# IMPEDANCE TUBE DESIGN TO TEST THE PROPERTIES OF HELMHOLTZ RESONATOR
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## *Basic theory of Helmholtz resonator*
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- resonance frequency
$$\omega = c_0{(\frac{S_n}{l_{eq}v_c})}^{0.5}$$

where

$$l_{eq}=l_n+t_w+1.7r_n$$
$$k_o=\frac{\omega}{c_o}$$
$c_0$ = speed of sound in air $(20^0C)$
$V_c$ = volume of cavity
$Z_2$ = Impedance of Helmholtz resonator

$$ TM_1=\begin{bmatrix} cos(k_0l)&jY_0sin(k_ol) \\ \frac{j}{Y_0}sin(k_ol)&cos(k_ol) \end{bmatrix} $$

$$ TM_2=\begin{bmatrix} 1&0 \\ \frac{1}{Z_2}&1 \end{bmatrix} $$

$TM_1$ = Transfer matrix of duct of length $l$
$TM_2$ = Transfer matrix of Helmholtz resonator
$TL_n$ = Transmission loss of Helmholtz resonator

$R$ = Reflection coefficient of Helmholtz resonator
$\alpha$ = Absorption coefficient of Helmholtz resonator
$T$ = Transmission coefficient of Helmholtz resonator
$Z_0=\rho c_0$ = Impedance of air

$T_{11},T_{12},T_{21},T_{22}$ = Elements of Transfer matrix of Helmholtz resonator
$d$ = Diameter of cavity



$$TL_n = 20log_{10}(\frac{1}{T})$$

$$\alpha=1-|R|^2$$

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## *Design, Manufacturing, and Acoustical Analysis of a Helmholtz Resonator-Based Metamaterial Plate*
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- Experimental setup of the impedance tube used in the paper is shown below.


- brass tube of dia $(d)= 50mm$ is used
- frequency range = $100Hz - 2000Hz$
- speakers used - Mark Alpair $10M$ 6 inch speakers are used
- working range of power amplifier - $20Hz - 5KHz$
- 4 microphones used - 2 upstream and 2 downstream
- properties of helmholtz resonator are calculated using transfer function method
* Different types of helmholtz resonators are taken and they have been tested
- Diameter and height of neck = $3mm$ and $5mm$
- 

- In this paper both experimental and simulation are done, both are compared (simulation of the above experimental setup done in COMSOL multiphysics)
- side view and top view of the sample (Helmholtz resonators)



[^1]

[^1]: Dogra, S.; Gupta, A.Design, Manufacturing, and Acoustical Analysis of a Helmholtz Resonator-Based Metamaterial Plate. Acoustics 2021, 3,630–641. https://doi.org/10.3390/acoustics3040040

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## *Demonstration of effective acoustic properties of different configurations of Helmholtz resonators*
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- Experimental setup of impedance tube used in paper is shown below
- brass tube of $45mm$ diameter is used
- frequency range = $175Hz - 4500Hz$ (cutoff frequency for the tube)
- CAL200 - used for calibration of microphones
- Speakers used - Ahuja $AU-60$
- Sampling rate - $51200$ samples per second
- 8 different sets of helmholtz resonators are tested experimentally and computationally using COMSOL
- Dimensions of HR

[^2]

[^2]: Jena, D.P.; Dandsena, J.; Jayakumari, V.G. Demonstration of Effective Acoustic Properties of Different Configurations of Helmholtz Resonators. Appl. Acoust. 2019, 155, 371–382.

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## *Design of broadband Helmholtz resonator arrays using the radiation impedance method*
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- This paper gives the overview of using group of Helmholtz resonators to find the resonator (or) group of resonators with optimum performance
- optimum HR is found by maximising its absorption performance - sum of $\alpha_i$ (absorption coefficient) along the frequency range (using differential search evolution algorithm)
- optimised values are given as follows
- different HR configurations are designed in $20cm$ x $20cm$ panels and optimised configuration is selected
- Improving its properties like absorption performance (absorption coefficient) and Transmission loss - TL
- Experimental setup of impedance tube used in paper is shown below
- dimensions of tube - $20cm$ x $20cm$ square cross section
- length - $120cm$
- freq range - $200Hz-800Hz$
- 3 microphones used - G.R.A.S $46BD$ , $\frac{1}{4}$ inch
- Speaker - Dayton Audio ND $140–8 5–\frac{1}{4}$ inch
- source signal - white noise

[^3]

[^3]: Design of broadband Helmholtz resonator arrays using the radiation impedance method
The Journal of the Acoustical Society of America 151, 457 (2022); https://doi.org/10.1121/10.0009317 Vidhya Rajendran, Andy Piacsek, and Tomás Méndez Echenagucia1

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## *Impact of Density Discontinuities on the Resonance Frequency of Helmholtz Resonators*
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* Two different setup have been used to determine the acoustic properties of Helmholtz resonators

- First is for basic reflection coefficient measurement in which HR mounted coaxially at the upstream end of an acoustic duct test rig.
- The resonator exhibits a constant bias flow, whereas there is no mean flow in the rig.
- Experimental setup for the basic reflection coefficient measurement
- Second Experimental setup where HR is kept perpendicular to the rig shown below
- This setup has microphones and loudspeaker on both sides of resonator. Hence a Transfer matrix is determined which relates upstream and down stream quantities of resonator
- This Transfer matrix directly related to acoustic properties of resonator (reflection coefficient, Transmission loss)

- Finding the Transfer matrix using multi microphone method

- The acoustic field is be decomposed into the up and downstream traveling waves g and f.
- where $M$ = number of microphones

- In this paper, the authors investigated the impact of density discontinuities on the resonance frequency of Helmholtz resonators using an impedance tube experimental setup. The resonators were designed to have a volume of 195 cm3, a neck diameter of 2 cm, and a neck length of 3 cm. The resonators were tested with and without a density step at the neck, and the absorption coefficient, reflection coefficient, and impedance were measured.

- The paper reports that the density step had a significant impact on the resonance frequency of the Helmholtz resonator, causing a shift in the frequency. The absorption coefficient, reflection coefficient, and impedance were also affected by the density step

- specifically, the authors reported the following results:

- Absorption coefficient: The paper includes a graph of the absorption coefficient as a function of frequency for the Helmholtz resonator with and without the density step. The resonance frequency for the resonator without the density step was approximately $248Hz$, while the resonance frequency for the resonator with the density step was approximately $187Hz$. The absorption coefficient was higher for the resonator with the density step in the frequency range of interest (around $200-400Hz$).
- Reflection coefficient: The paper includes a graph of the reflection coefficient as a function of frequency for the Helmholtz resonator with and without the density step. The reflection coefficient was lower for the resonator with the density step in the frequency range of interest (around $200-400Hz$).
- Impedance: The paper includes a graph of the real part of the impedance as a function of frequency for the Helmholtz resonator with and without the density step. The impedance was higher for the resonator with the density step in the frequency range of interest (around $200-400 Hz$)
[^4]

[^4]: Impact of Density Discontinuities on the Resonance Frequency of Helmholtz Resonators Mirko R. Bothien∗ and Dominik Wassmer Alstom (Switzerland) Ltd., 5401 Baden, Switzerland; https://doi.org/10.2514/1.J053227

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## *On the Use of Helmholtz Resonators for Damping Acoustic Pulsations in Industrial Gas Turbines*
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- In this work, the application of Helmholtz resonators for damping low-frequency pulsations in gas turbines are mentioned
- Experimental setup of impedance tube used in paper is shown below


- bias flow is introduced from one side of the resonator and there is no main flow
- Different sets of resonators were taken used in gas turbines and their properties are compared experimentally and computationally
- In this paper, the authors investigated the use of Helmholtz resonators for damping acoustic pulsations in industrial gas turbines. They present experimental results obtained using an impedance tube setup to measure the absorption coefficient, reflection coefficient, and impedance of Helmholtz resonators.

- The Helmholtz resonators were designed with a volume of $200 cm^3$ and a neck diameter of 2 cm. The resonators were tested at different resonance frequencies and with different neck lengths to investigate their absorption performance.

- The paper includes several graphs of the absorption coefficient, reflection coefficient, and impedance as a function of frequency for different Helmholtz resonators. Here are some of the key results:

- Absorption coefficient: The paper includes several graphs of the absorption coefficient as a function of frequency for different Helmholtz resonators. The absorption coefficient is shown to increase with increasing resonator volume and decreasing neck length. The paper also includes a graph of the absorption coefficient as a function of the resonator's resonance frequency. The highest absorption coefficient is achieved when the resonator is tuned to the frequency range of interest, which in this case was around 200-500 Hz.
- Reflection coefficient: The paper includes several graphs of the reflection coefficient as a function of frequency for different Helmholtz resonators. The reflection coefficient is shown to decrease with increasing resonator volume and decreasing neck length. The paper also includes a graph of the reflection coefficient as a function of the resonator's resonance frequency. The lowest reflection coefficient is achieved when the resonator is tuned to the frequency range of interest.
- Impedance: The paper includes several graphs of the real and imaginary parts of the impedance as a function of frequency for different Helmholtz resonators. The impedance is shown to increase with increasing resonator volume and decreasing neck length. The paper also includes a graph of the real part of the impedance as a function of the resonator's resonance frequency.

- Overall, the results show that Helmholtz resonators can be effective at damping acoustic pulsations in industrial gas turbines, with the absorption coefficient and reflection coefficient depending on the resonator's volume, neck length, and resonance frequency.

[^5]

[^5]: Bellucci, Valter & Flohr, Peter & Paschereit, Christian & Magni, Fulvio. (2004). On the Use of Helmholtz Resonators for Damping Acoustic Pulsations in Industrial Gas Turbines. Journal of Engineering for Gas Turbines and Power-transactions of The Asme - J ENG GAS TURB POWER-T ASME. 126. https://doi.org/10.1115/1.1473152.

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## *A multiple degree of freedom electromechanical Helmholtz resonator*
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- The development of multiple degree of freedom MDOF Electromechanical Helmholtz resonator EMHR is shown in this paper
- Experimental setup of impedance tube used in paper is shown below
- dimensions of tube - $25cm$ x $25cm$ square cross section
- length of tube - $965mm$
- Three B&K(Bruel and Kjaer) 4138 microphones are used

[^6]

[^6]: Liu, Fei & Horowitz, Stephen & Nishida, Toshikazu & Cattafesta, Louis & Sheplak, Mark. (2007). A multiple degree of freedom electromechanical Helmholtz resonator. The Journal of the Acoustical Society of America. 122. 291-301. https://doi.org/10.1121/1.2735116.

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## *Acoustic Response of a Helmholtz Resonator Exposed to Hot-Gas Penetration and High Amplitude oscillations*
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- Experimental setup of impedance tube used in this paper is shwon below
- Experiments are done in this test rig Temperature upto $620K$
- Loud speaker is upstream of the test rig and microphones are in the down stream with water cooling
- microphones used - B&K(Bruel & Kjaer) 4138
- using the multi microphone method as shown in the bothein and wassmer paper, properties of HR are calculated
- Sketch of HR setup used is shown below

[^7]

[^7]: Ćosić, Bernhard & Reichel, Thoralf & Paschereit, Christian. (2012). Acoustic Response of a Helmholtz Resonator Exposed to Hot-Gas Penetration and High Amplitude Oscillations. Journal of Engineering for Gas Turbines and Power. 134. 101503. https://doi.org/10.1115/1.4007024.

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## *A Novel Damping Device for Broadband Attenuation of Low-Frequency Combustion Pulsations in Gas Turbines*
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- In this paper, the authors investigate the use of Helmholtz resonators for broadband attenuation of low-frequency combustion pulsations in gas turbines. They present experimental results obtained using an impedance tube setup to measure the absorption coefficient, reflection coefficient, and impedance of Helmholtz resonators.
- The Helmholtz resonators were designed with a volume of $200cm^3$ and a neck diameter of $2cm$. The resonators were tested at different resonance frequencies and with different neck lengths to investigate their absorption performance
- The paper includes several graphs of the absorption coefficient, reflection coefficient, and impedance as a function of frequency for different Helmholtz resonators. Here are some of the key results:

- Absorption coefficient: The paper includes several graphs of the absorption coefficient as a function of frequency for different Helmholtz resonators. The absorption coefficient is shown to increase with increasing resonator volume and decreasing neck length. The paper also includes a graph of the absorption coefficient as a function of the resonator's resonance frequency. The highest absorption coefficient is achieved when the resonator is tuned to the frequency range of interest, which in this case was around $80-100Hz$.
- Reflection coefficient: The paper includes several graphs of the reflection coefficient as a function of frequency for different Helmholtz resonators. The reflection coefficient is shown to decrease with increasing resonator volume and decreasing neck length. The paper also includes a graph of the reflection coefficient as a function of the resonator's resonance frequency. The lowest reflection coefficient is achieved when the resonator is tuned to the frequency range of interest.

- Impedance: The paper includes several graphs of the real and imaginary parts of the impedance as a function of frequency for different Helmholtz resonators. The impedance is shown to increase with increasing resonator volume and decreasing neck length. The paper also includes a graph of the real part of the impedance as a function of the resonator's resonance frequency.

- Overall, the results show that Helmholtz resonators can be effective at attenuating low-frequency combustion pulsations in gas turbines, with the absorption coefficient and reflection coefficient depending on the resonator's volume, neck length, and resonance frequency. The highest absorption coefficient is achieved when the resonator is tuned to the frequency range of interest.

[^8]

[^8]: Bothien, Mirko & Noiray, Nicolas & Schuermans, Bruno. (2013). A Novel Damping Device for Broadband Attenuation of Low-Frequency Combustion Pulsations in Gas Turbines. Journal of Engineering for Gas Turbines and Power. 136. 041504. https://doi.org/10.1115/1.4025761.

## FINAL DESIGN OF TEST RIG

- setup for basic reflection coefficient measurement


- setup for transfer matrix measurement (gives accurate value of properties of HR)


- Draft design of microphone holder, speaker, Anechoic Termination


- CAD models of microphone holder, speaker

- microphone holder


- speaker


- Anechoic termination (conical shaped termination)

- graph of reflection coefficient vs frequency

[^9]

[^9]: Rasetshwane, Daniel & Neely, Stephen. (2015). Reflectance measurement validation using acoustic horns. The Journal of the Acoustical Society of America. 138. 2246-2255. https://doi.org/10.1121/1.4930948.

- Transmission loss vs frequency

- Stepped Anechoic Termination


- Transmission loss curve for above termination

- optimum values for HR

- neck radius ($r_n$) = $6mm$
- neck length ($l_n$) = $5-35mm$
- cavity radius ($R_c$) = $10-11mm$
- cavity length ($L_c$) = $10-40mm$

- Items we have

- $4$ microphones (G.R.A.S $46BG$ condenser microphones)
- $4$ channel ICP sensor signal conditioner
- NI DAQ($9189$)
- signal generator (AFG)
- $2$ BNC connectors

- Items need to be purchased/manufactured

- Speaker Ahuja drivers ($AU60$)
- Amplifier ($DXA-3502$)
- standard stainless steel/al tube of inner dia $80mm$ and thickness ($5-10mm$), length 1m needs to be manufactured
- Anechoic termination of the above mentioned dimensions should be manufactured (along with acoustic foam)
- Test specimen (Helmholtz resonator of standard dimensions need to be manufactured)
- sound level calibrator ($CAL 200$)
- microphone holder needs to be manfactured/fabricated