Tuning a semi-active Helmholtz resonator

S. Singha, C. Q. Howardb, C. H. Hansenc

School of Mechanical EngineeringThe University of Adelaide

Adelaide, S.A.Australia, 5005

ABSTRACTAdaptive Helmholtz resonators are used to reduce tonal noise propagating as planewaves in ducts. Optimal tuning of the resonator has previously been achieved by usinga pressure sensor located in the duct downstream of the resonator. The work describedhere is concerned with the development of a cost function that could be used by a con-troller to optimally tune the Helmholtz resonator without any in-duct pressure sensor.The cost function that was developed is based on the phase difference between thepressure at the top of the closed end of the cavity of the Helmholtz resonator and thepressure at the neck wall, close to the neck duct interface. Damping in the system istaken into account using a correction factor applied to the cost function.

1 INTRODUCTION

1.1 Motivation

The problem of low-frequency tonal noise is inherent in industries using internal com-bustion (IC) engines, compressors, fans, blowers, power transformers, gearboxes etc. Thehumming nature of a tonal noise not only causes annoyance to workers within the industrybut also to the surrounding community. Depending upon the type of application, existingset up and cost constraints, tonal noise transmission can be controlled in many possibleways like by installing reactive silencers, barriers, side branch elements and active noisecontrol devices. The work described here is concerned with the attenuation of tonal noisetransmission in ducts by using side branch resonators.

Passive Helmholtz resonators (HRs) are specifically designed to achieve their optimalperformance at one frequency only, and are only effective over a very narrow frequencyband. Any slight change in the frequency, change in temperature which changes the speedof sound and hence the wavelength of the noise will decrease the effectiveness of theresonator. Resonators incorporating the provision for altering their geometrical parame-ters in real-time in order to adapt themselves to the environmental or operating conditionchanges offer an obvious solution. Such an adaptive system is referred to as asemi-activesystem in which a change in the physical parameters of the passive element is causedby an active control system. Because of the benefits of semi-active systems over any ex-clusive active and passive systems [1, 2], semi-active systems are gaining popularity inindustry. Some of the industrial applications of the semi-active systems are highlighted inthe next section.

aEmail address:sarabjeet.singh@adelaide.edu.aubEmail address:carl.howard@adelaide.edu.aucEmail address:colin.hansen@adelaide.edu.au

1.2 Previous Work

There have been many previous studies that report on the successful implementation ofadaptive Helmholtz resonators for attenuating the noise transmission in ducts. Laman-cusa [3] demonstrated the use of a volume variable Helmholtz resonator for attenuatingthe firing frequency noise of an IC engine. Matsuhisaet al. [4, 5] conducted experimentswith two types of resonators, first, in which the cavity volume was varied and second, inwhich the neck cross-sectional area was varied. Tuning of the resonator was facilitated us-ing an algorithm that was based on the phase difference between the pressure in the cavityof the resonator and the pressure in the duct. Bedoutet al. [6] also developed an adaptiveHelmholtz resonator which was tuned by using the signal from a microphone located inthe duct downstream of the resonator. Radcliffet al. [7] reported on the development ofa semi-active Helmholtz resonator for which tuning was achieved by locating two pres-sure sensors, first, in the cavity of the resonator and second, in the duct downstream ofthe resonator. Recent development of an adaptive Helmholtz resonator was reported byEst̀eve et al. [8] for reducing broadband noise transmission into a rocket payload fair-ing. Many more developments on adaptive Helmholtz resonators can be found in refer-ences [9, 10, 11, 12, 13, 14, 15, 16], which demonstrate their use in industrial applicationsfor attenuating the intake and exhaust noise of IC engines, and exhaust noise produced bygas turbines and combustors.

Previous work has shown that the optimal tuning of HRs has been achieved by usingthe information from one or more pressure sensors located in the duct downstream ofthe resonator. However, in many cases, especially in industrial exhaust stacks where thestacks serve as a passage for the exhaust gases to be driven out to the environment, itis neither desirable nor practical to mount microphones in a duct. This is because of thepossibility that signals so obtained are often contaminated with unrelated noise or affecteddue to physical problems with the microphones such as [17]:

• accumulation of exhaust particulates- the response of the pressure sensors can besignificantly affected by the exhaust particulates which may accumulate on them,

• heating- the possibility of high temperature of the exhaust gases can also directlyaffect the operation of the sensors,

• mean flow- presence of mean flow generates turbulence and adds to spurious noise,and

• inappropriate location- variations in the disturbing frequency may result in thelocation of the pressure sensor being close to a pressure node and unable to properlyobtain a measure of the pressure. However, some researchers have overcome thisproblem by using multiple microphones along the duct [17, 18].

In order to avoid the above mentioned potential risks and also, for reasons of convenienceand practicality, it is highly desirable to have a self-contained adaptive Helmholtz res-onator that does not need any external measures of quantities outside the resonator. Thisinvolves locating the pressure sensors in the resonator’s cavity and/or neck. The output ofthese sensors can be fed to a controller to optimally adjust the physical dimensions of theHR, which will result in the minimisation of the sound propagating downstream of the

HR.When a HR is modelled as a stand-alone device, its resonance frequency can be ac-

curately predicted by using the classical formula developed by Helmholtz when the res-onator is small compared to a wavelength of sound. For the cases where the resonator maynot be small compared to a wavelength, others have developed equations for predictingthe resonance frequency of a cylindrical HR [19, 20]. However, when a HR is mountedonto a duct, a coupled system is created whose resonance frequency is different to thatfor the stand-alone HR. More importantly, the cost function for tuning the HR which hasbeen developed here is not related to the resonance frequency of either of the two systems.

Another important issue is the end correction factor used to estimate the effectivelength of the resonator neck. Even though only plane waves are propagating down theduct, the sound field in the duct adjacent to the neck opening is not planar due to the nearfield generated by the neck opening. Thus the end correction factor for the neck at thisinterface is not very well understood. Inaccuracies in estimating the neck-duct interfaceend correction factor lead to inaccuracies in predicting the resonance frequency of ductmounted HR.

2 EXPERIMENTS

A variable volume cylindrical HR was mounted onto a 3 m long circular duct at 0.5m from the source end. At the source end, a loudspeaker backed by an air tight cavitywas fixed to the duct. A schematic of a duct mounted HR is shown in figure1. Twomicrophones, ’mic. 1’ and ’mic. 2’, were flush mounted in the duct downstream of theHR in order to estimate the net acoustic power transmission down the inside of the duct.For the purpose of designing a self-contained adaptive HR without any transducer remotefrom the resonator, one microphone, microphone ’A’, was located at the top of the closedend of the cavity and the other microphone, microphone ’B’, was located at the neck wallclose to the neck-duct interface. Another microphone was fixed in the speaker’s backingcavity for the purpose of measuring the volume velocity of the speaker. The response ofall the microphones was normalised by the input volume velocity of the loudspeaker. Theloudspeaker was excited by random noise from 0 to 400 Hz. A multi-channel B&K Pulsesystem was used to measure and record the pressure from all the microphones’ locationsin the system.

The results of experiments are divided into two sections:(2.1) broadband frequencyanalysis, in which the results of net acoustic power transmission, acoustic pressure andtransfer function are plotted as a function of frequency, and(2.2) single frequency analy-sis, in which all of the above stated results are plotted as a function of the resonator’s ge-ometry. The purpose of the first analysis was to illustrate the difference in the frequenciescorresponding to maximum acoustic pressure at microphone ’A’, maximum amplitude ofthe pressure transfer function between microphone ’A’ and microphone ’B’ (microphone’A’ / microphone ’B’), and maximum reduction of the net acoustic power transmission inthe duct downstream of the HR. The purpose of the second analysis was to highlight thedifferences in the net acoustic power transmission in the duct as a result of changing thelength of the cavity.

Figure 1:A schematic of a duct mounted HR showing the locations of the microphones

2.1 Broadband Analysis

The dimensions of the HR are: cavity length = 0.070 m, cavity diameter = 0.131 m, phys-ical neck length = 0.093 m and neck diameter = 0.0525 m. Figure2 shows the plot of netacoustic power transmission in the duct downstream of the HR along with the net acousticpower transmission in the duct without the HR. The net acoustic power transmission wascalculated by using the two-microphone technique developed by Chung and Blaser [21]and extended by Åbom [22]. When the HR was attached to the duct, the net acousticpower transmitted in the duct downstream of the HR was reduced by 18 dB at 226 Hz, asshown by the thick line in figure2.

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Figure 2:Net acoustic power transmission in duct with and without the HR asa function of frequency.

The following two figures highlight the difference in the frequencies corresponding tothe maximum acoustic pressure at the top of the closed end of the cavity (microphone ’A’),maximum amplitude of the pressure transfer function between microphone ’A’ and mi-crophone ’B’ (microphone ’A’ / microphone ’B’), and maximum reduction of net acousticpower in the duct downstream of the HR. The vertical line at 226 Hz in both the plots in-dicate the frequency at which the net acoustic power transmission in the duct downstreamof the HR was minimised.

Figure3 shows the normalised acoustic pressure at microphones ’A’ and ’B’. As seenfrom the figure, the pressure at the top of the closed end of the cavity (microphone ’A’)was maximised at 218 Hz, which is 8 Hz lower than the frequency where the maximumreduction of net acoustic power in the duct occurred. Also, the pressure at the neck wall(microphone ’B’) at 226 Hz was neither a maximum nor a minimum and hence cannot beused to tune the HR.

The pressure transfer function between microphone ’A’ and ’B’ (microphone ’A’ /microphone ’B’) was measured and its amplitude and phase plots are shown in figure4.The maximised response of the amplitude of the pressure transfer function occurs at 246Hz which is again different from 226 Hz, where the net acoustic power in the duct isminimised.

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Figure 3:Acoustic pressure at the top of the closed end of the cavity (micro-phone ’A’) and neck wall in proximity to the neck-duct interface (microphone’B’) as a function of frequency. The vertical line at 226 Hz corresponds to thefrequency of maximum reduction of the acoustic power transmission in theduct.

Figures3 and 4 show that neither the pressure at the top of the closed end of thecavity nor the pressure transfer function between microphone ’A’ and microphone ’B’can be used as cost functions to tune the HR to minimise the acoustic power transmissionin the duct.

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Figure 4:Amplitude and phase plots of the pressure transfer function betweenmicrophone ’A’ and microphone ’B’ as a function of frequency. The verticalline at 226 Hz corresponds to the frequency of maximum reduction of theacoustic power transmission in the duct.

2.2 Single Frequency Analysis

In industrial applications where a centrifugal fan or blower is installed at one end of theexhaust stack in order to drive the exhaust gases out to the environment, noise is oftengenerated at the blade passage frequency (BPF) of a fan, which is determined by thenumber of blades and the fan’s rotation speed. The noise is tonal and hence resonators areoften required to attenuate a single frequency rather than broadband noise.

Experiments were conducted using the same duct-HR apparatus described in section2, where the length of the cavity was altered. The HR used in these experiments resemblesa piston, where the diameter of the cavity is fixed but the length can be varied to changeit’s volume. The loudspeaker was excited by random noise from 0 to 400 Hz and thelength of the cavity was changed from 60 mm to 90 mm in increments of 1 mm. With theaim to attenuate noise at 226 Hz, the experimental results at 226 Hz were extracted fromthe experimental set of 0 to 400 Hz.

Figure5 shows the variation in the net acoustic power in the duct measured down-stream of the HR corresponding to 226 Hz as a function of the cavity length of the HR.As evident from the plot, the optimal length of the cavity required to attenuate the noiseat 226 Hz, is 70 mm.

The acoustic pressure at microphone ’A’ and microphone ’B’ at 226 Hz was plottedas a function of the cavity length of the HR and is shown in figure6. The pressure at thetop of the closed end of the cavity was maximised at the cavity length of 63 mm and was7 mm off the optimal length required to achieve the maximum reduction of net acousticpower at 226 Hz.

Figure 7 shows the amplitude and phase of the pressure transfer function betweenmicrophone ’A’ and microphone ’B’ (microphone ’A’ / microphone ’B’) as a function ofthe cavity length of the HR. The maximum amplitude of the transfer function occurs at a

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Figure 5:Net acoustic power transmission in the duct downstream of the HRcorresponding to 226 Hz, plotted as a function of cavity length.

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Figure 6:Acoustic pressure at the top of the closed of the cavity (microphone’A’) and neck wall close to the neck-duct interface (microphone ’B’) corre-sponding to 226 Hz, as a function of cavity length. The vertical line at 70 mmcorresponds to the cavity length required for the maximum reduction of acous-tic power transmission in the duct at 226 Hz.

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Figure 7:Amplitude and phase of the pressure transfer function between mi-crophone ’A’ and microphone ’B’ corresponding to 226 Hz, as a function ofcavity length. The vertical line at 70 mm corresponds to the cavity length re-quired for the maximum reduction of acoustic power transmission in the ductat 226 Hz.

cavity length of 84 mm which is 14 mm off 70 mm, where the net acoustic power in theduct is minimised.

Figures6 & 7 show that the maximised response of the pressure at the top of the closedend of the cavity and the pressure transfer function between microphone ’A’ and micro-phone ’B’, respectively, do not occur at the same volume of the HR which corresponds tothe minimisation of net acoustic power transmission in the duct.

Hence, these results show that the conventional understanding of trying to achievea resonance condition in the side branch HR, does not result in the minimisation of netacoustic power transmission in the duct. Instead, an alternative is needed to determinewhen the acoustic power transmission in the duct is minimised, and this is described inthe next section.

3 COST FUNCTION

A cost function was developed utilising the measured damping of the system and thevalue of phase difference between microphone ’A’ and microphone ’B’ corresponding tothe frequency for which the maximum reduction of net acoustic power transmission in theduct is achieved as measured by microphones 1 and 2.

Experiments were conducted to measure the net acoustic power transmission in theduct as a function of the volume of the HR. The cavity diameter and neck length through-out the experiments were kept constant and three different neck diameters (0.0405 m,0.0525 m and 0.0675 m) were tested. For each different neck size, the cavity length wasvaried between 0.050 m and 0.170 m corresponding to a volume variation of 6.74×10-4

m3 to 2.29×10-3 m3. For each configuration of the HR, the value of the phase differencebetween microphone ’A’ and microphone ’B’ corresponding to the frequency, for which

the maximum reduction of net acoustic power was achieved, was noted and the dampingratio of the system was calculated. The plot of damping ratio versus phase difference forall the three neck sizes is shown in figure8. It reveals that as the damping of the systemincreases, the phase difference decreases.

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Figure 8: Damping ratio versus phase difference corresponding to the fre-quency where maximum reduction of the acoustic power transmission in theduct occurs, for three different neck sizes.

An empirical curve fitting relation between the phase difference and the damping ratiowas found, using a second order polynomial (with a correlation coefficient of 0.882), andis given by:

phase difference= 0.94(damping ratio)2 − 8.139(damping ratio) + 5.1 (1)

Equation (1) can be referred to as a cost function that can be used by an electroniccontroller to optimally tune the HR for minimising the acoustic power transmission in theduct downstream of the HR. The damping present in the system, which needs to be pre-determined, can be substituted in equation (1), which will then yield the desired value ofthe phase difference. The length of the cavity of the HR will then be varied until the phasedifference between ’microphone A’ and ’microphone B’ reaches the desired/ calculatedvalue. This calculated value of the phase difference will approximately correspond to theoptimal length of the cavity required to minimise the net acoustic power transmission inthe duct.

Figure9 shows the normalised net acoustic power transmission in the duct at 226 Hzas a function of cavity length, for several cost functions. The markers indicate the lengthof the cavity which corresponds to the:

• maximum value of the amplitude of the pressure transfer function between micro-phone ’A’ and microphone ’B’ - (’maximum transfer function’),

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Figure 9:Net acoustic power in the duct downstream the HR correspondingto 226 Hz plotted as a function of cavity length, with markers indicating theacoustic power level for various cost functions.

• maximum value of the acoustic pressure at the top of the closed end of the cavity(microphone ’A’) - (’maximum pressure in cavity’),

• value of the phase difference, calculated by using equation (1), needed to be achievedby varying the cavity length for minimising the net acoustic power transmission inthe duct downstream of the HR - (’new cost function’), and

• minimum acoustic power measured by mounting two microphones in the duct down-stream of the HR - (’actual minimum power’).

Figure9 shows that in order to achieve the maximum reduction of the acoustic powertransmission in the duct, the newly proposed cost function gives the best possible cavitylength when compared to the actual cavity length needed to optimally tune the adaptiveHR.

4 CONCLUSION

A cost function has been presented for minimising the net acoustic power transmissionin a duct to which a Helmholtz resonator is attached, using only pressure measurementsinside the resonator. The cost function is based on the damping present in the system andthe phase difference between the pressure at the top of the closed end of the cavity and thepressure at the neck wall close to the neck-duct interface. It was shown experimentallythat neither the pressure at the top of the closed end of the cavity nor the pressure transferfunction can be used to optimally tune the HR, whereas minimisation of the cost functionproposed here will result in minimisation of the in-duct acoustic power transmission.

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