Tonal noise attenuation in ducts by optimising adaptive Helmholtz resonators

Transcription

1 Tonal noise attenuation in ducts by optimising adaptive Helmholtz resonators Sarabjeet Singh School of Mechanical Engineering The University of Adelaide Adelaide, S.A AUSTRALIA Dissertation submitted for the award of the degree of Master of Engineering Science on the 25 th of September, Qualified on the 28 th of November, 2006.

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3 Dedicated to my parents, Sardar Jagmehar Singh and Sardarni Pawinder Kaur, and my siblings, Arvinder and Gurwinder

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5 Abstract Tonal noise propagating in ducts and radiating from their outlets is a common problem in situations where a fan or a blower is used to drive exhaust gases through the exhaust duct out to the environment. It is also a problem in the exhausts of large diesel engines such as those used to power large marine vessels. One way of attenuating tonal noise propagating in ducts is to use one or more side branch resonators, each of which is specifically designed for optimal performance at a particular frequency. One of the major problems associated with the use of side branch resonators is that any slight change in excitation frequency decreases the effectiveness of the resonators. The change in excitation frequency can be caused by a change in the speed of the engine, fan or blower, or change in temperature in the duct, which changes the speed of sound, and hence the wavelength of the noise. Resonators incorporating a provision for altering their geometry in real-time in order to adapt to environmental or operating condition changes is one approach that has been used by previous researchers. In particular, adaptive Helmholtz resonators have received considerable attention in the literature. Previous work has involved the use of one or more pressure sensors located in the duct downstream of the resonator to provide a cost function to be minimised by an electronic control system which alters the geometry of the resonator. However, in many cases, especially where the duct serves as a passage for exhaust gases to be driven out to the environment, it is not desirable to mount microphones in the duct. Also, microphones located remote from the resonator introduce wiring problems as well as the need to mount the microphones at the correct location in the duct, which will change as the wavelength of the tonal noise in the duct changes as a result of changes in operating or environmental conditions. It is highly desirable to have a completely self-contained Helmholtz resonator (HR) which can be attached to the duct and for which the only external wiring needed is the power supply. The work described in this thesis is concerned with the development of a selfcontained adaptive HR which can be optimally tuned by using signals from two miv

6 crophones located in the cavity and neck of the resonator, respectively. The primary focus of the work is the development of a novel cost function, which can be used by an electronic controller to optimally tune the HR. The scope of the analysis has been restricted here to the no mean flow condition. The theoretical and numerical analysis of the duct-hr system is first conducted using the well known transfer matrix method and finite element analysis (FEA) software package ANSYS, respectively. The net acoustic power transmission in the duct downstream of the HR is estimated by using the two-microphone method. Analysing the duct-hr system with the transfer matrix method mandates the incorporation of three end-correction factors which are related to the unflanged open end of the duct, neck-cavity interface and neck-duct interface. However, because of the complexity in estimating the end-correction factor of the neck at the neck-duct interface due to the generation of a complex sound field in the vicinity of the neck opening, the transfer matrix method only approximates the in-duct net acoustic power transmission. This implies that changing the value of the neck-duct interface end-correction factor changes the calculated frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs. On the other hand, ANSYS does not require the inclusion of any kind of end-correction factors apart from the actual physical dimensions of the system, and is thus much more accurate than the transfer matrix method. To minimise the in-duct net acoustic power transmission downstream of the HR, a number of different cost functions that were related to the net acoustic power transmission were investigated theoretically, numerically and experimentally. These all involved either the acoustic pressure at the top of the closed end of the cavity of the HR or at the neck wall of the HR close to the neck-duct interface or the amplitude of the pressure transfer function between two microphones located in the resonator. The two potential cost functions which were initially considered to be maximised for indicating the minimisation of the in-duct net acoustic power transmission downstream of the resonator vi

7 were: (a) the pressure at the top of the closed end of the cavity, and (b) the amplitude of the pressure transfer function between the pressure at the top of the closed end of the cavity and the pressure at the neck wall close to the neck-duct interface. It was found that the location of the microphone in the neck was extremely important, with the best location being at the centre of the duct adjacent to the neck opening. However, this location was not considered practical because a microphone in the duct can obstruct the mean flow of gas in the duct. The best location for mounting the microphone in the neck was found to be at the neck wall as close as possible to the neck-duct interface. The results are shown in two different ways: (1) broadband analysis, whereby the in-duct net acoustic power transmission downstream of the HR, the pressure at the top of the closed end of the cavity and the pressure transfer function between the pressure at the top of the closed end of the cavity and at the neck wall close to the neckduct interface are plotted as a function of frequency, and (2) single frequency analysis, whereby all the aforementioned results are plotted as a function of the cylindrical cavity length (for a fixed cavity diameter) for a single, tonal frequency. For broadband analysis, the numerical (ANSYS) results showed that the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs differs from the frequencies which correspond to the maximum responses of cost functions (a) and (b) described above. For single frequency analysis, when trying to optimise the performance of a duct-mounted HR at a particular frequency by altering its volume, the optimal dimensions of the HR required to attain the maximum reduction of in-duct net acoustic power transmission at that frequency differ from the dimensions of the HR which correspond to the maximised responses of the cost functions (a) and (b). These results were validated experimentally using a 3 m long circular duct of m diameter with an attached cylindrical HR. During the experimental work, only plane waves were propagating down the duct and there was no mean flow in the duct. vii

8 Instead of only focusing on the amplitude of the pressure transfer function between the pressures at the top of the closed end of the cavity and the pressure at the neck wall close to the neck-duct interface, the phase difference between the same locations in the HR was also considered. It was found that the phase difference depends on the quality factor (or damping) of the entire acoustic system. Experiments were conducted with varying dimensions of the HR and two novel cost functions were empirically derived. Both cost functions, which does not include any kind of measurement remote from the HR, are based on the damping (or the quality factor) of the duct-hr system and the phase difference between the pressure at the top of the closed end of the cavity and the pressure at the neck wall close to the neck-duct interface. The effectiveness and performance of both cost functions were found to be excellent for minimising the in-duct net acoustic power transmission downstream of the HR. However, the second cost function is preferred because the procedure involved for measuring the system damping is more convenient from the practical point of view than the procedure for the first one. The quality factor of the duct-mounted HR, at the frequency at which noise needs to be attenuated, was determined by tuning the length of the cavity of the HR so as to maximise the amplitude of the pressure transfer function of the HR. This estimated quality factor was found to be directly related to the transfer function phase which corresponds to the minimum in-duct net acoustic power transmission at the tonal frequency. Once this optimum transfer function phase is known, an active control system can be used to drive a motor to adjust the cavity length of the HR to achieve the optimum phase. viii

9 Statement of Originality To the best of my knowledge and belief, all the material presented in this thesis, except where otherwise referenced, is my own original work, and has not been published previously for the award of any other degree or diploma in any university. If accepted for the award of the degree of Master of Engineering Science, I consent that this thesis be made available for loan and photocopying. Sarabjeet Singh Date: ix

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11 Acknowledgements I would like to acknowledge the efforts of all the people who have contributed towards the work in this thesis. I am deeply indebted to both of my research advisors, Professor Colin Hansen and Dr. Carl Howard, without whom this thesis would have never been completed. Apart from appreciating their erudite suggestions on the research matters and their will to sharing the knowledge from their repository, I am extremely thankful to them for entertaining my unscheduled intrusions. A note to appreciating all their help and support would be incomplete without praising their professionalism (kudos to the spirit of their work ethics!). I am also thankful to the dextrous instrumentation and workshop personnel, George Osborne, Silvio De leso, Bill Finch, Ron Jager and Bob Dyer, for the construction of the rigs used in this research work. I am extremely grateful to Karen Adams for helping me in writing with her English language expertise. I owe my deepest and sincere thanks from the abyss of my inner self to my parents, Sardar Jagmehar Singh and Sardarni Pawinder Kaur, for all the sacrifices they made to get me a quality education, for the financial support, constant encouragement and patience. Thanks also go to my siblings, Arvinder and Gurwinder, for their support, love and understanding. I would also like to express my honest gratitude to Mr. Pomi Sandhu and Mrs. Jas Sandhu for their extended support and help while finishing. Finally, a big thanks to my friends, Melinda, Manjinder, Sahil, Shonali, Adrian, Chris, Jackie and Sam, for sharing many laughs and tolerating my frustrations while writing. Special thanks go to Manjinder, Sahil and Shonali for helping me at the finishing phase. Special thanks also go to Melinda for buying me many smoothies and a T-shirt with an awesome print of Don t ask me about my thesis. Waheguru Ji Ka Khalsa, Waheguru Ji Ki Fateh (The Khalsa belongs to God, victory belongs to God) xi

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13 Contents Abstract v Statement of Originality ix Acknowledgements xi Contents xiii List of Figures xix List of Tables xxxi 1 Introduction Introduction and Significance Aims Outline of the Thesis Literature Survey Introduction Stand-alone Helmholtz Resonator Resonance Frequency Effects of Geometry on the Resonance Frequency of HRs Duct with an Attached Helmholtz Resonator Effects of Geometry on the Resonance Frequency and Transmission Loss of Duct-mounted HRs xiii

14 CONTENTS Neck-Duct Interface End-Correction Factor Acoustic Performance of HRs In-Duct Acoustic Power Transmission Effects of Flow on the Resonance Frequency and Acoustic Performance of HRs Applications of HRs Adaptive Helmholtz Resonator Summary of Gaps Theoretical Modelling Introduction Stand-alone Helmholtz Resonator Equivalent Circuit Analysis End-Corrections Models to Predict Resonance Frequencies of HRs Analysing the Models Summary Circular Duct with an Attached Helmholtz Resonator Transfer Matrix Method Uniform Circular Duct Cut-on Frequency / Plane Wave Assumption Radiation Impedance Transfer Matrix of a Circular Duct Acoustic Pressure and Velocity at the Duct Exit Acoustic Pressure and Velocity at Arbitrary Locations Modal Decomposition of Sound Field In-Duct Net Acoustic Power Transmission Duct-HR System Neck-Duct Interface End-Correction Factor xiv

15 CONTENTS Transfer Matrix of a HR Input Point Impedance of a HR Complete Transfer Matrix of a Duct-HR System Acoustic Pressure and Velocity at the Duct Exit Acoustic Pressure at the Top Closed End of the Cavity Pressure Transfer Function of a HR In-Duct Net Acoustic Power Transmission Downstream of the HR Effects of Varying the End-Correction Factors Optimal Location of HRs Conclusion Numerical Modelling Introduction Uniform Circular Duct Building the Model Meshing the Model Applying Boundary Conditions Rigid-Wall Boundary Condition Open-End Boundary Condition Applying Loads Solving the Model Results In-Duct Net Acoustic Power Transmission Resonance Frequencies Summary Circular Duct with an Attached Helmholtz Resonator Introduction Meshing the Critical Regions xv

16 CONTENTS Implementing Damping in the HR Harmonic Analysis Results Acoustic Pressure Downstream of the HR Acoustic Pressure at the Top Closed End of the Cavity Acoustic Pressure in Critical Regions Pressure Transfer Function of the HR In-Duct Net Acoustic Power Transmission Downstream of the HR Effect of Different Mesh Densities Sensitivity Analysis Conclusion Experimental Verification Introduction Stand-alone Helmholtz Resonator Experimental Setup Experimental Results Resonance Frequencies Pressure Transfer Functions Summary Circular Duct with an Attached Helmholtz Resonator Experimental Setup Broadband Analysis Acoustic Pressure at the Top Closed End of the Cavity Acoustic Pressure at the Neck Opening Pressure Transfer Function of the HR In-Duct Net Acoustic Power Transmission Downstream of the HR xvi

17 CONTENTS Discussion Single Frequency Analysis In-Duct Net Acoustic Power Transmission Downstream of the HR Acoustic Pressure in the HR Pressure Transfer Function of the HR Discussion Conclusion Cost Function Introduction Approach Damping Cost Function The First Cost Function Approximating Damping at a Single Frequency The Second Cost Function Comparison of the Two Cost Functions Performance of the Two Cost Functions Limitations of the Two Cost Functions Conclusion Summary and Conclusion Summary and Conclusion Recommendations for Future Work References 191 A Transition of the Transfer Matrix Method Elements and Sequence of the State Variables Adopted For This Study 201 xvii

18 CONTENTS B List of Symbols 205 C Publications Originating from this Thesis 209 xviii

19 List of Figures 3.1 A Helmholtz resonator Comparison of the resonance frequencies of cylindrical HRs predicted using the classical, Panton and Miller s and Li s models A schematic of a uniform circular duct modelled as being driven by a constant amplitude piston source mounted at one (left) end and open at the other (right) A schematic of a uniform duct depicting the end-correction at the unflanged open end of the duct Theoretical (transfer matrix method) prediction for the net acoustic power transmission in a uniform circular duct driven by a constant amplitude piston source mounted at one (left) end and open at the other (right) A schematic of a duct-hr system for estimating the expression for the input point impedance of the HR A schematic of a duct-hr system showing its division into three elements Theoretical (transfer matrix method) prediction for the acoustic pressure at a point located at the top centre of the closed end of the cavity of the HR mounted onto a duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR xix

20 LIST OF FIGURES 3.9 Theoretical (transfer matrix method) prediction for the pressure transfer function amplitude of the HR Theoretical (transfer matrix method) prediction for the pressure transfer function phase of the HR Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Theoretical (transfer matrix method) predictions for the pressure transfer function amplitudes of the HR using different exterior end-correction models Theoretical (transfer matrix method) predictions for the pressure transfer function phases of the HR using different exterior end-correction models Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR using different exterior correction models. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR The results from figure 3.14 expanded over the frequency range from 205 Hz to 235 Hz Sound pressure level along the duct depicting a standing wave plot corresponding to a tonal (excitation) frequency of 197 Hz Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR at 197 Hz for varying the HR mounting location along the duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR xx

21 LIST OF FIGURES 3.18 Theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR at 197 Hz for the HR mounted at a distance of 0.45 m (a pressure node) and 0.88 m (an antinode) from the source end of the duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Sound pressure level along the duct depicting a standing wave plot corresponding to a tonal (excitation) frequency of 224 Hz Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR at 224 Hz for varying the HR mounting location along the duct. The duct was being driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR for the HR mounted at a distance of 0.75 m (a pressure node), 0.37 m (an antinode) and 0.56 m (an optimal location) from the source end. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Sound pressure level along the duct depicting a standing wave plot corresponding to a tonal (excitation) frequency of 211 Hz Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR at 211 Hz for varying the HR mounting location along the duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR xxi

22 LIST OF FIGURES 3.24 Theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR for the HR mounted at a distance of 0.61 m (a pressure node), 1.02 m (an antinode) and 0.91 m (an optimal location) from the source end. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR A schematic of the geometry of a FLUID29 element Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the net acoustic power transmission in a uniform circular duct driven by a constant amplitude piston source mounted at one (left) end and open at the other (right) A schematic of the duct-hr system showing four different regions to which the system was divided A three-dimensional finite element model of the duct-hr system displaying coarse mesh A three-dimensional finite element model of the duct-hr system displaying fine mesh Selected regions, critical duct section, cavity section and neck section, of the three-dimensional finite element model shown in figure Selected regions, critical duct section, cavity section and neck section, of the three-dimensional finite element model shown in figure A schematic of a duct-hr system showing the locations at which the acoustic pressures were calculated using ANSYS Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the acoustic pressure at a point located in the duct wall at 2.2 m from the source end (location 1 ) downstream of the HR. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR xxii

23 LIST OF FIGURES 4.10 The results from figure 4.9 expanded over the frequency range from 190 Hz to 260 Hz Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the acoustic pressure at a point located at the top center of the closed end of the cavity (location A ) of the HR mounted onto a duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR The results from figure 4.11 expanded over the frequency range from 190 Hz to 260 Hz Numerical (ANSYS) prediction for the real part of the acoustic pressure within the duct-hr system at 226 Hz Numerical (ANSYS) prediction for the imaginary part of the acoustic pressure within the duct-hr system at 226 Hz Numerical (ANSYS) predictions for the acoustic pressures at locations E and T and theoretical (transfer matrix method) results Numerical (ANSYS) predictions for the acoustic pressures at locations G and H and theoretical (transfer matrix method) results Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the pressure transfer function amplitudes of the HR Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the pressure transfer function phases of the HR Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR xxiii

24 LIST OF FIGURES 4.20 Numerical (ANSYS) predictions showing the affect of varying mesh densities on the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Numerical (ANSYS) predictions showing the effects of positive phase errors of a few degrees on the in-duct net acoustic transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Numerical (ANSYS) predictions showing the effects of negative phase errors of a few degrees on the in-duct net acoustic transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Numerical (ANSYS) predictions showing the effects of different sound velocities on the in-duct net acoustic transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR The results from figure 4.23 expanded over the frequency range from 190 Hz to 260 Hz A picture of a HR placed on a table inside an anechoic chamber A different view of the HR pictured in figure 5.1 showing the microphones mounted at the opening of the neck A schematic of a stand-alone cylindrical HR showing the locations of the microphones xxiv

25 LIST OF FIGURES 5.4 Experimentally measured resonance frequencies of stand-alone cylindrical HRs along with the predictions from several theoretical models Experimentally measured pressure transfer function amplitude between microphone A and microphone E. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR Experimentally measured pressure transfer function phase between microphone A and microphone E. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR Experimentally measured pressure transfer function amplitudes between microphones A and C, A and B, and A and D. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR Experimentally measured pressure transfer function phases between microphones A and C, A and B, and A and D. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR A picture of the experimental rig showing a circular duct with an attached HR along with the B&K Pulse system and amplifiers A picture of a duct showing a circular section welded on the duct at a distance of 0.5 m from the source end A picture of the PVC sections A schematic of the duct-hr system showing the locations of the microphones Experimental, numerical (ANSYS) and theoretical (transfer matrix method) results for the acoustic pressure at microphone A, as a function of frequency The results from figure 5.13 expanded over the frequency range from 190 Hz to 290 Hz xxv

26 LIST OF FIGURES 5.15 Experimentally measured acoustic pressures at the neck opening microphones B, C, D and T, as a function of frequency The results from figure 5.15 expanded over the frequency range from 190 Hz to 290 Hz Numerical (ANSYS) predictions for the acoustic pressures at the locations analogous to the neck opening microphones B, C, D and T, as a function of frequency The results from figure 5.17 expanded over the frequency range from 190 Hz to 290 Hz Experimentally measured pressure transfer function amplitudes between microphones A and B, A and C, A and D, and A and T, as a function of frequency Experimentally measured pressure transfer function phases between microphones A and B, A and C, A and D, and A and T, as a function of frequency Numerical (ANSYS) predictions for the pressure transfer function amplitudes between locations A and B, A and C, A and D, and A and T, as a function of frequency Numerical (ANSYS) predictions for the pressure transfer function phases between locations A and B, A and C, A and D, and A and T, as a function of frequency Numerical (ANSYS) predictions of the pressure transfer function amplitudes between locations A and B, A and C, A and D, and A and T for the location of the HR as 1.2 m from the source end of the duct, as a function of frequency xxvi

27 LIST OF FIGURES 5.24 Numerical (ANSYS) predictions of the pressure transfer function phases between locations A and B, A and C, A and D, and A and T for the location of the HR as 1.2 m from the source end of the duct, as a function of frequency Experimental, numerical (ANSYS) and theoretical (transfer matrix method) results for the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Experimental, numerical (ANSYS) and theoretical (transfer matrix method) results for the in-duct net acoustic power transmission without the HR for the duct driven by a constant amplitude piston source mounted at one (left) end and open at the other (right) Experimentally measured in-duct net acoustic power transmission without the HR and with the HR attached to the duct for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR Numerical (ANSYS) predictions for the in-duct net acoustic power transmission without the HR and with the HR attached to the duct for the duct driven by a constant amplitude piston source mounted at the end upstream of the resonator and open at the end downstream of the HR Experimentally measured acoustic pressures at microphone A, B and T, as a function of frequency. The vertical line at 226 Hz corresponds to the frequency of the maximum reduction of the in-duct net acoustic power transmission downstream of the HR Experimentally measured pressure transfer function amplitudes between microphones A and B, and A and T, as a function of frequency xxvii

28 LIST OF FIGURES 5.31 Experimentally measured pressure transfer function phases between microphones A and B, and A and T, as a function of frequency Experimental and numerical (ANSYS) results for the in-duct net acoustic power transmission downstream of the HR corresponding to 226 Hz, as a function of the HR cavity length Experimentally measured acoustic pressure at microphones A, B and T corresponding to 226 Hz, as a function of the HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of in-duct acoustic power transmission downstream of the HR at 226 Hz Experimentally measured pressure transfer function amplitudes between microphones A and B, and A and T corresponding to 226 Hz, as a function of the HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of in-duct net acoustic power transmission downstream of the HR at 226 Hz Experimentally measured pressure transfer function phases between microphones A and B, and A and T corresponding to 226 Hz, as a function of HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of in-duct net acoustic power transmission downstream of the HR at 226 Hz Frequency response of a system showing bandwidth and half-power points Experimentally measured critical damping ratios of the duct-hr system for various HRs Critical damping ratio versus phase difference corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs, for three different neck diameters and many cavity volume sizes A schematic representation of a single degree-of-freedom system xxviii

29 LIST OF FIGURES 6.5 Experimentally measured ratio of pressures at microphone A and microphone B corresponding to 226 Hz, as a function of the HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of the in-duct net acoustic power transmission downstream of the HR at 226 Hz Experimentally measured ratio of pressures at microphone A and microphone B corresponding to the cavity length of 84 mm, as a function of frequency. The vertical line at 226 Hz corresponds to the frequency of maximum in-duct net acoustic power reduction downstream of the HR Qualtiy factors of several HRs obtained using two different methods: (1) by measuring the maximum pressure ratio of microphone A to microphone B, and (2) by applying the half-power point bandwidth method Quality factor versus phase difference corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs, for three different neck diameters Experimentally measured in-duct net acoustic power transmission downstream of the HR corresponding to 226 Hz plotted as a function of the HR cavity length; markers indicate the acoustic power level for various cost functions Numerical (ANSYS) predictions for the in-duct net acoustic power transmission downstream of the HR corresponding to 226 Hz as a function of the HR cavity length with no damping included; markers indicate the acoustic power level for various cost functions xxix

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31 List of Tables 3.1 Criterion for selecting an appropriate theoretical model for calculating the resonance frequencies of cylindrical HRs Theoretical (Blevin s formula and transfer matrix method) predictions for the first seven acoustic resonance frequencies of a piston driven-open duct system Numerical (ANSYS) and theoretical (Blevin s formula and transfer matrix method) predictions for the acoustic resonance frequencies of a piston driven-open duct system Error between the acoustic resonance frequencies tabulated in table Summary of the elements per wavelength at 226 Hz for dividing the model of the duct-hr system Experimentally measured resonance frequencies of stand-alone cylindrical HRs along with the predictions from several theoretical models xxxi

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33 Chapter 1 Introduction 1.1 Introduction and Significance The problem of low-frequency tonal noise is inherent to industries that use machinery such as internal combustion engines, compressors, fans, blowers, power transformers and gearboxes. The need is ever-present for industry to reduce the annoyance caused by the humming nature of tonal noise, not only to workers within an industry but also to the surrounding community. Depending upon the type of application and cost constraints, tonal noise transmission can be controlled in a number of ways, such as by the installation of reactive silencers, barriers, side branch elements and active noise control devices. The work described in this study is concerned with the attenuation of tonal noise transmission in ducts by using side branch resonators. Tonal noise propagating in ducts and radiating from their outlets is common in industries where a fan, blower or internal combustion engine, installed at one end of the duct, is used to drive exhaust gases through the exhaust duct out to the environment. One of the most effective ways to reduce tonal noise transmission in ducts is to use side branch resonators, which are specifically designed for their optimal performance at a particular frequency and work effectively over a very narrow frequency band close to their respective resonance frequencies. Although they are capable of providing high noise reduction, one of the major drawbacks associated with the use of side branch 1

34 1.1. Introduction and Significance resonators is that even a slight change in excitation frequency decreases their effectiveness. The change in excitation frequency can be caused by a change in the speed of the engine, fan or blower, or change in temperature of the gas in the duct. This changes the speed of sound and hence the wavelength of the noise. The frequency range of the resonators for which they are effective can be broadened by lining their inner surfaces with an acoustically absorptive material, such as fiberglass, rockwool, mineral wool and porous foams; however, this will in turn compromise the maximum noise reduction capabilities of the resonators. More importantly, lining the resonators is not preferred when they are used in dusty environments, such as in hot exhaust stacks where the exhaust particulates can be moist and sticky. The porous nature of absorptive materials renders them susceptible to contamination, moisture and clogging with the in-duct flow-stream particulates, as a result of which absorptive material gradually loses its effectiveness in terms of acoustic absorption and requires routine replacement. This increases the maintenance and operating cost. As an alternative to lining the resonators, many researchers have adopted an approach of incorporating some provision for altering the geometry of the resonators in real-time in order to retain their optimum effectiveness as excitation frequency changes. Such an adaptive system has been referred to as a semi-active (or adaptive-passive) system in which a change in the physical parameters of the passive element is caused by an active control system. Bernhard et al. [1, 2] and Eriksson et al. [3] outlined the benefits of semi-active systems over any exclusive active or passive systems. Implementation of semi-active systems has increased in both commercial applications and small-scale laboratory setups. In particular, adaptive Helmholtz resonators (HRs) have been a subject of investigation by many researchers and have received considerable attention in the literature. Many previous studies have reported on the successful implementation of adaptive HRs for attenuating tonal noise transmission in ducts. Previous work has involved the use of one or more pressure sensors located in the duct downstream of the resonator to 2

35 Chapter 1. Introduction provide a cost function to be minimised by an electronic control system, which alters the geometry of the resonator. However, in many cases, especially where the duct serves as a passage for the exhaust gases to be driven out to the environment, it is not desirable to mount microphones in the duct. This is because it is often possible for signals so obtained to be contaminated with unrelated noise or affected due to physical problems with the microphones such those listed below [4]. Accumulation of exhaust particulates - The response of the pressure sensors can be significantly affected by the in-duct exhaust particulates which may accumulate on them. Heating - The possibility of increase in temperature of the exhaust gases during the operation, can also directly affect the operation of the sensors. Mean flow - The presence of mean flow generates turbulence and adds to spurious noise resulting in contamination of actual acoustic pressure measurements. Inappropriate location - Variations in the excitation frequency due to the aforementioned reasons may result in the location of the pressure sensor being close to a pressure node. As a result a pressure sensor would be unable to properly obtain a measure of the pressure. However, some researchers have overcome this problem by using multiple microphones along the duct [5]. Another issue associated with the in-duct mounting of pressure sensors, which is not related to the quality of pressure measurement but is related to convenience, is the wiring problem. Pressure sensors located in the duct need additional wires to be connected to the electronic control unit. For a noise control device attached to the duct, ideally the only external wiring would be an electrical power supply. In order to avoid the abovementioned potential risks and also, for reasons of convenience and practicality, it would be highly desirable to have a completely self-contained adaptive HR that does not need any external measurement remote from its structure. 3

36 1.2. Aims This would involve locating the pressure sensors in the cavity and/or neck of the resonator. The output from these sensors could be fed to an electronic controller to optimally adjust the dimensions of the HR in order to minimise the in-duct net acoustic power transmission downstream of the resonator. Thus what is needed is some measurement, using the physical sensors in the resonator, which is directly proportional to sound power transmitted down the duct to which the resonator is attached. 1.2 Aims The aims of this study are to develop a cost function, which can be used by an electronic controller, to minimise the in-duct net acoustic power transmission downstream of the HR, to find suitable locations for mounting the pressure sensors to provide the required cost function. 1.3 Outline of the Thesis A literature review is presented in chapter 2 which describes the previous work in the areas related to this study. Chapter 3 contains a description of the theoretical modelling of HRs as a standalone device and a HR attached to a duct. The latter case is more often referred to as a duct-hr system in this thesis. In addition to the classical formula developed by Rayliegh [6], two other existing mathematical equations [7, 8] were used to predict the resonance frequencies of cylindrical HRs as stand-alone devices. The estimates of the resonance frequencies obtained using the three formulas were compared, and the differences in the results were explained. The transfer matrix method was used for the theoretical analysis of the duct-hr system. Building the transfer matrix equations of a uniform circular duct and a circular 4

37 Chapter 1. Introduction duct with an attached cylindrical HR are described. An overview is given of the end-correction factor of the neck of the HR at the neck-duct interface. The acoustic pressures at various locations within the duct-hr system, pressure transfer function of the HR and in-duct net acoustic power transmission downstream of the resonator were estimated. The in-duct net acoustic power transmission without and with the HR facilitated the estimation of the acoustic performance of the duct-mounted HR. Using the transfer matrix method mandates the need to incorporate the endcorrection factors in addition to the actual dimensions of the system. Due to the unknown neck-duct interface end-correction factor, three different estimates of the neck-duct interface end-correction factor were used and the pertinent differences in the results were shown. The applicability of the transfer matrix method being limited to planar sound fields made this method unfeasible for analysing the duct-hr system due to the presence of complex sound fields at the opening of the resonator. Chapter 3 concludes by summarising the results of the theoretical modelling of a stand-alone HR and the duct-hr system, and highlights the limitations of the transfer matrix method in analysing the duct-hr system. In order to overcome the limitations of the transfer matrix method, numerical modelling of the duct-hr system was conducted using the finite element analysis software package ANSYS, and is described in chapter 4. A three-dimensional finite element model of a cylindrical HR mounted onto a circular duct was built and solved to analyse the sound field inside the duct and HR. The terminating end of the duct was modelled as being open and radiating into free space. The damping in the resonator, which mainly occurs due to the movement of fluid particles in the neck, was also included during the analysis. The performance of the duct-mounted HR was estimated by comparing the net acoustic power transmission in the duct without the HR and with the HR. A comparison was made of the ANSYS modelling results with theoretical predictions. The numerical modelling for the analysis of the duct-hr system proved to be ad- 5

38 1.3. Outline of the Thesis vantageous over the theoretical modelling because the end-correction factors are automatically incorporated in the model, whereas in the transfer matrix method, the end-correction factors have to be estimated. Chapter 5 contains the experimental results of stand-alone HRs and the duct-hr system. The resonance frequencies of stand-alone HRs were measured and compared with the theoretical predictions. For the circular duct with an attached HR, the induct net acoustic power transmission downstream of the HR was measured using the two-microphone method. The experimental results pertinent to the duct-hr system were presented in two ways: (1) by broadband frequency analysis, and (2) by single frequency analysis. It was found that the experimental results favourably matched the theoretical and numerical predictions. With an aim to optimise the HR on the basis of the acoustic pressure measurements, one microphone was located at the top of the closed end of the cavity and the other was located at the neck wall as close as possible to the neck-duct interface. The experimental results showed that neither the pressure at the above stated locations nor the pressure transfer function between them was a maximum or a minimum at the frequency which corresponds to the minimum in-duct net acoustic power transmission downstream of the HR. In order to achieve the optimal tuning of the HR on the basis of pressure measurements in the resonator only, a cost function was desired and its derivation is described in chapter 6. The last chapter summarises the work presented in this thesis along with recommendations for future research. 6

39 Chapter 2 Literature Survey 2.1 Introduction This chapter presents a review of previous research on Helmholtz resonators (HRs). The contents of this chapter are divided into two broad sections: (1) HRs as standalone devices and (2) HRs as mounted on a duct. The initial discussion on stand-alone HRs will present research into the design aspects and the mathematical formula used for calculating their resonance frequencies. The current study will consider one of the most common applications of HRs, which is the reduction of noise transmission in ducts. The second part of this review will discuss research on the aspects related to evaluating the acoustic performance of HRs, which is related to the reduction of in-duct acoustic power transmission. 2.2 Stand-alone Helmholtz Resonator A stand-alone HR refers to an acoustic device consisting of a cavity (volume) and a neck (opening), that is not connected to any acoustic system. Because the purpose of a HR is to provide sound attenuation at a narrow band of frequencies in close proximity to its resonance frequency, it is important to be able to accurately predict its resonance frequency. The classical formula for calculating the resonance frequency 7

40 2.2. Stand-alone Helmholtz Resonator of HRs has been modified by several authors as a result of the reported discrepancies in the experimental and theoretical results. These modifications are reported in the following sections Resonance Frequency Helmholtz resonators were first described in the literature by Hermann Ludwig Ferdinand von Helmholtz in Helmholtz [9] described the first mathematical theory for cavity resonators (volume) having a circular opening, and presented a mathematical formula for calculating their resonance frequencies. The formula was based on the volume of the cavity and the radius of the opening, and is given by f r = c 2r 2π V (2.1) where, c is the speed of sound, r is the radius of the opening and V is the volume of the cavity. Later, Rayleigh [6] presented a simplified theory of HRs. He stated that in practice, the length of the opening (or neck) is not merely constrained to the thickness of the cavity wall. He suggested that the fluid particles in close proximity to the opening take part in the induced motion of the fluid oscillating inside the neck. Therefore, an additional length should be added to the actual length of the neck (or cavity wall thickness) in order to include the mass loading of the fluid at the two ends of the neck. This additional length is more commonly referred to as an end-correction in many acoustic text books [10, 11, 12]. Rayleigh [6] made clear that two end-correction factors, one for each end, must be added to the physical length of the neck. These two endcorrection factors are: (1) interior end-correction factor (δ i ), which corresponds to the neck-cavity interface, and (2) exterior end-correction factor (δ e ), which corresponds to the opening face of the neck in communication with the environment. This is illustrated in figure 3.1, in the next chapter of this thesis. Rayleigh [6] derived the expression for the exterior end-correction factor of the neck 8

41 Chapter 2. Literature Survey using the model of a circular piston radiating from an infinite baffle [10, 11, 12], and assuming a constant velocity profile of the fluid over the neck cross-sectional area. The expression for the exterior end-correction factor was also used as a basis to calculate the interior end-correction factor of the neck. Rayleigh [6] also developed the formula for calculating the resonance frequencies of HRs which, in addition to the volume of the cavity and cross-sectional area of the opening, also includes the effective length of the neck, l eff. The formula, which is more commonly referred to as the classical formula [10, 11, 12], is based on the assumption that all the fluid particles in the neck oscillate at same velocity and phase, and also applies when a short tube of the same external diameter as the opening is placed in the opening, and is given by f r = c πr 2 2π l eff V (2.2) where, l eff is the cavity wall thickness (or length of the tube, if applicable) plus the two end-correction factors. In 1953, Ingard [13, 14, 15] presented a series of work on resonators covering a wide range of topics. Pertinent to the interior end-correction factor of the neck, which was assumed to be equal to the exterior end-correction as proposed by Rayleigh [6], Ingard [13] found that the application of Rayleigh s formula for calculating the interior end-correction is only valid for the cases where the dimensions of the neck are very small compared to the dimensions of the cavity, and that the formula can lead to significant errors when this condition is not satisfied. Ingard [13] investigated several geometries of neck and cavity of the HR, which include circular and square crosssectional orifices located in either circular, square or rectangular cavities. He derived different mathematical equations for the interior (neck-cavity interface) end-correction factor with respect to different geometries, modelling the neck as a piston oscillating into an expanded pipe of infinite length. Ingard [13] was the first person to report the effects of geometry on the resonance frequency of resonators using different expressions for the neck-cavity interface end- 9

42 2.2. Stand-alone Helmholtz Resonator correction factor as a function of the resonator geometry in the formula developed by Rayleigh (equation (2.2)) [6]. However, his investigations on the effects of geometry on the resonance frequency were limited to the neck-cavity interface end-correction factor. Many researchers have reported on the effects of geometry on the resonance frequency of resonators, and the relevant literature is presented in the next section Effects of Geometry on the Resonance Frequency of HRs Alster [16] observed that the error in the experimental measurements and theoretical calculations of the resonance frequencies of resonators obtained using Rayleigh s [6] formula can be upto 30%. He hypothesised that the cause for the discrepancies can be due to the simplicity of the theoretical formula, which fails to account for the shape of the resonator. In fact, the formulas derived by Helmholtz and Rayleigh for calculating the resonance frequencies of resonators are independent of the shape of the cavity. Rayleigh [6], in his book The Theory of Sound, also mentioned that the experiments conducted by Liscovious and Sondhauss revealed that the resonance frequency of a resonator does not depend on its shape. However, Alster [16] claimed that the shape of a resonator has a significant effect on its resonance frequency. He presented a different formula for calculating the resonance frequency of a HR, also applicable to quarter wave resonators, which accounts for the shape of the resonator cavity. Unlike classical theory [6], which assumed that all the fluid particles in the neck have same velocity and phase, the derivations accomplished by Alster were based on variable fluid velocity amplitude in the resonator having highest amplitude at the beginning of the neck and zero at the end of the cavity. However, the fluid particles were considered to be in phase. He presented different expressions referred to as a form-factor which accounts for different shapes of the resonator cavities. The resonance frequency of a resonator was calculated by incorporating respective form-factor in the formula he developed. He also stated that the validity of the formula is limited to the resonators for which the dimensions 10

43 Chapter 2. Literature Survey do not exceed a quarter of the wavelength at the resonance frequency and the height is not smaller than the diameter of the opening. For cylindrical resonators, Tang and Sirignano [17] presented a sophisticated analysis assuming one-dimensional wave propagation in both the neck and the cavity of the resonator. They developed a general formulation and applied it to quarter-wave resonators and HRs. The length of the neck and cavity in their analysis was comparable to the wavelength at the resonance frequency, which, contrasts the assumptions in the classical [6] and Alster s formula [16], where the length is assumed to be much smaller than a wavelength. They also reported on certain other aspects, such as size and weight of the resonators for consideration in engineering practice. Later, Panton and Miller [7] demonstrated that the classical formula, developed by Rayleigh [6], for calculating the resonance frequency of HRs is not valid if any dimension of the resonator exceeds 1/16 of the wavelength at the resonance frequency. They presented a different formula for calculating the resonance frequency of cylindrical HRs. It was assumed that the length of the cylindrical cavity was large compared to the opening. The formula was derived by considering one-dimensional wave propagation in the cavity but the neck was treated as a lumped mass. Chanaud [18, 19] examined the influence of neck and cavity geometry on the resonance frequency of HRs. He investigated several geometries of HRs and developed the neck-cavity interface end-correction models for various combinations of the neck and cavity geometries. In his first publication on the series of Effects of geometry on the resonance frequency of HRs, Chanaud [18] presented a detailed analysis for calculating the end-correction factors for a rectangular parallelepiped cavity with an orifice that was either circular, rectangular or cross-shaped. One of the investigations accomplished by Ingard [13] was related to the calculation of the neck-cavity interface end-correction factor of an orifice (opening) eccentrically located in the cylindrical cavity. However, the expression was only calculated for a single geometry of the HR corresponding to the ratio of orifice radius to cavity radius 11

44 2.2. Stand-alone Helmholtz Resonator of Later, Chanaud [19] extended his own work [18] and Ingard s work [13], and developed the interior end-correction model for an arbitrarily located circular orifice on a cylindrical cavity. He checked the validity of his theory with the experimentally measured resonance frequencies of cylindrical HRs (having a centrally located circular orifice) reported by Panton and Miller [7]. He concluded that for a fixed cavity volume and orifice size, the location of an orifice has a substantial effect on the resonance frequency of the resonator, whereas the orifice geometry is not significant if the largest dimension of the resonator is small compared to the wavelength of the resonance frequency. Chanaud [19] did not comment on the work accomplished by Alster [16]. Recently, as an extension to the work accomplished by Panton and Miller [7], Li [8] proposed another mathematical model for predicting the resonance frequencies of cylindrical HRs. His derivation was more generalised than the model derived by Panton and Miller [7] in the sense that one-dimensional wave propagation was considered in both cavity and neck of the resonator, whereas in Panton and Miller s model, wave propagation was only considered in the cavity. As cylindrical HRs were used in this study, the formulas developed by Panton and Miller [7], and Li [8] were analysed for their validity and limitations. The experimentally measured and theoretically calculated resonance frequencies of cylindrical HRs obtained using the formulas in references [7, 8] will be compared in the next chapter of this thesis: Theoretical Modelling. Over the past few years, Selamet and his co-workers [20, 21, 22, 23, 24] have conducted interesting work related to HRs. In addition to investigating the effects of geometry on the resonance frequency of HRs, they also reported on the noise attenuation performance of HRs mounted onto a duct. Their work is described in the next section. 12

45 Chapter 2. Literature Survey 2.3 Duct with an Attached Helmholtz Resonator One of the most important applications of HRs is to reduce noise propagation in ducts. When a resonator is mounted on a duct, it creates a change in the impedance of the duct at a point where the resonator is located [11, 25]. The change in the impedance causes the propagating acoustic waves to be reflected to the origin of noise source. The acoustic suppression performance of HRs is usually evaluated by calculating one of the three acoustical parameters, namely insertion loss (IL), transmission loss (TL) and noise reduction (NR) or level difference (LD) [26]. These parameters are described in section of the next chapter of this thesis. The greater the value of these parameters, the greater the effectiveness of the resonators. Selamet and his co-workers demonstrated the effects of geometry on the resonance frequency and transmission loss of cylindrical HRs mounted onto a cylindrical duct and this work is described in the next section Effects of Geometry on the Resonance Frequency and Transmission Loss of Duct-mounted HRs The literature presented so far in this chapter shows that the orifice and cavity geometries affect the resonance frequency of HRs. Different shapes of the resonator geometries have been considered but the extremes of the cavity geometry, high and low cavity length-to-diameter ratio, have not been investigated. Selamet et al. [20] focused their work on the effects of cavity dimensions (length-to-diameter ratio) on the resonance frequency and transmission loss of cylindrical HRs mounted onto a duct. A mismatch between the one-dimensional (axial) theoretical and experimental results at low cavity length-to-diameter ratio was reported [20] which was later addressed by presenting a limit to the validity of the theory for low length-to-diameter ratios (< 1) [21]. Later, Selamet et al. [22] developed a two-dimensional theoretical model to improve 13

46 2.3. Duct with an Attached Helmholtz Resonator the theoretical predictions of the resonance frequency and transmission loss. A twodimensional wave propagation in the complete duct-hr system, except at the neckduct interface, was considered. Due to an area discontinuity at the resonator mounting location on a duct, a three-dimensional sound field exists at the neck-duct interface. Selamet et al. [22] assumed this three-dimensional sound field to be one-dimensional in the derivation of the theoretical model. They also developed a three-dimensional boundary element model (BEM) to predict the transmission loss of a HR mounted on a duct. A favourable match between the experimental and BEM results was reported but deviation of the two-dimensional theoretical results from the experimental results was reported when the cavity length-to-diameter ratio ranged from 0.1 to 3. Selamet et al. extended their previous investigations of axisymmetric cylindrical HRs [20, 21, 22] to asymmetric cylindrical HRs [23], which are obtained by eccentrically locating the neck in the cavity. They developed a three-dimensional theoretical model for calculating the resonance frequency and transmission loss of duct-mounted cylindrical asymmetric HRs. As an extension to Ingard s [13] work, they also developed the interior end-correction factor at the neck-cavity interface for an eccentrically located neck and claimed this to be better than Ingard s [13] model. The comparison of the three-dimensional theoretical results with the respective BEM calculations was in good agreement. The decrease in the resonance frequency of the resonator with increasing the eccentricity of the neck was illustrated, which is similar to that of Chaunaud s findings [19]. Finally, Selamet et al. [24] analysed the case of the neck extending into the cavity of the HR. The extension increased the overall length of the neck which reduced the resonance frequency of the resonator. The neck extensions included circular tubes of constant cross-section, diverging and converging conical extensions and perforations in the tube of constant cross-section. This way of reducing the resonance frequency of a resonator is quite useful, particularly, in applications where space constraint is an issue, such as HVAC installation. 14

47 Chapter 2. Literature Survey None of the work reported by Selamet and his co-workers compares the resonance frequency of a HR and the frequency at which the maximum transmission loss occurs when it is mounted on a duct. However, the formula based on the classical theory (lumped parameter analysis) and the formulas developed by Selamet et al. [20, 21, 22] calculate the maximum transmission loss at a frequency which is considered to be the resonance frequency of a resonator. This thesis shows that there is a difference in the resonance frequency of a HR (as a stand-alone device) and the frequency at which the maximum reduction of in-duct acoustic power transmission downstream of the resonator occurs. Selamet et al. [23, 24] tested the validity of a three-dimensional theoretical model for predicting the resonance frequency and transmission loss of HRs by comparing the respective theoretical predictions with BEM calculations. However, the threedimensional complex sound field at the neck-duct interface was assumed to be onedimensional during the derivation of the theoretical model. This simplification leads to inaccuracies in predicting the acoustic performance of a resonator, especially the frequency at which the in-duct acoustic power transmission is minimised. Although the three-dimensional complex sound field at the neck-duct interface was demonstrated using BEM by Selamet and his co-workers, no estimate of the neck-duct interface endcorrection factor to be included in the theoretical models was suggested. However, two authors have addressed the issue of this end-correction factor and their investigations are reported in the next section Neck-Duct Interface End-Correction Factor Sudden area discontinuities are commonly seen in various acoustic systems. One of the classic examples of an area discontinuity is an expansion chamber muffler where area discontinuities occur at the expansion of an inlet tube into a chamber and at the contraction of a chamber into an outlet tube. The above mentioned discontinuities are well documented in the literature [13, 26, 27, 28, 29]. 15

48 2.3. Duct with an Attached Helmholtz Resonator For a HR as a stand-alone device, an area discontinuity is formed at the neck-cavity interface, and its effects on the resonance frequency and transmission loss have been investigated by others [13, 19, 23]. However, the physics of a discontinuity formed by mounting a side branch resonator to a main duct is still not well understood in terms of the formation of a three-dimensional complex sound field at the resonator-duct interface. Nevertheless, it is important to know an exact estimate of the end-correction factor at this interface. In this study, such a discontinuity is referred to as a neck-duct interface. Onorati [30] suggested that a correction factor of 0.3a, where a denotes the radius of the main duct, should be added at the interface of three pipes, such as at the interface of a column resonator and a main duct or a HR and a main duct. His suggested value was solely dependent on the diameter of the main duct, and the details of the experimental rig provided by the author in his paper show that the neck and duct diameter were comparable to each other, and were 50.4 mm and 73 mm, respectively. There is no evidence that this correction factor of 0.3a is valid for neck-duct diameter ratios much different to unity. Ji [31], derived two curve-fitted expressions for calculating the end-correction factor at the interface of a cylindrical tube perpendicularly attached to another cylindrical tube using BEM. The expressions depend on the ratio of the two tubes diameter. He then applied the appropriate expression to the case of a duct-mounted HR system and calculated the transmission loss of the resonator. His BEM and experimental estimations of the transmission loss of the HR were in good agreement. In contrast to the end-correction factor suggested by Onorati [30], Ji s [31] expressions depend on both the diameter of the neck and the main duct. By using different neck-duct interface end-correction models, different theoretical values for the acoustic performance of a duct-mounted HR system can be calculated. However, no mathematical model currently exists that gives the correct measure of the neck-duct interface end-correction factor, which is important for accurately calculating the TL of a HR. 16

49 Chapter 2. Literature Survey Acoustic Performance of HRs There exists a large amount of literature on mufflers that can be found in many journals and acoustic text books. This section is focused on the literature pertinent to the duct- HR system. As stated earlier in section 2.3, the acoustic performance of a HR mounted on a duct is generally evaluated by calculating or measuring one of the three acoustic performance parameters, namely insertion loss (IL), transmission loss (TL) and noise reduction (NR) or level difference (LD) [26]. Prasad and Crocker [32] discussed the pros and cons related to theoretically calculating and experimentally measuring the above described acoustic parameters. However, the ultimate selection of an evaluation criterion is based on trade-offs between the desired accuracy in the measurements and the amount of available resources. The acoustic performance of a HR can be evaluated by theoretically estimating the TL of the resonator. This is because the TL of an acoustic element is a characteristic solely of the element and does not depend on the source and termination impedances [26, 32]. The derivation of the TL of a HR mounted on a duct was presented in one of the studies which describes a very detailed and systematic analysis of mufflers [33]. The derivation was based on a number of assumptions, such as plane wave propagation in the duct-hr system, dimensions of the resonator small compared to a wavelength of sound and no mean flow. The formula presented in reference [33] calculates a maximum TL at the resonance frequency of a HR. As discussed in section 2.3.1, Selamet et al. [20, 21] developed an expression for estimating the TL of duct-mounted cylindrical HRs with one-dimensional wave propagation. Although two- and three-dimensional theoretical models for predicting the TL were developed, these models did not include the effects of the complex sound field at the neck-duct interface [22, 23, 24], and hence the models could not be used to accurately evaluate the performance of the resonator. 17

50 2.3. Duct with an Attached Helmholtz Resonator Chen et al. [34] also reported on the TL of a duct-mounted HR and claimed that the TL could be improved by adding HRs. Munjal [26] discusses a scheme to predict the TL of HRs using the transfer matrix method. All the existing theoretical models for calculating the acoustic performance of resonators lead to a conclusion that the frequencies corresponding to the resonance of a resonator and the maximum TL are same. Also, the theoretical models overestimate the acoustic power reduction levels due to the neglect of the viscous losses which occurs in the neck. However, losses (or damping) can be incorporated in the transfer matrix method using complex wave numbers [26]. Optimal Performance of a HR by Detuning As stated earlier, the work presented in this thesis shows that there is a difference in the resonance frequency of a HR (as a stand-alone device) and the frequency at which the maximum reduction of in-duct acoustic power transmission downstream of the resonator occurs. This implies that a HR should be detuned in order to attain its optimum acoustic performance. A similar kind of findings were also reported by Fuller et al. [35]. They demonstrated analytically that a global reduction of the total acoustic potential energy inside the cylindrical shell, modelled as a representative of a fuselage structure, occurs at a frequency which does not correspond to the natural frequency of the vibration absorbers mounted on the shell to reduce the sound transmission into it. Carneal et al. [36] also reported on the achievement of better performance of vibration absorbers when they were detuned for minimising sound radiating from plates. Experimentally, the TL of an acoustic element is calculated by measuring the acoustic pressure at two locations one upstream and one downstream of the element. This method is more commonly known as the two-microphone method and the respective literature is described in the next section. 18

51 Chapter 2. Literature Survey In-Duct Acoustic Power Transmission In this thesis, the net acoustic power transmission in the duct is estimated by using the well known two-microphone method [37]. The two-microphone technique was introduced by Seybert and Ross [38] as an alternative to the standing wave ratio method for determining the acoustic properties pertinent to sound propagation in ducts. From measurements of auto- and cross-spectral densities between two microphones located in the duct, they were able to separate the left- and right-travelling wave spectra. Later, Chung and Blaser [37] developed a similar two-microphone method using a measurement of the transfer function between the two microphones for determining the in-duct acoustic properties. Although the approach adopted by Seybert and Ross [38], and Chung and Blaser [37] for developing the two-microphone method is different, they essentially give the same results. As the measurement of transfer function is standard on all two-channel spectrum analysers, Chung and Blaser s method has been made the standard test method [39, 40, 41]. Åbom [42] extended the two-microphone method developed by Chung and Blaser [37], which is restricted to plane wave propagation in ducts, to account for higher-order duct modes. The in-duct modal decomposition scheme provided by Åbom is used in this study to determine the net acoustic power transmission, as it is convenient to use, even when only plane waves propagate. Influence of errors on the two-microphone method One of the most influential factors which affects acoustic power or intensity measurements using the two-microphone method is a phase mismatch between the microphones. The variation in the acoustic intensity measurements with and without correcting the microphone phase mismatches has been investigated previously [37, 43, 44]. Seybert and Ross [38] suggested a method to calibrate the gain and phase of the microphones by mounting the microphones in a rigid termination placed at the end of the duct. 19

52 2.3. Duct with an Attached Helmholtz Resonator A circuit-switching technique for correcting the gain and phase mismatch errors was reported by Chung [44] and Chung and Blaser [37]. This technique involves recording and processing the data by interchanging the instrumentation channels (microphones, preamplifiers, cables) and doubles the time for processing a single set of data. Seybert and Graves [45] compared these two calibration techniques and claimed that Seybert and Ross s technique [38] is better than Chung s [44, 37]. Other types of factors which affect the measurements recorded using the twomicrophone method include microphone spacing, type of microphones, tube attenuation and mean flow. Bodén and Åbom [46] discussed the bias and random errors in the measured transfer function, and suggested ways to minimise the effects of these errors. Because of the grazing and near-field effects around the microphones, the exact location of the duct cross-section for measuring the acoustic pressure cannot be known. This leads to a difference between the actual microphone separation distance and the acoustic pressure measuring (microphone) separation distance. Bodén and Åbom [46] also investigated the influence of microphone spacing on the transfer function measurement. Chu [47] investigated the effects of tube attenuation on the transfer function measurement, which were ignored by Bodén and Åbom [46]. Later, Åbom and Bodén [48] presented a complete error analysis for the in-duct two-microphone measurements, and included the effects of flow in the duct Effects of Flow on the Resonance Frequency and Acoustic Performance of HRs In this thesis, the scope of investigations is restricted to sound propagation with no mean flow. Therefore, only a limited literature survey is presented here related to the duct-hr systems with flow. When a HR is mounted on a duct, air flow can significantly affect its performance. In the experiments conducted by Meyer et al. [49] with turbulent flow in the main duct lined with undamped HR, 20% increase in the resonance frequency of the resonator 20

53 Chapter 2. Literature Survey was reported as the flow velocity reached approximately 70 m/s. Anderson [50] also reported a similar kind of shift in the resonance frequency of the resonator from 306 Hz to 380 Hz as the flow with velocity increased to 52.8 m/s. Panton et al. [51] also discussed the shift on the excitation of the HR by a turbulent boundary layer and attributed the reason to the interaction of the turbulence affecting the fluid moving back and forth in the neck. As a result, the effective exterior end-correction factor is changed, which subsequently changes the resonance frequency Applications of HRs HRs are used in air intake and exhaust systems to reduce noise from internal combustion engines. Munjal [26] discusses the application of HRs and other reactive (passive) elements with respect to vehicle exhaust systems. Tang and Sirignano [17] presented an overview of the use of HRs in reducing oscillations inside jet engines and furnaces. Other applications of HRs involve damping acoustic vibrations in combustors [52], gas turbines [53] and discharge system of compressors. HRs are commonly used in the construction of industrial exhaust systems to attenuate sound propagation in the exhaust. Howard et al. [54] demonstrated the use of HRs along with quarter wave resonators in designing a reactive silencer for the 980 MW power station. Recently the application of HRs was reported by Howard et al. [55] and Estève [56] for reducing sound transmission through payload fairings of launch vehicles. 2.4 Adaptive Helmholtz Resonator Because HRs are only effective over a narrow frequency band in close proximity to their resonance frequencies, any change in the excitation frequency reduces the effectiveness of the resonators. Hence, adaptive HRs have been developed in order to improve their effectiveness as the excitation frequency changes. 21

54 2.4. Adaptive Helmholtz Resonator There have been many previous studies that report on the successful implementation of adaptive HRs for attenuating the noise transmission in ducts. Niese and Koopman [57, 58] presented an overview about the phenomenon by which a centrifugal fan produces a blade passage frequency tone. They demonstrated the use of an adjustable quarter-wave resonator for reducing the intensity of the blade passage frequency tone propagating in a duct attached to the fan. The resonator was tuned by changing its cavity length with a piston. A microphone located in the duct was used to track the frequency spectra of sound pressure level inside the duct. No discussion was provided on the procedure used for tuning the resonator, but a reduction of up to 29 db was reported at the blade passage frequency tone of the fan. Lamancusa [59] developed a volume variable HR for attenuating the firing frequency noise of an IC engine. Two designs of the volume variable HR were proposed and tested: first, in which the cavity length was varied by moving a piston using a lead screw and a DC motor, and second, in which closable partitions were used to achieve a certain volume. Although it was stated that the resonators were adjusted with the engine speed, neither the details about the control mechanism nor the locations of error sensors were discussed. It was reported that an insertion loss of up to 30 db was achieved. For the purpose of suppressing variable tonal noise propagation in ducts, Matsuhisa et al. [60, 61] illustrated the use of adjustable HRs. Experiments were conducted with two types of resonators: first, in which the cavity volume was varied and second, in which the neck cross-sectional area was varied. The two microphones were located within the system (duct-hr) and were used to tune the resonators to the fundamental tone generated by a fan. The microphones were located in the resonator cavity and in the duct, respectively. The authors suggested that the tuning algorithm was based on achieving a phase difference of 90 degrees between the pressure in the cavity of the resonator and the pressure in the duct. It was reported that the sound pressure level in the duct reduced by 30 db for a speaker driven system, and 20 db for a fan driven 22

55 Chapter 2. Literature Survey system, with the difference probably a result of the mean flow associated with the fan. Bedout et al. [62] also developed an adaptive HR for minimising tonal noise propagation in ducts. The variation in the volume of the HR was facilitated by rotating an internal radial wall inside the resonator cavity with respect to a fixed internal wall. The control algorithm for tuning the resonator was based on the sound pressure level in the duct which was measured by a microphone located in the duct downstream of the resonator. Radcliffe et al. [63] reported on the development of a semi-active HR for which tuning was achieved by locating two pressure sensors, first, in the cavity of the resonator and second, in the duct downstream of the resonator. Recent development of an adaptive HR was reported by Estève et al. [56, 64, 65] for reducing broadband noise transmission into a rocket payload fairing. Experiments were conducted on a cylinder (diameter = 2.8 m and height = 2.5 m) that was representative of a payload fairing. Eight adaptive HRs were used to reduce broadband noise transmission in the cylinder at its third acoustic mode. The algorithm used to tune the resonators was based on the dot-product of two microphone responses located at the top of the closed end of the cavity and at the opening of the resonator. The resonators were considered to be tuned when the dot-product was zero. Many authors were granted patents for their work on the development of adaptive HRs. The usage of adaptive resonators presented in patents is mainly focused on attenuating the intake and exhaust noise of IC engines [66, 67, 68, 69, 70]. A few patents discuss the implementation of adaptive resonators in commercial industrial applications for reducing acoustic vibrations in combustors [52] and gas turbines [53]. Although all these patents describe which dimension of the resonator was varied, none of them describe the control mechanism for adjusting the resonator, or the cost function that was used to tune the resonator. 23

56 2.5. Summary of Gaps 2.5 Summary of Gaps It is widely believed that the optimum performance of a HR occurs at its resonance frequency i.e. when a resonator is mounted on a duct, the maximum reduction of the acoustic power transmission in the duct downstream of the resonator occurs at its resonance frequency, which is defined as the frequency corresponding to the maximum acoustic pressure at the top of the closed end of the resonator cavity [18]. Previous authors have presented different theoretical models (lumped [33], one-dimensional [20, 21], two-dimensional [22] and three-dimensional [23, 24] wave propagation) of duct-hr systems for calculating the transmission loss of duct-mounted HRs but all the theoretical models neglected the complex sound field at the interface of neck and duct. The difficulty in incorporating the complex sound field effects at the neck-duct interface led to the use of finite element modelling techniques for predicting the TL of duct-mounted HRs with greater accuracy compared to what is possible using existing theoretical models. Even with the estimation of TL using BEM, the frequency corresponding to the maximum TL was considered to be the resonance frequency of a HR. No attempt has been made to investigate if the frequency at which the maximum TL of a duct-mounted HR occurs equals the resonance frequency of the HR. It is shown in this study that a difference exists between the two frequencies. Even the resonance frequency of a duct-hr system, which corresponds to the maximum acoustic pressure in the cavity, differs from the frequency of maximum TL of the duct-mounted HR. Because the optimum performance of a HR when mounted on a duct (maximum reduction of in-duct acoustic power transmission) does not occur at its resonance frequency, some kind of provision is needed for adjusting the resonator dimension(s) so as to minimise the acoustic power transmission. Although adaptive HRs have been developed, the optimal tuning of the resonators was achieved by using the signal from a microphone located in the duct downstream of the resonator. Using the signal from a microphone located in the duct is the most convenient way to keep track of the changes in the excitation frequency. However, there are a number of problems which are asso- 24

57 Chapter 2. Literature Survey ciated with the in-duct location of the microphones and these have been described in chapter 1, section 1.1 [4]. No attempt has been made prior to the work described in this thesis to tune the resonator without using a pressure sensor in the duct to which the resonator is attached. This study reports on the tuning of a HR using sensors located only within the resonator, and is novel. The resonator is adjusted for its optimal performance using two microphones located in the resonator: (1) at the top of the closed end of the cavity, and (2) at the neck wall in close proximity to the neck-duct interface. A similar technique was adopted by Estève et al. [64, 65, 56] for controlling broadband noise transmission. The scheme for adjusting HRs was based on the dot-product of two microphone responses, first, located at the top of the closed end of the cavity, and second, located at the opening of the resonator. The neck opening of the resonators was adjusted until the dot-product was zero. A zero dot-product at a frequency corresponding to the maximum transfer function between two microphones can be achieved if the phase between them is 90 degrees. However, in the case of a HR or a duct-hr system, the frequency which corresponds to the maximum transfer function between two microphones located in the resonator cavity and neck changes with the location of the neck microphone, which consequently results in a zero dot-product at different frequencies, which do not necessarily correspond to the frequency of minimum in-duct acoustic power transmission. This is due to the presence of a complex sound field at the opening of the resonator. More importantly, achieving a phase difference of 90 degrees at the frequency corresponding to the maximum transfer function between the cavity and one of the neck microphones is not possible due to the influence of system damping. Hence, this technique does not guarantee optimum performance of a HR mounted on a duct with regard to controlling tonal excitation. A few authors have claimed the successful implementation of optimally adjusting the resonator on the basis of matching its resonance frequency with the excitation 25

58 2.5. Summary of Gaps frequency. A problem associated with this approach is that when a resonator is attached to a duct, the resonance frequency of the duct-hr system is different from the resonance frequency of the stand-alone HR. Moreover, it is shown in chapter 4 and 5 that the minimisation of acoustic power transmission in the duct occurs neither at the resonance frequency of the stand-alone HR nor at the resonance frequency of the duct-hr system. A cost function is developed in chapter 6 for optimally tuning the HR for changes in the excitation frequency. The cost function is based on the damping of the duct- HR system and the phase difference between the microphones located at the top of the closed end of the cavity and at the neck wall in close proximity to the neck-duct interface. 26

59 Chapter 3 Theoretical Modelling 3.1 Introduction This chapter discusses the theoretical modelling of a stand-alone HR and a circular duct with an attached HR (a coupled duct-hr system). It begins with equivalent circuit analysis of stand-alone HRs which ultimately leads to the classical formula developed by Rayleigh [6] for calculating their resonance frequencies. In addition to the classical formula, two other existing mathematical equations presented by Panton and Miller [7] and Li [8] for calculating the resonance frequencies of cylindrical HRs are discussed. Apart from stating the limitations and conditions related to each of the three mathematical formulas, the differences in the calculated resonance frequencies obtained using the three equations are also highlighted. The theoretical modelling of a circular duct with an attached HR was accomplished by using the well known transfer matrix method [26]. Providing the background to the basic theory behind the transfer matrix method, the section on the transfer matrix method begins with a description of the transfer matrix of a uniform circular duct. The net acoustic power transmission in the duct, which was modelled as being driven by a piston at one end and open to free space at the other, was estimated by two different approaches: (1) using the estimates of acoustic pressure and velocity at the terminating open end (duct exit), and (2) decomposing the sound field in the duct, 27

60 3.2. Stand-alone Helmholtz Resonator also referred to as the two-microphone method developed by Chung and Blaser [37] and extended by Åbom [42]. The analysis of a duct-hr system is presented and the complete procedure from building the transfer matrix of the duct-hr system to estimating the in-duct net acoustic power transmission downstream of the HR is described. The performance of the HR was evaluated by comparing the in-duct net acoustic power transmission without and with the HR. An overview is given of the end-correction factor of the neck of the HR at the neckduct interface, when the resonator is mounted on a duct. As in the transfer matrix method, the inclusion of both end-correction factors of the neck of the HR plays a vital role in the analysis of the duct-hr system. Due to the lack of procedures to accurately calculate the neck-duct interface end-correction factor, the effects of changing this end-correction factor on the acoustic pressure at different locations within the duct-hr system, pressure transfer function of the HR and in-duct net acoustic power transmission downstream of the HR are also shown. Finally, the importance of mounting the HR at the correct location on the duct is illustrated by showing the effects of HR mounting location on the in-duct net acoustic power transmission downstream of the resonator. 3.2 Stand-alone Helmholtz Resonator A stand-alone HR refers to an independent passive noise control device which is not attached to any acoustic system Equivalent Circuit Analysis A HR is principally composed of two physical parts: (1) a rigid-walled cavity of volume V, and (2) an opening (port) or neck of radius r and length l n. The simplest geometry of the HR is shown in figure

61 Chapter 3. Theoretical Modelling Figure 3.1: A Helmholtz resonator Using the assumption that all the characteristic dimensions of the resonator are smaller than an acoustic wavelength, the sections of the resonator can be divided into three acoustic elements: (1) the fluid, which reciprocates inside the neck, (2) the opening, which radiates sound and (3) the fluid, which is compressed inside the cavity [25]. Mechanical Analogue A HR can also be considered as an equivalent mechanical system where the previously stated three acoustic elements correspond to a mass element, a resistance element and a stiffness element, respectively. The dynamic behaviour of the system can be derived by considering a piston driving the fluid in the neck. The displacement of the fluid in the neck, ξ, can cause the volume of the fluid inside the cavity to be adiabatically compressed. The governing linear differential equation for the displacement of the fluid in the neck is given by [25] where, M d2 ξ dt 2 + C dξ dt + Kξ = p eπr 2 (3.1) p e is the instantaneous acoustic pressure at the opening of the neck, 29

62 3.2. Stand-alone Helmholtz Resonator r M K C is the radius of the neck, is the effective mass of the fluid in the neck, is the stiffness of the volume of the fluid in the cavity, is the viscous damping, which depends on the dissipation of the resonator which occurs due to heat conduction and viscous effects in the neck. The mass of the air in the neck and the stiffness of the air in the cavity models are given by [25] M = ρπr 2 l eff (3.2) K = ρc2 (πr 2 ) 2 V (3.3) where, ρ c is the density of the fluid medium, is the speed of sound in the fluid medium, l eff is the effective length of the neck which is longer than the actual (physical) length. It includes the two end-corrections for each side of the neck, and further discussed in section A HR can be interpreted as a forced harmonic oscillator behaving as a single degreeof-freedom system with one natural frequency. This natural frequency is termed as its resonance frequency, f r, and is given by [25] f r = 1 K 2π M = c πr 2 2π l eff V (3.4) The above equation represents the undamped resonance frequency of the Helmholtz resonator of volume V, neck radius r and neck length l n. 30

63 Chapter 3. Theoretical Modelling Electroacoustical Analogue In terms of electroacoustic analogies, inductance is referred to as inertance and capacitance as compliance. It is equivalent to say that the mass of fluid in the neck of the resonator represents a lumped intertance and the acoustic spring stiffness of the cavity represents a lumped compliance and these are expressed as follows [12]: M e = ρl eff πr 2 (3.5) C e = V ρc 2 (3.6) Ignoring the dissipation in the resonator, its resonance frequency is given by f r = 1 1 = c πr 2 2π Me C e 2π l eff V (3.7) End-Corrections The term end-correction is a length additional to the actual length of a tube or an orifice that accounts for the entrained mass of fluid that vibrates outside the tube s or orifice s opening. When a small tube or an orifice is exposed to a large volume or an open space, the fluid present inside the tube experiences a mass loading by the fluid just outside the opening. This additional mass of fluid moves with the same velocity as that of the fluid in the tube. In the case of a HR, the lumped mass of fluid in the neck incorporates the movement of those fluid particles which are in close proximity to both its ends. This implies that two end-corrections must be added to the actual length of the neck in order to calculate the effective length, l eff, which actively takes part in the induced motion of the fluid particles present inside the neck [6]. These end-correction models depend on the geometry of the two ends of the neck and can be flanged or unflanged. Rayleigh [6] presented a well known expression for the exterior end-correction of the neck of the 31

64 3.2. Stand-alone Helmholtz Resonator HR, δ e, radiating into free space which is given by [6] δ e = 8r 3π (3.8) The derivation of equation (3.8) is based on a piston radiating in an infinite baffle [11, 12, 25]. This expression was indiscriminately used for calculating the interior endcorrection of the neck as well until Ingard [13] came up with a new expression. He stated that equation (3.8) is only valid for the cases where the dimensions of the neck are very small as compared to the dimensions of the cavity and can lead to significant errors when this condition is not satisfied. He proposed a new model for the interior end-correction factor, δ i, which was based on a piston radiating into a tube (in this case, the cavity) of radius R. It is given by δ i 0.48 ( πr r ) 8r ( r ) R 3π R (3.9) However, Ingard [13] documented that equation (3.9) is only valid when r R < 0.4 and when this is not the case, a design chart [13] should be used to approximate the interior end-correction of the neck of the HR Models to Predict Resonance Frequencies of HRs Classical Model Because a HR can provide a significant reduction in noise levels at a frequency which is close to its resonance frequency, it is of paramount importance for its optimal performance that its resonance frequency is accurately calculated. The classical formula for predicting the resonance frequency of a HR developed by Rayleigh [6], is given by equation (3.4) or (3.7). The derivation of these equations was based on an assumption that the dimensions of the resonator are small compared to an acoustic wavelength of interest. As described in chapter 2, section 2.2.2, the formula for calculating the resonance 32

65 Chapter 3. Theoretical Modelling frequencies of HRs has been modified by many researchers [7, 8, 16, 17]. However, this study will consider the formulas presented by Panton and Miller [7] and Li [8]. The reason being that cylindrical HRs were used in this study and the formulas in references [7, 8] were exclusively developed for cylindrical HRs. These formulas are described below. Improved Models Panton and Miller [7] showed that the classical formula (equation (3.4)) is not valid if the dimensions of the resonator exceed 1/16 of the wavelength. They presented an improved formula for calculating the resonance frequencies of cylindrical HRs as follows: f r = c A n 2π l eff V c L2 c A n (3.10) where, An Vc Lc is the cross-sectional area of the opening, is the volume of the cavity, is the length of the cavity and l eff is the effective length of the neck. As described in chapter 2, section 2.2.2, equation (3.10) was derived assuming onedimensional wave propagation in the cavity but the neck was treated as a lumped mass. Unlike the assumptions for the classical formula, equation (3.10) is accurate for cavity lengths which may be similar to or longer than a wavelength but the cavity diameter and neck dimensions must be smaller than a wavelength. Panton and Miller s [7] investigations highlighted certain aspects pertinent to the discrepancies between theoretical and experimental measures of the resonance frequencies of resonators with dimensions exceeding 1/16 of the wavelength. 33

66 3.2. Stand-alone Helmholtz Resonator The mathematical model presented by Li [8] is given by f r = c 2π 3l effa c + L c A n 2leff 3 A c + ( 3leff A c + L c A n 2l 3 eff A c ) 2 + 3A n l 3 eff L ca c (3.11) where, Ac is the cross-sectional area of the cavity and all the other variables have the same meaning as they do in equation (3.10). His derivation was also based on the wave-tube theory but was more generalised than the model derived by Panton and Miller [7]. Equation (3.11) can be used for calculating the resonance frequencies of cylindrical HRs when the dimensions of the neck and the cavity length are comparable to or longer than a wavelength, but the diameter of the cavity must be smaller than a wavelength Analysing the Models The three different models described above by equations (3.4), (3.10) and (3.11) result in different values for the resonance frequency of the same HR. The predicted resonance frequencies using these three models will now be compared for one example resonator. The dimensions of the HR were chosen randomly as: cavity diameter = m, neck diameter = m and physical neck length = m. In order to obtain and compare a few different values of the resonance frequencies of the resonator, the cavity length was varied from m to m. The exterior and interior end-correction factors were calculated using equations (3.8) and (3.9), respectively. Resonance frequencies from all the models were calculated and plotted as a function of their cavity lengths, and are shown in figure 3.2. It is very clear from the figure that selecting a correct model for calculating the resonance frequencies of the HR is extremely important. Here, the predictions obtained from Li s model (equation (3.11)) are correct because all the dimensions of the analysed HRs conformed with the conditions associated with Li s model. As the dimensions of the HR were greater than 1/16 of the wavelength, the estimates obtained using the classical model (equation (3.4)) are incorrect. Figure 34

67 Chapter 3. Theoretical Modelling classical Panton & Miller Li resonance frequency (Hz) cavity length (m) Figure 3.2: Comparison of the resonance frequencies of cylindrical HRs predicted using the classical, Panton and Miller s and Li s models. 3.2 shows that the frequencies predicted using the classical model (equation (3.4)) and Li s model (equation (3.11)) approximately follow a similar trend with almost a constant difference between their estimates, except for higher frequencies, at which the assumption related to Li s model, that the cavity diameter should be smaller than the wavelength, was not valid. The frequencies predicted using Panton and Miller s model (equation (3.10)) shift from Li s to the classical model s predictions as the estimates approach towards higher frequencies. This occurs for two reasons, which are related to the derivation of equation (3.10) by Panton and Miller [7]. 1. the neck dimensions and the cavity diameter were not smaller than 1/16 of a wavelength, as were assumed during the derivation, and 2. the length of the neck was very small compared to the cavity as was assumed during the derivation. As the length of the neck becomes comparable to or greater than the cavity length, predictions become less reliable. By selecting the appropriate mathematical model with respect to the dimensions of the resonator, the resonance frequencies of HRs can be accurately predicted. The selection 35

68 3.2. Stand-alone Helmholtz Resonator criterion is described in table 3.1. The aforementioned models are valid when a HR is considered as a stand-alone device, for which both the end-corrections of the neck are well known. Conditions related to the dimensions of cylindrical HRs All the HR dimensions should be < 1/16 of a wavelength Cavity length can be > a wavelength, but the neck dimensions and cavity diameter should be < 1/16 of a wavelength Cavity length and neck dimensions can be > a wavelength, but the cavity diameter should be < a wavelength Mathematical model to use for calculating the resonance frequencies of cylindrical HRs classical model (equation (3.4)) Panton and Miller s model (equation (3.10)) Li s model (equation (3.11)) Table 3.1: Criterion for selecting an appropriate theoretical model for calculating the resonance frequencies of cylindrical HRs Summary So far in this chapter, the theoretical models for predicting the resonance frequencies of HRs have been discussed. The importance of selecting an appropriate model is demonstrated with the help of a graph and shows that there can a difference of up to 9.7% in the predicted resonance frequencies, depending on the model used. The criterion for selecting an appropriate theoretical model for calculating the resonance frequencies of cylindrical HRs was also described. The next section describes the theoretical modelling of a circular duct with an attached cylindrical HR in order to evaluate the acoustic performance of the HR. 36

69 Chapter 3. Theoretical Modelling 3.3 Circular Duct with an Attached Helmholtz Resonator When a HR is mounted on a duct, a coupled system is created whose resonance frequency is different to that of the stand-alone HR. A HR works by causing an impedance change in the acoustic system at its point of insertion. HRs act like passive bandstop filters barring the transmission of acoustic power past their location at frequencies in close proximity to their resonance frequencies. In this thesis, the evaluation of the acoustic power reduction in a duct downstream of a HR is referred to as an acoustic performance of the HR mounted onto the duct, which indicates the frequency at which the maximum reduction of in-duct net acoustic power transmission occurs. The transfer matrix method was used to theoretically evaluate the acoustic performance of a duct-mounted HR, and is described in the next section Transfer Matrix Method The transfer matrix method (also known as transmission matrix or four-pole parameter representation) for analysis of acoustic duct systems has been extensively explored by other researchers [26, 71, 72]. It has been widely accepted as a tool to analyze complex systems due to its computational efficiency and flexibility. Snowdon [73] documented the basic concept of four poles of a dynamic system. He elaborated the four-pole parameters of a linear mechanical system which comprised either lumped or distributed elements or a combination of any such elements. Munjal [26] describes the use of the transfer matrix method applied to acoustic systems. The essence of the transfer matrix method is its ability to break the system into discrete elements which can be modelled by using basic acoustic principles. This is facilitated by taking two state variables, acoustic pressure p and acoustic mass velocity q, at the input and output sides of an element and relating them by defining a

70 3.3. Circular Duct with an Attached Helmholtz Resonator matrix [26]. p r q r = A 11 A 12 A 21 A 22 p r 1 q r 1 (3.12) Equation (3.12) shows the relation between two state variables at either side of the element r. The elements of the transfer matrix A 11, A 12, A 21 and A 22 are determined by applying the following relationships to the pressure and mass velocity to the element of interest: A 11 = p r p r 1 q r 1 = 0, A 12 = p r q r 1 p r 1 = 0, A 13 = q r p r 1 q r 1 = 0, A 14 = q r q r 1 p r 1 = 0 As evident from the above relationships, each element gives the unique response of a system depending on the terminating boundary condition(s). For example, A 11 is the ratio of the upstream pressure to the downstream pressure, provided the downstream end is rigidly terminated. Similarly, A 12 is the ratio of the upstream pressure to the velocity at the downstream end, which is assumed to be totally free or unrestrained, and so on. The elements A 11 and A 22 are dimensionless: A 12 has the dimensions of impedance and A 21 the dimensions of admittance [73]. The transfer matrix method is very useful in evaluating the effectiveness of acoustic filters (or mufflers) by estimating their acoustical performance parameters. These performance parameters are: (1) Insertion Loss (IL), (2) Transmission Loss (TL) and (3) Level Difference (LD) or Noise Reduction (NR). An overview of these parameters is given below [26]. Insertion Loss, IL This is defined as the difference in the acoustic power radiated without a filter and with a filter. It is expressed as 38

71 Chapter 3. Theoretical Modelling IL = W 1 W 2 where, subscripts 1 and 2 denote systems without filter and with filter, respectively. Estimation of insertion loss generally requires mounting microphone(s) at a point(s) downstream of the filter. It is also of paramount importance that the source and radiation impedances are known before hand. Transmission Loss, TL This is the difference between the acoustic power incident on the filter, W i, and the acoustic power transmitted, W t, through it. It is expressed as TL = W i W t Estimation of transmission loss requires mounting microphones upstream and downstream of the filter. Transmission loss of a filter is an exclusive characteristic of the filter, independent of source impedance, and requires an anechoic termination at the downstream end to be measured effectively. Level Difference, LD or Noise Reduction, NR This is the difference between the sound pressure levels upstream and downstream of the filter. It is expressed as LD = p u p d where, p u and p d denote the measure of pressure upstream and downstream of the filter, respectively. Keeping in mind that in industry, one of the potential applications of HRs is to attenuate tonal noise transmission generated by a fan or a blower or an internal combustion engine, which propagates in an exhaust duct, the downstream radiating end 39

72 3.3. Circular Duct with an Attached Helmholtz Resonator of the duct must be modelled as an open end radiating into free space. This eliminates the option of using TL, which mandates the need for anechoic termination at the downstream end, as an acoustical parameter for evaluating the HR s performance. More importantly, because TL is a sole property of a filter, it does not tells a designer the actual sound reduction achieved in an acoustic system after the filter is installed. In contrast, IL, which involves measuring the acoustic power in a system before and after the installation of a filter, is much more useful from a practical point of view. In this study, IL is used as the evaluation parameter of the HR. However, results are presented as acoustic power transmitted down the duct before and after the installation of the HR. This is analogous to IL which is the difference between the two. Unlike the case of an anechoic termination which does not reflect a travelling wave back to the source, an open end almost fully reflects the incident wave with opposite phase; the remainder is radiated to the free space. The total pressure inside the duct at any location under such reflective conditions is the contribution from the incident and reflected waves. Modal decomposition can be used to determine the characteristics of incident and reflected waves, and ultimately, the net acoustic power transmission in the duct Uniform Circular Duct A duct is said to be uniform when its cross-sectional area along its whole length remains constant. Figure 3.3 shows a circular duct of length l, radius a and cross-sectional area S. The left end of the duct, referred to as a source end, was modelled as being driven by a piston with a unit acoustic volume velocity and the right end, referred to as a duct exit, was modelled as open end radiating into free space. Sections to describe a step-by-step procedure from building the transfer matrix of a uniform circular duct to estimating the in-duct net acoustic power transmission. The in-duct net acoustic power transmission was estimated by using two methods: (1) the product of acoustic pressure and volume velocity at the duct exit, and 40

73 Chapter 3. Theoretical Modelling Figure 3.3: A schematic of a uniform circular duct modelled as being driven by a constant amplitude piston source mounted at one (left) end and open at the other (right). (2) the in-duct modal decomposition of the sound field. As the modal decomposition of a sound field requires the estimate of the acoustic pressure at two different locations, the method for calculating the acoustic pressure at any arbitrary location in the duct is also described Cut-on Frequency / Plane Wave Assumption The transfer matrix method works on a principle which assumes that only plane waves propagate inside the duct. The validity of the plane wave assumption is dependent upon the dimensions of the duct with respect to the frequency of interest. The frequency below which only plane waves propagate inside the duct, of radius a, is known as cut-on frequency and is given by [10] f c = c 2πa (3.13) where, c is the speed of sound and a is the radius of the duct. For the duct studied here, which has a diameter of m and length of 3 m, the cut-on frequency is 1292 Hz. Thus, the plane wave mode, also denoted by (0,0), is the only mode which propagates below 1292 Hz. 41

74 3.3. Circular Duct with an Attached Helmholtz Resonator Radiation Impedance In order to model the end of a duct as open and radiating into free space, the calculation of radiation impedance was required. Radiation impedance is the impedance which is imposed by the atmosphere to the sound waves radiating from the open end of a duct. It is approximated by assuming a circular piston located in an infinite baffle and has been documented in many acoustic text books (for example see references [10, 11, 12, 25, 26]). It is the ratio of the force exerted by the piston on the acoustic field to the velocity of the piston. It is a complex quantity and expressed as Z r = R + jx (3.14) The real part of radiation impedance, R, is termed radiation resistance and represents the energy radiated away from the open end in the form of sound waves. The imaginary part, X, is termed radiation reactance and represents the mass loading of the fluid (air) just outside the open end. It is well known that the theoretical expression for the impedance of an unflanged open duct, with plane waves propagating inside it, is given by [25, 26] Z l = ρc S [ (ka) j (0.6)ka ] (3.15) where, Z l is the radiation impedance, ρ is the density of the fluid medium, c is the speed of sound, k = ω c is the wave number and a is the radius of the duct. 42

75 Chapter 3. Theoretical Modelling Transfer Matrix of a Circular Duct The transfer matrix of a circular duct, also referred to as a distributed element (denoted by r), of uniform cross-sectional area S and length l, is given by [26] p r q r = cos(ˆkl) j sin(ˆkl) Y r jy r sin(ˆkl) cos(ˆkl) p r 1 q r 1 (3.16) where, Y r = c S is the characteristic impedance, and ˆk = k (1 + iη) is the complex wave number. Complex wave numbers were introduced to incorporate damping in the system. η is a loss factor which is related by a factor of 2 to the critical damping ratio. The value of η was kept as for all the transfer matrix method analyses conducted in this study. This value was selected in order to match the theoretical calculations with those of the numerical (ANSYS) and experimental results, which are presented in chapters 4 and 5, respectively. p r, p r 1 and q r, q r 1 are the acoustic pressures and acoustic mass velocities at the extreme ends of the duct, respectively (input and output sides). Note that these quantities can be rms values, amplitudes or instantaneous, provided all the quantities are of the same type. Rewriting equation (3.16) by relating acoustic volume velocity and acoustic pressure instead of acoustic mass velocity and acoustic pressure respectively, gives v r p r = cos(ˆkl) j ρc S sin(ˆkl) j S ρc sin(ˆkl) v r 1 (3.17) cos(ˆkl) p r 1 where v r and v r 1 represent acoustic volume velocities at input and output sides of element r, respectively. 43

76 3.3. Circular Duct with an Attached Helmholtz Resonator This transition affected the definition of the elements of the transfer matrix compared to those defined by Munjal [26] (in equation (3.16)), which relates acoustic mass velocity q and acoustic pressure p at the input and output sides of the duct. The reason for implementing this transition is that acoustic pressure can be normalised by the input acoustic volume velocity of the source (piston). This enables a comparison of the theoretical predictions with those of the numerical and experimental results. Also, the sequence of the state variables at input and output sides of the element was changed for the convenience of the analysis. The derivation of equation (3.17) using equation (3.16) is described in Appendix A. According to figure 3.3, v 0 and p 0 are acoustic volume velocity and pressure at the input side of the duct. Similarly, v l and p l have the same definition at the output side. Relating these state variables gives the transfer matrix of the duct shown in figure 3.3, which is expressed as v 0 p 0 = cos(ˆkl) j S ρc sin(ˆkl) j S ρc sin(ˆkl) cos(ˆkl) v l p l (3.18) In order to make the utilisation of equation (3.18) easier for future calculations, the elements of the transfer matrix were assigned alphabetical names as shown below v 0 p 0 = A l C l B l D l v l p l (3.19) Once the transfer matrix of a duct is generated, it can be used to estimate the state variables at any location and of most importance, the net acoustic power transmission in the duct Acoustic Pressure and Velocity at the Duct Exit This section details the procedure to estimate the acoustic pressure and volume velocity at the duct exit. Estimation of these two variables will in turn facilitate the estimation 44

77 Chapter 3. Theoretical Modelling of in-duct net acoustic power transmission. From equation (3.19), we get v 0 = A l v l + B l p l (3.20) By using the fact that Z l = p l v l, equation (3.20) can be written as Solving equation (3.21) for p l gives v 0 = A l p l Z l + B l p l (3.21) p l v 0 = Z l A l + B l Z l (3.22) Equation (3.22) gives the measure of the ratio of acoustic pressure p l at the duct exit to the input volume velocity v 0. Z l is the radiation impedance of an unflanged open end of a duct and can be calculated by using equation (3.15). Similarly, solving equation (3.20) for v l, which incorporates the use of equation (3.22), we get v l v 0 = ( 1 B lz l ) A l + B l Z l (3.23) A l Equation (3.23) gives the measure of the ratio of acoustic volume velocity v l at the duct exit to the input volume velocity v 0. If v 0 is set to unity (1 m 3 /sec), then p l and v l would be p l = Z l A l + B l Z l (3.24) v l = ( 1 B ) lz l A l + B l Z l (3.25) A l These two equations (3.24) and (3.25) represent the estimates of acoustic pressure 45

78 3.3. Circular Duct with an Attached Helmholtz Resonator and volume velocity at the duct exit driven by a piston at the source end with a unit volume velocity. Knowledge of these two variables is sufficient to estimate the net acoustic power transmission in the duct, which is discussed in section (see equations (3.49) and (3.50)). Also, another method which is used in this study for estimating the in-duct net acoustic power transmission is the modal decomposition of the sound field in a duct [42]. This is also referred to as the two-microphone technique developed by Chung and Blaser [37]. In addition to using the modal decomposition scheme for the theoretical analysis of a uniform circular duct and a circular duct with an attached HR, this scheme was also used for the numerical and experimental work, which will be discussed further in chapters 4 and 5, respectively. However, in-duct modal decomposition of the sound field requires an estimate of the acoustic pressure at two different locations inside a duct. The next section details the procedure for estimating the acoustic pressure at different locations inside a duct Acoustic Pressure and Velocity at Arbitrary Locations Figure 3.4 shows a schematic of a uniform duct of length l depicting the end-correction, l 0, at the open end of the duct. Suppose that acoustic pressure and acoustic volume velocity at location x in figure 3.4 need to be estimated and are denoted by p x and v x, respectively. As evident from equations (3.24) and (3.25), the value of acoustic impedance at location x is required and calculation of this is detailed in the following section. Acoustic Impedance at Arbitrary Location Inside a Duct From standard acoustic text books [11, 12, 25, 26], we know that specific acoustic impedance at any location inside a duct can be expressed as Z x = ρc S a + exp jkx +a exp +jkx (3.26) a + exp jkx a exp +jkx 46

79 Chapter 3. Theoretical Modelling Figure 3.4: A schematic of a uniform duct depicting the end-correction at the unflanged open end of the duct. where, a +, a sign +sign x are the modal amplitudes of incident and reflected acoustic waves, represents the propagation of acoustic wave in the +ve x direction, represents the propagation of acoustic wave in the -ve x direction, and is the location along the duct. By using equation (3.26), the values of acoustic impedance at location x=0 and x=l are given by Z 0 = ρc S a + + a a + a (3.27) Z l = ρc S a + exp jkl +a exp +jkl (3.28) a + exp jkl a exp +jkl Eliminating constants a + and a facilitates a relationship between Z 0 and Z l, which is given by Z 0 = Z l + j ρc S tan(kl) 1 + j S ρc Z l tan(kl) (3.29) 47

80 3.3. Circular Duct with an Attached Helmholtz Resonator If the impedance at the termination of a duct is known, the value of the acoustic impedance at any point inside the duct can be calculated by using equation (3.29). In order to distinguish the source and radiation impedance, equation (3.29) was rewritten with a slight modification in terms of nomenclature. Denoting the acoustic impedance at x=0 by Z s (source impedance) and x=l by Z r (radiation impedance), we get Z s = Z r + j ρc S tan(kl) 1 + j S ρc Z r tan(kl) (3.30) Finally, the value of acoustic impedance at location x looking into the tube from the source end (x=0) can be given by (utilising equation (3.30)) Z x = Z s j ρc S tan(kx) 1 j S ρc Z s tan(kx) (3.31) The alternate expression for calculating the value of Zx, by using the value of Zr instead of Zs, is Z x = Z r + j ρc S tan(k(l x)) 1 + j S ρc Z r tan(k(l x)) (3.32) Equation (3.32) shows the acoustic impedance at location x looking into the tube from the open end (x=l). End-Corrections The vital point which is worth noting here, before using equations (3.31) and (3.32) for calculating the value of Z x, is to account for the end-correction of an unflanged open end of a duct. The expression for the end-correction of an unflanged open end of a duct radiating into free space is given by [10, 12, 25, 26] l 0 = 0.6a (3.33) 48

81 Chapter 3. Theoretical Modelling where, a is the radius of the duct. After incorporating the value of end-correction, l 0, in equations (3.31) & (3.32), Z x becomes Z x = Z s j ρc S tan(k(x l 0)) 1 j S ρc Z s tan(k(x l 0 )) (3.34) Z x = Z r + j ρc S tan(k(l x + l 0)) 1 + j S ρc Z r tan(k(l x + l 0 )) (3.35) Once the value of acoustic impedance at a particular location in the duct is known, the acoustic pressure and velocity can be calculated by using equations (3.24) and (3.25), respectively. The acoustic pressure at location x with respect to unit v 0 is given by p x = where, A x = cos(ˆkx) and B x = j S ρc sin(ˆkx). by Z x A x + B x Z x (3.36) Similarly, the acoustic volume velocity at location x with respect to unit v 0 is given v x = ( 1 B ) xz x A x + B x Z x (3.37) A x The acoustic pressure and volume velocity at any location inside the duct can be estimated similarly. Knowledge of the acoustic pressure at two different locations, appropriately spaced is required to estimate the in-duct net acoustic power transmission using the modal decomposition scheme which is discussed in the next section. 49

82 3.3. Circular Duct with an Attached Helmholtz Resonator Modal Decomposition of Sound Field As stated earlier at the end of section , the in-duct modal decomposition of the sound field was used to estimate the theoretical, numerical and experimental results of the net acoustic power transmission in the duct. Modal decomposition is a technique for determining the amplitudes of the acoustic waves propagating each way inside the duct. It facilitates the separation of modes into incident and reflected parts. Once the acoustic power associated with the incident and reflected waves is determined, the net acoustic power transmission is simply their difference. Åbom [42] extended the two-microphone technique developed by Chung and Blaser [37] and documented the scheme for in-duct modal decomposition. Generally, for a uniform straight duct with rigid walls and an arbitrary crosssectional shape carrying an axial mean flow, the sound field in the duct can be expressed as [42, 74] where, ˆp = [â+ exp ik +nx +â exp ] +ik nx Ψ n (3.38) n=0 ˆp â +, â is the acoustic pressure, are the modal amplitudes of acoustic pressure associated with the incident (positive x direction) and reflected waves (negative x direction), ˆk +n, ˆk n n are the axial wave numbers in the positive and negative directions, is the mode number and Ψ n is the eigenfunction for mode n. For the case of a uniform circular duct with rigid-walls and carrying no axial flow, with only plane waves propagating inside it, equation (3.38) can be written as: ˆp = â + exp ikx +â exp +ikx (3.39) 50

83 Chapter 3. Theoretical Modelling The terms â + and â can be calculated by measuring the pressure at two locations in the duct and using the two-microphone technique [37]. Expressing the results of such a measurement in a matrix formulation yields [42] ˆp = Mâ (3.40) where, ˆp is a [2 1] column vector containing the measures of the acoustic pressures at two different locations, M â is a [2 2] modal matrix containing the propagation terms, and is a [2 1] column vector containing the unknown modal amplitudes. Writing vector ˆp in terms of transfer function between the two pressure measuring (microphone) locations, yields [42] ˆp = ˆp r H (3.41) where, ˆp r H is the acoustic pressure at one of the two locations, and is the [2 1] column vector containing the transfer function between the two locations. By using equation (3.41), equation (3.40) becomes ˆp r H = Mâ (3.42) and solving equation (3.42) for â, gives â = ˆp r M 1 H (3.43) 51

84 3.3. Circular Duct with an Attached Helmholtz Resonator â + â = ˆp r 1 1 exp iks exp +iks 1 H 12 (3.44) For this study, ˆp r was measured at the first (microphone) location, H 12 represents the pressure transfer function between two microphones and s is the axial distance between the two pressure measuring locations. Solving equation (3.44), for the estimates of â + and â, yields â + = ˆp ( ) r exp +iks H 12 (3.45) exp +iks exp iks â = ˆp ( r H12 exp iks) (3.46) exp +iks exp iks The acoustic power associated with the incident and reflected waves is given by [42] W + = S â + 2 2ρc W = S â 2 2ρc The net acoustic power transmitted in x direction is estimated as the difference between W + and W and is expressed as W net = W + W (3.47) W net = S [ â+ 2 â 2] (3.48) 2ρc In-Duct Net Acoustic Power Transmission Sound power is the rate at which energy is radiated by a sound source. The net acoustic power transmission in the duct is estimated by two methods: 1. by the in-duct modal decomposition of the sound field as described above, and 2. by using the estimates of acoustic pressure, p l and acoustic volume velocity, v l 52

85 Chapter 3. Theoretical Modelling at the duct exit. The net acoustic power flux associated with a plane wave inside the duct is given by [26] W net = p rms v rms (3.49) where p rms = p 2 and v rms = v 2, where p and v represent pressure and volume velocity amplitudes, respectively. Therefore, W net = 1 pv (3.50) 2 If we assume that the analysis in the preceding section used amplitude quantities then we can substitute the values of p l and v l from equations (3.24) and (3.25), respectively in equation (3.50). This gives the estimate of net acoustic power transmission in the duct when the fluid (air) present in the duct is assumed to be driven by a constant amplitude piston source with a unit volume velocity. Figure 3.5 shows a plot for the normalised net acoustic power transmission in the duct. As described in section , the acoustic volume velocity (v 0 ) at the end of the duct, which was assumed to be driven by a piston, was considered to be unity. Consequently, the calculated net acoustic power resulted in being normalised by the square of acoustic volume velocity. The reason for normalising the acoustic power with the square of the volume velocity is to enable the comparison of the theoretical (transfer matrix) results with those of the numerical (ANSYS) and experimental results. The frequencies corresponding to the peaks (high sound power level) in this plot indicate the acoustic resonances of the duct driven by a piston source at one end and open at the other. These resonance frequencies are closely related to a closed-open duct configuration, the solution for which is given by Blevins [75] f r = nc 4l (3.51) 53

86 3.3. Circular Duct with an Attached Helmholtz Resonator normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) frequency (Hz) Figure 3.5: Theoretical (transfer matrix method) prediction for the net acoustic power transmission in a uniform circular duct driven by a constant amplitude piston source mounted at one (left) end and open at the other (right). Mode no. Resonance frequency (Hz) Error (%) Transfer matrix Blevin s formula for formulation with zero-pressure radiation impedance boundary condition boundary condition Table 3.2: Theoretical (Blevin s formula and transfer matrix method) predictions for the first seven acoustic resonance frequencies of a piston driven-open duct system. 54

87 Chapter 3. Theoretical Modelling where, n is the mode number, c is the speed of sound and l is the length of the duct. Table 3.2 compares the acoustic resonances of the duct estimated using the transfer matrix method and Blevin s formula, equation (3.51). The results from the transfer matrix method are accurate because during the formulation of the transfer matrix of the duct, both radiation impedance and end-correction of the unflanged open end of the duct were considered. The discrepancies in the estimates of the resonance frequencies shown in table 3.2 are because Blevin s solution accounts for neither the radiation impedance nor the end-correction of the open end of the duct. This point will be discussed further in chapter 4; Numerical Modelling. The following section describes the transfer matrix analyses for a HR attached to a duct Duct-HR System Mounting the resonator on a duct will drastically reduce the noise transmission at a particular frequency which is a characteristic of the resonator s geometry. Comparison of the acoustic power transmission inside the duct without and with the resonator, i.e. the IL, can be used to evaluate its performance. The following sections discuss the transfer matrix formulation of a duct with an attached HR with a view to obtaining expressions for estimating the acoustical performance of the HR Neck-Duct Interface End-Correction Factor When a HR is mounted onto a duct, the end-correction model, which was described in section as an exterior end-correction (δ e ) and taken as 8r, is no longer 3π appropriate. This is because the end of the neck of the HR attached to the duct no longer simulates the geometry of an infinite baffle and connection to free space, which was the fundamental basis for deriving the exterior end-correction model [6]. This endcorrection model, which from now on will be referred to as a neck-duct interface endcorrection factor, has not been very well understood due to the difficulty in interpreting 55

88 3.3. Circular Duct with an Attached Helmholtz Resonator the non-planar sound field in the regions of the duct adjacent to the neck opening. Even though only plane waves propagate down the duct, the non-planar sound field in the region of the duct adjacent to the neck opening is caused by the near field generated at the neck opening. Selamet et al. [23, 24] demonstrated the complex sound field in the regions in close proximity to the neck-duct interface using BEM. As described in chapter 2, section 2.3.2, Onorati [30] suggested that a correction factor of 0.3a should be added at the junction/interface of three pipes, such as at the interface of a column resonator and a main duct or a HR and a main duct, where a denotes the radius of the main duct. He claimed that his theoretical predictions of the transfer function between two microphones, one located upstream of the HR in the duct and the other, at the downstream of the HR in an open environment matched closely with the experimental results. Ji [31] suggested two curve-fitted expressions for the neck-duct interface end-correction factor which can be used for a circular duct-hr system. The equations were based on the ratio of neck diameter to duct diameter and are given by ( ) ( ) δ e r r 2 r = r 0.694, a a a 0.4 (3.52) ( ) δ e r r = , a r a > 0.4 (3.53) where, r is the radius of the neck of the HR, a is the radius of the duct to which the HR is mounted and δ e denotes the exterior end-correction factor at the neck-duct interface. For the analyses carried out in this study, three different values of the neckduct interface end-correction factor were used and its effect on the in-duct net acoustic power transmission and pressure transfer function of the HR are shown in section Neck-Cavity Interface End-Correction Factor The neck-cavity interface end-correction factor is also referred to as interior endcorrection factor. As described in section , this end-correction factor was denoted 56

89 Chapter 3. Theoretical Modelling by δ i and is given by [13] δ i = 8r ( r ) 3π R Transfer Matrix of a HR The transfer matrix of a duct-hr system requires the calculation of the transfer matrix of the HR in addition to the duct and is described below. Figure 3.6 shows a schematic of a uniform duct with an attached HR, as a side branch, to one of its walls at location x. The complex amplitude of the acoustic volume velocity entering the duct, before the HR s mounting, is denoted by v 0. Similarly, it is v 1 in the section of the duct after the resonator s mounting and v HR flowing into the HR. Figure 3.6: A schematic of a duct-hr system for estimating the expression for the input point impedance of the HR. From the continuity condition, the acoustic volume velocity and pressure are locally conserved at the junction x [12, 25]. Therefore, v 0 (x) = v HR (x) + v 1 (x) (3.54) p 0 (x) = p HR (x) = p 1 (x) (3.55) Putting equations (3.54) and (3.55) in matrix form 57

90 3.3. Circular Duct with an Attached Helmholtz Resonator v 0 p 0 = v HR + v 1 p 1 (3.56) v 0 p 0 = 1 v HR p v 1 p 1 (3.57) p HR From equation (3.55), p 1 (x) = p HR (x) and also, = Z HR. Here, Z HR is the v HR acoustic impedance just outside the opening of the resonator, which can be referred to as an input point impedance. Therefore, we can write v 0 p 0 = 1 1 Z HR 0 1 v 1 p 1 (3.58) Finally, the matrix resonator. 1 1 Z HR 0 1 represents the transfer matrix of the Helmholtz Input Point Impedance of a HR The impedance at the opening of the HR, i.e. Z HR, can be calculated by applying the transfer matrix equation to the neck and cavity of the HR. Let us say that the neck section of the resonator is denoted by s1 having cross-sectional area A n and length l 1, and the cavity section is denoted by s2 having cross-sectional area A c and length l 2. The length l 1 represents the effective length of the neck, which includes two endcorrections at each side of the neck. Relating the acoustic volume velocity and pressure at the opening of the neck v s1, p s1 and end of the cavity v s2, p s2 (see figure 3.6), the transfer matrix for the HR is v s1 p s1 = 2 2 transfer matrix of a neck 2 2 transfer matrix of a cavity v s2 p s2 (3.59) 58

91 Chapter 3. Theoretical Modelling or v s1 p s1 = A s1 C s1 B s1 D s1 A s2 C s2 B s2 D s2 v s2 p s2 (3.60) which can be written as: v s1 p s1 = A s C s B s D s v s2 p s2 (3.61) Because the end of the resonator is rigidly terminated, there will be no flow at the end of the resonator, hence, v s2 = 0. Substituting the value of v s2 = 0 in equation (3.61), the acoustic pressure and volume velocity at the opening of the neck becomes v s1 = B s p s2 (3.62) p s1 = D s p s2 (3.63) Dividing equation (3.63) by (3.62), gives the estimate of the input point impedance of the HR p s1 v s1 = Z HR = D s B s (3.64) Complete Transfer Matrix of a Duct-HR System In order to calculate the acoustic performance of a HR mounted onto a duct, the transfer matrix equation of the duct-hr system is required and described below. A schematic of a duct-hr system illustrating the division of the system into three elements is shown in figure 3.7. Consider that the duct is divided into two elements (1) element 0 - upstream of the HR and (2) element 1 - downstream of the HR. Let the first section of the duct be denoted by subscript 0 and the second one by subscript 1. The transfer matrix of the complete system is then obtained by inserting the transfer 59

92 3.3. Circular Duct with an Attached Helmholtz Resonator Figure 3.7: A schematic of a duct-hr system showing its division into three elements. matrix of the HR between the transfer matrices of the two sections of the duct. Thus, v 0 p 0 = 2 2 transfer matrix matrix of element 0 upstream of the HR 2 2 transfer matrix of element HR 2 2 transfer matrix matrix of element 1 downstream of the HR v 2 p 2 or v 0 p 0 = A 0 B 0 C 0 D Z HR 0 1 A 1 B 1 C 1 D 1 v l p l (3.65) which can be rewritten as: v 0 p 0 = A B C D v l p l (3.66) Equation (3.66) represents the resultant matrix of the complete duct-hr system, which can be used to evaluate the in-duct net acoustic power transmission downstream of the HR. This estimation was facilitated by using the estimates of the acoustic pressure and volume velocity at the duct exit of the duct-hr system. The next section shows the utilisation of the resultant matrix of the duct-hr system (equation (3.66)) 60

93 Chapter 3. Theoretical Modelling for calculating the acoustic pressure and volume velocity at the duct exit Acoustic Pressure and Velocity at the Duct Exit The acoustic pressure and acoustic volume velocity at the duct exit, x=l, can be calculated as explained earlier in section (see equations (3.24) and (3.25)) and are given by (using equation (3.66)) p l = Z l A + BZ l (3.67) v l = ( 1 BZ ) l A + BZ l A (3.68) The product of equations (3.67) and (3.68) calculates the in-duct net acoustic power transmission downstream of the HR, and is shown in section Acoustic Pressure at the Top Closed End of the Cavity Chanaud [18] experimentally determined the resonance frequency of a stand-alone HR by measuring the acoustic pressure at the top of the closed end of its cavity. It was expected that the mounting of a HR on a duct will result in the maximum reduction of the in-duct net acoustic power transmission downstream of the resonator at a frequency corresponding to the maximum acoustic pressure at the top of the closed end of the cavity. For the purpose of verifying this hypothesis, the acoustic pressure at the top of the closed end of the cavity was estimated and the pertinent procedure is explained below. It involves two steps: 1. Calculating the acoustic pressure at location x = 1 (see figure 3.7). This solely considers the transfer matrix of element 0. 61

94 3.3. Circular Duct with an Attached Helmholtz Resonator 2 2 v 0 = v 1 transfer matrix of (3.69) p 0 p 1 element 0 v 0 p 0 = p 1 is given by (refer equation (3.24)) A 0 B 0 v 1 C 0 D 0 p 1 (3.70) p 1 = Z 1 A 0 + B 0 Z 1 (3.71) 2. Using the transfer matrix of the HR, developed in equation (3.63), we can write p s2 = p s1 D s (3.72) From the continuity condition at location x = 1, p 1 = p s1. Here p s2 represents the acoustic pressure at the top of the closed end of the cavity and is given by p s2 = p ( 1 Z 1 1 = D s A 0 + B 0 Z 1) D s (3.73) As an example, consider a HR with dimensions as: cavity diameter=0.131 m, cavity length=0.070 m, neck diameter= m and actual neck length=0.093 m. The two end-corrections at both sides of the neck were calculated as δ i = 8r ( r ) 3π R and δ e = 8r 3π where, δ i is the interior end-correction at the neck-cavity interface, δ e is the exterior end-correction at the neck-duct interface, r is the radius of the neck, and 62

95 Chapter 3. Theoretical Modelling R the radius of the cavity. As stated earlier for the case of the duct-hr system, the exterior end-correction of the neck will be referred to as neck-duct interface end-correction factor. However, its value was taken as that of the exterior end-correction of the unflanged neck corresponding to the stand-alone HR (radiating into free space). Knowledge of the correct estimate of a neck-duct interface end-correction factor is important for evaluating a duct-hr system; however, the purpose of the current and the next two analyses is not to accurately evaluate the duct-hr system but to compare several frequencies. These frequencies correspond to 1. the maximum acoustic pressure at the top of the closed end of the cavity of the HR, 2. the maximum pressure transfer function of the HR, and 3. the maximum reduction of in-duct net acoustic power transmission downstream of the HR. Figure 3.8 shows the plot for the acoustic pressure at a point located at the centre of the top of the closed end of the cavity of the HR. The damping in the transfer matrix equation was included using the complex wave numbers. The value of the loss factor, η, was set to and kept unchanged for all the transfer matrix method analyses conducted in this study. The results for all the acoustic pressure calculated, p cal, using the transfer matrix method was normalised by the input volume velocity of the piston source, v 0, as shown below. The reason for normalising the acoustic pressure with the acoustic volume velocity is to enable the comparison of the theoretical results with those of the numerical (ANSYS) and experimental results. ( ) pcal p norm = 20 log 10 v 0 (3.74) 63

96 3.3. Circular Duct with an Attached Helmholtz Resonator normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz frequency(hz) Figure 3.8: Theoretical (transfer matrix method) prediction for the acoustic pressure at a point located at the top centre of the closed end of the cavity of the HR mounted onto a duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. Figure 3.8 shows that the frequency which corresponds to the maximum acoustic pressure is 215 Hz (apart from one low frequency duct resonance which is not associated with the HR). Before checking the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs, the pressure transfer function between the cavity and neck of the HR was estimated and is described in the next section, as a possible alternative indicator of the minimum in-duct net acoustic power transmission Pressure Transfer Function of a HR The pressure transfer function of the HR, also referred to as its pressure amplification [25], is defined as the ratio of the acoustic pressure at the top of the closed end of the cavity to the acoustic pressure at the opening of the resonator. The pressure transfer function was estimated in order to compare the frequency of its maximum amplitude with that of the maximum acoustic pressure at the top of the closed end of the cavity and the maximum in-duct net acoustic power reduction. By using equation (3.63), the pressure amplification of the HR can be written as 64

97 Chapter 3. Theoretical Modelling p s2 p s1 = 1 D s (3.75) The dimensions of the duct-hr system, including both the end-correction factors of the neck, were kept the same as described in the previous section. The amplitude (= ( ) ps2 20 log 10 ) and phase plots for the pressure transfer function of the HR are shown p s1 in figures 3.9 and 3.10, respectively. The frequency corresponding to the maximum pressure transfer amplitude can also be considered as the resonance frequency of the HR as a stand-alone device. This is because: 1. equation (3.63) developed in section does not have any involvement of the duct to which the HR is attached, and 2. the transfer function method is only applicable for plane-wave conditions. It assumes that the sound field in the regions close to the neck opening is planar and this is its main weakness. Comparison of figures 3.8 and 3.9 shows a difference of 7 Hz (2.7%) between the frequencies corresponding to the maximum acoustic pressure at the top of the closed end of the cavity of the HR and the resonance of the stand-alone HR. Prediction Models and Transfer Matrix Method When comparing the resonance frequency of the HR estimated using the transfer matrix method (pressure transfer function) with those of the estimates obtained using all the three prediction models (equations (3.4), (3.10) and (3.11)), it was observed that the transfer matrix method (equation ( 3.75)) results were close to that of Li s results (equation (3.11)). The validity of the transfer matrix method for calculating the resonance frequencies of HRs will be discussed further in chapter 5, Experimental Verification. 65

98 3.3. Circular Duct with an Attached Helmholtz Resonator Hz pressure transfer function (db) frequency (Hz) Figure 3.9: Theoretical (transfer matrix method) prediction for the pressure transfer function amplitude of the HR. 0 pressure transfer function phase (degrees) frequency (Hz) Figure 3.10: Theoretical (transfer matrix method) prediction for the pressure transfer function phase of the HR. 66

99 Chapter 3. Theoretical Modelling In-Duct Net Acoustic Power Transmission Downstream of the HR Keeping all the dimensions of the duct-hr system unchanged, including the two endcorrections of the neck, the net acoustic power transmission in the duct downstream of the HR was estimated by substituting values of p l and v l, from equations (3.67) and (3.68), in equation (3.50). A plot for the net acoustic power transmission in the duct downstream of the HR when it is mounted on the duct at the distance of 0.5 m from the source end is shown in figure This figure also includes the plot for the net acoustic power transmission in the duct without the resonator. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz without HR with HR frequency (Hz) Figure 3.11: Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. The effect of mounting the HR onto a duct, as evident from the plot, minimised the net acoustic power transmission by 18 db at a particular frequency, which is 221 Hz in this case. This frequency matched the frequency which corresponds to the maximum value of the pressure transfer function amplitude of the HR (see figure 3.9) but differed from the frequency corresponding to the maximum acoustic pressure at the top of the closed end of the cavity (see figure 3.8). 67

100 3.3. Circular Duct with an Attached Helmholtz Resonator Effects of Varying the End-Correction Factors As stated earlier, the neck-duct interface end-correction factor cannot be predicted easily due to the generation of the complex sound field in the region of the duct close to the neck opening. In all the analyses pertinent to the duct-hr system, which have been carried out in this chapter, the value used for the neck-duct interface end-correction has been the exterior end-correction of the neck of the stand-alone HR. This section will now investigate the effects of using different neck-duct interface end-correction factors on the pressure transfer function of the HR and the in-duct net acoustic power transmission downstream of the HR. Pressure Transfer Function of the HR All the dimensions of the resonator described in the earlier sections, including the interior end correction (δ i - neck-cavity interface), were retained, with the exception of the exterior end correction factor (δ e - neck-duct interface). Two different values of the exterior end-correction factor were tested as per: (1) Onorati s calculation [30], and (2) Ji s expression (equation (3.52)) [31]. The pressure transfer function of the HR was calculated by using equation (3.75) and the corresponding amplitude and phase plots are shown in figures 3.12 and 3.13, respectively. The results are only shown over the frequency range from 205 Hz to 235 Hz for a clear distinction of the differences in the curves. Also, the previous pressure transfer function plots (figures 3.9 and 3.10) obtained by using the neck-duct interface end-correction factor of δ e = 8r are included in the current plots for comparison. As 3π evident from the plots, the result of using different end-correction models causes a change in the response of the pressure transfer function of the HR. In other words, the peak value of the amplitude of the pressure transfer function was shifted from 221 Hz for δ e = 8r 3π to 220 Hz and 224 Hz for δ e as per Onorati s [30] and Ji s (equation (3.52)) [31], respectively. 68

101 Chapter 3. Theoretical Modelling Hz 221 Hz 224 Hz pressure transfer function (db) unflanged open end Onorati Ji frequency (Hz) Figure 3.12: Theoretical (transfer matrix method) predictions for the pressure transfer function amplitudes of the HR using different exterior end-correction models. 0 pressure transfer function phase (degrees) unflanged open end 160 Onorati Ji frequency (Hz) Figure 3.13: Theoretical (transfer matrix method) predictions for the pressure transfer function phases of the HR using different exterior end-correction models. 69

102 3.3. Circular Duct with an Attached Helmholtz Resonator In-Duct Net Acoustic Power Transmission When the net acoustic power transmission in the duct was calculated for the neck-duct interface end-correction factor, δ e as per Onorati s [30] and Ji s [31] expressions, the frequencies at which the maximum reduction of the in-duct net acoustic power transmission downstream of the HR occurred were 220 Hz and 224 Hz, respectively. Figures 3.14 and 3.15 (expanded scale) show the comparison of the three in-duct net acoustic power transmissions for different values for the neck-duct interface end-correction factor. Figures 3.12 and 3.15 show that no matter which end-correction factors are used, the transfer matrix method does not show any distinction between the resonance frequency of a HR, determined from the (cavity/neck) pressure transfer function calculation, and the frequency at which the maximum reduction of in-duct net acoustic power transmission occurs Optimal Location of HRs In order to maximise the attenuation of in-duct noise transmission at any particular frequency, the location of the HR needs to be optimised. This means that the resonator must be mounted at the location of maximum sound pressure level (pressure antinodes), corresponding to a particular frequency, along the duct. For the investigations pertinent to the duct-hr system, which were discussed in this chapter, the location of the HR was chosen to be 0.5 m from the source end. It has been stated in the literature [11] that if a HR is mounted on a duct near one of the pressure antinodes corresponding to the excitation frequency along the duct, greater reduction in the acoustic power transmission downstream of the resonator can be achieved compared to mounting the HR at any other. In the work reported in this thesis, this statement is not true for all the cases and strongly depends on the frequency separation between the excitation frequency and the closest duct resonance frequency. The following three (theoretical) examples demonstrate the optimal mounting locations 70

103 Chapter 3. Theoretical Modelling normalised net acoustic power (db re 1watt/(1m 3 /sec)) unflanged open end Onorati Ji frequency (Hz) Figure 3.14: Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR using different exterior correction models. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. normalised net acoustic power (db re 1watt/(1m 3 /sec)) Hz 221 Hz 224 Hz unflanged open end Onorati Ji frequency (Hz) Figure 3.15: The results from figure 3.14 expanded over the frequency range from 205 Hz to 235 Hz. 71

104 3.3. Circular Duct with an Attached Helmholtz Resonator of the HR for three excitation frequencies. These frequencies correspond to a resonance frequency of the duct, an antiresonance frequency of the duct and a frequency which lies between a resonance and an antiresonance frequency of the duct. In-duct net acoustic power reduction downstream of the HR at a frequency which corresponds to one of the resonance frequencies of the duct The first example is related to the case for which the maximum reduction of the in-duct net acoustic power transmission downstream of the HR occurs when the resonator is mounted at one of the pressure antinodes, corresponding to the excitation frequency, along the duct. For this case an excitation frequency of 197 Hz was selected as the frequency at which noise is to be attenuated. This frequency corresponds to one of the acoustic resonance frequencies of the duct (see table 3.2 for the transfer matrix method results). The neck diameter, actual neck length and cavity diameter were kept the same as described in section (below equation (3.73)). The HR cavity length was set to m as that is the length corresponding to maximum in-duct acoustic power reduction at 197 Hz. The effective length of the neck, l eff, was calculated using the expressions for the interior (neck-cavity interface) and exterior (neck-duct interface) end-correction factors as per equations (3.9) and Ji s (equation (3.52)) [31] expression, respectively. The damping was included using the loss factor, η, of Figure 3.16 shows a plot of the pressure distribution along the duct at 197 Hz. The circular and square markers indicate the pressure nodes and antinodes, respectively. Figure 3.17 shows the in-duct net acoustic power transmission downstream of the HR at 197 Hz for various mounting locations of the HR along the duct. As can be seen from figure 3.17, the maximum reduction of the in-duct acoustic power occurs when the resonator was mounted at 0.88 m, 1.75 m and 2.61 m, indicated by the diamondshaped markers. The locations corresponding to the diamond-shaped markers will be referred to as optimal locations in the remaining text of this example and the next two 72

105 Chapter 3. Theoretical Modelling sound pressure level (db re 20uPa) m 1.75 m 2.61 m 0.45 m 1.31 m 2.18 m 110 antinodes nodes distance (m) Figure 3.16: Sound pressure level along the duct depicting a standing wave plot corresponding to a tonal (excitation) frequency of 197 Hz. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) optimal location antinode node 0.88 m 1.75 m 2.61 m HR mounting location (m) Figure 3.17: Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR at 197 Hz for varying the HR mounting location along the duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. 73

106 3.3. Circular Duct with an Attached Helmholtz Resonator examples. The circular and square markers were plotted in figure 3.17 at the locations corresponding to those of the pressure nodes and antinodes, respectively shown in figure It can be seen from figure 3.17 that the values corresponding to the optimal HR locations (diamond-shaped markers) are equal to those obtained with the HR at the pressure antinode (square markers) locations. This implies that mounting the HR at one of the locations corresponding to the pressure antinodes will result in maximum reduction of the in-duct net acoustic power transmission. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz no HR HR at node HR at antinode frequency (Hz) Figure 3.18: Theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR at 197 Hz for the HR mounted at a distance of 0.45 m (a pressure node) and 0.88 m (an antinode) from the source end of the duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. Figure 3.18 shows the in-duct net acoustic power transmissions downstream of the HR with its mounting locations as 0.45 m (a pressure node) and 0.88 m (an antinode, which also corresponds to the optimal HR location on the duct), along with the in-duct net acoustic power transmission without the HR. A 3 db reduction in the in-duct net acoustic power transmission occurred when the HR was mounted at 0.45 m. On the other hand, a reduction of approximately 50 db is evident when the location of the HR was changed to 0.88 m. It should be noted that such a high reduction in practical applications is not achievable due to the presence of system damping. As it was not 74

107 Chapter 3. Theoretical Modelling possible by theoretical means to estimate the duct-hr system damping, complex wave numbers were used to include damping in the transfer matrix formulations. This was equivalent to using a loss factor, η, of In-duct net acoustic power reduction downstream of the HR at a frequency which corresponds to the antiresonance frequency of the duct The second example shows that the maximum reduction of the in-duct net acoustic power transmission does not necessarily occur if the resonator is mounted at one of the locations corresponding to an antinode of the duct response. An excitation frequency of 224 Hz is selected as the frequency at which the acoustic power transmission downstream of the HR needs to be attenuated. Here, 224 Hz corresponds to one of the acoustic antiresonance frequencies of the duct (see figure 3.5). All the dimensions of the duct-hr system were kept the same as in the previous example but the cavity length was set to m as that is the length corresponding to maximum in-duct net acoustic power reduction at 224 Hz. The pressure distribution along the duct at 224 Hz and the in-duct net acoustic power transmission downstream of the HR at 224 Hz for various mounting locations of the HR are shown in figures 3.19 and 3.20, respectively. The circular and square markers in both figures correspond to the pressure nodes and antinodes, respectively for 224 Hz. Figure 3.20 shows that the maximum reduction of the in-duct net acoustic power occurs for the HR mounting locations of 0.56 m, 1.33 m, 2.10 and 2.86 m, indicated by the diamond-shaped markers. It can be seen from figure 3.20 that mounting the HR at the locations corresponding to those of the pressure antinodes will result in less acoustic power reduction compared to the optimal HR mounting locations (diamondshaped markers). Figure 3.20 shows that approximately 18 db higher reduction in the in-duct net acoustic power transmission is obtained for the HR mounting locations corresponding to the optimal locations compared to those of the pressure antinodes. Figure 3.20 also shows that the acoustic power reduction for the HR mounting locations 75

108 3.3. Circular Duct with an Attached Helmholtz Resonator sound pressure level (db re 20uPa) antinodes nodes 0.37 m 1.13 m 1.89 m 2.66 m 0.75 m 1.51 m 2.28 m distance (m) Figure 3.19: Sound pressure level along the duct depicting a standing wave plot corresponding to a tonal (excitation) frequency of 224 Hz. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) optimal location antinode node 0.56 m 1.31 m 2.1 m 2.86 m HR mounting location (m) Figure 3.20: Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR at 224 Hz for varying the HR mounting location along the duct. The duct was being driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. 76

109 Chapter 3. Theoretical Modelling corresponding to the troughs occurring between the optimal locations is comparable to the reduction related to the optimal locations. All the troughs in figure 3.20 are related to each other in the sense that they are odd multiples of 1/8 of the wavelength from the open end of the duct. The acoustic power reduction at the locations indicated by the circular markers, which corresponds to the pressure nodes as shown in figure 3.19, is comparable to that at the antinodes (square markers), which is a worst case situation. Unlike the previous case for the excitation frequency of 197 Hz for which approximately 50 db reduction in the acoustic power transmission was obtained, only a maximum of 20 db reduction was predicted for the excitation frequency of 224 Hz. This is because 224 Hz corresponds to an antiresonance frequency of the duct at which the acoustic power level is less compared to that of at 197 Hz which corresponds to a resonance frequency of the duct. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz no HR HR at node HR at antinode HR at optimal location frequency (Hz) Figure 3.21: Theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR for the HR mounted at a distance of 0.75 m (a pressure node), 0.37 m (an antinode) and 0.56 m (an optimal location) from the source end. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. Figure 3.21 shows the in-duct net acoustic power transmissions downstream of the HR with its mounting locations as 0.75 m (a pressure node), 0.37 m (an antinode) and 77

110 3.3. Circular Duct with an Attached Helmholtz Resonator 0.56 m (an optimal location), along with the in-duct net acoustic power transmission without the HR. Almost no reduction in the in-duct net acoustic power transmission occurred when the HR was mounted at 0.75 m (a nodal location). On the other hand, reduction of approximately 20 db is evident when the location of the HR was changed to 0.56 m compared to 10 db when the HR was placed at 0.37 m (at an antinode). In-duct net acoustic power reduction downstream of the HR at a frequency which lies between the resonance and antiresonance frequencies of the duct This final example demonstrates the optimal locations for the HR mounting for attenuating the in-duct acoustic power transmission at the excitation frequency of 211 Hz which lies between a resonance and an antiresonance frequency of the duct. The cavity length was set to m and all the dimensions of the duct-hr system were kept the same as in the two previous examples. Figure 3.22 shows a plot of the pressure distribution along the duct at 211 Hz. The circular and square markers indicate the pressure nodes and antinodes, respectively. The calculated in-duct net acoustic power transmission downstream of the HR at 211 Hz for various mounting locations of the HR, along the duct shown in figure 3.23, where it can be seen that the maximum reduction of the in-duct acoustic power occurs when the resonator is mounted at locations 0.1 m, 0.91 m, 1.73 m and 2.54 m, indicated by the diamond-shaped markers and denoted by the optimal locations. Although the optimal mounting locations are not at the pressure antinodes, the acoustic power level difference corresponding to the two different locations differs by only 2 db. In a practical application, a 2 db difference is small enough to neglect as the acoustic power reduction can range from 15 to 30 db. Figure 3.24 shows the in-duct net acoustic power transmissions downstream of the HR with its mounting locations at 0.61 m (a pressure node), 1.02 m (an antinode) and 0.91 m (an optimal location), along with the in-duct net acoustic power transmission without the HR. The in-duct net acoustic power transmission was reduced by 27 db 78

111 Chapter 3. Theoretical Modelling sound pressure level (db re 20uPa) m 1.02 m 1.83 m 2.64 m 1.42 m 2.23 m 110 antinodes 0.61 m nodes distance (m) Figure 3.22: Sound pressure level along the duct depicting a standing wave plot corresponding to a tonal (excitation) frequency of 211 Hz. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) m 0.91 m optimal location antinode node 1.73 m 2.54 m HR mounting location (m) Figure 3.23: Theoretical (transfer matrix method) prediction for the in-duct net acoustic power transmission downstream of the HR at 211 Hz for varying the HR mounting location along the duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. 79

112 3.3. Circular Duct with an Attached Helmholtz Resonator when the HR was placed at 0.91 m which is approximately equivalent to that achieved with a HR mounting location at 1.02 m. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) no HR HR at node HR at antinode HR at optimal location 211 Hz frequency (Hz) Figure 3.24: Theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR for the HR mounted at a distance of 0.61 m (a pressure node), 1.02 m (an antinode) and 0.91 m (an optimal location) from the source end. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. The three examples shown above demonstrated that the maximum reduction of the in-duct net acoustic power transmission downstream of the HR does not always occur at the pressure antinodes corresponding to the excitation frequency along the duct. However, in practical applications where the presence of damping reduces the acoustic performance of the resonators, the difference of the in-duct acoustic power reduction between the different resonator mounting locations can be of less significance as long as the resonator is mounted near a pressure antinode. The only case where the antinode location gives much lower attenuation than the optimal location is when the excitation frequency corresponds to an antiresonance frequency in the duct, where the power transmission is much smaller. 80

113 Chapter 3. Theoretical Modelling 3.4 Conclusion Three theoretical models, the classical (equation (3.4)), Panton and Miller s (equation (3.10)) and Li s (equation (3.11)) formulas, from the literature were presented to predict the resonance frequencies of HRs as stand-alone devices, and the differences were shown with the help of a graph. A circular duct with an attached HR was analysed using the transfer matrix method. A detailed scheme for the in-duct modal decomposition of sound field was presented. In addition to calculating the in-duct net acoustic power transmission with the modal decomposition scheme, it was also calculated using the product of acoustic pressure and volume velocity at the duct exit. A step-by-step procedure for building the transfer matrix of the duct-hr system was described. The results of using the transfer matrix method showed that the frequencies corresponding to the maximum acoustic pressure at the top of the closed end of the cavity and the maximum pressure transfer function amplitude of the HR were different. The results also showed that the maximum reduction of the in-duct net acoustic power transmission downstream of the HR occurred at the frequency which corresponds to the maximum pressure transfer function amplitude of the HR. The key issue in all the analyses using the transfer matrix method was the incorporation of the end-correction factors in addition to the actual dimensions of the elements. These end-correction factors are related to the unflanged open-end of a duct radiating into free space and also to the neck of the HR. The end-corrections for the open end of a duct and the neck-cavity interface are well understood and documented. However, the end-correction factor of the neck at the neck-duct interface has not been well documented. This is due to the presence of the complex sound field in the regions of the duct in close proximity to the resonator s opening. The effects of using different values of the neck-duct interface end-correction factor on the pressure transfer function of the HR and the in-duct net acoustic power transmission downstream of the HR were also shown. 81

114 3.4. Conclusion Two limitations of the transfer matrix method pertinent to the duct-hr system which were observed in this chapter are described below. The transfer matrix method can only be used for plane-wave conditions. However, non-planar sound fields exist in the regions of the duct which are in close proximity to the resonator s opening, and hence the transfer matrix method is not accurate. As the HRs are used to attenuate noise at a single frequency, the transfer matrix method cannot be used to accurately estimate the performance of a duct-mounted HR (frequency corresponding to the maximum in-duct net acoustic power transmission) because of no exact knowledge of the neck-duct interface end-correction factor. Due to the aforementioned limitations of the transfer matrix method, a better approach to investigate the duct-hr system was required. For this reason, numerical discretization technique was considered and is described in the next chapter. 82

115 Chapter 4 Numerical Modelling 4.1 Introduction In the previous chapter, the transfer matrix method was used to estimate the resonance frequency of the HR and the acoustic performance of the HR attached to the duct. The results of the transfer matrix method analyses showed that the frequencies corresponding to the resonance of the HR (analogous to the maximum amplitude of the pressure transfer function of the HR) and the maximum reduction of in-duct net acoustic power transmission downstream of the resonator were similar. The important part in all the transfer matrix method analyses was the incorporation of the two end-correction factors of the neck of the HR in addition to the actual dimensions of the duct-hr system. Because of the associated uncertainty in the neck-duct interface end-correction factor estimate, it was not possible by theoretical means to accurately evaluate the acoustic performance of the duct-mounted HR, which corresponds to the frequency attaining maximum reduction of the in-duct acoustic power transmission downstream of the HR. Modelling of the duct-hr system using the finite element software package ANSYS eliminates the need to account for the end-corrections of the neck and hence, provides a reliable verification reference, provided that the model is representative of the system under investigation. The only limitation with the modelling of the duct-hr system in ANSYS is associated with difficulty in obtaining an accurate 83

116 4.1. Introduction estimate of system damping to include in the analysis. This chapter discusses the finite element modelling/numerical analysis of a uniform circular duct and a uniform circular duct with an attached cylindrical HR. A circular duct was modelled as a rigid-walled structure being driven by a piston at one end and radiating into free space at the other end. A two-dimensional finite element model of the duct was constructed and all the pertinent steps for conducting the analysis are detailed in order to obtain the estimates of the in-duct acoustic power transmission. The open-end of the duct was modelled as radiating into free space by applying a radiation impedance boundary condition at the end of the duct. The net acoustic power transmission in the duct was estimated by using the two-microphone method [37, 42]. An important step in any finite element analysis is dividing the model into a sufficient number of elements. Thus, the effect of varying the mesh density (or element size) on the in-duct net acoustic power transmission and the acoustic resonances of the duct was also investigated here. A three-dimensional finite element model of a circular duct with an attached cylindrical HR was developed and solved for the pressure distribution over the frequency range from 0 to 400 Hz. One end of the duct was modelled as being driven by a piston and the other end was modelled as open and radiating into free space. Dissipation/damping of the HR, which mainly occurs due to the viscous losses of the air moving back and forth in the neck [13], was also incorporated in the finite element duct-hr model. The acoustic performance of the HR was evaluated by comparing the in-duct net acoustic power transmission without and with the HR. The regions of the duct in close proximity to the opening of the resonator and the regions of the neck exhibit a non-planar sound field, and are referred to as critical regions in this study. Thus, the effect of varying the mesh density, especially in the critical regions, on the in-duct net acoustic power transmission was also investigated. Finally, the results from the ANSYS models are compared with results from the 84

117 Chapter 4. Numerical Modelling transfer matrix method. Objectives The objectives of the research presented in this chapter are: to develop an accurate finite element model of the duct-hr system so that a system with varying geometry can be analysed for future work, to verify the validity and accuracy of the transfer matrix method by comparing the results from the transfer matrix method with results from the ANSYS modelling, to investigate the limitations of the transfer matrix method, and to investigate the effects of varying the mesh density on the acoustic resonance frequencies of the duct and the in-duct net acoustic power transmission. 4.2 Uniform Circular Duct A uniform circular duct having the same geometry as the one modelled in the previous chapter (diameter = m and length = 3 m), driven by a piston at one end (left end) and the other end (right end) radiating into free space, was modelled. Four basic steps involved in any ANSYS related analysis are: (1) build the model, (2) mesh the model, (3) apply boundary conditions, and (4) solve the model. All these steps, as they relate to this study, are detailed in the following sections. For the uniform circular duct with no HR attached, a two-dimensional modelling using the two-dimensional elements was conducted as a uniform circular duct represents an axis-symmetric structure. The use of an axis-symmetric model greatly reduces the modelling and analysis time compared to that of an equivalent three-dimensional model. In contrast, modelling of a circular duct with an attached cylindrical HR should be conducted as a three-dimensional object. The reason being in ANSYS, only those 85

118 4.2. Uniform Circular Duct rectangular geometries of a HR can be analysed as two-dimensional objects which have a depth of 1 m. This is because the ANSYS software package assumes unit depth for any two-dimensional model. However, the HRs considered in this study are circular and do not have a constant depth when analysed in 2-D Building the Model Element Definition The relevant acoustic elements available in the ANSYS library are FLUID29 for twodimensional and FLUID30 for three-dimensional analysis [76]. These elements are pressure formulated elements which model the pressure variation in the element associated with an acoustic wave [77]. Figure 4.1 shows a schematic of the geometry of a FLUID29 element. The FLUID29 element has four corner nodes and three degrees of freedom per node: translations (or displacements) in nodal x and y directions and pressure. Figure 4.1: A schematic of the geometry of a FLUID29 element. The parameter KEYOPT(2), in the model definition, is used to specify the absence or presence of the structure. Keeping KEYOPT(2) = 1 models FLUID29 elements as pure acoustic elements with no fluid-structure interaction. Pure acoustic elements have a single degree-of-freedom at each node, which is pressure (PRES) [76]. In contrast, defining KEYOPT(2) = 0 denotes the presence of structure at the element interface. These elements have both pressure and displacement degrees-of-freedom at each node. A two-dimensional model of a duct, which has an area of m 3 m, was built using the three element types described below: 86

119 Chapter 4. Numerical Modelling 1. FLUID29 elements having a pressure degree-of-freedom at each node (KEYOPT (2) = 1), defined as pure acoustic elements 2. FLUID29 elements having both pressure and displacement degrees-of-freedom at each node (KEYOPT (2) = 0), defined as acoustic elements with structure 3. SURF153 elements, defined as surface effect elements Pure acoustic elements were used to model the fluid medium inside the entire duct except at the right end, which should have a radiation impedance boundary condition imposed in order to achieve the condition of an open end radiating into free space. As acoustic impedance is defined as the ratio of acoustic pressure to particle (or volume) velocity, the displacement of fluid at the right end of the duct must be considered. Therefore, at the right end of the duct, FLUID29 elements having both pressure and displacement degrees-of-freedom were used in order to account for the fluid displacement. SURF153 elements, which are surface effect elements, were used to model the complex radiation impedance boundary condition at the right end. The utilisation of SURF153 elements in building the finite element model of the duct is discussed in section For the FLUID29 elements, the reference pressure (PREF), density (DENS) and speed of sound (SONC) were input as N/m 2, 1.21 m/sec 2 and 344 m/sec, respectively. The values of the input parameters for SURF153 elements are described later in section Meshing the Model Meshing the model means dividing the solid model into a number of discrete elements. This step is important as the accuracy of the analysis depends on using a sufficient number of elements. In general, a minimum of 6 elements per wavelength (EPW) is sufficient to analyse the acoustic space in which ANSYS acoustic type plane waves 87

120 4.2. Uniform Circular Duct propagate [77]. However, the model of the duct was divided as per 6, 10, 20 and 85 EPW at 400 Hz in order to investigate the effects of varying the mesh density on the acoustic resonance frequencies of the duct and the in-duct net acoustic power transmission. The two-dimensional acoustic space of the model was uniformly meshed by using quadrilateral-shaped FLUID29 elements. Analyses with different EPWs (element sizes) as described in the preceding paragraph were conducted and comparison of the results are shown in section Applying Boundary Conditions The two-dimensional finite element model of the duct was modelled as a rigid-walled duct with its left end being driven by a piston and right end being open and radiating into free space. The rigid-walled and open end boundary conditions are discussed in the next two sections Rigid-Wall Boundary Condition In ANSYS, a rigid-walled duct is modelled if no boundary condition is applied on the boundary of pure acoustic element mesh Open-End Boundary Condition The open end of the duct radiating into free space was modelled using the radiation impedance condition for an unflanged open end of a duct, which is given by equation (3.15). As the radiation impedance is a complex value, it is necessary to model both real and imaginary frequency dependent components in ANSYS. Imaoka [78] describes a method to model the complex impedance boundary condition for the open end of the duct, and was used in this analysis. Surface effect elements (SURF153 for two-dimensional and SURF154 for threedimensional analysis) were used to model complex impedance boundary conditions. 88

121 Chapter 4. Numerical Modelling They can be overlaid on most of the two-dimensional structural solid elements available in ANSYS [76]. As already stated in section 4.2.1, FLUID29 elements at the right end of the duct must have nodal displacement degrees-of-freedom in order to account for the displacement of the fluid. Therefore, after meshing the entire model with FLUID29 elements having only a pressure degree-of-freedom at each node, the acoustic elements at the right end of the duct having only a pressure degree-of-freedom (FLUID29 with KEY- OPT(2) = 1) were modified to the acoustic elements having both pressure and displacement degrees-of-freedom (FLUID29 with KEYOPT(2) = 0). Thereafter, a layer of SURF153 elements was laid over the FLUID29 elements having both pressure and displacement degrees-of-freedom at each node. For modelling the real part of the radiation impedance, the parameter VISC can be used, which is a material property of SURF153 element. In contrast, the imaginary part can be modelled by using ADMSUA (added mass per unit area), which is a real constant of SURF153 element [78]. The expressions for modelling the real and the imaginary parts of the radiation impedance of an unflanged open end of a duct are given by VISC = Re(Z r ) = ρa2 ω 2 4c (4.1) ADMSUA = imaginary(z r) ω = 0.61ρa (4.2) where, ρ is the density of the fluid (air) medium, ω is the angular frequency, c is the speed of sound in the fluid (air) medium and a is the radius of the duct. NOTE: In evaluating the values for the parameters VISC and ADMSUA, the crosssectional area of the duct was ignored because ANSYS automatically takes the area into account while processing. The reason being that division of a solid model in ANSYS into a finite number of elements results in generation of nodes, each of which has an associated area. During the solution phase, ANSYS calculates the basic quantities (as 89

122 4.2. Uniform Circular Duct per the type of analysis) for each node which takes into account the area associated with the node Applying Loads In order to simulate the duct being driven by a piston, unit volume acceleration (denoted by label FLOW) was applied to the left end of the finite element model of the duct. The ANSYS acoustic tutorial [77] states that the Nodal fluid loads can be considered as fluid mass acceleration and is equivalent to the negative fluid particle acceleration times the effective surface area associated with the node times the mean fluid density [77]. The equivalent volume velocity is calculated as v v = FLOW jρω (4.3) where, v v is the acoustic volume velocity of the piston source exciting the fluid, and FLOW is the volume acceleration associated with each node Solving the Model As stated in section 4.2.2, in order to investigate the effects of mesh density, the acoustic space of the model was divided as per 6, 10, 20 and 85 EPW at 400 Hz. For each configuration of the EPW, a harmonic analysis was conducted over the frequency range from 0 to 400 Hz in 1 Hz increments. The purpose of conducting the harmonic analysis was to estimate the net acoustic power transmission in the duct. As the radiation impedance is a frequency dependent term, it is necessary to change the definition of the impedance elements (SURF153) at each analysis frequency. 90

123 Chapter 4. Numerical Modelling Results For each harmonic analysis, the real and imaginary parts of the pressure at two duct cross-sections spaced 0.3 m apart axially were written to a text file, and processed in MATLAB to estimate the in-duct net acoustic power transmission [37, 42] In-Duct Net Acoustic Power Transmission The net acoustic power, W net, in the duct was estimated by decomposing the sound field into forward (+ve x direction) and backward (-ve x direction) travelling waves [37, 42]. The scheme for the in-duct modal decomposition of the sound field has already been described in chapter 3, section The calculated net acoustic power transmission was normalised with respect to the input volume velocity, v v, as shown below. ( ) Wnet W norm = 10 log 10 vv 2 where, W norm denotes the normalised net acoustic power transmission. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) EPW 10 EPW EPW 85 EPW theory frequency (Hz) Figure 4.2: Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the net acoustic power transmission in a uniform circular duct driven by a constant amplitude piston source mounted at one (left) end and open at the other (right). Figure 4.2 shows four different plots for the in-duct net acoustic power transmission 91

124 4.2. Uniform Circular Duct corresponding to a uniform division of the acoustic space of the duct as per 6, 10, 20 and 85 EPW at 400 Hz. Also, the plot for the in-duct net acoustic power obtained using the transfer matrix method is included in figure 4.2. Excellent agreement between the ANSYS results corresponding to the uniform division of the acoustic space as per 20 and 85 EPW, and the theoretical results is evident from figure 4.2. A marked distinction between the ANSYS results corresponding to different EPWs described above, especially at frequencies higher than 200 Hz, shows the importance of dividing the finite element model into a sufficient number of elements. The differences in the ANSYS results with respect to various element sizes shown in figure 4.2 are discussed in the next section in terms of the acoustic resonance frequencies of the duct Resonance Frequencies The peaks (high sound power level) in figure 4.2 correspond to the acoustic resonances of the duct. Table 4.1 summaries the estimates for the acoustic resonance frequencies of the duct predicted using ANSYS with respect to different EPWs. Also, the theoretical estimates for the resonance frequencies of the duct obtained by using the transfer matrix method and Blevin s formula [75] are included in table 4.1. Blevin s formula has already been described in chapter 3, section , equation (3.51), and the resonance frequencies from the transfer matrix method correspond to the peak values in the plot of in-duct net acoustic power shown in figure 4.2. Of all the results tabulated in table 4.1, the results from the transfer matrix method are accurate because both radiation impedance and end-correction factor for the unflanged open end of the duct were included during the estimation of the acoustic power. Table 4.2 shows the error (%) between the theoretical and ANSYS results. For the first seven acoustic modes of the duct, the approximate error of 1.55% was found between Blevin s estimates and the transfer matrix method results. The reason for the error is that in Blevin s formula, the pressure at the open end of the duct is assumed to be zero. In fact, for a duct radiating into free space, the open end of the duct has a 92

125 Chapter 4. Numerical Modelling Mode No. Blevin s formula with zeropressure condition for the open end Transfer matrix with radiation impedance condition for the open end Frequency (Hz) ANSYS with radiation impedance boundary condition for the open end Hz Hz Hz Hz Table 4.1: Numerical (ANSYS) and theoretical (Blevin s formula and transfer matrix method) predictions for the acoustic resonance frequencies of a piston driven-open duct system. Mode No. Error (%) ANSYS Blevin s ANSYS (10 formula (6 EPW) EPW) and and and transfer transfer transfer matrix matrix matrix method method method ANSYS (20 EPW) and transfer matrix method ANSYS (85 EPW) and transfer matrix method Table 4.2: Error between the acoustic resonance frequencies tabulated in table

126 4.2. Uniform Circular Duct radiation impedance which does not correspond to zero-pressure at the open end. The last four columns in table 4.2 show the errors between the transfer matrix method and ANSYS results. An error of 3.18% in the seventh acoustic mode of the duct was found between the ANSYS prediction corresponding to the mesh size of 6 EPW and the transfer matrix method results. This error was reduced to 0.03% as the mesh size of the finite element model of the duct was refined as per 85 EPW. Error of less than 0.4% exist between the transfer matrix method results and ANSYS predictions corresponding to 20 and 85 EPW at 400 Hz. The magnitude of error decreases as the mesh size of the ANSYS model is refined. Figure 4.2 and table 4.2 show the importance of meshing the finite element model with regard to the frequency range over which the analysis has to be conducted Summary A two-dimensional finite element model of a uniform duct was built and solved in order to estimate the in-duct net acoustic power transmission. The right end of the duct was modelled as open and radiating into free space. The in-duct net acoustic power transmission obtained using the ANSYS modelling was compared with the transfer matrix method results. Excellent agreement between the transfer matrix method and ANSYS modelling results was achieved when the finite element model of the duct was discretised into a sufficient number of elements. The acoustic resonance frequencies of the duct obtained using the ANSYS modelling and the transfer matrix method results were also compared with Blevin s estimates. It was found that in Blevin s formula, the boundary condition at the open end of the duct was incorrectly assumed to be of zero-pressure rather than the radiation impedance condition. The remaining sections describe the finite element model of a uniform circular duct with an attached cylindrical HR. 94

127 Chapter 4. Numerical Modelling 4.3 Circular Duct with an Attached Helmholtz Resonator It has already been stated that analysing stand-alone HRs or duct-hr systems in ANSYS eliminates the need to account for both end-corrections of the neck of the HR. For this study, it became one of the main advantages of the finite element modelling because the neck-duct interface end-correction factor has not been very well understood and still remains an ongoing topic of research. Also, it was stated in the previous chapter that the transfer matrix method cannot be used to analyse the complex sound field in the regions of the duct in proximity to the opening of the resonator. The transfer matrix method works on a principle of lumped element theory, and therefore cannot reveal the pressure distribution between different locations on a particular (vertical) cross-section of a duct. Finite element analysis does not suffer from such restrictions and can be used to investigate complex three-dimensional sound fields Introduction The duct-hr system was modelled as a three-dimensional object in ANSYS. The reason for not analysing the duct-hr system as a two-dimensional object has already been stated in section 4.2. A three-dimensional model of a circular duct with an attached cylindrical HR was built in the ANSYS software package. The elements used were: (1) 3-D acoustic elements with only a pressure degree of freedom at each node (FLUID30 with KEY- OPT(2) = 1), (2) 3-D acoustic elements with both pressure and displacement degrees of freedom at each node (FLUID30 with KEYOPT(2) = 0), and (3) 3-D surface effect elements (SURF154). For the acoustic elements, reference pressure (PREF), density (DENS) and speed of sound in air (SONC) were set to N/m 2, 1.21 m/sec 2 and 344 m/sec, respectively. In order to model the end of the duct as open and radiating into free space, the parameters VISC and ADMSUA for SURF154 elements were 95

128 4.3. Circular Duct with an Attached Helmholtz Resonator set in accordance to equation (4.1) and (4.2), respectively. Like the previously described analysis of the uniform duct, one end (left) of the duct upstream of the HR was modelled as being driven by a constant amplitude piston source by applying a load of unit volume acceleration (denoted by label FLOW) to the left end of the duct. The other (right) end of the duct downstream of the HR was modelled as open and radiating into free space by applying the radiation impedance boundary condition [78]. The basic procedure for conducting a three-dimensional analysis is similar to that of a two-dimensional analysis and the steps pertinent to the two-dimensional analysis of the duct were listed in section 4.2. However, two steps related to the finite element analysis of the duct-hr system which were significant in terms of attaining an accurate solution are listed below, and described in sections and Meshing the regions of the non-planar sound field, and 2. Dissipation in the HR Meshing the Critical Regions In this chapter, critical regions are referred to those regions where the sound field was non-planar. In the duct-hr system investigated here, the regions of the duct in close proximity to the opening of the resonator and the regions of the neck were critical. It was therefore necessary to mesh such critical regions with sufficient mesh density in order to model the sound field variations with sufficient accuracy. It was also necessary so that near-field effects in the vicinity of the neck-duct interface be accurately modelled. However, it is not possible to provide a definitive EPW recommendation for the critical regions. The required EPW in the critical regions is a function of the geometry and the spatial gradients of the pressure. In order to investigate the effect of mesh size in different regions within the duct-hr system, the model of the duct-hr system was divided into four main regions, namely the duct section, the cavity section, the neck section and the critical duct section. Figure 4.3 shows a schematic of the duct-hr system highlighting the above mentioned regions. 96

129 Chapter 4. Numerical Modelling Each region of the duct-hr system was uniformly meshed by using the hexahedralshaped (brick) FLUID30 elements. Figure 4.3: A schematic of the duct-hr system showing four different regions to which the system was divided. Figures 4.4 and 4.5 display the pattern of uniform division of the duct-hr model with different mesh densities. A clear distinction between the mesh densities shown in figures 4.4 and 4.5 can be seen in figures 4.6 and 4.7, respectively which show the views of a few selected regions of the duct-hr system. Analyses varying the EPW in all the above described four regions of the duct- HR model were conducted. The effects of low and high mesh densities on the net acoustic power transmission in the duct downstream of the HR were investigated and the relevant results are presented in section Implementing Damping in the HR The finite element model of the uniform duct, analysed in section 4.2, was modelled as a rigid-walled structure with no damping included. However, in the case of duct- HR system, the damping of the HR needs to be accounted for, otherwise the response would be infinite at the resonance frequencies. More importantly, modelling the duct- HR system without damping would result in unreasonably high reductions in the in- 97

130 4.3. Circular Duct with an Attached Helmholtz Resonator Figure 4.4: A three-dimensional finite element model of the duct-hr system displaying coarse mesh. Figure 4.5: A three-dimensional finite element model of the duct-hr system displaying fine mesh. 98

131 Chapter 4. Numerical Modelling Figure 4.6: Selected regions, critical duct section, cavity section and neck section, of the three-dimensional finite element model shown in figure 4.4. Figure 4.7: Selected regions, critical duct section, cavity section and neck section, of the three-dimensional finite element model shown in figure

132 4.3. Circular Duct with an Attached Helmholtz Resonator duct net acoustic power transmission downstream of the HR which is not realistic in practical duct-hr systems due to the presence of damping. Therefore, including the effects of damping within the HR was important in order to develop a finite element model of the duct-hr system. If no damping (or absorptive) material is placed inside the HR, damping mainly occurs due to the viscous losses of the air moving back and forth in the neck. Another cause for dissipation in the resonator is the heat conduction losses which take place at its surfaces. As the major factor responsible for damping in the resonator was assumed to be the viscous losses occurring in the neck [13], it was decided to model the damping by using dissipation elements to represent the surfaces of the neck. The procedure for implementing damping in the resonator is described below. The damping of the HR was included by using the boundary admittance coefficient. In ANSYS, the boundary admittance coefficient is defined as the ratio of the real component of specific acoustic impedance of the material at the boundary to the fluid s characteristic impedance. It is given by [76]: β = MU = Re(Z) ρc (4.4) where, β is the boundary admittance coefficient and Z is the specific acoustic impedance of the material at the boundary. The value of β is input by using the material property parameter MU which ranges from 0 to 1, corresponding to no sound absorption and maximum sound absorption, respectively. In terms of electroacoustic analogies, the acoustic impedance of a HR consists of a capacitance, inductance and resistance. The capacitance is due to the stiffness of the fluid in the cavity, and inductance and resistance are due to motion of the mass of the fluid in the neck [11]. The capacitance and inductance are imaginary components of the impedance of the HR but on the contrary, resistance is real and directly related to the damping. From equation (4.4), it is evident that only the real part i.e. resistance can be modelled in ANSYS. Therefore, the resistance of the fluid in the neck of the 100

133 Chapter 4. Numerical Modelling HR was estimated according to the expression [11] R A = ρc [ ( ) ktdln (γ 1) A n 2A n 3γ ( ) 4An kt log 10 + ε A nk 2 ] πh 2 2π + M (4.5) where, A n is the cross-sectional area of an orifice, D is the cross-sectional perimeter of the duct which has the orifice drilled into its wall, l n is the length of an orifice or a tube, t is the viscous boundary thickness and is equivalent to 2µ ρω µ is the gas viscosity and for air, at 20 o C, is equivalent to kg m -1 s -1, γ is the air of specific heats and for air is equivalent to 1.4, h is either the orifice radius or viscous boundary layer thickness, whichever is bigger. For our analysis, orifice edge radius (neck radius) is bigger than the viscous boundary layer thickness. Therefore, equals to neck radius, ε is either equal to 0 when the orifice or tube radiates into spaces of dimensions the wavelength of sound, or 0.5 when the orifice or tube radiates into a free space without a flange, or 1 when the orifice or tube radiates into a free space with a flange. As in our analysis, neck radiates into a cavity, therefore ε is set to 0, and M is the Mach number of mean flow through the orifice or tube and is taken as 0 as the system investigated here is assumed to have no mean flow. 101

134 4.3. Circular Duct with an Attached Helmholtz Resonator The dimensions of the duct-hr system were: duct diameter = m, duct length = 3 m, cavity diameter = m, cavity length = m, neck diameter = m and neck length = m. The calculated value of R A for the specified dimensions of the neck is R A = ρc A n ( ω ) (4.6) It was also stated earlier in section that in ANSYS, the cross-sectional area is automatically incorporated, therefore ignoring the cross-sectional area of the neck, the value of MU for this case (using equation (4.4)) is MU = ( ω ) (4.7) As the resistance, R A, is a frequency dependent term, the value of R A (or MU) was changed for each frequency. The impedance label (IMPD = 1) was activated at the surfaces of the neck in order to activate the sound absorption capabilities (or dissipation) in the finite element model Harmonic Analysis Harmonic analyses were conducted over the frequency range from 0 to 400 Hz in 1 Hz increments. The EPW calculated at 400 Hz corresponding to the mesh size in the previously described regions of the duct-hr system was: duct section = 17, cavity section = 50, neck section = 88 and in the critical duct section = 88. For each frequency, the real and imaginary parts of the pressure at few selected nodes within the finite element model of the duct-hr system were written to a text file which was then processed in MATLAB in order to generate the plots of pressure transfer function, normalised acoustic pressure and normalised acoustic power transmission. 102

135 Chapter 4. Numerical Modelling Results Figure 4.8 shows a schematic of the duct-hr system illustrating the locations where the pressure was estimated using ANSYS. Specific numerical and alphabetical characters were assigned to these locations, and below is the description of all the locations followed by the purpose of selecting these locations. Figure 4.8: A schematic of a duct-hr system showing the locations at which the acoustic pressures were calculated using ANSYS. 1. location 1 - point located on the duct wall at 2.2 m from the source end of the duct, 2. location A - point located at the top of the centre of the closed end of the cavity of the HR, 3. location T - point located at the intersection of the centreline of the duct and centreline of the HR, 4. location E - point located centrally at the immediate opening of the resonator along the centreline of the HR, 5. location G - point located at one of the neck-duct junctions, and 6. location H - point located on the axis of the duct, vertically aligned with above stated neck-duct junction. 103

136 4.3. Circular Duct with an Attached Helmholtz Resonator Locations 1 and A were selected to verify the applicability of the transfer matrix method in the regions of planar sound field by comparing the results from ANSYS and the transfer matrix method. Another reason for selecting the location A was to compare the frequencies corresponding to the maximum acoustic pressure at the top of the closed end of the cavity and the maximum in-duct net acoustic power reduction downstream of the HR. The transfer matrix method results did not show any difference between these two frequencies. Locations E, G, T and H were selected to demonstrate the presence of complex sound field at the opening of the resonator, which cannot be shown using the transfer matrix method due to its use being restricted to plane wave conditions Acoustic Pressure Downstream of the HR Figures 4.9 and 4.10 (expanded scale) show the plot for the acoustic pressure at location 1 predicted using ANSYS along with three different plots for the acoustic pressure calculated using the transfer matrix method. Three different theoretical estimates for the acoustic pressure at location 1 were obtained by using three different values for the neck-duct interface end-correction factor (δ e ) in the transfer matrix equation. Two values of δ e (described in chapter 3, section ) were taken as per Onorati s [30] and Ji s ((equation (3.52))) [31] models, and the third one is described in the paragraph following the next. As seen from figures 4.9 and 4.10, the transfer matrix method estimates for the acoustic pressure corresponding to the value of δ e as per Onorati s [30] and Ji s [31] models do not match the ANSYS predictions over the frequency range from 210 Hz to 250 Hz. Using the value of δ e as per Onorati s [30] model in the transfer matrix equation of the duct-hr system resulted in the local minimisation of the acoustic pressure at location 1 at a frequency of 220 Hz. The theoretical estimate corresponding to the frequency of a local minimum of the acoustic pressure at location 1 became 224 Hz when the value of δ e was used as per Ji s [31] model. In contrast, ANSYS results in a 104

137 Chapter 4. Numerical Modelling normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) ansys theory ext. end corr. = Ji theory ext. end corr. = Onorati theory ext. end corr. = new frequency (Hz) Figure 4.9: Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the acoustic pressure at a point located in the duct wall at 2.2 m from the source end (location 1 ) downstream of the HR. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz 224 Hz ansys theory ext. end corr. = Ji theory ext. end corr. = Onorati theory ext. end corr. = new 226 Hz frequency (Hz) Figure 4.10: The results from figure 4.9 expanded over the frequency range from 190 Hz to 260 Hz. 105

138 4.3. Circular Duct with an Attached Helmholtz Resonator local minimum of the acoustic pressure at 226 Hz. As stated earlier, because ANSYS automatically incorporates end-correction factors in the finite element model based on first principles, the ANSYS results are considered more reliable than the transfer matrix method results. Therefore, in order to obtain a match between the ANSYS and the transfer matrix method results, the value of the neck-duct interface end correction factor was adjusted to 0.6r (r is the radius of the neck) in the transfer matrix equation. The plot with circular markers shows the transfer matrix method estimates for the acoustic pressure obtained by using δ e = 0.6r in the transfer matrix equation, and these matched favourably with the ANSYS results. Coincidentally, 0.6r also corresponds to the end-correction factor for an unflanged open end of a circular duct. Nevertheless, it is likely that the value of neck-duct interface end-correction factor may change for different duct and resonator dimensions. However, this new value for the neck-duct interface end-correction was not investigated further due to time constraints Acoustic Pressure at the Top Closed End of the Cavity Figures 4.11 and 4.12 (expanded scale) show a comparison of the acoustic pressure at the top centre of the closed end of the cavity (location A ) predicted using ANSYS with that obtained using the transfer matrix method. As described in the previous section, three different values of the neck-duct interface end-correction (δ e ) were used in the transfer matrix equation. The transfer matrix method results corresponding to δ e = 0.6r matched favourably with the ANSYS results. On the other hand, the transfer matrix method estimates corresponding to the value of δ e as per Onorati s and Ji s (equation (3.52)) models differ from ANSYS results over the frequency range from 210 Hz to 250 Hz. The theoretical estimates for the frequencies at which the acoustic pressure at location A was a maximum were 214 Hz, 218 Hz and 220 Hz (apart from one low frequency duct resonance which is not associated with the HR), and correspond to the value of δ e as per Onorati s, Ji s and the new estimate, respectively. 106

139 Chapter 4. Numerical Modelling 130 normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) ansys theory ext. end corr. = Ji 60 theory ext. end corr. = Onorati theory ext. end corr. = new frequency (Hz) Figure 4.11: Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the acoustic pressure at a point located at the top center of the closed end of the cavity (location A ) of the HR mounted onto a duct. The duct was driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. 130 normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz 218 Hz 220 Hz 70 ansys 60 theory ext. end corr. = Ji theory ext. end corr. = Onorati theory ext. end corr. = new frequency (Hz) Figure 4.12: The results from figure 4.11 expanded over the frequency range from 190 Hz to 260 Hz. 107

140 4.3. Circular Duct with an Attached Helmholtz Resonator Figures 4.9 and 4.11 revealed an excellent match between ANSYS and the transfer matrix method results, provided the value of the neck-duct interface end-correction factor in the transfer matrix equation was accurately adjusted. The sound field was planar in the duct downstream of the HR (location 1 ) and at the top of the closed end of the cavity (location A ), for which the results were presented above. An excellent match between the ANSYS and the transfer matrix method results validated the usefulness of the transfer matrix method in the regions of planar sound fields. The next section demonstrates the inapplicability of the transfer matrix method in the regions of complex sound fields Acoustic Pressure in Critical Regions When a HR is mounted on a duct, the impedance in the duct, especially near the opening of the resonator becomes sufficiently complicated not to be predictable using the same principles that were adopted for the case of a uniform duct (see chapter 3, section ). This because the sound field in the regions of the duct in close proximity to the neck opening does not depict planar behaviour. Figures 4.13 and 4.14 show the real and imaginary parts of the pressure in the selected regions of the duct-hr system at 226 Hz, respectively. The pressure contours are presented at 226 Hz because for the dimensions of the duct-hr system tested in this chapter (described above equation (4.6)), the in-duct net acoustic power transmission downstream of the HR, which will be further discussed in section , minimises at 226 Hz. The results show that the real part of the pressure in the neck of the HR increases from approximately Pa at the neck-duct interface to Pa at the neck-cavity interface, which represents an increase of 78%. Similarly, the imaginary part of the pressure increases by 74%. Although the results presented in figures 4.13 and 4.14 are not normalised, this will not alter the percentage of the pressure increase upon normalisation. Figure 4.15 shows the ANSYS estimations for the acoustic pressure at locations 108

141 Chapter 4. Numerical Modelling Figure 4.13: Numerical (ANSYS) prediction for the real part of the acoustic pressure within the duct-hr system at 226 Hz. Figure 4.14: Numerical (ANSYS) prediction for the imaginary part of the acoustic pressure within the duct-hr system at 226 Hz. 109

142 4.3. Circular Duct with an Attached Helmholtz Resonator E and T along with the predictions obtained using the transfer matrix method. The value of the neck-duct interface end-correction factor (δ e ) in the transfer matrix equation was set to 0.6r. In the previous sections and , it was shown that the transfer matrix method results for the acoustic pressure at locations 1 and A corresponding to the value of δ e calculated according to Onorati s and Ji s estimates did not match the ANSYS results. Therefore, the transfer matrix method results with δ e as per Onorati s and Ji s estimates are not included in figure As points E and T lie on the same cross-section of the duct, the difference in the pressure distribution, predicted using ANSYS, marks the presence of a complex sound field. It is not possible to show such a distinction by using the transfer matrix method due to its applicability being limited to planar sound fields. However, the estimates for the acoustic pressure obtained using the transfer matrix method matched the ANSYS results corresponding to that of location T. Similarly, figure 4.16 shows the results for the acoustic pressure at locations G and H obtained using ANSYS and the transfer matrix method. The value of δ e in the transfer matrix equation of the duct-hr system was set to 0.6r as described previously. Figure 4.16 also shows that the transfer matrix method results matched the ANSYS results corresponding to that of location H. Figures 4.15 and 4.16 show the presence of a complex sound field in the regions of the duct in close proximity to the resonator s opening. The pressure variation predicted using ANSYS at different locations on the same cross-section of the duct was not possible to calculate with the transfer matrix method. This is one of the weaknesses of the transfer matrix method when used to analyse a duct-hr system Pressure Transfer Function of the HR The definition and calculation of the pressure transfer function of the HR was described in chapter 3, section Because the transfer matrix method cannot be used to analyse the complex sound field at the opening of the resonator, only a single theoretical 110

143 Chapter 4. Numerical Modelling normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) loc. E (ansys) loc. T (ansys) theory frequency (Hz) Figure 4.15: Numerical (ANSYS) predictions for the acoustic pressures at locations E and T and theoretical (transfer matrix method) results. normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) loc. G (ansys) loc. H (ansys) theory frequency (Hz) Figure 4.16: Numerical (ANSYS) predictions for the acoustic pressures at locations G and H and theoretical (transfer matrix method) results. 111

144 4.3. Circular Duct with an Attached Helmholtz Resonator estimate for the pressure transfer function of the HR was obtained. However, with the ANSYS modelling, different estimates for the pressure transfer function of the duct-mounted HR can be obtained depending upon the pressure measurement location outside the resonator. This is due to a non-planar sound field in the region of the duct in proximity to the opening of the HR, which has already been shown in figures 4.15 and Figure 4.17 contains three different curves for the amplitude of the pressure transfer function of the HR. The curves with broken (dashed) and solid thick line are the results of ANSYS modelling which correspond to the ratio of the pressure at location A to the pressure at location E, and the pressure at location A to the pressure at location T, respectively. The third curve with solid markers shows the result of the transfer matrix method estimation. The value of δ e was kept unchanged to 0.6r. Their corresponding phase information is shown in figure For the reason described in section , the transfer matrix method results with δ e as per Onorati s and Ji s models are not included in figures 4.17 and However, the transfer matrix method results for the pressure transfer function of the HR with δ e as per Onorati s and Ji s models were discussed in chapter 3, section , and can be seen in figures 3.12 and 3.13 of chapter 3. Figures 4.17 and 4.18 show that depending upon the pressure measuring location outside the resonator, different responses for the pressure transfer function of the HR can be obtained. For the duct-hr system investigated in this chapter, the ANSYS results showed a difference of 24 Hz (9.6%) between the frequencies corresponding to the peaks of the amplitude of the pressure transfer function with respect to the pressure measuring locations, E and T, outside the resonator. The pressure transfer function calculated using the transfer matrix method (with δ e = 0.6r) matched the ANSYS results for the pressure response between location A and location T (see figure 4.8 for the locations). 112

145 Chapter 4. Numerical Modelling loc. A / loc. E (ansys) loc. A / loc. T (ansys) theory 250 Hz pressure transfer function (db) Hz frequency (Hz) Figure 4.17: Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the pressure transfer function amplitudes of the HR pressure transfer function phase (degrees) loc. A / loc. E (ansys) 180 loc. A / loc. T (ansys) theory frequency (Hz) Figure 4.18: Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the pressure transfer function phases of the HR. 113

146 4.3. Circular Duct with an Attached Helmholtz Resonator In-Duct Net Acoustic Power Transmission Downstream of the HR The net acoustic power transmission in the duct downstream of the HR was calculated by using the two-microphone method [37, 42]. Figure 4.19 shows the ANSYS results for the in-duct net acoustic power transmission downstream of the HR along with the transfer matrix method results. The plots for the net acoustic power transmission in the duct without the HR obtained using ANSYS and the transfer matrix method are also included in figure An excellent match between the transfer matrix method and ANSYS estimations was achieved. The value of the neck-duct interface end-correction factor (δ e ) was kept unchanged to 0.6r as described earlier. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz no HR ansys with HR ansys no HR theory with HR theory frequency (Hz) Figure 4.19: Numerical (ANSYS) and theoretical (transfer matrix method) predictions for the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. The ANSYS results show that the frequency at which the maximum reduction of induct net acoustic power transmission downstream of the HR occurred is 226 Hz. This frequency equaled the frequency of the maximum amplitude of the pressure transfer function of the HR between the pressures at location A and location T (see figure 4.17). Comparison of figures 4.17 and 4.19 shows that maximising the pressure transfer 114

147 Chapter 4. Numerical Modelling function (location A /location T ) between the pressure at the top of the closed end of the cavity and the pressure in the centre of the duct adjacent to the neck opening indicates the frequency which corresponds to the minimum in-duct net acoustic power transmission. However, it is not practical to mount a microphone in the duct as it will obstruct the flow stream of the exhaust gases. Therefore, the next best location somewhere in the neck of the resonator has to be chosen and is described in chapter 5, Experimental Verification. The results presented in sections to show the limitations of the transfer matrix method in analysing the duct-hr system. Also, with the transfer matrix method, the regions of the planar sound fields within the duct-hr system can only be analysed provided the end-corrections of the neck are accurately known Effect of Different Mesh Densities The purpose of this section is to show the importance of meshing the regions with sufficient mesh density in the finite element model of the duct-hr system where the sound field is complex. It was shown in section that the three-dimensional model of the duct-hr system was divided into 4 regions, namely the duct section, the cavity section, the neck section and the critical duct section. Five different configurations of uniformly dividing the three-dimensional duct-hr model were obtained by varying the mesh density (or EPW) in all the above mentioned four regions. The summary of EPW at 226 Hz in the four regions is tabulated in table 4.3. As seen from figure 4.19, because 226 Hz was the frequency achieving the maximum reduction of in-duct net acoustic power, the quantity EPW was calculated at 226 Hz in order to investigate the effects of EPW on in-duct net acoustic power at 226 Hz. For each configuration, a harmonic analysis was conducted over the frequency range from 200 to 250 Hz in 1 Hz increments. This frequency range was chosen in order to see the effect of different mesh densities on the frequency (226 Hz) at which the maximum 115

148 4.3. Circular Duct with an Attached Helmholtz Resonator Region EPW at 226 Hz Configuration Duct section Cavity section Neck section Critical duct section Frequency of min. power Table 4.3: Summary of the elements per wavelength at 226 Hz for dividing the model of the duct-hr system. reduction of in-duct net acoustic power transmission downstream of the HR occurs. Figure 4.20 shows five plots for the net acoustic power transmission in the duct downstream of the HR for the above tabulated five configurations. For the configuration 1, the maximum reduction of the in-duct net acoustic power transmission occurred at 230 Hz. In the second configuration, the element size in the critical duct section was halved from its value in configuration 1, which resulted in doubling the EPW, while the element size in the duct, cavity and neck section was kept unchanged. For the configuration 2, the maximum in-duct net acoustic power reduction was achieved at 229 Hz as compared to 230 Hz for the configuration 1. Similarly, the configuration 3 was achieved by further refining the element size in the critical duct section so that the EPW was three times its value in configuration 1. Corresponding to the configuration 3, the maximum acoustic power reduction occurred at 228 Hz, which is 2 Hz lower than the frequency of minimum acoustic power for configuration 1. The element size in the critical duct section was further reduced so that the number of EPW became four times its value in configuration 1. However, this did not change the estimate of the frequency of minimum acoustic power from its previous estimate of 228 Hz corresponding to the configuration 3. Next, the element size in the neck section of the duct-hr model, which was kept constant in all the first three tabulated configurations, was reduced by approximately five times its value in the previous configurations. The refinements in the critical duct and neck sections described in this paragraph describe configuration 4, and resulted in 227 Hz as the frequency of minimum acoustic power. 116

149 Chapter 4. Numerical Modelling normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) configuration 1 configuration 2 configuration 3 configuration 4 configuration frequency (Hz) Figure 4.20: Numerical (ANSYS) predictions showing the affect of varying mesh densities on the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. Further to the refinements carried out in the duct-hr model in order to achieve configuration 4, the element size in the cavity section was reduced by five times its value in the previous configurations. As a result, the frequency of minimum acoustic power was calculated as 226 Hz. Any further refinement in any of the regions of the duct-hr model was not possible after configuration 5 with the ANSYS software package used for this study. This is because the ANSYS software package used for this study has a limitation of the number of allowable elements for descretising a model. Also, in order to study the effect of mesh size in the duct section, the number of EPW in the duct section were increased after each configuration, except configuration 5, which was not possible due to the above stated limitation of the ANSYS software package. However, this did not affect the in-duct acoustic power transmission. From figure 4.20 and table 4.3, it can be observed that reducing the element size, which results in increasing the number of elements in the HR and the region of the duct in close proximity to the HR mounting, affects the in-duct acoustic power transmission. However, no specific guidelines for meshing the duct-hr model was devised from this 117

150 4.4. Sensitivity Analysis analysis. It is very clear that there must be sufficient elements in the HR and the non-planar sound field regions in order to accurately model the pressure variations. 4.4 Sensitivity Analysis The aim of this section is to show the influence of measurement errors, which exist in real/practical applications on the in-duct acoustic power transmission downstream the HR. Some of the possible causes for these errors are: inaccuracy in measuring the pressure transfer function between two microphones mounted onto the duct downstream of the HR in order to decompose the sound field, and variation in the speed of sound. As the estimation of the in-duct net acoustic power transmission using the twomicrophone technique [37, 42] involves measuring the pressure transfer function between them, any inaccuracy in this measurement can directly affect the acoustic power transmission estimates. In order to observe the effect of phase inaccuracies on the acoustic power transmission estimates, the original phase estimates corresponding to one of the two pressure measurements downstream of the HR obtained using ANSYS was changed by few degrees, and the net acoustic power transmission in the duct was calculated. Figures 4.21 and 4.22 show the effect of phase errors, corresponding to the pressure transfer function between the two pressure measurements in the duct, on the in-duct net acoustic power transmission downstream of the HR. Figure 4.21 shows that an error of +5 o overestimated the acoustic performance of the HR at 226 Hz by approximately 15 db. In contrast, figure 4.22 shows a reduction of 3 db in the acoustic performance of the HR at 226 Hz. Thus, in a practical application, it is very important to accurately calibrate the microphones for both amplitude and phase in order to obtain accurate 118

151 Chapter 4. Numerical Modelling normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) deg 1 deg 2 deg 3 deg 4 deg 5 deg frequency (Hz) Figure 4.21: Numerical (ANSYS) predictions showing the effects of positive phase errors of a few degrees on the in-duct net acoustic transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) deg 1 deg 2 deg 3 deg 4 deg 5 deg frequency (Hz) Figure 4.22: Numerical (ANSYS) predictions showing the effects of negative phase errors of a few degrees on the in-duct net acoustic transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. 119

152 4.4. Sensitivity Analysis normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) c = 340 m/sec c = 344 m/sec c = 348 m/sec frequency (Hz) Figure 4.23: Numerical (ANSYS) predictions showing the effects of different sound velocities on the in-duct net acoustic transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz 226 Hz 228 Hz c = 340 m/sec c = 344 m/sec c = 348 m/sec frequency (Hz) Figure 4.24: The results from figure 4.23 expanded over the frequency range from 190 Hz to 260 Hz. 120

153 Chapter 4. Numerical Modelling estimates of the net acoustic power transmission. It is important to note that the frequency of minimum power transmission is not affected much by the phase errors. The speed of sound is another important parameter which needs to be carefully set for the accurate analysis of systems. In the ANSYS analyses conducted here, the speed of sound was defined during the element definition of the model. The duct-hr system was analysed for three different values of the speed of sound, and the corresponding induct net acoustic power transmissions downstream of the HR are shown in figures 4.23 and The figures show that the variation in the speed of sound can significantly affect the amplitude of the acoustic power transmission in the duct as well as the frequency corresponding to the minimum power transmission. 4.5 Conclusion A two-dimensional finite element model of a circular cross-sectional duct with no HR attached was built and solved for the pressure distribution inside it. One end of the duct was modelled as being driven by a piston and the other was modelled as open and radiating into free space. Four different configurations of uniform division of the model of the duct were obtained by varying the element size (or mesh density or EPW). For each configuration, the net acoustic power transmission in the duct was estimated and the acoustic resonance frequencies of the duct were extracted from the acoustic power plots. The results from the ANSYS predictions were compared with results from the transfer matrix method and excellent agreement was demonstrated, provided that the AN- SYS model was discretised into a sufficient number of elements. A three-dimensional finite element model of a circular duct with an attached cylindrical HR was also developed and solved for investigating the sound field inside the duct and the HR. As for the uniform duct, the duct-hr system was modelled as a piston driven-open system. Damping within the HR was also incorporated in finite element model to approximately account for the losses which occur in the resonator. 121

154 4.5. Conclusion The estimates for the acoustic pressure at a few locations within the duct-hr system were presented. These locations were at the duct wall downstream of the HR, at the top of the closed end of the cavity of the HR and in the centre of the duct adjacent to the opening of the HR. The presence of complex sound field in the duct in close proximity to the resonator s opening was shown. Different measures for the pressure transfer function of the HR were shown with respect to the pressure measurement location outside the resonator. The acoustic performance of the duct-mounted HR was evaluated by comparing the in-duct acoustic power transmission in the duct with and without the HR. Comparison of the results from the ANSYS modelling with those of the transfer matrix method showed that the latter approach can only be used to analyse the duct- HR system if estimates of all the end-corrections are accurate. In the case of duct- HR system, the neck-duct interface end-correction factor was not clearly known and therefore, was adjusted. Once this end correction was set, an excellent match between the estimates obtained from the transfer matrix method and the ANSYS solution was achieved in the planar sound field regions. However, the transfer matrix method is unable to be used to show the pressure variations on a particular (vertical) crosssection of a duct having non-planar sound field, which was an important limitation of the transfer matrix method when used for the case of a HR attached to a duct. To conclude, the transfer matrix method is not a viable option for analysing the duct-hr system. 122

155 Chapter 5 Experimental Verification 5.1 Introduction This chapter presents experimental findings pertinent to a stand-alone Helmholtz resonator and a circular duct with an attached cylindrical Helmholtz resonator. Following the approach set out in the previous two chapters, Theoretical Modelling and Numerical Modelling, this chapter also begins with a section related to stand-alone HRs. Experiments were conducted to measure the resonance frequencies of stand-alone cylindrical HRs and these results were compared to theoretical predictions. The resonance frequency of a HR can be defined as the frequency corresponding to the maximum acoustic pressure at the top of the closed end of the cavity. The resonance frequency of a HR can also be determined by measuring the pressure transfer function between two microphones: the first located at the top of the closed end of the cavity and the second located at the centre of the opening of the neck of the HR. The resonance frequency is then defined as the peak value of this transfer function. Results obtained using both methods were compared with each other and with theoretical predictions. In addition to locating the microphone at the top of the closed end of the cavity, three microphones were flush mounted along the periphery of the neck wall as close as possible to the neck opening. Two reasons for mounting the microphones on the neck wall rather than in the centre of the duct adjacent to the neck opening are: 123

156 5.1. Introduction 1. to facilitate the development of a completely self-contained HR, which does not require any kind of measurement remote from the device, and 2. to avoid the potential risk of an obstruction to the mean flow of the (exhaust) gases due to the location of a microphone in or outside the opening of the neck The pressure transfer functions of the HR between the top of the closed end of the cavity and at all the three neck wall microphone locations were measured. Experiments were then carried out on a circular duct with an attached cylindrical HR, and the results are presented in the two different ways: (1) broadband analysis, whereby the acoustic pressure at the top of the closed end of the cavity, different pressure transfer functions of the HR (measured with respect to three different neck wall microphones) and the induct net acoustic power transmission downstream of the HR are plotted as a function of frequency, and (2) single frequency analysis, whereby all the aforementioned results are plotted for a single frequency as a function of the cavity length of the HR. Of the three microphones mounted flush along the periphery of the neck wall, one was selected for future pressure transfer function measurement of the HR. The reasons for selecting that microphone are discussed further in the chapter. In the case of broadband analysis, the results showed that the frequency at which the greatest reduction of in-duct net acoustic power transmission downstream of the HR occurs differs from the frequencies corresponding to the maximum acoustic pressure at the top of the closed end of the cavity and the maximum amplitude of pressure transfer function of the HR. In the case of single frequency analysis, the above stated results imply that the dimensions of the HR required to attain maximum in-duct net acoustic power reduction downstream of the HR differ from the dimensions of the HR corresponding to the maximum acoustic pressure at the top of the closed end of the cavity and the maximum amplitude of the pressure transfer function of the HR. All the experimental results of the duct-hr system were compared with those of the theoretical (transfer matrix method) and numerical (ANSYS) modelling predictions. Good agreement between the experimental, numerical and theoretical results was found. 124

157 Chapter 5. Experimental Verification This chapter concludes by summarising the experimental findings and highlighting the need for a suitable cost function based on the pressure measurements inside the resonator only, which can be used by an electronic controller to optimally tune the HR. 5.2 Stand-alone Helmholtz Resonator Steel tubes were used to build cylindrical HRs with a variable volume. A 0.2 m long cavity with an inside diameter of m was cut from a standard steel tube with a median wall thickness of 4.8 mm. Three different neck sections each of length m, having inside diameters of m, m and m, were also cut from standard steel tubes of the same median wall thickness as stated above. Three cylindrical HRs of different neck diameters ( m, m and m) with constant cavity diameter and neck length were tested. In order to alter the volume of the HRs, an air tight circular piston made from PVC was inserted in the cavity of the resonators as shown in figure 5.1. Figure 5.1: A picture of a HR placed on a table inside an anechoic chamber Experimental Setup An experiment was conducted in an anechoic chamber to measure the resonance frequencies of several cylindrical HRs. Figure 5.1 shows a part of the experimental setup 125

158 5.2. Stand-alone Helmholtz Resonator Figure 5.2: A different view of the HR pictured in figure 5.1 showing the microphones mounted at the opening of the neck. Figure 5.3: A schematic of a stand-alone cylindrical HR showing the locations of the microphones. 126

159 Chapter 5. Experimental Verification where the HR was placed on a table inside the anechoic chamber. The other equipment, microphone amplifier, power amplifier and B&K Pulse system, was placed in the room adjoining the anechoic chamber. The length of the neck for all the neck sizes was extended from m to m by adding PVC sections having inside diameters the same as those of neck inside diameters. The reason for adding the PVC sections is described in section The neck opening corresponding to the PVC section was cut into a saddle joint to allow the cylindrical HR to be attached to a circular duct. The saddled shape of the neck opening can be seen in figure 5.1. A different view of the same HR pictured in figure 5.1 is shown in figure 5.2 depicting the microphones located at the neck opening. The locations of the microphones are described below: microphone E - located centrally in the plane of the neck opening, and microphones B, C and D - located at the neck walls at a distance of 5 mm from the plane of neck opening. Another microphone which is not visible in either of the two pictures but has been sketched in figure 5.1 was flush mounted in the piston and is denoted by microphone A. As the piston forms the top closed end of the HR cavity, microphone A always remains at the top closed end of the HR cavity for various cavity lengths. A schematic of a cylindrical HR depicting the locations of all the microphones described above are shown in figure 5.3. As seen from figure 5.1, the neck of the HR was orientated so that it pointed away from the surface of the table to ensure that the pressure measurements at microphones B, C, D and E were not affected by possible reflections of acoustic waves from the top surface of the table. A 125 mm diameter loudspeaker was placed inside the anechoic chamber at a distance of 2.5 m (approximately) from the HR. This distance was sufficient for the development of plane acoustic waves with sufficient accuracy in the frequency range of interest. The loudspeaker facing the opening of the HR was 127

160 5.2. Stand-alone Helmholtz Resonator excited by random noise from 0 to 400 Hz. A multi-channel B&K Pulse system was used to measure and record the pressure at all microphone locations listed above. All the experimental measurements were recorded with a frequency resolution of 1 Hz Experimental Results Resonance Frequencies For these experimental measurements, the resonance frequency of a stand-alone HR was defined as the frequency at which the acoustic pressure at the top of the closed end of the cavity (microphone A ) was a maximum [19]. The fixed dimensions of the cylindrical HR were: cavity diameter = m, neck diameter = m and (physical) neck length = m. The volume of the HR was varied by changing the length of its cavity in 10 mm increments using a piston. Table 5.1 compares the experimentally measured resonance frequencies of one of three cylindrical HRs with the theoretical predictions obtained using the three theoretical models along with the transfer matrix method estimates. The three theoretical models, which have been discussed in chapter 3, section , correspond to the classical (equation (3.4)), Panton and Miller s (equation (3.10)) and Li s (equation (3.11)) models. The plot of the tabulated results, except the estimates obtained using the transfer matrix method, is also shown in figure 5.4. The predicted resonance frequencies obtained using the classical and Panton and Miller s models are well above those measured experimentally. As stated earlier in chapter 3, the classical equation is not valid if the dimensions of the HR are greater than 1/16 of a wavelength. The dimensions of the cylindrical HRs which were tested in this study were greater than 1/16 of the wavelength; hence there are significant differences between the resonance frequencies calculated using the classical model and the experimental results. In constrast, the theoretical predictions obtained using Panton and Miller s model are slightly better than those of the classical model predictions. This is because Panton and Miller s model was derived assuming one-dimensional wave 128

161 Chapter 5. Experimental Verification HR cavity length, (mm) classical, equation (3.4), (Hz) Panton & Miller s, equation (3.10), (Hz) Li s, equation (3.11), (Hz) transfer matrix method, equation (3.63), (Hz) experiment (Hz) Table 5.1: Experimentally measured resonance frequencies of stand-alone cylindrical HRs along with the predictions from several theoretical models. resonance frequency (Hz) classical Panton & Miller Li exp cavity length (mm) Figure 5.4: Experimentally measured resonance frequencies of stand-alone cylindrical HRs along with the predictions from several theoretical models. 129

162 5.2. Stand-alone Helmholtz Resonator propagation in the cavity as compared to lumped mass assumption in the classical theory. However, the predictions from Panton and Miller s model deviates from the experimental results due to two reasons: 1. the neck dimensions and the cavity diameter of tested HRs were not smaller than 1/16 of the wavelength, as were assumed during the derivation of Panton and Miller s model, and 2. while deriving the model it was assumed that the length of the neck was very small compared to the cavity length. As the length of the neck becomes comparable to or greater than the cavity length, predictions start deteriorating. The predictions from Li s model match favourably with the experimental results because Li s model did not make any assumption during its derivation. But, as the derivation of the model was based on the wave-tube theory (one-dimensional wave propagation in the cavity and neck), the diameter of the cavity must be smaller than the acoustic wavelength of interest for the results to be accurate [8]. A transfer matrix equation of the HR was developed by using the concept of infinite impedance at the top of the closed end of the cavity. As seen from the tabular comparison, the resonance frequencies predicted using the transfer matrix method are close to those of Li s predictions Pressure Transfer Functions A few researchers [8, 64, 79] have demonstrated that the resonance frequency of a HR can be determined by measuring the pressure transfer function between two microphones: first, located at the top of the closed end of the cavity, and second, located at the opening of the neck. The resonance frequency was defined as the frequency at which the amplitude of the pressure transfer function was a maximum. However, the authors in references [8, 64, 79] did not specify the distance of the second microphone from the opening of the resonator. The microphone at the opening of the resonator 130

163 Chapter 5. Experimental Verification in the environment must be positioned carefully so that its response is not affected by the sound field in proximity to the neck opening. The neck opening generates a near-field which results in the formation of a complex sound field in the regions close in proximity to the neck opening. The amplitude of the pressure transfer function between the cavity microphone and the opening microphone will be a maximum at a frequency corresponding to the maximum acoustic pressure at the cavity microphone, if the response of the microphone located near the neck opening is not affected by the near-field of the neck. Hence, in this study, the resonance frequency of the HR was not measured by the above described pressure transfer function method as used by the authors in references [8, 64, 79]. However, several pressure transfer function measurements between microphone A and E, A and B, A and C, and A and D were taken for two reasons: 1. to demonstrate the near-field effects at the opening of the resonator, and 2. to check whether the frequency corresponding to the above described maximum pressure transfer function amplitudes be equal to the frequency of the maximum in-duct net acoustic power transmission when the HR is mounted on a duct. The experimental results for the above mentioned pressure transfer functions are described in the following paragraphs. The length of the cavity was fixed at m and all the other dimensions of the HR were kept the same as described in section The pressure transfer function between microphone A and microphone E (microphone A /microphone E ) was measured and the pertinent amplitude and phase plots are shown in figures 5.5 and 5.6, respectively. As stated earlier in section and shown in figure 5.3, microphone E was located centrally in the plane of the neck opening. The vertical line in both figures corresponds to the measured resonance frequency of the HR described in section , table 5.1 (see the frequency corresponding to a cavity length of m). The amplitude of the pressure transfer function was a maximum at 236 Hz. However, the 131

164 5.2. Stand-alone Helmholtz Resonator 30 mic. A / mic. E Hz pressure transfer function (db) Hz frequency (Hz) Figure 5.5: Experimentally measured pressure transfer function amplitude between microphone A and microphone E. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR. 0 mic. A / mic. E pressure transfer function phase (degrees) Hz frequency (Hz) Figure 5.6: Experimentally measured pressure transfer function phase between microphone A and microphone E. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR. 132

165 Chapter 5. Experimental Verification experimentally measured resonance frequency of the HR was 222 Hz for the geometry of the HR stated above. The 14 Hz difference in the frequencies between the resonance of the HR and the maximum amplitude of the pressure transfer function highlights the importance of carefully positioning the microphone at the opening of the resonator. As stated earlier in chapter 1, microphones cannot be left suspended in the HR, such as at the location of microphone E for two reasons: 1. to eliminate any potential risk of causing an obstruction to the mean flow of fluid, such as exhaust gases or air in practical applications, 2. to develop a completely self-contained adaptive HR which does not require any pressure measurement remote from the device. Considering the above two reasons, the pressure transfer functions between microphone A and all the microphones located at the neck wall in proximity to the neck opening (microphones B, C and D ) were measured. Figures 5.7 and 5.8 show the amplitude and phase plots of the pressure transfer functions between microphones A and B, A and C, and A and D, respectively. The arrow pointing towards one of the peaks in the amplitude plot (figure 5.7) corresponds to the maximum pressure transfer function between microphones A and B which occurs at 246 Hz. On the other hand, the amplitudes of the transfer functions between microphones A and C, and A and D were a maximum at 248 Hz. The 2 Hz difference between the frequencies corresponding to the above described maximum pressure transfer function amplitudes is possibly due to the shape of the neck opening which was not planar but saddled. Comparison of figures 5.5 and 5.7 shows a difference of more than 10 Hz (4%) between the frequencies corresponding to the maximum amplitude of the pressure transfer functions between microphones A and E, and the other three transfer functions shown in figure 5.7. This difference in the frequencies indicates the presence of the complex sound field at the opening of the resonator. Another reason for the differences in the frequencies is the diffraction pattern which occurs at the opening of the neck. 133

166 5.2. Stand-alone Helmholtz Resonator pressure transfer function (db) Hz mic. A / mic. C mic. A / mic. B mic. A / mic. D Hz frequency (Hz) Figure 5.7: Experimentally measured pressure transfer function amplitudes between microphones A and C, A and B, and A and D. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR. pressure transfer function phase (degrees) Hz mic. A / mic. C mic. A / mic. B mic. A / mic. D frequency (Hz) Figure 5.8: Experimentally measured pressure transfer function phases between microphones A and C, A and B, and A and D. The vertical line at 222 Hz corresponds to the experimentally measured resonance frequency of the HR. 134

167 Chapter 5. Experimental Verification Summary So far in this chapter, the experimental results for the resonance frequencies of standalone cylindrical HRs and various transfer function measurements are presented. Comparison of the experimentally measured resonance frequencies with theoretical predictions showed the importance of selecting an appropriate theoretical model for calculating the resonance frequencies of HRs. The difference in the frequencies corresponding to the maximum amplitudes of the pressure transfer function between the cavity microphone and the neck opening microphones indicates the presence of a complex sound field at the opening of the resonator. Also, if the pressure transfer function method is to be utilised for measuring the resonance frequency of a HR, one needs to carefully position the microphone at the opening of the resonator so that its response is minimally affected by the near-field generated at the neck. Of course, it may not be possible to obtain an accurate measurement regardless of the location of the neck opening microphone unless it is well away from the opening. 5.3 Circular Duct with an Attached Helmholtz Resonator Experiments were conducted to measure the sound power reduction in the duct with an attached HR. The following sections describe the experimental setup followed by experimental results and discussion Experimental Setup Figure 5.9 shows a picture of the experimental setup displaying a part of the experimental rig, a steel circular duct with an attached cylindrical HR, along with other equipment (B&K Pulse system and amplifiers). A 3 m long circular steel duct with an inside diameter of m and a wall thickness of 6.3 mm was used in this study. A steel circular section, also cut from the standard steel tube, of diameter m 135

168 5.3. Circular Duct with an Attached Helmholtz Resonator Figure 5.9: A picture of the experimental rig showing a circular duct with an attached HR along with the B&K Pulse system and amplifiers. Figure 5.10: A picture of a duct showing a circular section welded on the duct at a distance of 0.5 m from the source end. Figure 5.11: A picture of the PVC sections. 136

169 Chapter 5. Experimental Verification was welded onto the duct at 0.5 m from the source end as shown in figure The purpose of welding this section was to enable mounting the three different necks of diameters m, and m. Because the diameter of the welded section was greater than the neck diameters, PVC sections, shown in figure 5.11, having their inside daimeters same as those to the neck inside diameters were inserted into the welded section. This was done to avoid the area discontinuity at the interface of the welded section and the neck. One of the three variable volume cylindrical HRs, described in section 5.2, was mounted onto the welded section having the appropriate PVC section inserted (see figure 5.9). A 125 mm diameter loudspeaker backed by an air tight cavity was mounted at the source end of the duct. The other end of the duct was open and radiating into free space. All the experiments were conducted in the laboratory where additional apparatus was also present. Figure 5.12: A schematic of the duct-hr system showing the locations of the microphones. A schematic of the experimental duct-hr system along with the microphone locations is shown in figure All the microphones (except microphone E ) used for the experimental work on stand-alone cylindrical HRs were retained, and four additional microphones were mounted within the duct-hr system. The description and purpose 137

170 5.3. Circular Duct with an Attached Helmholtz Resonator of these additional microphones are listed below. Two microphones, 1 and 2, were flush mounted onto the duct wall downstream of the HR in order to measure the net acoustic power transmission down the inside of the duct. Microphone 1 was 1.2 m from the mounting location of the HR, and the distance between microphones 1 and 2 was 0.3 m. Microphone T was placed in the centre of the duct immediately below the neck opening. As has already been stated, in practice it is not desirable to locate a microphone at any location within the duct-hr system which may obstruct the mean flow of exhaust gases in the duct. Thus, microphone T was included to enable a comparison of the acoustic pressure in the duct and neck. Moreover, for the experimental work conducted in this study, there was no mean flow in the duct. Microphone S (not shown) was located in the backing cavity of the loudspeaker for the purpose of measuring the volume velocity of the speaker. The response of all the microphones was normalised by the input volume velocity of the loudspeaker, v sp, which is given by [11] v sp = iωv spp sp ρc 2 (5.1) where, V sp is the volume of the backing cavity of the loudspeaker, p sp is the acoustic pressure inside the backing cavity of the loudspeaker, ω is the angular velocity, ρ is the density of the fluid medium (air) and c is the speed of sound. The dimensions of the HR were kept as: cavity length = m, cavity diameter = m, (physical) neck length = m and neck diameter = m. The loudspeaker was excited by random noise from 0 to 400 Hz and a multi-channel B&K Pulse system was used to measure and record the pressure at all the microphone locations within the duct-hr system. All the experimental measurements were recorded with a frequency resolution of 1 Hz. 138

171 Chapter 5. Experimental Verification The results of experiments are discussed in the following sections Broadband Analysis The experimentally measured acoustic pressure, pressure transfer functions and in-duct net acoustic power transmission were plotted as a function of frequency and will be discussed in following four sub sections. The purpose of presenting the results as a function of frequency is to illustrate the difference in the frequencies corresponding to the maximum acoustic pressure at the top of the closed end of the cavity, maximum amplitude of the pressure transfer functions and maximum reduction of in-duct net acoustic power transmission downstream of the HR Acoustic Pressure at the Top Closed End of the Cavity The experimentally measured acoustic pressure at the top of the closed end of the cavity (microphone A ) is shown in figures 5.13 and 5.14 (expanded scale) along with the theoretical (transfer matrix method) and numerical (ANSYS) modelling predictions. The theoretical predictions obtained using the transfer matrix method were described in chapter 3, Theoretical Modelling. As shown in chapter 4, Numerical Modelling, because ANSYS results are more reliable than those of the transfer matrix method results, the end-correction factor at the neck-duct interface was adjusted to 0.6r (r denotes the radius of the HR neck) in the transfer matrix formulation in order to achieve a favourable match between the transfer matrix method and ANSYS results. As also described in chapter 3, the damping in the transfer matrix formulations was incorporated using the complex wave numbers (see section ). The value of the loss factor used was The experimentally measured acoustic pressure at microphone A was a maximum at 218 Hz (apart from one low frequency duct resonance which is not associated with the HR). In contrast, the theoretical (transfer matrix method) and numerical (ANSYS) results corresponding to the maximum acoustic pressure at the top of the closed end of 139

172 5.3. Circular Duct with an Attached Helmholtz Resonator normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz 220 Hz exp ansys theory frequency(hz) Figure 5.13: Experimental, numerical (ANSYS) and theoretical (transfer matrix method) results for the acoustic pressure at microphone A, as a function of frequency. normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz 220 Hz 60 exp ansys theory frequency(hz) Figure 5.14: The results from figure 5.13 expanded over the frequency range from 190 Hz to 290 Hz. 140

173 Chapter 5. Experimental Verification the cavity were a maximum at 220 Hz. Here, the resonance frequency of the duct-hr system, which was defined as the frequency corresponding to the maximum acoustic pressure at the top of the closed end of the cavity, is 218 Hz and 4 Hz less than the resonance of the stand-alone HR (see table 5.1 for the frequency corresponding to a cavity length of m). The numerical (ANSYS) and theoretical (transfer matrix method) predictions for the acoustic pressure shown in figure 5.13 compare favourably with the experimental results except at the peaks and below 20 Hz. The reason for the differences of numerical and theoretical predictions from the experimental results at the peaks is the inaccuracy in incorporating damping in the numerical and theoretical models of the duct-hr system. Below 20 Hz, the experimental results differ from the numerical and theoretical predictions due to the poor quality of the signal generated by the loudspeaker at low frequencies, leading to significant noise contamination of the signal Acoustic Pressure at the Neck Opening Figures 5.15 and 5.16 (expanded scale) show the experimentally measured acoustic pressure at microphones B, C, D and T. It is evident from figure 5.15 that there are no significant differences between the acoustic pressure measurements at microphones B, C and D as they are closely mounted along the periphery of the circular neck. On the other hand, the acoustic pressure at microphone T shows marked difference in the frequency band of 150 Hz to 250 Hz as compared to the other microphones responses. Unlike the measures of the acoustic pressure at microphones B, C and D, the acoustic pressure at microphone T has a local minimum at 225 Hz before attaining the peak at approximately 260 Hz, which conforms with those of the neck-wall microphones responses (microphones B, C and D ). The difference in the pressure responses between the neck-wall microphones and microphone T is because the neck-wall microphones experience near-field effects generated at the neck opening. In 141

174 5.3. Circular Duct with an Attached Helmholtz Resonator normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz mic. B mic. C mic. D mic. T frequency (Hz) Figure 5.15: Experimentally measured acoustic pressures at the neck opening microphones B, C, D and T, as a function of frequency. normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz mic. B mic. C mic. D mic. T frequency (Hz) Figure 5.16: The results from figure 5.15 expanded over the frequency range from 190 Hz to 290 Hz. 142

175 Chapter 5. Experimental Verification normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz loc. B loc. C loc. D loc. T frequency (Hz) Figure 5.17: Numerical (ANSYS) predictions for the acoustic pressures at the locations analogous to the neck opening microphones B, C, D and T, as a function of frequency. normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz loc. B loc. C loc. D loc. T frequency (Hz) Figure 5.18: The results from figure 5.17 expanded over the frequency range from 190 Hz to 290 Hz. 143

176 5.3. Circular Duct with an Attached Helmholtz Resonator contrast, the pressure response at microphone T, which was placed in the centre of the duct, was not significantly affected by the near-field. ANSYS was used to predict the acoustic pressure at the locations analogous to those of microphone locations B, C, D and T shown in figure The variation in the experimentally measured acoustic pressure in the duct and neck shown in figures 5.15 and 5.16 was also seen in the ANSYS results shown in figures 5.17 and 5.18 (expanded scale). The predicted acoustic pressure using ANSYS at the centre of the duct adjacent to the neck opening (location T ) had a local minimum at 226 Hz. Although the ANSYS results were not included in figure 5.15 to avoid congestion, a favourable match between the experimental and ANSYS results is evident by comparing figures 5.15 and Pressure Transfer Function of the HR Four pressure transfer functions between microphone A and B, A and C, A and D, and A and T were measured and the corresponding amplitude and phase plots of these pressure transfer functions are shown in figures 5.19 and 5.20, respectively. Figure 5.19 shows that the experimentally measured frequencies corresponding to the maximum amplitude of the pressure transfer functions between microphones A and B, A and C, and A and D are close to each other, and correspond to 247 Hz, 252 Hz and 250 Hz, respectively. The reason for the proximity in the measures of the pressure transfer function amplitude is due to similar responses of the acoustic pressure at microphones B, C and D as shown in figure The comparison of the pressure transfer function amplitudes for the case of the duct-hr system described above (figure 5.19) and the stand-alone HR (figure 5.7) shows a slight difference in all the three measures of the frequencies corresponding to their respective maximum responses. The frequencies corresponding to the pressure transfer function between microphones A and B, A and C, and A and D for the duct-mounted HR differ by 1 Hz, 4 Hz and 2 Hz, respectively from the corresponding 144

177 Chapter 5. Experimental Verification mic. A / mic. B mic. A / mic. C mic. A / mic. D mic. A / mic. T 225 Hz pressure transfer function (db) frequency (Hz) Figure 5.19: Experimentally measured pressure transfer function amplitudes between microphones A and B, A and C, A and D, and A and T, as a function of frequency. 0 pressure transfer function phase (degrees) mic. A / mic. B mic. A / mic. C mic. A / mic. D mic. A / mic. T frequency (Hz) Figure 5.20: Experimentally measured pressure transfer function phases between microphones A and B, A and C, A and D, and A and T, as a function of frequency. 145

178 5.3. Circular Duct with an Attached Helmholtz Resonator pressure transfer function (db) loc. A / loc. B loc. A / loc. C loc. A / loc. D loc. A / loc. T 226 Hz frequency (Hz) Figure 5.21: Numerical (ANSYS) predictions for the pressure transfer function amplitudes between locations A and B, A and C, A and D, and A and T, as a function of frequency. 0 pressure transfer function phase (degrees) loc. A / loc. B loc. A / loc. C loc. A / loc. D loc. A / loc. T frequency (Hz) Figure 5.22: Numerical (ANSYS) predictions for the pressure transfer function phases between locations A and B, A and C, A and D, and A and T, as a function of frequency. 146

179 Chapter 5. Experimental Verification pressure transfer function (db) loc. A / loc. B loc. A / loc. C loc. A / loc. D loc. A / loc. T 226 Hz frequency (Hz) Figure 5.23: Numerical (ANSYS) predictions of the pressure transfer function amplitudes between locations A and B, A and C, A and D, and A and T for the location of the HR as 1.2 m from the source end of the duct, as a function of frequency. 0 pressure transfer function phase (degrees) loc. A / loc. B loc. A / loc. C loc. A / loc. D loc. A / loc. T frequency (Hz) Figure 5.24: Numerical (ANSYS) predictions of the pressure transfer function phases between locations A and B, A and C, A and D, and A and T for the location of the HR as 1.2 m from the source end of the duct, as a function of frequency. 147

180 5.3. Circular Duct with an Attached Helmholtz Resonator frequencies pertinent to the stand-alone HR. This difference is due to the interaction of the duct response with the HR response. In contrast, the maximum amplitude of the pressure transfer function between microphones A and T for the duct-hr system occurred at 225 Hz as shown in figure This is because the acoustic pressure at microphone T had a local minimum at 225 Hz as shown in figure Although the location of microphone T cannot be considered for practical/real applications, in this study this pressure transfer function was measured to show how it compared with several other transfer function measurements. The difference in the various pressure transfer functions presented in figure 5.19 shows the importance of locating the microphone at the opening of the resonator. For the duct-hr system tested, the frequency corresponding to the maximum amplitude of the pressure transfer function between microphone A and T differs by more than 20 Hz (8%) from all the three frequencies corresponding to the pressure transfer functions between microphones A and B, A and C, and A and D. Also, it is evident from figures 5.19 and 5.20 that the pressure transfer functions between microphones A and C, A and D, and A and T are not as smooth as compared to that between microphones A and B, which is shown by a thick line. Figures 5.21 and 5.22 show the numerical predictions for the same pressure transfer functions obtained using ANSYS. The experimental and numerical results are not shown in a single graph with a view to avoid congestion, but a favourable match between the results can be seen. The spikes in the experimental and numerical results approximately at 67 Hz and its corresponding harmonics at 135 Hz, 207 Hz, 268 Hz and 335 Hz are related to the mounting location of the HR being close to a pressure node in the duct acoustic field. The acoustic half-wavelength corresponding to the frequency of 67 Hz, which is approximately 2.5 m, equals the distance of the HR mounting location from the open end of the duct. This claim was confirmed by solving the duct-hr model using ANSYS with varying the HR location. The location of the HR was changed to 1.2 m from its previous location of 0.5 m from the source end. The predicted pressure 148

181 Chapter 5. Experimental Verification transfer function amplitude and phase using ANSYS are shown in figures 5.23 and 5.24, respectively. The spikes in the responses, similar to those shown in figures 5.19 (5.21) and 5.20 (5.22), approximately at 95 Hz and its corresponding harmonics at 190 Hz, 285 Hz and 380 Hz are related to the mounting location of the HR being close to a pressure node in the duct acoustic field. The acoustic half-wavelength of 1.8 m at 95 Hz corresponds to the distance of the resonator from the open end of the duct. As the acoustic pressure at one of the neck-wall microphones, B, C and D, in addition to the pressure at the top of the closed end of the cavity (microphone A ), has to be considered for optimally tuning the resonator, the location corresponding to that of microphone B was chosen over C and D. This is because the pressure transfer function between microphones A and B was smooth (free from the spikes) compared to those between microphones A and C, and A and D. The reason for the smoothness of the pressure transfer function between microphones A and B is that the antiresonance frequencies corresponding to the pressure response at microphone A (68 Hz, 135 Hz, 204 Hz, 272 Hz and 340 Hz) shown in figure 5.13 exactly match the antiresonance frequencies which corresponds to the response at microphone B except for 247 Hz shown in figure As a result, the pressure transfer function was a smooth curve throughout the tested frequency range with a peak at 247 Hz (see figure 5.19). In constrast, a mismatch in the antiresonance frequencies corresponding to the responses at C and D from those of microphone A resulted in the spikes in their respective transfer function responses In-Duct Net Acoustic Power Transmission Downstream of the HR The net acoustic power transmission in the duct downstream of the HR was estimated from the acoustic pressure measurements at microphones 1 and 2 using the twomicrophone technique developed by Chung and Blaser [37] and extended by Åbom [42]. The technique has already been described in chapter 3, section Figure 5.25 shows the experimentally measured in-duct net acoustic power trans- 149

182 5.3. Circular Duct with an Attached Helmholtz Resonator normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz exp ansys theory frequency (Hz) Figure 5.25: Experimental, numerical (ANSYS) and theoretical (transfer matrix method) results for the in-duct net acoustic power transmission downstream of the HR for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) exp ansys theory frequency (Hz) Figure 5.26: Experimental, numerical (ANSYS) and theoretical (transfer matrix method) results for the in-duct net acoustic power transmission without the HR for the duct driven by a constant amplitude piston source mounted at one (left) end and open at the other (right). 150

183 Chapter 5. Experimental Verification normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz without HR with HR frequency (Hz) Figure 5.27: Experimentally measured in-duct net acoustic power transmission without the HR and with the HR attached to the duct for the duct driven by a constant amplitude piston source mounted at the end upstream of the HR and open at the end downstream of the HR. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz without HR with HR frequency (Hz) Figure 5.28: Numerical (ANSYS) predictions for the in-duct net acoustic power transmission without the HR and with the HR attached to the duct for the duct driven by a constant amplitude piston source mounted at the end upstream of the resonator and open at the end downstream of the HR 151

184 5.3. Circular Duct with an Attached Helmholtz Resonator mission downstream of the HR along with the theoretical (transfer matrix method) and numerical (ANSYS) modelling predictions. The value of the neck-duct interface end-correction factor and loss factor used in the transfer matrix formulations were kept unchanged from those described in section As shown earlier in chapter 4 (Numerical Modelling), section , the in-duct net acoustic power transmission was normalised by the square of the acoustic volume velocity of a constant amplitude piston source, given by equation (4.3). Similarly, the experimentally measured in-duct net acoustic power transmission was normalised by the square of the acoustic volume velocity of the loudspeaker, given by equation (5.1). The theoretical predictions were also normalised on a similar basis. The experimental, numerical and theoretical in-duct net acoustic power transmission estimates without the HR are shown in figure The experimental and numerical (ANSYS) results for the in-duct net acoustic power reduction achieved by mounting the HR on the duct are shown in figures 5.27 and 5.28, respectively. The theoretical (transfer matrix method) results are not shown separately because they were similar to the numerical results. The experimental results show a reduction of 18 db at 226 Hz in the in-duct net acoustic power transmission as a result of mounting the HR on the duct. A reduction of 23.5 db at 226 Hz was predicted by using ANSYS and the transfer matrix model. The numerical (ANSYS) and theoretical (transfer matrix method) estimates for the net acoustic power transmission shown in figures 5.25 and 5.26 compare favourably with the experimental results, except at the peaks, at 226 Hz and below 120 Hz. The reason for the differences of the numerical and theoretical estimates from the experimental results at peaks at 226 Hz is the inaccuracy in incorporating damping in the theoretical and numerical models of the duct-hr system. Below 120 Hz, the experimental results differ from the numerical and theoretical results due to the poor quality of the signal generated by the loudspeaker, leading to significant noise contamination of the signal and subsequent large errors in the measurements of phase between microphones 1 and 2, which translates to large errors in the power transmission estimates. 152

185 Chapter 5. Experimental Verification Discussion Figure 5.29 summarises and describes the results shown in figures 5.13, 5.15 and The experimentally measured acoustic pressure at the top of the closed end of the cavity (microphone A ) was a maximum at 218 Hz (apart from one low frequency duct resonance which is not associated with the HR), which is 8 Hz less than the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurred. The difference of 3.5% in the two frequencies mentioned above implies that maximising the pressure at the top of the closed end of the cavity does not result in a maximum reduction of the in-duct net acoustic power transmission downstream of the resonator. normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) Hz 225 Hz 226 Hz mic. A mic. B mic. T frequency (Hz) Figure 5.29: Experimentally measured acoustic pressures at microphone A, B and T, as a function of frequency. The vertical line at 226 Hz corresponds to the frequency of the maximum reduction of the in-duct net acoustic power transmission downstream of the HR. The acoustic pressure at the neck wall microphones, B, C and D, at 226 Hz was neither a maximum nor minimum as seen from the results in figure However, the acoustic pressure at microphones C and D are not included in figure 5.29 because the results at microphones B, C and D were very similar to each other and can be seen in figure Moreover, in order to develop a completely self-contained HR, which does not require any measurement remote from its structure, microphones must be located 153

186 5.3. Circular Duct with an Attached Helmholtz Resonator in the resonator only. In other words, the acoustic pressure measurements neither at microphone A nor at microphones B, C and D can be exclusively used for tuning the HR without applying any kind of correction factor or arithmetic manipulation to either of them. Figure 5.29 also shows the experimentally measured acoustic pressure at microphone T. Although in practice it is not desirable for a pressure sensor to be located within the duct (microphone T ), it is noteworthy that the acoustic pressure at microphone T showed a local minimum at 225 Hz. This implies that the acoustic pressure in the centre of the duct adjacent to the neck opening has a local minimum at the frequency which approximately equals the frequency of the maximum in-duct net acoustic power reduction downstream of the HR. The experimentally measured pressure transfer functions shown in figures 5.19 and 5.20 are summarised in figures 5.30 and 5.31, respectively. The amplitude of the pressure transfer functions between microphones A and B was a maximum at 247 Hz, which is greater than 226 Hz by 21 Hz (8.5%). This also implies that the pressure transfer function between the microphones located at the top of the closed end of the cavity and neck wall of the HR cannot be used to achieve optimal tuning of the resonator without using a correction factor. In contrast, the pressure transfer function amplitude between microphones A and T was a maximum at 225 Hz. So far from the experimental and numerical results, it can be concluded that one of the simplest ways to achieve the optimal tuning of the resonator is to maximise the pressure transfer function between the pressure at the top closed end of the cavity and the pressure in the centre of the duct adjacent to the neck opening. The above summarised experimental results presented in this chapter for the duct- HR system indicate that the dimensions of the HR which correspond to the maximum acoustic pressure at the top of the closed end of the cavity (microphone A ) and the pressure transfer function (microphone A /microphone B ) are different to those which correspond to the maximum in-duct net acoustic power reduction. The following 154

187 Chapter 5. Experimental Verification mic. A / mic. B mic. A / mic. T Hz 247 Hz pressure transfer function (db) Hz frequency (Hz) Figure 5.30: Experimentally measured pressure transfer function amplitudes between microphones A and B, and A and T, as a function of frequency. 0 pressure transfer function phase (degrees) mic. A / mic. B mic. A / mic. T 226 Hz frequency (Hz) Figure 5.31: Experimentally measured pressure transfer function phases between microphones A and B, and A and T, as a function of frequency. 155

188 5.3. Circular Duct with an Attached Helmholtz Resonator section shows experimental results of the duct-hr system when only a single frequency was used, and the dimensions of the HR were varied Single Frequency Analysis In industrial applications where a centrifugal fan or blower is installed at one end of an exhaust duct in order to drive the exhaust gases out to the environment, noise is often generated at the blade passage frequency (BPF) of a fan, which is determined by the number of blades and the fan s rotational speed. The noise is tonal and hence resonators are often required to attenuate a single frequency rather than broadband noise. Experiments were conducted using the same duct-hr apparatus described in section The HR used in these experiments was cylindrical with a fixed cavity diameter with variable neck length which can be adjusted to change the cavity volume. As shown in figure 5.25, the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurred at 226 Hz. The following three subsections show experimental results at 226 Hz as the cavity length of the HR was varied. The loudspeaker was excited by random noise from 0 to 400 Hz and the length of the cavity was changed from 60 mm to 90 mm in increments of 1 mm. The experimental results corresponding to 226 Hz were extracted from the experimental set over a range from 0 to 400 Hz and processed in MATLAB In-Duct Net Acoustic Power Transmission Downstream of the HR Figure 5.32 shows the variation in the experimentally measured net acoustic power transmitted in the duct downstream of the HR corresponding to 226 Hz as a function of the cavity length of the HR. The pertinent results from the numerical (ANSYS) modelling are also included in figure Experimental results agree well with predictions from the numerical model. It is evident from figure 5.32 that the optimal length of the cavity required to attenuate the noise at 226 Hz is 70 mm. 156

189 Chapter 5. Experimental Verification normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) exp ansys cavity length (mm) Figure 5.32: Experimental and numerical (ANSYS) results for the in-duct net acoustic power transmission downstream of the HR corresponding to 226 Hz, as a function of the HR cavity length. normalised acoustic pressure (db re 1Pa/(1m 3 /sec)) optimum cavity length mic. A mic. B mic. T cavity length (mm) Figure 5.33: Experimentally measured acoustic pressure at microphones A, B and T corresponding to 226 Hz, as a function of the HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of in-duct acoustic power transmission downstream of the HR at 226 Hz. 157

190 5.3. Circular Duct with an Attached Helmholtz Resonator Acoustic Pressure in the HR As stated earlier in section , the location of microphone B was selected as a suitable location for mounting the microphone in the neck. Figure 5.33 shows the experimentally measured acoustic pressure at microphones A, B and T at 226 Hz plotted as a function of the cavity length of the HR. The optimum cavity length is 70 mm, which results in the minimum net acoustic power transmission at 226 Hz, and a vertical line is drawn in figure The acoustic pressure measured at the top of the closed end of the cavity (microphone A ) and neck wall (microphone B ) were both a maximum at the cavity length of 63 mm, which was 7 mm off the optimal cavity length. The acoustic pressure at microphone B was a minimum at 84 mm which is 14 mm greater than the optimal cavity length. The acoustic pressure measured at microphone T was a minimum when the cavity length was 70 mm, which corresponds to the optimal cavity length. However, due to practical issues, the microphone cannot be located in the duct Pressure Transfer Function of the HR Figures 5.34 and 5.35 show the experimentally measured pressure transfer function amplitudes and phases, respectively, between the microphones A and B (microphone A /microphone B ), and microphones A and T (microphone A /microphone T ) corresponding to 226 Hz, as a function of the cavity length of the HR. The amplitudes of these pressure transfer functions were a maximum at cavity lengths of 84 mm and 70 mm, respectively. The cavity length of the HR which corresponds to the maximum amplitude of the pressure transfer function between microphones A and B is 14 mm greater than the optimum cavity length. On the other hand, the amplitude of the pressure transfer function between microphones A and T was a maximum at the cavity length of 70 mm. This is consistent with the minimisation of the acoustic pressure at microphone T at a cavity length of 70 mm, as seen from figure

191 Chapter 5. Experimental Verification pressure transfer function (db) optimum cavity length mic. A / mic. B mic. A / mic. T cavity length (mm) Figure 5.34: Experimentally measured pressure transfer function amplitudes between microphones A and B, and A and T corresponding to 226 Hz, as a function of the HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of in-duct net acoustic power transmission downstream of the HR at 226 Hz. pressure transfer function phase (degrees) optimum cavity length mic. A / mic. B mic. A / mic. T cavity length (mm) Figure 5.35: Experimentally measured pressure transfer function phases between microphones A and B, and A and T corresponding to 226 Hz, as a function of HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of in-duct net acoustic power transmission downstream of the HR at 226 Hz. 159

192 5.4. Conclusion Discussion Figures 5.32 to 5.34 showed that the maximised response of the pressure at the top of the closed end of the cavity (microphone A ) and at the neck wall close to the neck duct interface (microphone B ), and the pressure transfer function between microphone A and microphone B do not occur at the same volume of the HR which corresponds to the minimisation of in-duct net acoustic power transmission. Hence, these results show that the conventional understanding of trying to achieve a resonance condition in the side branch HR, does not result in the minimisation of net acoustic power transmission in the duct. Instead, an alternative measure or cost function is needed, in order to determine when the acoustic power transmission in the duct is minimised, and this is described in the next chapter. 5.4 Conclusion The resonance frequencies of stand-alone cylindrical HRs were measured and compared with several theoretical models. Significant differences in the experimental measures and theoretical predictions from two of the theoretical models showed that selecting the appropriate model is important if an accurate estimate of the resonance frequencies of a HR is to be obtained. As the resonance frequency of the HR can also be measured by measuring the pressure transfer function between the pressure at the top of the closed end of the cavity and the pressure at the opening of the resonator, four microphones were mounted at the opening of the neck at different locations. Results showed that the location for the microphone at the opening of the resonator was critical, and must be chosen so that the response of the microphone is not affected by the near-field at the neck opening. Differences of more than 20 Hz from the actual HR resonance frequency were observed while measuring the resonance frequency of the HR by measuring the acoustic pressure at the top of the closed end of the cavity, and by measuring the pressure transfer function between the cavity microphone and the microphone located 160

193 Chapter 5. Experimental Verification at the immediate opening of the resonator. The acoustic performance of the HR was evaluated by experimentally measuring the net acoustic power transmission in the duct downstream of the resonator. The results were presented in two different ways: (1) broadband analysis, whereby the in-duct net acoustic power transmission downstream of the HR, the pressure at the top of the closed end of the cavity and the pressure transfer function between the pressure at the top of the closed end of the cavity and at the neck wall close to the neck-duct interface were plotted as a function of frequency, and (2) single frequency analysis, whereby all the aforementioned results were plotted as a function of the cylindrical cavity length (for fixed cavity diameter and neck dimensions) for a single, tonal frequency. For broadband analysis, the experimental results showed that the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs differs from the frequencies which correspond to: (a) the maximum acoustic pressure at the top of the closed end of the cavity, and (b) the maximum pressure transfer function between the pressure at the top of the closed end of the cavity and at the neck wall close to the neck-duct interface. For single frequency analysis, when trying to optimise the performance of a duct-mounted HR at a particular frequency by altering its volume, the optimal dimensions of the HR required to attain the maximum reduction of in-duct net acoustic power transmission at that frequency differ from the dimensions of the HR which correspond to the two above-mentioned maximum responses (a) and (b). The results presented in this chapter show that these two methods (a and b) cannot be used to tune a HR without applying a correction factor to their actual responses. Hence, a need exists to determine how to determine and implement such a correction factor in order to derive a suitable cost function. The next chapter describes the development of such a cost function which can be used by an electronic controller to minimise the in-duct net acoustic power transmission. 161

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195 Chapter 6 Cost Function 6.1 Introduction This chapter describes the derivation of the cost function required to minimise the net acoustic power transmission in the duct downstream of the HR by using two pressure measurements inside the resonator: (1) at the top of the closed end of the cavity (microphone A ), and (2) at the neck wall in close proximity to the neck-duct interface (microphone B ). Two cost functions were empirically derived, and both were related to the damping (or the quality factor) of the duct-hr system and the phase difference between the above described two pressure measuring locations (microphone A and microphone B ). This chapter begins with a discussion of the well known half-power point bandwidth method for estimating the damping (or the quality factor) of the duct-hr system. As described in the previous chapter, the resonance frequency of the duct-hr system was considered to be that which corresponds to the maximum acoustic pressure at the top of the closed end of the cavity (microphone A ). Therefore, for the first cost function, the damping of the duct-hr system was estimated using the half-power point bandwidth method on the frequency response curve of the cavity pressure (microphone A ). As practical usage of HRs often involves attenuating the intensity of the blade passage frequency tone of a fan or blower or the tone in an engine exhaust, information 163

196 6.2. Approach at any other frequency, except the frequency of a disturbing tone, is not available. As a result, it is not convenient from the practical point of view to estimate the damping (or the quality factor) of such a duct-hr system using the half-power point bandwidth method because it needs information at the frequencies corresponding to the halfpower points (or 3 db down points) bounding the system resonance. Therefore, in this chapter, a second, an alternative method, is reported, which only uses information at a single (excitation) frequency, to determine the damping (or the quality factor) of the duct-hr system. The theoretical justification for the validity of this method is also presented, followed by a comparison of the quality factors obtained using the half-power point bandwidth method and the alternative method. A step-by-step method for developing both cost functions (the one based on the half-power point bandwidth and the one based on the alternative damping measure) along with steps for the potential use of the cost functions by an electronic controller is also described. Finally, the usefulness of both cost functions is discussed. 6.2 Approach The experimental and numerical (ANSYS) results have shown that the best possible way to indicate the frequency of maximum in-duct acoustic power reduction downstream of the HR without actually measuring the acoustic power transmission in the duct is to measure a pressure transfer function amplitude between the pressure at the top of the closed end of the cavity (microphone A ) and the pressure in the duct adjacent to the neck opening (microphone T ). This implies that the best location for mounting a microphone at the opening of the resonator lies at the centre of the duct adjacent to the neck opening, which in this study corresponds to the location of microphone T. However, it was decided to move the neck opening microphone to the neck wall (microphone B ) to avoid any obstruction to the mean flow of (exhaust) gases and more importantly, to achieve a self-contained HR. For an illustration of the locations 164

197 Chapter 6. Cost Function of the microphones mentioned above, see figure The experimental and numerical (ANSYS) results have also shown that neither the acoustic pressures at microphone A and microphone B nor the amplitude of the pressure transfer function between microphone A and microphone B (microphone A /microphone B ) was a maximum or minimum at the frequency which corresponds to the maximum reduction of in-duct net acoustic power transmission. As described in the previous chapter, there exists a difference of 3.5% between the frequencies corresponding to the maximum acoustic pressure at microphone A and maximum in-duct net acoustic power reduction downstream of the HR. A difference of 8% was observed between the frequencies corresponding to the maximum pressure transfer function amplitude (microphone A /microphone B ) and maximum in-duct net acoustic power reduction. Thus, it was decided to investigate the phase difference between microphone A and microphone B. Although the phase between microphone A and microphone B was suddenly changed by approximately 90 degrees at a frequency close to the frequency of the maximum pressure transfer function amplitude, there was no sudden shift at the frequency of the maximum in-duct net acoustic power reduction. It may be expected that the resonance frequency of the HR on the duct would correspond to a pressure transfer function phase shift of 90 degrees. Thus, it is apparent that the frequency of minimum in-duct acoustic power transmission does not correspond to the resonance frequency of the duct-hr system. For example, from the experimental results of the duct-hr system presented in chapter 5 figures 5.19 and 5.20, the value of phase difference between microphone A and microphone B was degrees at the frequency of the maximum transfer function amplitude (246 Hz) and -5.5 degrees at the frequency of the maximum in-duct acoustic power reduction (226 Hz). The reason for these numbers not being -90 degrees and 0 degrees is the influence of the duct-hr system damping. It is well understood that the presence of damping in a system affects its frequency response. In the case of a duct-hr system, the ideal way to estimate the system 165

198 6.3. Damping damping is to measure the sharpness of the frequency response curve of the acoustic pressure at the top of the closed end of the cavity (microphone A ). Thus, the phase difference between microphone A and microphone B corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission occurs was investigated with respect to the damping present in the duct-hr system. Two cost functions were empirically derived which are related to the duct-hr damping and the phase difference between microphone A and microphone B. The next section describes the scheme to estimate the damping (or the quality factor) of the duct-hr system. 6.3 Damping Figure 6.1 shows a schematic of a typical frequency response curve of a single degreeof-freedom (SDOF) system in the vicinity of the system resonance. The solid circular R max half power points amplification factor, R R max /sqrt[2] Q bandwidth f 1 f f n 2 frequency ratio (f/f n ) Figure 6.1: Frequency response of a system showing bandwidth and half-power points. markers at either sides of the resonance peak denote the frequencies which correspond 1 to half-power points. These half-power points correspond to of the maximum 2 response, R max. The damping and the corresponding quality factor of a system are given by 166

199 Chapter 6. Cost Function Q = 1 2ζ = f n f 2 f 1 (6.1) where, Q ζ is the quality factor of the system, is the critical damping ratio of the system, f n is the resonance frequency of the system, and f 1, f 2 are the frequencies corresponding to the half-power points. The above described scheme for estimating the damping of a system is referred to as the half-power point bandwidth method and was used to estimate the damping of the duct-hr system. As the resonance frequency of the duct-hr system is considered to be the frequency corresponding to the maximum acoustic pressure at the top of the closed end of the cavity of the HR (microphone A ), the half-power point bandwidth method was applied on the frequency response curve of the acoustic pressure at microphone A. For the case of the duct-hr system, f n was the frequency corresponding to the maximum acoustic pressure at microphone A, and f 2 and f 1 were the (half-power point bandwidth) frequencies corresponding to 1 of the maximum value of the acoustic 2 pressure at microphone A. Curve (a) of figure 6.2 shows the experimentally measured critical damping ratios of the duct-hr system estimated according to the above described scheme (the halfpower point bandwidth method), as a function of the cavity length. Curve (b) will be discussed later in this section. The fixed dimensions of the duct-hr system corresponding to the results shown in figure 6.2 were: duct diameter = m, duct length = 3 m, neck diameter = m, actual neck length = m, cavity diameter = m. The length of the cavity was varied from 60 mm to 170 mm in increments of 5 mm. For each cavity length, the corresponding resonance frequency and the halfpower point bandwidth frequencies were determined. It is evident from the trend of 167

200 6.3. Damping damping ratio (%) a b cavity length (mm) Figure 6.2: Experimentally measured critical damping ratios of the duct-hr system for various HRs. curve (a) shown in figure 6.2 that the measurement of the damping of the duct-hr system is strongly affected by the acoustic field in the wave guide and the variation with frequency of the pressure at the entrance of the resonator neck. The maximum difference of 30% in the damping measures as the cavity length varies from 90 mm to 130 mm is an artifact of the measurement procedure rather than a real phenomenon. It is a result of the strong variation as a function of frequency of the acoustic pressure standing wave in the duct, resulting in a strong variation in the pressure at the entry to the resonator as a function of frequency. This results in a distortion of the frequency response curve. Thus, it is desirable to normalise the cavity pressure measurement with a measure of the acoustic pressure at the entrance to the resonator by taking a transfer function where the pressure at location A (microphone A ) is divided by that measured at location B (microphone B ). Therefore, the half-power point bandwidth method was applied to the frequency response curve of the pressure transfer function between microphone A and microphone B. The pertinent results give a much more consistent and accurate measure of the critical damping ratio as illustrated by curve (b) of figure

201 Chapter 6. Cost Function The next section describes two cost functions which were derived for achieving the optimal tuning of the HR. 6.4 Cost Function In order to tune the HR by using only acoustic pressure measurements inside the resonator (microphone A and microphone B ), two cost functions were empirically derived. These two cost functions were mathematical equations related to the duct- HR system damping and the phase difference between microphone A and microphone B. The two cost functions are the same except that the procedure for measuring the system damping is different. Both cost functions were derived in such a way as to determine a value of the phase difference which corresponds to the minimum induct net acoustic power transmission. This phase difference is related directly to the system damping which can be experimentally measured. This estimated value of the phase difference corresponds to the minimum in-duct net acoustic power transmission downstream of the HR. In order to achieve the estimated phase difference, the cavity length of the HR needs to be varied until the phase difference between microphone A and B approximately equals the estimated phase difference. Once, the estimated phase difference is achieved, HR is considered to be tuned The First Cost Function The first cost function was developed using the measured critical damping of the duct- HR system as indicated by the results shown by curve (a) of figure 6.2 and the value of phase difference between microphone A and microphone B corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission was measured by microphones 1 and 2. A step-by step procedure adopted for developing the first cost function is described following the next paragraph, which describes the geometries of the HR used. 169

202 6.4. Cost Function Experiments were conducted with varying dimensions of the HR. The cavity diameter and neck length throughout the experiments were kept constant and three different neck diameters ( m, m and m) were tested. For each different neck size, the cavity length was varied between m and m corresponding to a volume variation of m 3 to m 3. Step-by-Step Procedure for Developing the First Cost Function The steps which were taken to develop the first cost function are detailed below: 1. For each configuration of the HR, the in-duct net acoustic power transmission downstream of the resonator was measured by using microphones 1 and For each configuration of the HR, the value of the phase difference between microphone A and microphone B, corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission occured was noted. 3. For each configuration of the HR, the experimental measure of the critical damping ratio of the duct-hr system was estimated using the half-power point bandwidth method on the frequency response of the acoustic pressure at microphone A, as described in section All the values of experimentally measured critical damping ratio of the duct-hr system and the phase differences between microphone A and microphone B were plotted and shown in figure An empirical curve fitting relation between the phase difference and the critical damping ratio was found, using a second order polynomial (with a correlation coefficient of 0.882), and is given by: phase difference = 0.94(damping ratio) (damping ratio) (6.2) 170

203 Chapter 6. Cost Function neck dia. = m neck dia. = m neck dia. = m curve fit phase (degrees) damping ratio (%) Figure 6.3: Critical damping ratio versus phase difference corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs, for three different neck diameters and many cavity volume sizes. Equation (6.2) can be referred to as a cost function that can be used by an electronic controller to optimally tune the HR for minimising the acoustic power transmission in the duct downstream of the HR. The damping present in the system, which needs to be predetermined, can be substituted in equation (6.2), which will then yield the desired value of the phase difference. The length of the cavity of the HR is then varied until the phase difference between microphone A and microphone B reaches the desired/calculated value. This calculated value of the phase difference will approximately correspond to the optimal length of the cavity required to minimise the net acoustic power transmission in the duct. A drawback associated with the cost function given by equation (6.2) is that the damping of the duct-hr system must be determined utilising the half-power point bandwidth method as described in section 6.3. Measuring the damping (or the quality factor) of a system using the half-power point bandwidth method requires a system to be excited by a broadband noise signal so that information at the band of frequencies (the half-power point frequencies) bounding the resonance is available. Considering the 171

204 6.4. Cost Function case of a duct-hr system in practice where a centrifugal fan or a blower is installed at one end of the exhaust duct, noise is often generated at the BPF of the fan, and is tonal in nature. The signals at all the other frequencies can either be spurious or meaningless. Hence, estimating the damping (or quality factor) of such a system using the halfpower point bandwidth method would involve some kind of provision to introduce a broadband signal in the duct, which seems unrealistic from the practical point of view. Thus, an alternative method, which uses information at a single (excitation) frequency, for estimating the damping (or quality factor) is needed, and is described in the next section Approximating Damping at a Single Frequency Figure 6.1 also shows that the height of the resonance peak is a function of damping in a system. This may be quantified by reference to a SDOF system. Figure 6.4 shows a mass-spring-damper system which represents a classical example of a SDOF system. The mass and spring elements are associated with storage of energy while the damper with dissipation of energy. The equation which represents a SDOF system is given by [80] Figure 6.4: A schematic representation of a single degree-of-freedom system. M d2 ξ dt + C dξ + Kξ = F(t) (6.3) 2 dt 172

205 Chapter 6. Cost Function where M, K and C denote the mass, stiffness and damping of the system, ξ is the harmonic displacement of the mass resulting from the application of the forcing function, F(t). For harmonic excitation of the system, the forcing function F(t) = F sin(ωt), and equation (6.3) becomes A solution to equation (6.4) is given by M d2 ξ dt + C dξ + Kξ = Fsin(ωt) (6.4) 2 dt ξ = Xsin(ωt φ) (6.5) where, X and φ are the amplitude and phase of the frequency response, respectively, and ω is the driving (or excitation) angular frequency. The value of X can be obtained by substituting equation (6.5) into equation (6.4) and is given by [80] X = F (K ω 2 M) 2 + (ωc) 2 (6.6) Rewriting equation (6.6) with slight manipulation gives X = F/K ( ) 2 ( 1 ω2 M ωc + K K ) 2 (6.7) Using the expressions K M = C ω2 n and ζ = 2, equation (6.7) can be arranged to KM give X F/K = 1 ( ( ) ) (6.8) ω 2 2 ( 1 + 2ζ ω ) 2 ωn ω n where, ω n is the natural angular frequency of the system, ζ is called the damping ratio, and 173

206 6.4. Cost Function X F/K is the amplification factor of the system, denoted by R as shown in figure 6.1. When the excitation frequency (ω) equals the natural frequency (ω n ) of the system, the amplification factor attains a maximum value. Upon the substitution of ω = ω n in equation (6.8), we get X F/K = R max = 1 2ζ = Q (6.9) As described in chapter 3 (Theoretical Modelling), a HR may also be represented as a SDOF system. If the pressure at the input to the resonator neck is considered to be the forcing function and the pressure in the cavity is the response, then the left side of equation (6.9) may be represented as the ratio of the two, more specifically referred to as the transfer function. This demonstrates that the transfer function between the acoustic pressure in the resonator cavity and the pressure at the entrance to the neck can be used to estimate the quality factor of the duct-hr system. The next step is to find the maximum value of the frequency response function given that only a fixed (single) excitation frequency is available. This is done by varying the cavity length (depth) until the maximum transfer function is obtained. From the previous set of the experimental results presented in chapter 5, figure 5.25 showed that the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurred at 226 Hz. Considering tonal noise propagation at 226 Hz in a duct which needs to be controlled by using a self-contained HR, or by measuring the acoustic pressure at microphones A and B, it would be convenient from the practical point of view to approximate the damping (or quality factor) of such a duct-hr system at 226 Hz only. Figure 6.5 shows experimentally measured pressure transfer function between microphone A and microphone B corresponding to 226 Hz as a function of the cavity length of the HR. The maximum value of the pressure transfer function is 56 and it occurs at a cavity length of 84 mm. The vertical line at a cavity length of 70 mm corresponds to the minimum acoustic power transmission in the duct and indicates 174

207 Chapter 6. Cost Function pressure ratio (mic. A / mic. B) cavity length (mm) Figure 6.5: Experimentally measured ratio of pressures at microphone A and microphone B corresponding to 226 Hz, as a function of the HR cavity length. The vertical line at 70 mm corresponds to the cavity length required to attain the maximum reduction of the in-duct net acoustic power transmission downstream of the HR at 226 Hz pressure ratio (mic. A/mic. B) frequency (Hz) Figure 6.6: Experimentally measured ratio of pressures at microphone A and microphone B corresponding to the cavity length of 84 mm, as a function of frequency. The vertical line at 226 Hz corresponds to the frequency of maximum in-duct net acoustic power reduction downstream of the HR. 175

208 6.4. Cost Function that the maximum value of the transfer function is in itself not a suitable cost function for an active noise control system to minimise the transmitted in-duct acoustic power. As a check on the validity of equation (6.9) for estimating the quality factor, the experimental measure of the ratio of pressures at microphone A and microphone B corresponding to the cavity length of 84 mm is plotted as a function of frequency in figure 6.6. The maximum value of the pressure ratio occurring at 226 Hz equals 56 which approximately corresponds to the quality factor of the duct-hr system. For verification purposes, the quality factor for the same duct-hr system was also estimated by using the half-power point bandwidth method on the frequency response curve of the pressure ratio of microphone A and microphone B, and was found to be equal to quality factor, Q half power point bandwidth maximum amplification factor configurations Figure 6.7: Qualtiy factors of several HRs obtained using two different methods: (1) by measuring the maximum pressure ratio of microphone A to microphone B, and (2) by applying the half-power point bandwidth method. For further verification, the quality factors of different HRs obtained by varying the neck diameter and cavity length were measured using the height of the resonance response and estimated using the half-power point bandwidth method, with the results shown in figure 6.7. The excellent agreement indicates that estimating the quality factor of a HR by using the maximum value of pressure ratio (microphone A /microphone B ) for a fixed frequency as the cavity volume is varied is a valid method and more 176

209 Chapter 6. Cost Function convenient than using the half-power point bandwidth method. Considering that pressure measurements at locations A and B are available at one frequency only at which the noise needs to be controlled, and there is provision to change the length of the cavity (necessary for the system to be adaptive), approximating the quality factor of the duct-hr system using the ratio of two measured pressures as a function of cavity length proves to be a very practical approach. The next section describes the second cost function which is related to the measured quality factor of the duct-hr system and the phase difference between microphone A and microphone B. Like the first cost function (equation (6.2)), the second cost function is also a mathematical equation derived to estimate the phase difference on the basis of the measured quality factor of the duct-hr system. The estimated phase difference, which is related to the minimum in-duct net acoustic power transmission, can be achieved by varying the cavity length as was done in the case of the first cost function. The second cost function is different to the first one in a sense that for the first cost function, the system damping was estimated using the half-power point bandwidth method, whereas for the second cost function, the system quality factor (or damping) was estimated using the pressure measurements at microphone A and microphone B at a single frequency only The Second Cost Function Step-by-Step Procedure for Developing the Second Cost Function The steps which were taken to develop the second cost function are detailed below: 1. A range of HR neck sizes and cavity volumes (HR configurations), corresponding to a range of resonance frequencies were tested. For each configuration of the HR, the in-duct net acoustic power transmission downstream of the resonator was measured by using microphones 1 and 2. For each configuration of the HR, the frequency which achieved the maximum reduction of in-duct net acoustic 177

210 6.4. Cost Function power transmission was recorded. For the tested range of the dimensions of the HRs, these frequencies varied from 120 to 270 Hz. 2. For each configuration of the HR, the value of the phase difference between microphone A and microphone B corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission occurred was noted. 3. For frequencies at approximately 5 Hz increments throughout the frequency range stated in step 1 (corresponding to the maximum in-duct net acoustic power reduction), the response curve of the ratio of the pressure measurements at microphone A and microphone B (microphone A /microphone B ) was plotted as a function of the cavity length in the vicinity of the maximum pressure ratio. The corresponding quality factor of the duct-hr was then approximated by using the height of the response curve, as shown in figure Values of experimentally measured quality factors of the duct-hr system and the phase differences between microphone A and microphone B, corresponding to the frequency at which the acoustic power transmission in the duct was a minimum, were plotted and are shown in figure An empirical curve-fitting relation between the phase difference and the quality factor was found, using a second order polynomial (with a correlation coefficient of 0.87), and is given by: phase difference = (quality factor) (quality factor) (6.10) Equation (6.10) can be referred to as a cost function that can be used by an electronic controller to optimally tune the HR for minimising the acoustic power transmission in the duct downstream of the resonator. The controller would adjust the resonator cavity length until the phase difference between signals from microphone A and microphone B matched the phase difference given by equation (6.10). As stated 178

211 Chapter 6. Cost Function phase (degrees) neck dia. = m neck dia. = m neck dia. = m curve fit quality factor Figure 6.8: Quality factor versus phase difference corresponding to the frequency at which the maximum reduction of in-duct net acoustic power transmission downstream of the HR occurs, for three different neck diameters. previously, the quality factor is determined by adjusting the cavity length until a peak in the A / B transfer function is obtained, and then it is equal to that peak value. With a view to reducing the scatter in the data in figure 6.8, attempts were made to normalise the data by multiplying the quality factor by non dimensional combinations of duct-hr system parameters. The parameters included neck diameter, duct diameter, cavity length and the frequency of the maximum reduction of the in-duct net acoustic power transmission. The parameters were arranged so as to yield a non-dimensional term on the abscissa. The values of the phase difference presented in figure 6.8 were plotted with the modified quality factor which was obtained by multiplying the actual quality factor by the non-dimensional term as described above. As the results did not show any appreciable reduction in scatter over the results presented in figure 6.8, they are not included here. It should be noted that the maximum scatter in figure 6.8 is about 1.3 degrees from the curve of best fit. This means that in many cases there will be a phase error associated with using the curve of best fit as the cost function. For a 1.3 degree error, the approximate compromise in the reduction in acoustic power transmission is 3 db which is acceptable as the total reduction can range from 15 to 179

212 6.4. Cost Function 25 db. Practical Implementation of the Second Cost Function The steps involved in utilising the developed cost function to minimise the tonal in-duct net acoustic power transmission are detailed below: 1. The length of the cavity is varied until the ratio of pressures at microphone A and microphone B (microphone A /microphone B ) reaches its maximum value at the frequency of tonal noise that is to be controlled. 2. The quality factor of the HR is then estimated by using the maximum value of the pressure ratio of microphone A and microphone B. 3. The measured quality factor is substituted in equation (6.10) to calculate a value of the phase difference between microphone A and microphone B. This calculated phase difference will approximately correspond to the optimal length of the cavity required to minimise the in-duct net acoustic power transmission. 4. Finally, the length of the cavity is tuned until the calculated phase difference is reached Comparison of the Two Cost Functions It was not possible to compare the two cost functions given by equations (6.2) and (6.10) on a single graph. This is because the basis used for estimating the duct- HR system damping for the first cost function and the quality factor for the second cost function was different. For the first cost function, the critical damping ratio was estimated by applying the half-power point bandwidth method to the pressure versus frequency curve corresponding to the acoustic pressure at the top of the closed end of the cavity (microphone A ). In contrast, for the second cost function, the quality factor was estimated by using the height of the frequency response curve corresponding to the ratio of the cavity pressure and neck pressure (microphone A /microphone B ). 180

213 Chapter 6. Cost Function Also, as stated earlier in section 6.3, the critical damping ratio of the duct-hr system based on the cavity pressure measurement is strongly influenced by the presence of standing waves in the duct. This could be a possible reason for the similarity of the damping measures for the neck sizes of m and m, as illustrated in figure 6.3. Although, a comparison of the two cost functions was not possible, their individual usefulness for tuning the HR to minimise the in-duct acoustic power transmission was excellent in both cases. The next section describes the effectiveness and performance of both cost functions along with two other cost functions which were assumed to minimise the in-duct acoustic power transmission without the need for any in-duct pressure sensors or microphones Performance of the Two Cost Functions The effectiveness of the two developed cost functions is demonstrated by indicating the length of the cavity required for the optimal tuning of the HR. In order to show the effectiveness of several cost functions, different markers were plotted on the curve of in-duct net acoustic power transmission at 226 Hz as a function of cavity length, and shown in figure 6.9. The markers indicate the length of the cavity which corresponds to the following: maximum value of the acoustic pressure at the top of the closed end of the cavity (microphone A ) - (labelled maximum pressure in cavity ), maximum value of the amplitude of the pressure transfer function between microphone A and microphone B - (labelled maximum transfer function ), value of the phase difference, calculated by using equation (6.2), needed to be achieved by varying the cavity length for minimising the net acoustic power transmission in the duct downstream of the HR - (labelled first cost function ), 181

214 6.4. Cost Function normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) acoustic power with HR maximum pressure in cavity maximum transfer function first cost function new cost function actual minimum power cavity length (mm) Figure 6.9: Experimentally measured in-duct net acoustic power transmission downstream of the HR corresponding to 226 Hz plotted as a function of the HR cavity length; markers indicate the acoustic power level for various cost functions. value of the phase difference, calculated by using equation (6.10), needed to be achieved by varying the cavity length for minimising the net acoustic power transmission in the duct downstream of the HR - (labelled second cost function ), and minimum acoustic power measured by mounting two microphones in the duct downstream of the HR - (labelled actual minimum power ). Figure 6.9 shows that in order to achieve the maximum reduction of the in-duct acoustic power transmission, both the developed cost functions, first cost function and second cost function, give the best possible cavity length when compared to the two alternative cost functions, maximum pressure in the cavity and maximum transfer function. The developed cost functions estimated the optimal cavity length as 71 mm in comparison to 70 mm which corresponds to the actual cavity length required to minimise the acoustic power transmission at 226 Hz. In contrast, the alternative two cost functions described above indicated the optimal cavity length as 63 and 84 mm, respectively. Although both of the cost functions that included damping estimates are capable of attaining the optimal tuning of the HR, the second cost function is preferred because 182

215 Chapter 6. Cost Function it involves the estimation of the duct-hr system damping (or the quality factor) using only information at a single frequency Limitations of the Two Cost Functions The cost functions given by equation (6.2) and (6.10) are only valid for the dimensions of the duct-hr system which were covered in this study. Due to the time constraints, the testing was not possible with various sizes of the duct. In fact, only a single duct of diameter m was used in this study to test different HRs. Although a finite element model of the duct-hr system was developed as described in chapter 4 (Numerical Modelling), obtaining a cost function for a wide range of dimensions of the duct-hr system was not possible. This is because accurate estimation of damping in the finite element model is a problem for systems for which no measurements are available. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) acoustic power with HR maximum pressure in cavity maximum transfer function first cost function second cost function actual minimum power cavity length (mm) Figure 6.10: Numerical (ANSYS) predictions for the in-duct net acoustic power transmission downstream of the HR corresponding to 226 Hz as a function of the HR cavity length with no damping included; markers indicate the acoustic power level for various cost functions. A figure similar to figure 6.9 was constructed using an ANSYS analysis for the same system with no damping, to demonstrate the effect of damping on the cost functions and the importance of estimating the damping or quality factor accurately. The results 183

216 6.5. Conclusion are shown in figure 6.10 where it can be seen that both the cost functions correspond to the cavity length that is much different to the one corresponding to the actual minimum in-duct acoustic power transmission. 6.5 Conclusion Two empirically derived cost functions for tuning the HR in order to minimise the net acoustic power transmission in the duct were presented. Both cost functions were related to the experimentally measured duct-hr system damping and the phase difference between microphone A and microphone B corresponding to the measured frequency of minimum in-duct acoustic power transmission. For the first cost function, the duct-hr system damping was estimated using the half-power point bandwidth method on the frequency response curve of the acoustic pressure at the top of the closed end of the cavity. It was shown that the critical damping measures of the duct-hr system estimated by using the cavity pressure measurement are strongly affected by the presence of standing waves in the duct. In order to minimise the influence of standing waves on the system damping measures, the pressure at the top of the closed end of the cavity was normalised by the pressure at the entrance to the resonator. The frequency response curve of the ratio of the cavity and neck pressures was used to approximate the system damping, which was then used for deriving the second cost function. The second cost function is better than the first one reported in this chapter because the quality factor (or damping) of the duct-hr system was determined at a single (fixed) frequency, at which noise needs to be attenuated. 184

217 Chapter 7 Summary and Conclusion 7.1 Summary and Conclusion The aim of the work presented in this thesis was to develop a cost function that an active control system could use to optimise adaptive HRs to maximally attenuate tonal sound transmission in a duct to which they are attached. Theoretically, the resonance frequency of a stand-alone HR was predicted using an existing mathematical equation. In order to evaluate the performance of a HR mounted on a duct, the transfer matrix method was used. The (transfer matrix method) results showed that the frequencies corresponding to the maximum acoustic pressure at the top of the closed end of the cavity and the maximum pressure transfer function of the HR were different. In contrast, no difference between the frequencies corresponding to the maximum reduction of the in-duct net acoustic power transmission downstream of the HR and the maximum pressure transfer function was observed. This was due to the limitation of the transfer matrix method to only analyse planar sound fields. Because non-planar sound fields exist in the regions of the duct which are in close proximity to the opening of the resonator, the transfer matrix method was not considered accurate for the evaluation of the frequency which corresponds to the maximum reduction of in-duct acoustic power transmission downstream of the resonator. Instead the finite element analysis software package ANSYS was used. Modelling of the duct-hr system 185

218 7.1. Summary and Conclusion using ANSYS eliminates the need to account for the end-corrections of the neck and hence, provides a reliable verification reference. A three-dimensional finite element model of a circular duct with an attached cylindrical HR was developed and used to analyse the sound field inside the duct and the HR. ANSYS results illustrated the existence of a complex sound field in the regions of the neck and duct in close proximity to the opening of the resonator. Several pressure transfer function predictions of the HR were presented using pressure measurement in the HR cavity and normalised by the pressure measurement outside the resonator. These differences did not exist for the transfer matrix method of analysis as that did not account for the complexity of the sound field near the neckduct interface. It was shown using ANSYS that the pressure transfer function between the pressures at the top of the closed end of the cavity and at the centre of the duct adjacent to the neck opening was a maximum at the frequency which corresponds to the frequency of minimum in-duct acoustic power transmission downstream of the HR. However, in practical applications, a pressure sensor cannot be located in the duct to avoid any obstruction to the mean flow of (exhaust) gases. Hence, a location for measuring the pressure at the neck wall close to the neck-duct interface was selected. A difference of 8% was predicted in the frequencies corresponding to the maximum transfer function amplitude between cavity and neck wall pressures, and cavity and duct centre pressures using ANSYS. Also, a difference of 3.5% was predicted in the frequencies corresponding to the maximum pressure at the top of the closed end of the cavity and minimum in-duct acoustic power reduction. Experiments were conducted on the duct-hr system and the acoustic performance of the HR was evaluated by measuring the net acoustic power transmission in the duct downstream of the resonator. An excellent match between the experimental and ANSYS results was observed. In addition to showing all the experimentally measured results, such as the acoustic pressure at several locations, pressure transfer functions and in-duct net acoustic power transmission, over a broadband frequency range, the 186

219 Chapter 7. Summary and Conclusion results were also plotted for a single (excitation) frequency as a function of the cavity length. Single frequency analysis was required because in industry a centrifugal fan or blower installed at one end of an exhaust duct is used to drive the exhaust gases out to the environment. The noise is often generated at the blade passage frequency (BPF) of a fan and is tonal. For the single frequency analysis, the experimentally measured results showed that the dimensions of the HR required to attain maximum in-duct net acoustic power reduction downstream of the HR differed from the dimensions of the HR which corresponds to: (a) the maximum acoustic pressure at the top of the closed end of the cavity, and (b) the maximum amplitude of the pressure transfer function between the pressure at the top of the closed end of the cavity and at the neck opening. A cost function was required which was compatible with a self-contained resonator with no pressure sensor located within the duct. Two cost functions that were considered to be an indicator of the minimisation of tonal in-duct acoustic power transmission were investigated and shown to be of use only if a correction factor was applied to them. These two cost functions were: (a) maximum acoustic pressure at the top of the closed end of the cavity, and (b) maximum pressure transfer function between the pressure at the top of the closed end of the cavity and at the neck opening. Case (b) was chosen because it compensated for the duct dynamics and the correction factor was determined in two different ways on the basis of experimentally measured damping characteristics of the duct-hr system and the phase difference between two microphones located in the resonator. Two cost functions were derived which can be used by an electronic controller for optimally tuning a HR mounted on a duct in order to minimise tonal in-duct acoustic power transmission downstream of the resonator. The performance of both cost functions proved to be effective but the second cost function was preferred over the first as it was simpler to implement. Also, it should be pointed out that the cost functions are strictly only valid for the dimensions of the duct-hr system which were 187

220 7.2. Recommendations for Future Work covered in this study. Due to time constraints, experimental testing was not possible with various sizes of the duct and only a single duct of diameter m was used. Although a finite element model of the duct-hr system was developed, obtaining a cost function for a wide range of dimensions of the duct-hr system was not possible. This is because accurate estimation of damping in the finite element model is a problem for systems for which no measurements are available. 7.2 Recommendations for Future Work As in this study only a single duct of diameter m was used to test different HRs, the effectiveness of the empirically derived cost functions are valid for the tested duct and HR dimensions. It is likely that the cost functions would not result in the optimal minimisation of the in-duct tonal noise transmission when tested on different ducts. Hence, it is recommended that a general cost function, independent of the duct and HR dimensions, be developed for minimising the in-duct tonal noise transmission. Because the scope of this study was restricted to the no mean flow condition in the duct, this research should be extended to investigate the effects of flow on the performance of the cost functions. It is most likely expected that the performance of a HR will decrease with an increase in the mean flow through the duct. This is because the turbulence of the flow affects the movement of the fluid oscillating in the neck of the resonator as a result of which the acoustic resistance of the neck increases. This, in turn, increases the dissipation in the resonator resulting in a decrease in the acoustic performance of the resonator. Another effect of turbulence exciting the fluid in the neck is a change in the end-correction factor at the neck-duct interface. This subsequently changes the resonance frequency of a HR, and the frequency of minimum in-duct acoustic power transmission. Once a general cost function is developed taking into consideration the above described factors, it should be implemented on a practical application for attenuating the BPF tone of a centrifugal fan propagating in an exhaust duct by mounting an adaptive 188

221 Chapter 7. Summary and Conclusion HR. This can be realised by designing an active control system which comprises a tachometer for measuring the rotational speed of the fan which facilitates calculating the BPF of the fan at which sound needs to be attenuated, microphones mounted in the resonator cavity and neck which facilitate measuring the system damping at the BPF and the phase difference, a digital controller for measuring the duct-hr system damping by changing the cavity length so as to maximise the pressure ratio of the two microphones located in the resonator and estimating the optimum phase difference between the HR microphones using the cost function, and a stepper motor for driving the piston inserted in the resonator cavity until the pressure ratio of two microphone responses is maximised and optimum (calculated) phase difference is achieved, instrumentation filters and amplifiers which are used to condition the electrical signal so they are suitable for the digital controller. 189

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223 References [1] R. J. Bernhard, H. R. Hall, and J. D. Jones. Adaptive-passive noise control. In Inter-noise 92, pages , [2] R. J. Bernhard. The state of the art of active-passive noise control. In Joseph M. Cuschieri, Stewart A. L. Glegg, and David M. Yeager, editors, Proceedings of Noise-Con 94, pages Fort Lauderdale, Florida, May [3] L. J. Eriksson and M. T. Zuroski. From passive to active: a family of silencing possibilities. In Proceedings of Noise-Con 97, pages 71 80, [4] C. H. Hansen, C. Q. Howard, K. A. Burgemeister, and B. S. Cazzolato. Practical implementaion of an active control system in a hot stack. In Proceedings of the Australian Acoustical Society Conference, [5] X. Li, X. Qiu, D. J. J. Leclercq, A. C. Zander, and C. H. Hansen. Implementation of active noise control in a multi-modal spray dryer exhaust stack. Applied Acoustics, 67(1):28 48, [6] J. W. S. Rayleigh. Theory of Sound. MacMillan and Company, [7] R. L. Panton and J. M. Miller. Resonant frequencies of cylindrical Helmholtz resonators. Journal of the Acoustical Society of America, 57(6): , [8] D. Li. Vibroacoustic behaviour and noise control studies of advanced composite structures. PhD thesis, University of Pittsburgh,

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225 REFERENCES [21] N. S. Dickey and A. Selamet. Helmholtz resonators: one-dimensional limit for small cavity length-to-diameter ratios. Journal of Sound and Vibration, 195(3): , [22] A. Selamet, P. M. Radavich, N. S. Dickey, and J. M. Novak. Circular concentric Helmholtz resonators. Journal of the Acoustical Society of America, 101(1):41 51, January [23] A. Selamet and Z. L. Ji. Circular asymmetric Helmholtz resonator. Journal of the Acoustical Society of America, 107(5): , may [24] A. Selamet and I. Lee. Helmholtz resonator with extended neck. Journal of the Acoustical Society of America, 113(4): , April [25] L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders. Fundamentals of Acoustics. John Wiley & Sons, Third edition, [26] M. L. Munjal. Acoustics of ducts and mufflers. John Wiley & Sons, [27] J. W. Miles. The analysis of plane discontinuties in cylindrical tubes. Part I and Part II. Journal of the Acoustical Society of America, 17(3): , January [28] F. C. Karal. The analogous acoustical impedance for discontinuties and constrictions of circular cross section. Journal of the Acoustical Society of America, 25, [29] A. D. Sahasrabudhe, M. L. Munjal, and S. Anantha Ramu. Analysis of interface due to the higher order mode effects in a sudden area discontinuity. Journal of Sound and Vibration, 185(3): , [30] A. Onorati. Prediction of the acoustical performances of muffling pipe systems by the method of characteristics. Journal of Sound and Vibration, 171(3): ,

226 REFERENCES [31] Z. L. Ji. Acoustic length correction of a closed cylindrical side-branched tube. Journal of Sound and Vibration, 283: , [32] M. G. Prasad and M. J. Crocker. Insertion loss studies on models of automotive exhaust systems. Journal of Acoustical Society of America, 70(1981): , November [33] D. D. Davis, G. M. Stokes, D. Moore, and G. L. Stevens. Theoretical and experimental investigation of mufflers with comments on engine-exhaust muffler design. Technical report, NACA Annual Report, [34] K. T. Chen, Y. H. Chen, K. Y. Lin, and C. C. Weng. The improvement on the transmission loss of a duct by adding Helmholtz resonators. Applied Acoustics, 54(1):71 82, [35] C. R. Fuller, J. P. Maillard, M. Mercadal, and A. H. von Flotow. Control of aircraft interior noise using globally detuned vibration absorbers. Journal of Sound and Vibration, 203(5): , [36] J. P. Carneal, F. Charette, and C. R. Fuller. Minimization of sound radiation from plates using adaptive tuned vibration absorbers. Journal of Sound and Vibration, 270: , [37] J. Y. Chung and D. A. Blaser. Transfer function method for measuring in-duct acoustic properties. I. Theory, II. Experiments. Journal of the Acoustical Society of America, 68(3): , [38] A. F. Seybert and D. F. Ross. Experimental determination of acoustic properties using a two-microphone random-excitation technique. Journal of the Acoustical Society of America, 61(5): , May [39] ASTM International E Standard test method for impedance and absorption of acoustic material using a tube, two microphones and a digital frequency analysis system. 194

227 REFERENCES [40] ISO :1996 Acoustics - Determination of sound absorption coefficient and impedance in impedance tubes - Part 1: Method using standing wave ratio. [41] ISO :1996 Acoustics - Determination of sound absorption coefficient and impedance in impedance tubes - Part 2: Transfer-function method. [42] M. Åbom. Modal decomposition in ducts based on transfer function measurements between microphone pairs. Journal of Sound and Vibration, 135(1):95 114, [43] G. Krishnappa. Cross-spectral method of measuring acoustic intensity by correcting phase and gain mismatch errors by microphone calibration. Journal of the Acoustical Society of America, 69(1): , January [44] J. Y. Chung. Cross-spectral method of measuring acoustic intensity without error caused by instrument phase mismatch. Journal of the Acoustical Society of America, 64(6): , December [45] A. F. Seybert and D. K. Graves. Measurement of phase mismatch between two microphones. In Proceedings of Noise-Con 85, pages The Ohio State University, June [46] H. Bodén and M. Åbom. Influence of errors on the two-microphone method for measuring acoustic properties in ducts. Journal of the Acoustic Society of America, 79(2): , February [47] W. T. Chu. Extension of the two-microphone transfer function method for impedance tube measurements. Journal of the Acoustic Society of America, 80(1): , July [48] M. Åbom and H. Bodén. Error analysis of two-microphone measurements in ducts with flow. Journal of the Acoustic Society of America, 83(6): , June

228 REFERENCES [49] E. Meyer, F. Mechel, and G. Kurtez. Experiments on the influence of flow on sound attenuation in absorbing ducts. Journal of the Acoustical Society of America, 30(3): , March [50] J. S. Anderson. The effect of an air flow on a single side branch Helmholtz resonator in a circular duct. Journal of Sound and Vibration, 52(3): , June [51] R. L. Panton and J. M. Miller. Excitation of a Helmholtz resonator by a turbulent boundary layer. Journal of the Acoustical Society of America, 58(4): , October [52] C. O. Paschereit, W. Weisenstein, P. Flohr, and W. Polifke. Apparatus for damping acoustic vibrations in a combustor. United States Patent Number 6,634,457, [53] G. Kudernatsch. Exhaust gas system with Helmholtz resonator. United States Patent Number 6,705,428, [54] C. Q. Howard, B. S. Cazzolato, and C. H. Hansen. Exhaust stack silencer design using finite element analysis. Noise Control Engineering Journal, 48(4): , [55] C. Q. Howard, C. H. Hansen, and A. Zander. Vibro-acoustic noise control treatments for payloa bays of launch vehicles: Discrete to fuzzy solutions. Applied Acoustics, 66: , [56] S. J. Estève. Control of sound transmission into payload fairings using distributed vibration absorbers and Helmholtz resonators. PhD thesis, Virginia Polytechnic Institute and State University, [57] W. Neise and G. H. Koopmann. Reduction of centrifugal fan noise by use of resonators. Journal of Sound and Vibration, 73: ,

229 REFERENCES [58] G. Koopmann and W. Neise. The use of resonators to silence centrifugal blowers. Journal of Sound and Vibration, 82:17 27, [59] J. S. Lamancusa. An actively tuned, passive muffler system for engine silencing. In Jiri Tichy and Sabih Hayek, editors, Proceedings of Noise-Con 87, pages The Pennsylvania State University, State College, Pennsylvania, June [60] S. Sato and H. Matsuhisa. Semi-active noise control by a resonator with variable parameters. In Proceedings of Inter-Noise 90, pages , [61] H. Matsuhisa, B. Ren, and S. Sato. Semiactive control of duct noise by a volumevariable resonator. Japan Society of Mechanical Engineers, International Journal, 35(2): , [62] J. M. de Bedout, M. A. Franchek, R. J. Bernhard, and L. Mongeau. Adaptivepassive noise control with self-tuning Helmholtz resonators. Journal of Sound and Vibration, 202(1): , [63] C. J. Radcliffe and C. Birdsong. An electronically tunable resonator for noise control. Society of Automotive Engineers, Noise & Vibration conference & Exposition, [64] S. J. Estève and M. E. Johnson. Development of an adaptive Helmholtz resonator for broadband noise control. In Proceedings of IMECE 2004, Anaheim, CA, November ASME International Mechanical Engineering Congress. [65] S. J. Estève and M. E. Johnson. Control of the noise transmitted into a cylinder using optimally damped Helmholtz resonators and distributed vibration absorbers. In Ninth International Congress on Sound and Vibration, pages Orlando, Florida, USA, July [66] H. Kotera and S. Okhi. Resonator type silencer. United States Patent Number 5,5283,398,

230 REFERENCES [67] I. R. McLean. Variably tuned Helmholtz resonator with linear response controller. United States Patent Number 5,771,851, [68] P. E. R. Stuart. Variable resonator. United States Patent Number 6,508,331, [69] J. D. Kotsun, L. N. Goenka, D. J. Moenssen, and C. E. Shaw. Helmholtz resonator. United States Patent Number 6,792,907, [70] M. S. Ciray. Exhaust processor with variable tuning system. United States Patent Number 6,915,876, [71] C. W. S. To and A. G. Doige. A transient testing technique for the determination of matrix parameters of acoustic systems, I: theory and principles. Journal of Sound and Vibration, 62(2): , [72] C. W. S. To and A. G. Doige. A transient testing technique for the determination of matrix parameters of acoustic systems, II: experimental procedures and results. Journal of Sound and Vibration, 62(2): , [73] J C. Snowdon. Mechanical four pole parameters and their applications. Journal of Sound and Vibration, 15(3): , [74] C. L. Morfey. Sound transmission and generation in ducts with flow. Journal of Sound and Vibration, 14(1):37 55, [75] R. D. Blevins. Formulas for natural frequency and mode shape. Krieger Publishing Company, [76] ANSYS Inc. ANSYS Release 9.0 Documentation. [77] D. B. Woyak. Acoustics and Fluid-Structure Interaction. Swanson Analysis Systems, Inc, Houston. [78] S. Imaoka. Acoustic elements and boundary conditions. Memo number STI:05/01B,

231 REFERENCES [79] P. L. Driesch. Active control in enclosures using optimally designed Helmholtz resonators. PhD thesis, The Pennsylvania State University, Department of Mechanical and Nuclear Engineering, [80] F. S. Tse, I. E. Morse, and R. T. Hinkle. Mechanical Vibrations Theory and Applications. Allyn and Bacon Series in Mechanical Engineering and Applied Mechanics. Allyn and Bacon, Inc., second edition,

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233 Appendix A Transition of the Transfer Matrix Method Elements and Sequence of the State Variables Adopted For This Study As described in chapter 3, section , the transfer matrix of a circular duct of uniform cross-sectional area S and length l, is given by [26] p r q r = cos(ˆkl) j sin(ˆkl) Y r jy r sin(ˆkl) cos(ˆkl) p r 1 q r 1 (A.1) where, Y r = c S ˆk is the characteristic impedance, and is the complex wave number. p r, p r 1 and q r, q r 1 are the acoustic pressures and acoustic mass velocities at the extreme ends of the duct, respectively (input and output sides). However, for this 201

234 study equation (A.1) was transformed to a different equation given below. v r p r = cos(ˆkl) j ρc S sin(ˆkl) j S ρc sin(ˆkl) cos(ˆkl) v r 1 p r 1 (A.2) where v r and v r 1 represent acoustic volume velocities at input and output sides of element r, respectively. This transformation was facilitated by relating acoustic volume velocity and acoustic pressure instead of acoustic mass velocity and acoustic pressure respectively. The theory behind the transition is shown below. The acoustic mass velocity, q r is given by q = Sρu (A.3) where, u is the particle velocity and is given by u = v S (A.4) Using the expressions for the acoustic mass and particle velocities, equations (A.3) and (A.4), respectively in equation (A.1), one gets p r ρv r = cos(ˆkl) j sin(ˆkl) Y r jy r sin(ˆkl) cos(ˆkl) p r 1 ρv r 1 (A.5) Substituting Y r = c S in equation (A.5), one gets p r ρv r = cos(ˆkl) j S c sin(ˆkl) j c S sin(ˆkl) cos(ˆkl) p r 1 ρv r 1 (A.6) Multiplying the R.H.S. of equation (A.6), one gets p r = cos(ˆkl)p r 1 + j ρc S sin(ˆkl)v r 1 (A.7) 202

235 Appendix A. Transition of the Transfer Matrix Method Elements and Sequence of the State Variables Adopted For This Study ρv r = j S c sin(ˆkl)p r 1 + ρ cos(ˆkl)v r 1 (A.8) Writing equations (A.7) and (A.8) in the matrix form, one gets p r v r = cos(ˆkl) j S ρc sin(ˆkl) j ρc S sin(ˆkl) cos(ˆkl) p r 1 v r 1 (A.9) Inverting the sequence of state variables, acoustic pressure and volume velocity, in equation (A.9) results in v r p r = cos(ˆkl) j ρc S sin(ˆkl) j S ρc sin(ˆkl) cos(ˆkl) v r 1 p r 1 (A.10) 203

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237 Appendix B List of Symbols a â +, â radius of the duct modal amplitudes of acoustic pressure associated with the incident (positive x direction) and reflected waves (negative x direction) â Ac An c C D [2 1] column vector containing the unknown modal amplitudes cross-sectional area of the cavity cross-sectional area of the opening speed of sound in the fluid medium damping coefficient cross-sectional perimeter of the duct which has the orifice drilled into its wall f 1, f 2 frequencies corresponding to the half-power points f n resonance frequency of a SDOF system f r resonance frequency of a HR (as a stand-alone device) F forcing function 205

238 h H either the orifice radius or viscous boundary layer thickness [2 1] column vector containing the transfer function between the two locations k ˆk ˆk +n, ˆk n K l wave number complex wave number axial wave numbers in the positive and negative directions stiffness of the volume of the fluid in the cavity length of the duct l 0 end-correction of an unflanged open end of a duct l n physical neck length of the HR l eff effective length of the neck Lc M M n length of the cavity effective mass of the fluid in the neck [2 2] modal matrix containing the propagation terms acoustic mode number p e instantaneous acoustic pressure at the opening of the neck p r, p r 1 ˆp acoustic pressures at the input side (source end) of the duct [2 1] column vector containing the measures of acoustic pressures at two different locations q r, q r 1 Q acoustic mass velocities at the output sides (open end) of the duct quality factor of the duct-hr system 206

239 Appendix B. List of Symbols r R s S t u radius of the neck of the HR radius of the cavity of the HR microphone spacing cross-sectional area of the duct viscous boundary thickness acoustic particle velocity v sp acoustic volume velocity of the loudspeaker V Vc volume of the HR volume of the cavity of the HR W net in-duct net acoustic power transmission W norm normalised in-duct net acoustic power transmission x X Y Z l, Z r Z 0, Z s sign +sign β ξ ω location along the duct amplitude of the frequency response of a SDOF system characteristic impedance radiation impedance source impedance represents the propagation of acoustic wave in +ve x direction represents the propagation of acoustic wave in -ve x direction boundary admittance coefficient displacement of the fluid in the neck angular excitation frequency 207

240 ζ φ ρ critical damping ratio phase of the frequency response of the SDOF system density of the fluid medium Ψ n eigenfunction for mode n µ gas viscosity and for air, at 20 o c, is equivalent to kg m -1 s -1 γ ratio of specific heats and for air is equivalent to 1.4 ε either equal to 0 when the orifice or tube radiates into spaces of dimensions the wavelength of sound, or 0.5 when the orifice or tube radiates into a free space without a flange, or 1 when the orifice or tube radiates into a free space with a flange 208

241 Appendix C Publications Originating from this Thesis 209

242 Tuning a semi-active Helmholtz resonator S. Singh a, C. Q. Howard b, C. H. Hansen c School of Mechanical Engineering The University of Adelaide Adelaide, S.A. Australia, 5005 ABSTRACT Adaptive Helmholtz resonators are used to reduce tonal noise propagating as plane waves in ducts. Optimal tuning of the resonator has previously been achieved by using a pressure sensor located in the duct downstream of the resonator. The work described here is concerned with the development of a cost function that could be used by a controller 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 the pressure at the top of the closed end of the cavity of the Helmholtz resonator and the pressure at the neck wall, close to the neck duct interface. Damping in the system is taken 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 combustion (IC) engines, compressors, fans, blowers, power transformers, gearboxes etc. The humming nature of a tonal noise not only causes annoyance to workers within the industry but also to the surrounding community. Depending upon the type of application, existing set up and cost constraints, tonal noise transmission can be controlled in many possible ways like by installing reactive silencers, barriers, side branch elements and active noise control devices. The work described here is concerned with the attenuation of tonal noise transmission in ducts by using side branch resonators. Passive Helmholtz resonators (HRs) are specifically designed to achieve their optimal performance at one frequency only, and are only effective over a very narrow frequency band. Any slight change in the frequency, change in temperature which changes the speed of sound and hence the wavelength of the noise will decrease the effectiveness of the resonator. Resonators incorporating the provision for altering their geometrical parameters in real-time in order to adapt themselves to the environmental or operating condition changes offer an obvious solution. Such an adaptive system is referred to as a semi-active system in which a change in the physical parameters of the passive element is caused by an active control system. Because of the benefits of semi-active systems over any exclusive active and passive systems [1, 2], semi-active systems are gaining popularity in industry. Some of the industrial applications of the semi-active systems are highlighted in the next section. a address: sarabjeet.singh@adelaide.edu.au b address: carl.howard@adelaide.edu.au c address: colin.hansen@adelaide.edu.au

243 1.2 Previous Work There have been many previous studies that report on the successful implementation of adaptive Helmholtz resonators for attenuating the noise transmission in ducts. Lamancusa [3] demonstrated the use of a volume variable Helmholtz resonator for attenuating the firing frequency noise of an IC engine. Matsuhisa et al. [4, 5] conducted experiments with two types of resonators, first, in which the cavity volume was varied and second, in which the neck cross-sectional area was varied. Tuning of the resonator was facilitated using an algorithm that was based on the phase difference between the pressure in the cavity of the resonator and the pressure in the duct. Bedout et al. [6] also developed an adaptive Helmholtz resonator which was tuned by using the signal from a microphone located in the duct downstream of the resonator. Radcliff et al. [7] reported on the development of a semi-active Helmholtz resonator for which tuning was achieved by locating two pressure sensors, first, in the cavity of the resonator and second, in the duct downstream of the resonator. Recent development of an adaptive Helmholtz resonator was reported by Estève et al. [8] for reducing broadband noise transmission into a rocket payload fairing. Many more developments on adaptive Helmholtz resonators can be found in references [9, 10, 11, 12, 13, 14, 15, 16], which demonstrate their use in industrial applications for attenuating the intake and exhaust noise of IC engines, and exhaust noise produced by gas turbines and combustors. Previous work has shown that the optimal tuning of HRs has been achieved by using the information from one or more pressure sensors located in the duct downstream of the resonator. However, in many cases, especially in industrial exhaust stacks where the stacks serve as a passage for the exhaust gases to be driven out to the environment, it is neither desirable nor practical to mount microphones in a duct. This is because of the possibility that signals so obtained are often contaminated with unrelated noise or affected due to physical problems with the microphones such as [17]: accumulation of exhaust particulates - the response of the pressure sensors can be significantly affected by the exhaust particulates which may accumulate on them, heating - the possibility of high temperature of the exhaust gases can also directly affect 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 the location of the pressure sensor being close to a pressure node and unable to properly obtain a measure of the pressure. However, some researchers have overcome this problem by using multiple microphones along the duct [17, 18]. In order to avoid the above mentioned potential risks and also, for reasons of convenience and practicality, it is highly desirable to have a self-contained adaptive Helmholtz resonator that does not need any external measures of quantities outside the resonator. This involves locating the pressure sensors in the resonator s cavity and/or neck. The output of these sensors can be fed to a controller to optimally adjust the physical dimensions of the HR, which will result in the minimisation of the sound propagating downstream of the

244 HR. When a HR is modelled as a stand-alone device, its resonance frequency can be accurately predicted by using the classical formula developed by Helmholtz when the resonator is small compared to a wavelength of sound. For the cases where the resonator may not be small compared to a wavelength, others have developed equations for predicting the resonance frequency of a cylindrical HR [19, 20]. However, when a HR is mounted onto a duct, a coupled system is created whose resonance frequency is different to that for the stand-alone HR. More importantly, the cost function for tuning the HR which has been 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 effective length of the resonator neck. Even though only plane waves are propagating down the duct, the sound field in the duct adjacent to the neck opening is not planar due to the near field generated by the neck opening. Thus the end correction factor for the neck at this interface is not very well understood. Inaccuracies in estimating the neck-duct interface end correction factor lead to inaccuracies in predicting the resonance frequency of duct mounted HR. 2 EXPERIMENTS A variable volume cylindrical HR was mounted onto a 3 m long circular duct at 0.5 m from the source end. At the source end, a loudspeaker backed by an air tight cavity was fixed to the duct. A schematic of a duct mounted HR is shown in figure 1. Two microphones, mic. 1 and mic. 2, were flush mounted in the duct downstream of the HR 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 remote from the resonator, one microphone, microphone A, was located at the top of the closed end of the cavity and the other microphone, microphone B, was located at the neck wall close to the neck-duct interface. Another microphone was fixed in the speaker s backing cavity for the purpose of measuring the volume velocity of the speaker. The response of all the microphones was normalised by the input volume velocity of the loudspeaker. The loudspeaker was excited by random noise from 0 to 400 Hz. A multi-channel B&K Pulse system was used to measure and record the pressure from all the microphones locations in the system. The results of experiments are divided into two sections: (2.1) broadband frequency analysis, in which the results of net acoustic power transmission, acoustic pressure and transfer function are plotted as a function of frequency, and (2.2) single frequency analysis, in which all of the above stated results are plotted as a function of the resonator s geometry. The purpose of the first analysis was to illustrate the difference in the frequencies corresponding to maximum acoustic pressure at microphone A, maximum amplitude of the pressure transfer function between microphone A and microphone B (microphone A / microphone B ), and maximum reduction of the net acoustic power transmission in the duct downstream of the HR. The purpose of the second analysis was to highlight the differences in the net acoustic power transmission in the duct as a result of changing the length of the cavity.

245 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 = m, cavity diameter = m, physical neck length = m and neck diameter = m. Figure 2 shows the plot of net acoustic power transmission in the duct downstream of the HR along with the net acoustic power transmission in the duct without the HR. The net acoustic power transmission was calculated 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 acoustic power transmitted in the duct downstream of the HR was reduced by 18 db at 226 Hz, as shown by the thick line in figure 2. normalised net acoustic power (db) Hz without HR with HR frequency (Hz) Figure 2: Net acoustic power transmission in duct with and without the HR as a function of frequency.

246 The following two figures highlight the difference in the frequencies corresponding to the 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 microphone B (microphone A / microphone B ), and maximum reduction of net acoustic power in the duct downstream of the HR. The vertical line at 226 Hz in both the plots indicate the frequency at which the net acoustic power transmission in the duct downstream of the HR was minimised. Figure 3 shows the normalised acoustic pressure at microphones A and B. As seen from 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 maximum reduction 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 be used 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 figure 4. The maximised response of the amplitude of the pressure transfer function occurs at 246 Hz which is again different from 226 Hz, where the net acoustic power in the duct is minimised Hz cavity neck wall normalised acoustic pressure (db) Hz frequency (Hz) Figure 3: Acoustic pressure at the top of the closed end of the cavity (microphone 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 the frequency of maximum reduction of the acoustic power transmission in the duct. Figures 3 and 4 show that neither the pressure at the top of the closed end of the cavity 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 transmission in the duct.

247 40 amplitude (db) Hz 246 Hz frequency (Hz) 0 phase (degrees) Hz frequency (Hz) Figure 4: Amplitude and phase plots of the pressure transfer function between microphone A and microphone B as a function of frequency. The vertical line at 226 Hz corresponds to the frequency of maximum reduction of the acoustic 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 the exhaust stack in order to drive the exhaust gases out to the environment, noise is often generated at the blade passage frequency (BPF) of a fan, which is determined by the number of blades and the fan s rotation speed. The noise is tonal and hence resonators are often required to attenuate a single frequency rather than broadband noise. Experiments were conducted using the same duct-hr apparatus described in section 2, where the length of the cavity was altered. The HR used in these experiments resembles a piston, where the diameter of the cavity is fixed but the length can be varied to change it s volume. The loudspeaker was excited by random noise from 0 to 400 Hz and the length of the cavity was changed from 60 mm to 90 mm in increments of 1 mm. With the aim to attenuate noise at 226 Hz, the experimental results at 226 Hz were extracted from the experimental set of 0 to 400 Hz. Figure 5 shows the variation in the net acoustic power in the duct measured downstream 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 noise at 226 Hz, is 70 mm. The acoustic pressure at microphone A and microphone B at 226 Hz was plotted as a function of the cavity length of the HR and is shown in figure 6. The pressure at the top of the closed end of the cavity was maximised at the cavity length of 63 mm and was 7 mm off the optimal length required to achieve the maximum reduction of net acoustic power at 226 Hz. Figure 7 shows the amplitude and phase of the pressure transfer function between microphone A and microphone B (microphone A / microphone B ) as a function of the cavity length of the HR. The maximum amplitude of the transfer function occurs at a

248 0 normalised net acoustic power (db) cavity length (mm) Figure 5: Net acoustic power transmission in the duct downstream of the HR corresponding to 226 Hz, plotted as a function of cavity length cavity neck wall normalised acoustic pressure (db) optimum length cavity length (mm) 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 ) corresponding to 226 Hz, as a function of cavity length. The vertical line at 70 mm corresponds to the cavity length required for the maximum reduction of acoustic power transmission in the duct at 226 Hz.

249 amplitude (db) optimum length cavity length (mm) 0 phase (degrees) optimum length cavity length (mm) Figure 7: Amplitude and phase of the pressure transfer function between microphone A and microphone B corresponding to 226 Hz, as a function of cavity length. The vertical line at 70 mm corresponds to the cavity length required for the maximum reduction of acoustic power transmission in the duct at 226 Hz. cavity length of 84 mm which is 14 mm off 70 mm, where the net acoustic power in the duct is minimised. Figures 6 & 7 show that the maximised response of the pressure at the top of the closed end of the cavity and the pressure transfer function between microphone A and microphone B, respectively, do not occur at the same volume of the HR which corresponds to the minimisation of net acoustic power transmission in the duct. Hence, these results show that the conventional understanding of trying to achieve a resonance condition in the side branch HR, does not result in the minimisation of net acoustic power transmission in the duct. Instead, an alternative is needed to determine when the acoustic power transmission in the duct is minimised, and this is described in the next section. 3 COST FUNCTION A cost function was developed utilising the measured damping of the system and the value of phase difference between microphone A and microphone B corresponding to the frequency for which the maximum reduction of net acoustic power transmission in the duct is achieved as measured by microphones 1 and 2. Experiments were conducted to measure the net acoustic power transmission in the duct as a function of the volume of the HR. The cavity diameter and neck length throughout the experiments were kept constant and three different neck diameters ( m, m and m) were tested. For each different neck size, the cavity length was varied between m and m corresponding to a volume variation of m 3 to m 3. For each configuration of the HR, the value of the phase difference between microphone A and microphone B corresponding to the frequency, for which

250 the maximum reduction of net acoustic power was achieved, was noted and the damping ratio of the system was calculated. The plot of damping ratio versus phase difference for all the three neck sizes is shown in figure 8. It reveals that as the damping of the system increases, the phase difference decreases neck dia. = m neck dia. = m neck dia. = m curve fit phase (degrees) damping ratio (%) Figure 8: Damping ratio versus phase difference corresponding to the frequency where maximum reduction of the acoustic power transmission in the duct occurs, for three different neck sizes. An empirical curve fitting relation between the phase difference and the damping ratio was found, using a second order polynomial (with a correlation coefficient of 0.882), and is given by: phase difference = 0.94(damping ratio) (damping ratio) (1) Equation (1) can be referred to as a cost function that can be used by an electronic controller to optimally tune the HR for minimising the acoustic power transmission in the duct downstream of the HR. The damping present in the system, which needs to be predetermined, can be substituted in equation (1), which will then yield the desired value of the phase difference. The length of the cavity of the HR will then be varied until the phase difference between microphone A and microphone B reaches the desired/ calculated value. This calculated value of the phase difference will approximately correspond to the optimal length of the cavity required to minimise the net acoustic power transmission in the duct. Figure 9 shows the normalised net acoustic power transmission in the duct at 226 Hz as a function of cavity length, for several cost functions. The markers indicate the length of the cavity which corresponds to the: maximum value of the amplitude of the pressure transfer function between microphone A and microphone B - ( maximum transfer function ),

251 normalised net acoustic power (db) acoustic power with HR maximum pressure in cavity maximum transfer function new cost function actual minimum power cavity length (mm) Figure 9: Net acoustic power in the duct downstream the HR corresponding to 226 Hz plotted as a function of cavity length, with markers indicating the acoustic 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 achieved by varying the cavity length for minimising the net acoustic power transmission in the duct downstream of the HR - ( new cost function ), and minimum acoustic power measured by mounting two microphones in the duct downstream of the HR - ( actual minimum power ). Figure 9 shows that in order to achieve the maximum reduction of the acoustic power transmission in the duct, the newly proposed cost function gives the best possible cavity length when compared to the actual cavity length needed to optimally tune the adaptive HR. 4 CONCLUSION A cost function has been presented for minimising the net acoustic power transmission in a duct to which a Helmholtz resonator is attached, using only pressure measurements inside the resonator. The cost function is based on the damping present in the system and the phase difference between the pressure at the top of the closed end of the cavity and the pressure at the neck wall close to the neck-duct interface. It was shown experimentally that neither the pressure at the top of the closed end of the cavity nor the pressure transfer function can be used to optimally tune the HR, whereas minimisation of the cost function proposed here will result in minimisation of the in-duct acoustic power transmission.

252 REFERENCES [1] R. J. Bernhard, H. R. Hall, and J. D. Jones, Adaptive-passive noise control, Inter- Noise 92, pp (1992). [2] L. J. Eriksson and M. T. Zuroski, From passive to active: a family of silencing possibilities, Proceedings of Noise- Con 97, pp (1997). [3] J. S. Lamancusa, An actively tuned, passive muffler system for engine silencing, Proceedings of Noise-Con 87, The Pennsylvania State University (The Pennsylvania State University, State College, Pennsylvania, 1987). [4] H. Matsuhisa and S. Sato, Semi-active noise control by a resonator with variable parameters, Proceedings of Inter-Noise 90, pp (1990). [5] H. Matsuhisa, B. Ren, and S. Sato, Semiactive control of duct noise by a volumevariable resonator, Japan Society of Mechanical Engineers, International Journal, 35, (1992). [6] J. M. de Bedout, M. A. Franchek, R. J. Bernhard, and L. Mongeau, Adaptive-passive noise control with self-tuning helmholtz resonators, Journal of Sound and Vibration, 202(1), (1997). [7] C. J. Radcliffe and C. Birdsong, An electronically tunable resonator for noise control (Society of Automotive Engineers, Noise & Vibration conference & Exposition, 2001). [8] S. J. Estève and M. E. Johnson, Development of an adaptive helmholtz resonator for broadband noise control, Proceedings of IMECE2004 (2004). [9] H. Kotera and S. Okhi, Resonator type silencer, United State Patent Number 5,5283,398 (1994). [10] C. R. Cheng, J. D. McIntosh, M. T. Zuroski, and L. J. Eriksson, Tunable acoustic system, United States Patent Number 5,930,371 (1999). [11] I. R. McLean, Variably tuned helmholtz resonator with linear response controller, United States Patent Number 5,771,851 (1998). [12] P. E. R. Stuart, Variable resonator, United States Patent Number 6,508,331 (2003). [13] C. O. Paschereit, W. Weisenstein, P. Flohr, and W. Polifke, Apparatus for damping acoustic vibrations in a combustor, United States Patent Number 6,634,457 (2003). [14] G. Kudernatsch, Exhaust gas system with helmholtz resonator, United States Patent Number 6,705,428 (2004). [15] J. D. Kotsun, L. N. Goenka, D. J. Moenssen, and C. E. Shaw, Helmholtz resonator, United States Patent Number 6,792,907 (2004). [16] M. S. Ciray, Exhaust processor with variable tuning system, United States Patent Number 6,915,876 (2005). [17] C. H. Hansen, C. Q. Howard, K. A. Burgemeister, and B. S. Cazzolato, Practical implementation of an active control system in a hot exhaust stack, Proceedings of the Australian Acoustical Society Conference (1996). [18] X. Q. Li, D. L. L. Leclercq, A. C. Zander, and C. H. Hansen, Implementation of active noise control in a multi-modal spray dryer exhaust stack, Applied Acoustics, 67(1), (2006). [19] R. L. Panton and J. M. Miller, Resonant frequencies of cylindrical helmholtz resonators, Journal of the Acoustical Society of America, 57(6), (1975).

253 [20] D. Li, Vibroacoustic behaviour and noise control studies of advanced composite structures, Ph.D thesis, University of Pittsburgh (2003). [21] J. Y. Chung and D. A. Blaser, Transfer function method for measuring in-duct acoustic properties. I. theory, II. experiments, Journal of the Acoustical Society of America, 68(3), (1980). [22] M. Åbom, Modal decomposition in ducts based on transfer function measurements between microphone pairs, Journal of Sound and Vibration, 135(1), (1989).

254 Proceedings of ACOUSTICS November 2006, Christchurch, New Zealand The elusive cost function for tuning adaptive Helmholtz resonators Sarabjeet Singh, Colin H. Hansen and Carl Q. Howard School of Mechanical Engineering, The University of Adelaide, Adelaide, Australia ABSTRACT One of the problems associated with the use of Helmholtz resonators to control tonal noise propagation inside a pipe or duct is that any slight frequency changes in the tonal noise as a result of environmental changes or load changes on the device generating the noise will severely compromise the performance of the resonator. Thus, it is desirable to use an adaptive resonator whose volume or neck length can be adjusted to maintain optimal tuning as the excitation frequency or environmental conditions change. The ideal cost function would be a measure of the sound power propagating down the duct so the control system could minimise this quantity by driving motors that change the geometry of resonator. In practice, it is highly desirable to have available a self-contained adaptive resonator that does not need any external inputs or measures of quantities outside the resonator package. A cost function based on pressure measurements in the resonator, which corresponds to sound power in the duct has been found and verified experimentally and numerically. The effect of resonator damping on the cost function and a method to correct for the effect is also discussed. INTRODUCTION Background The work presented in this paper is an extension to the work reported by the authors in a previous paper (Singh, Howard and Hansen 2006). The motivation for the ongoing investigations is the desire for the development of a self-contained adaptive Helmholtz resonator, which does not require any external sensors, to attenuate tonal noise propagation in ducts. The current paper is focused on the development of a cost function that uses only measurements from transducers located on or in the resonator to determine a quantity to be minimised. Many authors have reported the successful implementation of adaptive Helmholtz resonators in small-scale laboratory setups. The optimal tuning of Helmholtz resonators has been achieved by using an electronic controller, which drives a motor in order to change the dimensions of either the cavity or neck of the resonator. The control algorithms reported in previous studies (Bedout et al. 1997) for changing the geometry of the resonators have used pressure measurements from microphones located in the duct downstream of the resonator. However, there are number of problems related to the in-duct mounting of the microphones and these have been discussed previously (Singh, Howard and Hansen 2006). A cost function based on pressure measurements at the top of the closed end of the cavity and at the neck wall in close proximity to the neck-duct interface was presented previously (Singh, Howard and Hansen 2006). The cost function was empirically derived by relating the measured damping of the duct-hr system and the phase difference between the two pressure measuring locations in the resonator described above. A new cost function is presented in this paper, which relates a different measure of the quality factor of the HR to the phase difference between pressure measurements at the top of the closed end of the cavity and at the neck wall in close proximity to the neck-duct interface. The quality factor, which is the reciprocal of twice the critical damping ratio, is typically estimated by using the bandwidth of frequencies bounding the resonance. Here the quality factor is estimated by using information at a single frequency only, at which the noise needs to be attenuated. This is a much more practical approach because in an actual system, the energy in the duct is dominated by the tonal noise to be controlled and it is often not convenient to introduce a loudspeaker to excite frequencies around the frequency to be controlled to enable the 3 db bandwidth to be measured. In the next section, the theoretical basis for estimating the resonance frequency and performance of a Helmholtz resonator is summarised. This is followed by a numerical analysis based on finite element analysis (FEA) using ANSYS software. Results obtained numerically are then compared with the theoretical results and the relationship between the pressures measured in the resonator and the in-duct sound power transmission is also discussed. Finally the cost function derived in our previous paper (Singh, Howard and Hansen 2006) is described and this is followed by the derivation and description of the new cost function. THEORETICAL ANALYSIS The following theoretical analysis is concerned with calculations of the resonance frequency of a HR as a stand-alone device and the acoustical performance of the HR mounted on a duct. Helmholtz resonators as stand-alone devices The resonance frequency, f r, of a HR can be accurately calculated by using the well known classical formula given by: 2 c πr f = (1) r 2π l V eff Acoustics

255 20-22 November 2006, Christchurch, New Zealand Proceedings of ACOUSTICS 2006 where, c is the speed of sound, r is the radius of the neck, l eff is the effective length of the neck, which includes two endcorrections, one at each end, and V is the volume of the cavity. However, Panton and Miller (Panton and Miller 1975) have shown that equation (1) is no longer valid when the dimensions of the HR exceed 1/16 of the wavelength of sound at the resonance frequency. Panton and Miller (Panton and Miller 1975) derived a new formula to calculate the resonance frequencies of cylindrical HRs, given by: c πr 2 fr = 2π 1 l V L 2 (2) eff + c A n 3 where, L C is the length of the cavity, and A n is the crosssectional area of the neck. Unlike the assumptions for the classical formula, equation (2) is accurate for a length of the cavity comparable to or longer than a wavelength but the cavity diameter and neck dimensions must be kept smaller than a wavelength. Also, for the derivation of equation (2), it was considered that the length of the neck was very small compared to the length of the cavity. As an extension to the work accomplished by Panton and Miller (Panton and Miller 1975), Li (Li 2003) proposed another model for calculating the resonance frequencies of cylindrical HRs. His derivation was also based on wave-tube theory but was more general than the model derived by Panton and Miller, and is given by: 2 f c 3Leff Ac + Lc A n 3Leff Ac + Lc A n 3An r = π 2L A + c L Ac L 3 eff 2 (3) eff efflc Ac where, A c is the cross-sectional area of the cavity, and all the other variables have the same meaning as they do in equation (2). As the dimensions of the HRs used in this study were all greater than 1/16 of the wavelength, the neck length was comparable or greater than the cavity length, and the other conditions related to the dimensions of the resonators conformed to those of Li s model, equation (3) was used to calculate the resonance frequencies. Helmholtz resonator attached to a duct When a Helmholtz resonator is mounted onto a duct, a coupled system is created whose resonance frequency is different to that of the stand-alone HR. HRs work by causing an impedance change in the acoustic system at its point of insertion. Thus, HRs act like passive bandstop filters barring the transmission of acoustic power past their location at frequencies in close proximity to their resonance frequencies. The transfer matrix method, also referred to as transmission matrix or four-pole parameter representation (Munjal 1987), was used to calculate the net acoustic power transmission in the duct to which the HR was attached. The complete transfer matrix equation of the duct-hr system was built by discretising the duct-hr model into three elements: (1) section of the duct upstream of the HR, (2) section of the duct downstream of the HR, and (3) the HR. Figure 1 shows a schematic of the duct-hr system illustrating the three elements desribed above, along with the neck-cavity and neck-duct interfaces. The key issue related to the theoretical analysis of the duct- HR system was the incorporation of the end-correction factors of the neck of the HR in addition to the actual dimensions of the elements. The end-correction factor at the neckcavity interface, also referred to as interior end correction factor, δ, is well known and is given by (Ingard 1953): i 8r r δ i = (4) 3π R where, r is the radius of the neck, and R is the radius of the cavity. However, there exists some uncertainty concerning the estimation of the end-correction factor at the neck-duct interface. Figure 1. A schematic of the duct-hr system highlighting the neck-cavity and neck-duct interfaces. Onorati (Onorati 1994) suggested that an end-correction factor of 0.3a, where a is the radius of the duct, should be added to the physical length of the neck in order to incorporate the mass loading of the fluid at the neck-duct interface. Ji (Ji 2005) presented an empirical value of the neck-duct interface end-correction factor that is related to the dimensions of the neck and the duct. Figure 2 shows two plots of the in-duct net acoustic power transmission downstream of a HR calculated by using two measures of the neck-duct interface end-correction factor as per Onorati s (Onorati 1994) and Ji s (Ji 2005) model. The dimensions of the duct-hr system were: duct diameter = m, duct length = 3 m, cavity diameter = m, cavity length = m, neck diameter = m and physical neck length = m. normalised net acoustic power (db re 1watt/(1m 3 /sec) 2 ) Hz 224 Hz Ji Onorati frequency (Hz) Figure 2. Theoretical results of the net acoustic power transmission in the duct downstream of the HR estimated by using estimates of the neck-duct interface end-correction factor as per Onorati s and Ji s models. The frequency at which the maximum reduction of in-duct net acoustic power transmission was calculated as 220 Hz when Onorati s estimate of the neck-duct interface end- 2 Acoustics 2006

256 Proceedings of ACOUSTICS 2006 correction factor was used in the transfer matrix equation of the duct-hr system. This frequency changed to 224 Hz when Ji s model was used to estimate the neck-duct interface endcorrection factor. It can be seen that using different values of the neck-duct interface end-correction factor results in different estimates of the in-duct net acoustic power transmission and in particular it leads to errors in the estimate of the frequency corresponding to the minimum sound power transmisson. Therefore, the transfer matrix method was not considered a reliable option for analysing the duct-hr system. To overcome the limitations of the transfer matrix method and the associated uncertanity in the neck-duct interface endcorrection estimate, a numerical analysis of the duct-hr system was conducted and this is described in the next section November 2006, Christchurch, New Zealand Figure 3 shows a schematic of a circular duct with an attached cylindrical HR, including the locations of the microphones A, B, 1 and 2 used for pressure estimates using ANSYS, which correspond to the locations used for the experimental work. The descriptions of the microphone locations are listed below. microphone A - located at the top of the closed end of the cavity, microphone B - located at the neck wall at a distance of 5 mm from the neck-duct interface, microphones 1 and 2 - flush mounted onto the duct wall downstream of the HR. Microphone 1 was located at a distance of 1.2 m from the mounting location of the HR and distance between microphones 1 and 2 was 0.3 m. NUMERICAL ANALYSIS Numerical analysis of the duct-hr system was facilitated by using the ANSYS FEA software package. Unlike the theoretical analysis in which the end-correction factors have to be calculated, ANSYS automatically determines and incorporates the end-correction factors during its solution phase. It will be seen that the effective end-correction factor determined using ANSYS is different to both theoretical estimates described previously and used in the transfer matrix analysis. One limitation with numerical analysis is associated with difficulty in obtaining an accurate estimate of system damping to include in the analysis. For the analysis undertaken here, the inclusion of damping is discussed following Equation (5). The dimensions of the duct analysed were identical to those used for the transfer matrix analysis and are stated in the previous section. In the ANSYS model, the source end of the duct was modelled as being driven by a piston by applying unit volume acceleration (denoted by the label FLOW in ANSYS). The other (right) end of the duct was modelled as open and radiating into free space. This was done by applying the frequency dependent complex radiation impedance boundary condition for the unflanged open end of a duct (Imaoka 2004). The theoretical expression used for the calculation of radiation impedance was that of the radiation impedance of an unflanged open duct with plane waves propagating inside it, and can be found in acoustic text books (Munjal 1987, Kinsler et al. 1982): 2 ρc ( ka) Z r = + j(0.6) ka S 4 Also, the frequency dependent viscous losses, which occur in the neck due to the oscillations of the fluid particles, were incorporated in the finite element model in order to model the damping in the HR. The estimate of the viscous losses was calculated by using the following expression for the resistance of the fluid in the neck (Bies and Hansen 2003): 2 ρc ktdw 5 4A Ak R A = 1+ ( γ 1) ktlog 10 + ε + M A 2A 3γ 2 πh 2π The physical meaning of each term and definitions of each variable can be found in Bies and Hansen (2003). (5) (6) Figure 3. A schematic of the duct-hr system showing pressure measurement locations. The sound pressures at these microphone locations were used to estimate power transmission in the duct (mic. 1 and mic. 2 ) and also the transfer function between locations A and B in the resonator, which is discussed in the next section. The duct and resonator dimensions used for the numerical and experimental analyses are identical to those used for the theoretical analysis in the previous section. RESULTS The experimental setup and procedure was described previously (Singh, Howard and Hansen 2006) and is not repeated here. Figure 4 shows the experimental, numerical and theoretical results of the in-duct net acoustic power transmission downstream of the HR along with the experimental net acoustic power in the duct without the HR. The net acoustic power transmission in the duct was calculated using the in-duct modal decomposition of the sound field by measuring the acoustic pressure at the downstream microphones 1 and 2 (Chung and Blaser 1980, Åbom 1989). The experimental results show a reduction of 18 db at 226 Hz in the in-duct net acoustic power transmission as a result of mounting the HR onto the duct. A reduction of 22 db at 226 Hz and 23 db at 224 Hz was predicted by using ANSYS and the transfer matrix model, respectively. The transfer matrix estimates of the in-duct net acoustic power shown in figure 4 were calculated by using the neckduct interface end-correction factor as per Ji s model (Ji 2005). As stated earlier, because ANSYS automatically incorporates end-corrections based on first principles, the ANSYS results are considered more reliable than the transfer matrix results. However, the transfer matrix estimates can be made to exactly match the ANSYS results in the vicinity of the resonance frequency by adjusting the value of the neck- Acoustics