10 0.13 0.16 0.19 0.22 0.25
185.71 186.48 185.81 183.96 181.06 177.09
Fig. 1. Schematic diagram of the Helmholtz resonator.
3. Some research on theoretical formula 3.1. Relation between Helmholtz resonant frequency and modal The resonant frequency of Helmholtz resonator has several orders, but the first order is studied generally. The resonant frequency calculated by theoretical formula corresponds to the first order natural frequency. Under the plane wave excitation, all plane wave modes can be excited by resonance. If using appropriate acoustic excitation, the other modes can also cause resonance, but the application of circumferential and radial modes is less. Fig. 2. Acoustic finite element model.
Example 1. The Helmholtz resonator with a right cylindrical cavity is selected. The length of the short tube is 0.02 m, the radius of the short tube is 0.01 m, the radius of the cylinder cavity is 0.05 m, the volume of the cylinder cavity is 0.001 m3 . In this example, the Helmholtz resonator is separate, and the resonant frequency calculated from the theoretical formula Eq. (1) is then 181.25 Hz. The acoustic finite element model is built, as shown in Fig. 2, the analyzing frequency is selected that the maximum size of the mesh is less than 1/6 of the minimum wave-
length. The acoustic finite element equation is
[KF + iωCF − ω2 MF ]{ p} = {FA }
(2)
In the formula, FA is the acoustic excitation, which is proportional to the normal velocity on the boundary condition. KF, MF, CF are the fluid stiffness, mass and damping matrix respectively, p is
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Table 4 Resonant frequency of Helmholtz resonator with different short tube radius. Radius of short tube (m)
Theoretical resonant frequency (Hz)
Numerical resonant frequency (Hz)
Error
0.006 0.010 0.014 0.018 0.022 0.026
115.88 181.25 239.84 293.14 342.18 387.72
116.99 184.34 245.57 301.85 353.62 401.00
0.96% 1.70% 2.39% 2.97% 3.34% 3.43%
Fig. 3. The mode shape of Example 1.
pressure.
{ p} =
n
αi {φi } = [φ ]{α}
(3)
i=1
In the formula, {α } is modal participation coefficient, [] is eigen vector. The first order natural frequency obtained from simulation is 184.18 Hz, and some modes is shown in Fig. 3. Apply plane wave excitation at the tube inlet of the Helmholtz resonator from 0–50 0 0 Hz, the frequency response is calculated as shown in the Fig. 4. In Table 1, ‘(2)’ means repeated root. Therefore, the mode shapes of plane wave or nearly plane wave could be excited by resonance, such as 1st, 2nd, 7th, 14th, 15th, 18th and 25th. The
application of the theoretical formula emphasizes the incidence of plane waves, and just the 1st resonant frequency. The higher order resonant frequency is out of the theoretical formula, but can be calculated by finite element method or others.
3.2. Premise of theoretical formula The Helmholtz resonator theoretical formula is confined to the plane wave incident vertically. For most engineering problems, the propagation of sound waves in the tube can satisfy the condition of plane wave, but for higher order waves, like oblique incident or nonlinear wave, we cannot use the theoretical formula. In the low frequency range, the resonant cavity geometry satisfies kh < 0.5, where k is wavenumber, and h is length of cavity. If
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the frequency is too high or the tube is too long, the plane wave cannot be satisfied, and the calculation error is not negligible. Example 2. The geometric parameters of the Helmholtz resonator is first selected as in Example 1, the resonant frequency is 181.25 Hz by theoretical formula, and 184.18 Hz by finite element method. So that
kh =
Fig. 4. Helmholtz resonator harmonic response curve of Example 1.
2π f h = 0.43 c0
(4)
Due to meet the prerequisite kh < 0.5, the error using the theoretical formula to calculate the resonant frequency is small. Then the volume of this example is changed to 0.005 m3 , which means the length of the cavity becomes 5 times longer than the original modal. The first order natural frequency is then 73.383 Hz calculated by finite element method, shown in Fig. 5, but 81.055 Hz calculated by theoretical formula. It is apparently that when kh > 0.5, deviating from the precondition of the theoretical formula, the calculation error is great. 3.3. Cutoff frequency of the Helmholtz resonator The cutoff frequency of the Helmholtz resonator corresponds to a mode, which is the mode of the first order that is not a plane wave. The cutoff frequency mode depends on the acoustic cavity and independent of the short tube. For cylindrical cavity, the plane wave cutoff frequency is
fcuto f f =
1.841c0 2π a
(5)
In the formula, a is the radius of cylindrical cavity. Example 3. The volume of the cavity is fixed at 0.001 m3 . The radius of the acoustic cavity is parameterized and the calculated numerical results are shown in Table 2. Modes below the cutoff frequency are plane wave modes. For the cylindrical cavity, because of its structural symmetry, the first mode which is not the plane wave is circumferential mode, and is the repeated root, shown in Fig. 6. Fig. 5. The 1st mode of Example 2 with cavity 0.005 m3 .
Fig. 6. Mode of cutoff frequency various radius.
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Fig. 7. Sound absorbing material added at the bottom of cylindrical cavity. Fig. 9. Sound absorbing material is added at the side of cylindrical cavity.
3.4. Effect of the shape of the cavity The resonant frequency is affected by the shape of the cavity and short tube but is not considered in the theoretical formula. If you want to design a good resonant cavity, only according to the theoretical formula is not enough. Example 4. The volume of the cavity is fixed at 0.001 m3 , Helmholtz resonators with right cylindrical cavities and rectangular cavities are selected. The short tube is fixed, the volume of the cavity is still fixed at 0.001 m3 , but the radius or the Underside length of the cavity is parameterized. Then the first order natural frequency is simulated as shown in Table 3.
is thinner, and the difference of the radius and width of the cavity is greater, the correction length will be more accurate, and the theoretical resonant frequency will be closer to actual. Example 5. The cylindrical cavity volume of the Helmholtz resonator is fixed at 0.001 m3 , the radius is 0.05 m, and the length of the short tube is 0.02 m, the Radius of the short tube is parameterized, and the resonant frequencies obtained from the theoretical formula and finite element simulation are shown in Table 4.
3.6. Effect of acoustic boundary on resonant frequency It is apparent that the resonant frequency of the Helmholtz resonator is related to the shape of the resonator. 3.5. Correction length of short tube The Correction length is obtained by approximating the end of tube as the infinite baffle. For the same cavity, when the short tube
The resonant frequency of the Helmholtz resonator is deduced at the hard boundary. If there are other kind of boundaries in the resonant cavity, the theoretical formula would not be used directly. In engineering practice, many walls are elastic. If the elastic of wall is considered, the resonant frequency is not the same as theory formula, which is researched by many people [6–10].
Fig. 8. Response curves with and without absorptive liner at the bottom of cavity.
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Fig. 10. Response curves with and without absorptive liner at the side of cavity.
When the sound absorbing material is added at the bottom of the cavity shown as the Fig. 7, the resonant frequency is not affected as shown in the Fig. 8. The conclusion is the same when the sound absorbing material is added at the bottom of the cavity, shown as the Figs. 9 and 10. When the sound absorbing material is added to the short tube shown as the Fig. 11, the resonance frequency is greatly affected, as shown in the Fig. 12. It is apparent that when the sound absorbing material is added to the short tube, the resonance will be weakened.
3.7. Effect of position of short tube on resonant frequency The position of the short tube is not considered in the theoretical formula. However, the resonant frequency of the Helmholtz resonator is effected by position of the short tube. More importantly, the premise of the theory formula on the porous plate is that the holes are evenly distributed, if the pore distribution is a lot of difference, it cannot be equivalent to the same parallel Helmholtz resonant cavity, and cannot use the formula to calculate the resonant frequency.
Fig. 11. Sound absorbing material is added at side of short tube.
Example 6. The geometric parameters are the same as Example 1, and the resonant cavity has sound absorbing material. This model uses a rather lightweight glass wool with ρ ap = 12 kg/m3 , dav = 10 μm, Rf = 1424.2 Pa. s/m2 , where ρ ap is the material’s apparent density, dav is the mean fiber diameter, and Rf is the flow resistivity by an empirical correlation [11]. The thickness of absorptive material is 0.002 m. In this example, the porous material is equivalent to Delany– Bazley–Miki model. The Delany–Bazley–Miki model is an empirical model used to describe fibrous materials such as rockwool or glass fiber. When 103 ≤ Rf ≤ 50 × 103 Pa s m−2 , the Delany–Bazley– Miki could be used as approximation for the porous material.
Example 7. The length of short tube is 0.02 m, the radius of short tube is 0.01 m, the radius of cylinder cavity is 0.08 m, and the volume is 0.001 m3 , d is the distance of the center of the short tube to center of the cavity. The numerical resonant frequency is shown as Fig. 13. From Fig. 13, we could see the short tube is farther away from the central axis of the cavity, lower the resonant frequency. If the short tube is far from the central axis of the cavity, the calculation of resonant frequency by theoretical formula will lead to great errors.
3.8. Effect of internal intubation on resonant frequency If the short tube is an internal intubation, the effective length of the short tube is about the total length of the inner tube plus the correction value, that is l + 0.85a.
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Fig. 12. Response curves with and without absorptive liner at the side of short tube.
Fig. 13. The effect of position of short tube on resonant frequency.
Example 8. Helmholtz resonator geometric parameters are set as in Example 1, except the short tube is 0.04 m with internal length 0.02 m. The model is shown as Fig. 14. The numerical resonant frequency is 142.68 Hz, and the theoretical value is calculated as 138.94 Hz. In addition, there are some researches on conical tubes, rotating tubes, etc. [12–15], which shows the influence of short tube shape on resonant frequency.
ently. Second, the flow encounters a sudden change in the section of resonant cavity, resulting in eddy current and generate noise, this noise will offset the transmission loss [16–18]. The mean background flow in the system is calculated with the SST turbulence model for Mach numbers Ma = 0, M = 0.05 and Ma = 0.1. For the consideration of the flow, Helmholtz resonator is set as a side branch, shown as Fig. 16, and the main pipeline transmission loss is calculated, shown as Fig. 17. As the flow rate increases, the resonance decreases, and resonant frequency shifts.
3.9. Effect of flow velocity on resonant frequency 4. Impedance tube test The effect of flow on the performance of the resonator is mainly through two paths: first, the overall motion of the flow affects the propagation of the acoustic waves, and the boundary conditions on the surface of the resonant cavity structure are changed accordingly, so the sound waves propagate in the resonant cavity differ-
The Helmholtz resonator is tested by impedance tube testing. Impedance tube measurement of sound absorption coefficient include two ways, one is the standing wave ratio method, and the other is the transfer function method. Transfer function method is
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Incident sound pressure PI and reflected pressure PR are respectively
PI = Pi e jkx
PR = Pr e− jkx
(6)
If the distance from microphone to material are x1 and x2 respectively, and the pressure of the two microphones are
P1 = Pi e jkx1 + Pr e− jkx1 P2 = Pi e jk2 x + Pr e− jk2 x
(7)
Pr = r Pi
(8)
Where r is the reflection coefficient. The transfer function of sound field H12 is
H12 = r=
P2 e jkx2 + r e− jkx2 = jkx P1 e 1 + r e− jkx1
H12 − HI 2 jkx1 e HR − H12
(9) (10)
The normal incidence sound absorption coefficient α is
α = 1 − |r|2 Fig. 14. Geometric model of internal intubation.
(11)
Fig. 19 is test photo. The acoustic source is white noise, and the data analysis software is Smart Office of Germany M + P International company. The specimen of acrylic plate has a central hole of 10 mm in diameter, whose outside diameter is 100 mm, thickness is 10 mm, shown as Fig. 20. The plate is put into the impedance tube whose location can be adjusted, so the hole and wall of impedance tube forms a Helmholtz resonator, the center hole as a short tube and the back cavity is the resonant cavity, shown as Fig. 21. The absorption coefficient of single hole plate specimens with different volumes of the back cavity is obtained by experiment, as shown in Fig. 22. The frequency of the maximum sound absorption coefficient corresponds to the resonant frequency. The resonant frequencies obtained from the experimental tests are not very different from those obtained by finite element analysis. The validity of the finite element numerical method is proved by experiments. 5. Conclusions In this paper, the theory of Helmholtz resonator is studied deeply. The influence factors of resonant frequency are analyzed by several finite element examples, such as geometric parameters, boundary conditions, and flow velocity.
Fig. 15. The first mode of Helmholtz resonator with internal intubation.
Fig. 16. The Helmholtz resonator as a side branch.
more effective compared to the standing wave ratio method, so the choice of measurement of the absorption coefficient is the transfer function method. Fig. 18 is impedance tube system diagram, and absorption coefficient is computed through Matlab by transfer function method.
(1) The theoretic resonant frequency corresponds to the first order natural frequency. Under the plane wave excitation, all plane wave modes can be excited by resonance. (2) The application of theoretic formula is limited to low frequency or short tube, and the product of the frequency and the length must be less than a certain value (3) The cutoff frequency of Helmholtz resonator depends on the acoustic cavity and independent of the short tube. (4) The resonant frequencies of Helmholtz resonator depend on the modes, and the modes depend on the geometry, so the resonant frequencies depend on shape. (5) The correction length is 0.85a when resonator is separate, and 1.7a when resonator is side branch. The difference of the short tube radius and the cavity radius is greater, the correction length will be more accurate, and the theoretical resonant frequency will be closer to actual. (6) The boundary conditions influence the resonant frequency, such as soft wall, elastic wall, and absorbing material. (7) The position of short tube influences the resonant frequency.
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Fig. 17. The effect of flow velocity on resonant frequency.
Fig. 18. Impedance tube test system diagram.
Fig. 20. Entity model of perforated panel with one hole. Fig. 19. Impedance tube and other experimental instruments.
(8) When the short tube is an internal intubation, the effective length of the short tube is about the total length of the inner tube plus the correction value, that is l + 0.85a. (9) As the flow rate increases, the resonance decreases, and resonant frequency shifts.
There are many restrictions on the application of the theoretical formula, and attention should be paid to applying the theoretical formula. Numerical calculations, such as finite element method yield more accurate results and do not have so many constraints.
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References
Fig. 21. Diagrammatic sketch of impedance tube test.
Fig. 22. Sound absorption coefficient of one hole panel with different back cavity volumes.
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Acknowledgments The authors would like to thank the National Natural Science Foundation of China (Grant No. 51505261), and the Natural Science Foundation of Shandong Province, China (Grant No. ZR2015AM013).
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Lijun Li, born in 1977, is currently an associate professor in Shandong University of Technology, China. She received her PhD degree from Shanghai Jiao tong University, China. Her research interests include the finite element analysis of structural and mechanical vibration and noise control. Yiran Liu, born in 1992, is currently a master candidate in School of Transportation and Vehicle Engineering, Shandong University of Technology, China. Fan Zhang, born in 1989, is currently a lecturer in School of Transportation and Vehicle Engineering, Shandong University of Technology, China. Zhenyong Sun, born in 1991, is currently a master candidate in School of Transportation and Vehicle Engineering, Shandong University of Technology, China.
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