Issue 
Acta Acust.
Volume 8, 2024



Article Number  22  
Number of page(s)  9  
Section  Environmental Noise  
DOI  https://doi.org/10.1051/aacus/2024013  
Published online  07 June 2024 
Scientific Article
Tunable silencer for rectangular ventilation duct based on composite Helmholtz resonators
^{1}
Tech X Academy, Shenzhen Polytechnic University, Shenzhen 518055, China
^{2}
Shenzhen Fantwave Tech. Co, Ltd, Shenzhen 518110, China
^{3}
Quantum Science and Technology Center and Advanced Materials Thrust, The Hong Kong University of Science and Technology (Guangzhou), Nansha 511466, Guangzhou, China
^{4}
Wave Functional Metamaterial Research Facility, The Hong Kong University of Science and Technology (Guangzhou), Guangzhou 511400, China
^{*} Corresponding author: phwen@ust.hk
Received:
18
October
2023
Accepted:
22
April
2024
To attenuate lowfrequency noise in square ventilation ducts, a tunable duct silencer consisting of composite Helmholtz resonators (HR) that precisely controlled by a step motor was proposed in this paper. An analytical model based on temporal coupled mode theory (TCMT) is applied to explain the sound absorption mechanism. The design breaks the limitation that the operating frequency of a silencer is fixed after manufacturing and avoids the degradation of ventilation conditions by attaching to the duct as a side branch. By using a step motor to drive the tunable cavity HR, the silencer can achieve accurate and continuous adjustment of the operating frequency over the entire operating frequency spectrum. In the experiment, the silencer has an absorption bandwidth of 250 Hz and ranges from 710 Hz to 960 Hz. Porous material is inserted into the cavities of HR to improve the damping properties. When integrated into the simulation model, it enables the rapid design of silencer dimensions for different sound absorption frequencies. The design demonstrates a new solution for low frequency sound attenuation in ventilation ducts and provides a potential technique for the future development of active controlled duct silencers.
Key words: Helmholtz resonator / Ventilation duct / Noise control / Sound absorption / Tunable
© The Author(s), Published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Since Liu et al. [1] introduced the concept of acoustic metamaterial, various types of designs have been proposed for applications such as noise control [2–4], waveguides [5, 6], vibration absorption [7–9] and energy harvesting [10].
Acoustic metamaterials with manually engineered structures offer alternative solutions for sound absorption at low frequencies [11]. By using the mechanism of high energy dissipation achieved through local resonances, the acoustic metamaterial can effectively absorb low frequency noise at a deep subwavelength thickness [12]. Sound absorption under ventilation conditions is one of the most important research interests in the field of noise control. The Helmholtz resonator (HR) is widely used for sound absorption in ventilation ducts [13–16]. Compared to other metastructures previously designed for the ventilation duct [12, 17, 18], the use of a HR has the advantage that the ventilation conditions are not affected by changing the crosssectional shape [19].
However, it has already been proved that the use of a single HR, which fulfils the critical coupling condition, can only lead to a maximum absorption of 50% in a twoport ventilation system [20]. To solve this problem, coupled HR pairs with asymmetric absorption mechanism are proposed to achieve perfect sound absorption [21]. In the design, two HRs are designated as reflector and absorber, respectively. The reflector HR has a high leakage rate when resonance, also called “bright” mode, is excited so that the incident wave is fully reflected [22]. The counterpart is denoted as the “dark mode”, which is formed by a HR that satisfies the critical coupling condition, meaning that its impedance matches the impedance of the surrounding medium (in this case, air). The absorber HR therefore has a highly dissipative resonance. The incident wave energy is converted into thermal energy by the friction between the air and the throat area and finally dissipated. The formation of the dark and bright modes is determined by the geometric parameters of the HR, such as the size of the neck [23]. Therefore, precise geometric design is required to ensure that the resonator satisfy the frequency demand and form bright/dark mode simultaneously. This complicates the design process of the silencer for actual applications.
Porous material has been used in silencers to improve sound attenuation properties [24–27]. Composite structures with porous material have been proposed for broadband sound absorption [28, 29]. It is also applied to HR to provide strong dissipation property [30]. With the insertion of porous material, HR can be directly converted into a strong dissipation mode [22] rather than using precise geometry design. In authors’ previous work, it is experimentally demonstrated that the use of porous material allows continuous adjustment of the peak absorption frequency while avoiding the repetitive trial and error of geometric parameters during the design process [31, 32].
In addition, in most of the acoustic metamaterials proposed so far, the working frequency is fixed after manufacturing and cannot be adjusted agilely. To overcome this limitation, the tuneability of metamaterial structures has attracted much attention and various designs have been proposed [33–38]. However, these tunable structures either need to change the crosssectional shape of the duct or require rigid back walls for reflection, thus affecting the ventilation status. Moreover, the hand tuning structures cannot provide precise position control, so continuous frequency tuning is difficult to realise. Some research has also attempted to expand the bandwidth by combining multiple HRs groups [13, 22, 39]. However, tuning the operating frequency by changing the HR silencer structure while avoiding affecting the crosssectional area is seldomly investigated. The introduction of potentials and the realisation method of conducting active control to HRbased silencer have also not been considered.
In this study, a duct silencer with electrically tunable structure is proposed and experimentally verified. In contrast to the silencers presented previously, this design allows the working frequency to be changed continuously and precisely. At the same time, it is designed as a sidebranch of a ventilation duct so as not to interfere with the ventilation conditions. A tuning range of 250 Hz has been numerically and experimentally demonstrated. An analytical model based on temporal coupled mode theory (TCMT) [40–42] is derived to explain the working mechanism and predict the sound absorption performance.
2 Structural design and theoretical model
The schematic diagram of the proposed silencer is illustrated in Figure 1a. HRs that forming absorbers and reflectors are placed symmetrically to enhance the absorption effect. The absorbers are placed at the side that is the closest to the incident wave and filled with porous material. For a regular HR structure, the resonance frequency f_{R} can be calculated through the formula [43]:
where c is the sound velocity, A is the crosssectional area of the neck, is the effective length of the neck with correction [44] and V is the volume of the cavity. According to the equation, the increase of cavity volume will shift the resonance frequency to lower range. Based on this mechanism, a reflector is equipped with slidable back plate that installed inside the cavity. The back plate is installed with a push rod, which is connected to a step motor for precise position control. The cavity volume can thus be adjusted when the back plate is sliding along the inner cavity wall. Since the linear travel per step of the step motor is 0.03175 mm, the location of back plate can be tuned elaborately. It allows the achievement of continuous adjustment of the reflector HR resonance frequency. The widths and heights of the absorbers and reflectors are defined to be equal, denoted as w_{c} and h_{c}, respectively. The length of the absorber is d. The maximum and minimum length of the reflector cavity are labelled as ∆d_{1} and ∆d_{2}, respectively. The tuning range is from 35 mm to 85 mm. The height, width and length of the neck are denoted as h_{n}, w_{n} and d_{n}. Distance D between the absorber and reflector should be a quarter wavelength of the peak absorption frequency, so the radiated wave from reflector will be able to cancel the incident wave [32]. In this study, the resonant frequency of the absorber is designed as 800 Hz, based on which, the corresponding quarter wavelength is D = 107 mm. The crosssection of the duct is an 80 × 80 mm square. The detailed geometrical parameters of the structure are given in Table 1.
Figure 1 (a) Schematic diagram of the proposed silencer. (b) Physical model diagram of the silencer structure. 
Geometrical parameters of the silencer (unit: mm).
To explain the working mechanism of the silencer, analytical model is derived based on the TCMT. The equivalent physical model of the silencer installed to a ventilation duct is sketched in Figure 1b. The fundamental resonance modes of the absorber and reflector are defined as φ_{A} and φ_{R}, respectively. In this case, single mode is assumed for each of the resonator. The two resonance modes are coupled with the duct. The incident wave pressure to absorber is represented by . Since only wave incidents from the left side is considered, it is defined that . According to this model, the transmission coefficient and reflection coefficient can be obtained by equations and , respectively. The following derivation aims to obtain the expression of these coefficients.
Based on the TCMT, the coupling equations of the absorber mode can be given as [45]:
For the reflector:
where s_{i+} represents the incident wave amplitudes to the resonance modes, and s_{i−} represents the radiated wave amplitudes from the modes to the duct (i = 1, 2, 3, 4). ω_{A} and ω_{R} are the resonance frequencies of the absorber and reflector resonance modes, and ω is the incident wave frequency. The resonance modes in HRs will lead to energy dissipation, which caused by thermal viscosity and friction. The dissipation rate is called intrinsic loss, denoted by and . Since the absorber and reflector possess different properties, the dissipation rate and intrinsic loss vary from each other. To differentiate, the subscript A and R are used to represent absorber and reflector mode, respectively. In addition, energy will be radiated to the coupled duct during resonance, the radiation rate is called leakage rate. The leakage rates are denoted by and , respectively. During the radiation, because the HR resonance is in symmetrical shape, the leakage rate towards left and right side of the duct are equal, so it satisfy the condition: [45]. Since the distance between the reflector and absorber modes are in a quarter wavelength, when the wave is transmitted to the reflector, a phase change will occur. Therefore, according to the transfer matrix function method of the acoustic waveguide [14], the amplitudes satisfy the following equations:
where , λ is the corresponding wavelength of the resonant frequency of the absorber. According to the TCMT, equation (1) can be converted to:
By substituting equation (9) into equation (3), the expression of s_{2−} can be obtained as:
Similarly, equation (4) can be derived as:
and as mentioned earlier, s_{4+} = 0. Substitute the above equation into equations (5) and (6):
where . Substitute equations (7), (8), (12) and (13) into equation (10), gives the transmission coefficient t:
where . The reflection coefficient r can be obtained as:
The absorption can then be calculated by:
When the incident wave frequency is the same or very close to the reflector/absorber resonance frequency, and with low reflector intrinsic loss, the transmission and reflection coefficient will be near zero. The analytical model reveals that by tuning operation frequency of the reflector HR, the absorption peak frequency will be changed accordingly.
3 Simulation and experiment setting
To investigate the sound absorption performance of the proposed silencer, numerical simulation and experimental tests are conducted. The simulation model is constructed in COMSOL Multiphysics. Figure 2 shows the schematic figure of the simulation model. The air domain of the silencer coupled duct is established in the model. Perfectly matched layer (PML) is defined at both sides of the model to avoid influence of the reflected waves. Since dissipation of wave energy is mainly caused by thermal viscosity property, “thermoviscous acoustic” field is applied to the neck and reflector cavity domains of the model, highlighted as blue in the figure. Plane wave is generated from the left side of the model by background pressure field setting, and acoustic pressure signals are detected from the two measure surfaces.
Figure 2 Schematic diagram of simulation model. 
To ensure the accuracy of simulation, mesh size is defined to be smaller than 1/6 of the smallest wavelength. Tetrahedral mesh is selected, and the model contains 28,406 domain elements. To simulate the effect of porous materials, “poroacoustic” domain condition is added to the absorber cavities, as labelled in the figure. The JohnsonChampouxAllard (JCA) model is selected for porous material simulation [46]. In JCA model, five material parameters are required for calculation: porosity, viscous characteristic length, thermal characteristic length, flow resistivity and tortuosity. Melamine foam is selected because of the sound dissipation capability. The detailed parameters of melamine are listed in Table 2. The length of reflector cavity is changed from 85 mm to 35 mm.
Experiments were conducted to investigate and verify the silencer’s acoustic property. Prototype is manufactured through 3D printing and manually assembled. The prototype and experiment setting are presented in Figure 3a. In the sample, the push rods of back plates in the reflector cavities are connected to a step motor. A 10V constant voltage power source is used to drive the step motor. The position of the motor rod can be adjusted through the step motor controller. Figures 3b and 3c shows the rod positions when the cavity length is changed from the maximum ∆d_{1} = 85 mm to the minimum ∆d_{2} = 35 mm. Reduction of reflector cavity volume leads to increase of reflector resonance frequency. Industrial Vaseline is slightly applied to the inner wall of reflectors cavities to provide lubrication. If control system and mechanism is incorporated, the cavity volume can be actively controlled automatically, providing potential for future active control system.
Figure 3 (a) A photograph of the experiment setup. Piston position for cavity volume changed from (b) maximum to (c) minimum. 
The sample is tested by a square impedance tube, with 100 mm inner side length. The standard fourmicrophone method based on transfer matrix method is employed [47, 48]. The cut off frequency of the impedance tube is about 1600 Hz, and the type of microphones is PCB 130F22 0.25inch. The measurement duration of each test is 20 s. Sampling frequency is set as 4096 S/s with Hanning window.
In addition, air flow speed tests were carried out to investigate the ventilation status of a duct structure with/without the installation of the proposed structure. The effective length of the proposed sample tube is 690 mm. For comparison, a regular PVC tube with square crosssection (80 × 80 mm) with 690 mm length is used as the control group sample, as shown in Figure 4a. As depicted in Figures 4b and 4c, two 500 mm – length PVC tubes are respectively connected to both sides of the sample tube and the control group PVC tube to construct a ventilation structure. A duct fan (airflow volume: 310 m^{3}/h) is installed to generate air flow, and air flow speed is detected at the other end with an anemometer (Aicevoos ASH6, accuracy 2% ± 0.5 m/s, measurement range 0.4–30 m/s).
Figure 4 (a) Photo of sample tube and regular PVC tube. Air flow speed test setting for (b) regular PVC tube and (c) sample tube. 
4 Results and discussion
The analytical, simulated, and measured results are compared in Figure 5.
Figure 5 The sound absorption coefficient of analytical, numerical, and experimental results of silencer’s reflector defined in different dimensions: (a) 85 mm, (b) 60 mm and (c) 35 mm. The right inset figures reveal the sound pressure field (colour map) and air velocity (arrow) of the model at peak absorption frequencies. 
For analytical model, three different cases of reflector parameters are defined. In the analytical model, the resonance frequency of the absorber HR is fixed at 800 Hz. The resonance frequency of the reflector is defined as 650 Hz, 800 Hz and 920 Hz, respectively. As shown in the figure, the peak absorption frequencies are Case A1: 658 Hz, Case A2: 800 Hz and Case A3: 906.5 Hz. In Case 1 & 3, the peak absorption frequencies deviate from the defined reflector frequencies because the absorption mainly relies on the absorber’s resonance frequency. The absorber provides more efficient sound absorption near its operation frequency. In Case 2, the resonance frequencies of absorber and reflector are the same and the distance D = 107 mm is exactly the quarter wavelength of 800 Hz sound wave (c = 343 m/s), therefore perfect absorption is achieved. In the other cases, since the reflector frequencies are shifted, the perfect absorption cannot be achieved anymore. But the reflected wave frequency is near the absorber resonance frequency, high sound absorption efficiency can still be achieved. The tuning range is selected to ensure the absorption coefficient to be >95%.
In simulated results, when the back cavity of the reflector is set as 35 mm, 60 mm and 85 mm, the peak absorption frequencies are 920 Hz, 800 Hz and 650 Hz, respectively. The results are consistent with the analytical model. The measured peak absorption frequencies of the three cases are about 960 Hz, 860 Hz and 710 Hz, indicating that through the design silencer mechanism, a 250 Hz absorption frequency tuning range can be achieved.
The discrepancies between the experimental and simulation results are mainly caused by the dimensional deviation in assembling. The 3D printed epoxy material will shrink slightly after finished and cooled down, leads to cavity volume reduction and increase of resonance frequency. In addition, the extra damping provided by the inner wall roughness also led to the expansion of absorption bandwidth [49], and higher sound absorption coefficient over the whole frequency spectrum. Absorption coefficient in the simulation and measured results of 35 mm and 85 mm cases are both higher than the prediction of analytical model. This phenomenon might result from the participation of higher order resonance that ignored in the analytical model. Only the main resonance is involved in the analytical model. But in simulation and actual situation, the higher order resonances in silencer will also affect the sound absorption through varies vibration modes of air in cavities, and therefore the sound absorption coefficient is enhanced and bandwidth broadened [50].
The air pressure filed and air velocity of the duct system when incident wave frequency is at the peak absorption frequency are shown in Figure 5. According to the figure, at peak absorption frequencies, air pressure accumulates at the HR cavity domain, results in fierce friction between air and the neck area of the HR when air flows in and out from the cavity. In 60 mm case, the perfect absorption is achieved, and no sound wave is transmitted to the right end of the duct, so blue arrows indicating air velocity are not revealed in the figure. In the other two cases, a small number of arrows appeared at the right end of the duct since certain sound waves are transmitted through, even though the ratio is low. Figures reveal that the silencer’s working mechanism is the same as predicted in the analytical model. Simulation model can be used to conduct fast design of silencer for various frequency demands, and thus the application potential is enhanced.
Figure 6 shows the absorption coefficient curve of the silencer when the reflector cavity length is tuned from 85 mm to 35 mm. The peak absorption frequency shift gradually along with the change of cavity volume, throughout the operation frequency range. The coefficient is reduced when the reflector resonance frequency is tuned further away from the absorber (when cavity length is <35 mm or >85 mm), as also predicted in the analytical model. Thus, the tuning range of the cavity length is designed to ensure the sound absorption coefficient >95% at the peak frequency.
Figure 6 The measured sound absorption coefficient curves when reflector cavity length is adjusted. 
Additionally, the air flow speed for sample tube and regular PVC tube are about 9.77 m/s and 9.72 m/s, respectively. The testing processes are revealed in Supplementary video 1. Therefore, it is proved that the installation of the proposed silencer will not influence the ventilation status.
By using the step motor as the actuator for the adjustable structure, simulation and measured results verify that the silencer can be rapidly modulated to handle different noise frequencies. Through the high accuracy position controllability of the step motor, the absorption frequency can be tuned elaborately to achieve near perfect absorption throughout the sound absorption frequency spectrum. The design also provides a platform that allows the silencer to integrate with active control system and realise realtime control based on incident wave frequency detection in the future. Since the simultaneous application of different pairs of Helmholtz resonators can enlarge the absorption bandwidth [32], the proposed structure applied with multiple tunable structure is also expected to achieve broadband sound absorption. In addition, experimental results indicate that the structure barely affect the ventilation condition in the duct. Such advantage gives the silencer good potential in application.
5 Conclusion
To conclude, a silencer designed for ventilation duct is proposed and verified in this work. Adjustable HR cavities are applied in the silencer. Experimental results demonstrate that it can be easily tuned to achieve a 250 Hz – wide sound absorption frequency bandwidth, ranges from about 670–710 Hz. The silencer is composed of weaklycoupled composite HRs. Analytical model is developed to explain the working mechanism. Simulation and experimental results prove the effectiveness of the analytical model. Insertion of porous material and application of simulation model allows fast design of the silencer for different sound absorption frequency ranges. Moreover, since the silencer is installed as sidebranch of the duct, ventilation condition is not hindered. It demonstrates feasibility in architectural acoustic application, such as air conditioning duct and ventilation window noise reduction. In addition, by utilising the step motor as actuator for structure control, the peak absorption frequency can be adjusted rapidly and continuously. In future research, if incorporated with a realtime sound signal analysis and feedback system, the operation frequency can then be adjusted accordingly. Moreover, in this study, the installation of the silencer has been proved to have basically no influence on ventilation status. However, for practical scale silencer deployment with larger opening, the corresponding effect on ventilation still requires further investigation. At last, the proposed silencer provides an effective solution for low frequency noise control in ventilation duct without interference of ventilation condition.
Funding
This research was supported by the Shenzhen Polytechnic University Research Fund (6024310008K) and Postdoctoral Laterstage Foundation Project of Shenzhen Polytechnic University (6023271040K), and the Department of Science and Technology of Guangdong Province (GDST20SC03) and the Project of Hetao ShenzhenHong Kong Science and Technology Innovation Cooperation Zone (HZQBKCZYB2020083). This research was also supported by National Natural Science Foundation of China (No. 12304348).
Conflicts of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability statement
Data are available on request from the authors.
Supplementary material
Video: Air flow speed test Access here
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Cite this article as: Gao C. Hu C. Hou B. Wu X. & Wen W. 2024. Tunable silencer for rectangular ventilation duct based on composite Helmholtz resonators Acta Acustica, 8, 22.
All Tables
All Figures
Figure 1 (a) Schematic diagram of the proposed silencer. (b) Physical model diagram of the silencer structure. 

In the text 
Figure 2 Schematic diagram of simulation model. 

In the text 
Figure 3 (a) A photograph of the experiment setup. Piston position for cavity volume changed from (b) maximum to (c) minimum. 

In the text 
Figure 4 (a) Photo of sample tube and regular PVC tube. Air flow speed test setting for (b) regular PVC tube and (c) sample tube. 

In the text 
Figure 5 The sound absorption coefficient of analytical, numerical, and experimental results of silencer’s reflector defined in different dimensions: (a) 85 mm, (b) 60 mm and (c) 35 mm. The right inset figures reveal the sound pressure field (colour map) and air velocity (arrow) of the model at peak absorption frequencies. 

In the text 
Figure 6 The measured sound absorption coefficient curves when reflector cavity length is adjusted. 

In the text 
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