ADAPTIVE-PASSIVE NOISE CONTROL WITH SELF-TUNING HELMHOLTZRESONATORSA ThesisSubmitted to the FacultyofPurdue UniversitybyJuan Manuel de BedoutIn Partial Ful�llment of theRequirements for the DegreeofMaster of Science in Mechanical EngineeringMay 1996

iiACKNOWLEDGMENTSI wish to express my appreciation for the guidance, support and expertise ofmy major professor, Doctor Matthew A. Franchek. Additionally, I am grateful tomy committee members, Professor Robert J. Bernhard, Professor Luc Mongeau andProfessor Patricia Davies, who have been in uential and helpful not only in the de-velopment of this thesis, but in the ful�llment of my masters degree as well. I amvery grateful for the support I have received from my family, especially my fatherJuan Ernesto, all of my grandparents, and my brother. Last, but by no means least,I wish to thank my girlfriend Michele, who has patiently tolerated and supported mybusy schedule.

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iiiTABLE OF CONTENTS PageLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viNOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Passive Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Active Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Hybrid Noise Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Adaptive-Passive Noise Control . . . . . . . . . . . . . . . . . . . . . 62.4.1 Previous Work in Adaptive-Passive Noise and Vibration Control 72.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. THEORETICAL ANALYSIS OF CONVENTIONAL HELMHOLTZ RES-ONATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.1 Conventional Helmholtz Resonator Model . . . . . . . . . . . . . . . 123.2 Frequency Response of Duct Systems with Sidebranch Helmholtz Res-onators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214. EXPERIMENTAL FACILITY MODEL DEVELOPMENT . . . . . . . . . 224.1 Adaptive Helmholtz Resonator . . . . . . . . . . . . . . . . . . . . . . 224.2 Duct and Noise Source Models . . . . . . . . . . . . . . . . . . . . . . 24

ivPage4.2.1 Loudspeaker Model . . . . . . . . . . . . . . . . . . . . . . . . 254.2.2 Determination Of System Impedance . . . . . . . . . . . . . . 264.2.3 Calculation of Sound Pressure at a Desired Location . . . . . 294.3 Analytical Model Results and Validation . . . . . . . . . . . . . . . . 304.4 Summary of System Modeling . . . . . . . . . . . . . . . . . . . . . . 355. ROBUST TUNING CONTROL LAW DEVELOPMENT . . . . . . . . . . 375.1 Tuning Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 385.1.2 Open Loop Tuning Algorithm . . . . . . . . . . . . . . . . . . 415.1.3 Gradient Based Feedback Tuning Control Law . . . . . . . . . 455.2 Summary of Robust Tuning Control Law . . . . . . . . . . . . . . . . 466. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.1 Steady State Response of Controlled System . . . . . . . . . . . . . . 486.2 Transient Response of Controlled System . . . . . . . . . . . . . . . . 486.3 Summary of Experimental Results . . . . . . . . . . . . . . . . . . . . 557. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK . . . . . . 587.1 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.3 Suggestions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 59LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61APPENDICESAppendix A: QuickBASIC 4.50 Adaptive-Passive Controller Algorithm . 64Appendix B: MATLAB Duct and Helmholtz Resonator System ModelingProgram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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vLIST OF TABLESTable Page4.1 Loudspeaker Model Variable Values . . . . . . . . . . . . . . . . . . . . 27

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viLIST OF FIGURESFigure Page3.1 Helmholtz Resonator and Vibration Absorber . . . . . . . . . . . . . . . 133.2 Simple Duct / Helmholtz resonator System and Electrical Circuit Analogyof System near Resonator Junction . . . . . . . . . . . . . . . . . . . . . 183.3 E�ect of Resonator Volume on Attenuation Bandwidth . . . . . . . . . . 203.4 E�ect of Resonator Damping on Attenuation Performance . . . . . . . . 214.1 Variable Volume Helmholtz Resonator . . . . . . . . . . . . . . . . . . . 234.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Equivalent Loudspeaker Circuit . . . . . . . . . . . . . . . . . . . . . . . 254.4 Simpli�ed Loudspeaker Equivalent Circuit . . . . . . . . . . . . . . . . . 264.5 Helmholtz Resonator Sound Pressure Spectrum Surface . . . . . . . . . 314.6 Top View of Helmholtz Resonator Sound Pressure Spectrum Surface . . 314.7 Comparison of Analytical and Measured Frequency Responses . . . . . . 334.8 Sound Pressure Spectrum of System with Helmholtz Resonator (fo=179Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.9 Sound Pressure Spectrum of System with Helmholtz Resonator (fo=124Hz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.10 Sound Pressure Spectrum of System with Helmholtz Resonator (fo=59 Hz) 345.1 Sound Pressure Level vs. Natural Frequency for Constant ExcitationFrequency of 80 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Experimental Setup and Controller Instrumentation . . . . . . . . . . . 39

viiFigure Page5.3 Conditioning Circuitry for FTV Excitation Frequency Determination . . 405.4 Theoretical and Measured Wall Angles as a Function of Natural Frequency 425.5 Di�erence Between Predicted and Measured Natural Frequencies as aFunction of Wall Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.6 Plot of the System Resonance and Antiresonance in the Resonator Oper-ating Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.7 Logic Flow Chart of Control Algorithm . . . . . . . . . . . . . . . . . . 476.1 Comparison of Treated and Untreated System Sound Pressure Spectra . 496.2 Comparison of Tuned Performance with Discrete Natural Frequencies . . 496.3 Transient Response of Controlled System to Excitation Change from 65to 160 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.4 Helmholtz Resonator Wall Angle for Excitation Change from 65 to 160 Hz 516.5 Transient Response of Controlled System to Excitation Change from 65to 100 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.6 Helmholtz Resonator Wall Angle for Excitation Change from 65 to 100 Hz 536.7 Transient Response of Controlled System to Two Excitation FrequencyChanges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.8 Helmholtz Resonator Wall Angle for Transient Test with Two ExcitationFrequency Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.9 Transient Response of Controlled System to Noise Magnitude Disturbance 556.10 Helmholtz Resonator Wall Angle for Noise Magnitude Disturbance Test 56

viiiNOMENCLATUREm = equivalent mass of Helmholtz resonator� = density of the medium in the H.R.Leff = e�ective neck length of Helmholtz resonatorL = physical neck length of Helmholtz resonatora = radius of Helmholtz resonator neckF = force applied at Helmholtz resonator neck� = particle displacement direction along H.R. neck longitudinal axis� = sti�ness of Helmholtz resonatorP = pressure at neck entrance of Helmholtz resonatorS = cross sectional area of Helmholtz resonator neckV = Helmholtz resonator cavity volume = ratio of speci�c heatsc = speed of soundk = wavenumberR = radiation resistance in Helmholtz resonator neck! = excitation frequency in rad/secRvisc = mechanical resistance in Helmholtz resonator neck due to viscous lossesRs = variable used to determine RviscZmsdm = mechanical impedance of mass-spring-damper system� = viscous damping in mass-spring-damper system� = mass element in mass-spring-damper systems = sti�ness element in mass-spring-damper system

ixZm = mechanical impedance of Helmholtz resonator!o = natural frequency of Helmholtz resonator (rad/sec)fo = natural frequency of Helmholtz resonator (Hz)Zres = acoustic impedance of Helmholtz resonatorPin = input pressure into duct systemPout = output pressure of duct systemU0� = volume velocity into point AU0+ = volume velocity out of point AUHR = volume velocity into Helmholtz resonatorZ(0+) = acoustic impedance of system downstream point AP(0�) = pressure immediately upstream point AP(0+) = pressure immediately downstream point AVin = input voltage into loudspeakerRe = loudspeaker coil resistanceRg = ampli�er resistanceB � l = product of magnetic ux density and coil wire length for loudspeakerRsp = damping of speaker suspensionMsp = lumped mass of speaker diaphragm and coilCsp = compliance of speaker suspensionSd = surface area of loudspeaker diaphragmZsys = acoustic impedance of duct system downstream of speakerCbk = compliance of speaker enclosureMbk = acoustic mass on back of speaker diaphragmZn = acoustic impedance at point n in the duct systemSp = cross sectional area of ductap = radius of ductL1 = length of duct separating termination from H.R.L2 = length of duct separating H.R. and speaker enclosureLenc = length of speaker enclosure duct

xSenc = cross sectional area of speaker enclosure ductPd = pressure in ductUd = volume velocity in ductVmike = low-pass �ltered microphone signalVfreq = low-pass �ltered FTV output signal�wall = angle between Helmholtz resonator radial wallsRcav = Helmholtz resonator cavity radiusLcav = Helmholtz resonator cavity lengthVwall = volume of Helmholtz resonator radial wallsf̂o = estimated resonator natural frequencyf̂noise = estimated excitation frequency�t = tolerance contingent upon model accuracyMi = microphone output voltage amplitude

xiABSTRACTde Bedout, Juan M., M.S.M.E., Purdue University,May 1996. Adaptive-Passive NoiseControl with Self-Tuning Helmholtz Resonators. Major Professor: Dr. Matthew A.Franchek, School of Mechanical Engineering.A tunable Helmholtz resonator integrated with a novel feedback based controllaw that achieves optimal resonator tuning for time-varying tonal noise control ap-plications is presented in this thesis. System models are �rst developed to quantifythe behavior of the resonator and facilitate the design of the control algorithm. Thelimitations in the accuracy of these models justify the need for feedback based tuning.The proposed tuning strategy combines an open loop tuning algorithm with a gra-dient descent approach that guarantees robust performance. Optimal tuning of theresonator is achieved despite system uncertainties such as variations in the excitationfrequency and environmental changes. An experimental veri�cation of the proposedcontrol law is included. Sound pressure level reductions of up to 30 decibels wereachieved in an experimental duct system with the tunable Helmholtz resonator.

11. INTRODUCTION1.1 MotivationHelmholtz resonators are often used to control steady, simple harmonic sound�elds that have tonal, narrow band spectrums. Since ancient Greece [Ing94], theacoustic properties of Helmholtz resonators have been exploited to enhance or at-tenuate sound �elds. They have found widespread use in reverberant spaces such aschurches [ABA93],as mu�ers in ducts and pipes [Mun87], and in many other appli-cations. One advantage of the Helmholtz resonator is its simplicity. It is comprisedof a cavity connected to the system of interest through one or several short narrowtubes. Among the limitations of Helmholtz resonators is the need for these devices tobe precisely tuned to achieve signi�cant noise attenuation. Therefore, the frequencyrange over which Helmholtz resonators are e�ective is relatively narrow. Changes inthe excitation frequency and environmental conditions in uence the tuning of thesedevices. In fact, a mistuned resonator can actually increase the noise in a system atcertain frequencies.Adaptive-passive devices are intelligent machines that adjust their passive param-eters to maintain optimal performance. By changing the passive parameters such asthe volume or neck length of a Helmholtz resonator, precise tuning of the device can bemaintained under changing environments and excitation frequencies. The advantagesof such an implementation over strictly passive noise control schemes are obvious:high levels of performance of the device are guaranteed despite the narrow attenu-ation frequency band. The advantages of adaptive-passive devices over active noisecontrol schemes include simple control algorithms and minimal power consumption.

21.2 Research ObjectiveThe research objectives of this thesis are:� To develop a robust feedback based control algorithm that guarantees optimaltuning of an adaptive Helmholtz resonator despite system uncertainties such asvariations in the excitation frequency and environmental conditions.� To experimentally implement and verify the feedback control algorithm on anadaptive Helmholtz resonator.1.3 Thesis OrganizationA robust control strategy for optimal tuning of an adaptive Helmholtz resonatorand an experimental veri�cation of the control algorithm are presented in this thesis.In Chapter 2, an overview of previous and related work in the area of adaptive-passivecontrol is detailed. In Chapter 3, a theoretical analysis of conventional Helmholtzresonators is presented. In Chapter 4, models of the tunable Helmholtz resonatorand the experimental setup are developed. A description of the control algorithm andcontroller implementation is given in Chapter 5. Chapter 6 contains steady-state andtransient results of the controller implementation. In Chapter 7, conclusions drawnfrom the work and proposed directions for future work in the area are presented.

32. BACKGROUNDAn overview of traditional noise control strategies is presented in this chapter.Bernhard et al. [BHJ92] have described four general categories for noise and vibra-tion control; active systems, passive systems, hybrid systems, and adaptive-passivesystems. Active noise control systems use actuators such as loudspeakers and shak-ers to load or unload the transmission path of the unwanted noise. Passive noisecontrol systems achieve sound attenuation by either utilizing reactive devices such asHelmholtz resonators and expansion chambers which load or unload the transmissionpath of the noise, or by using resistive materials such as acoustic linings and porousmembranes which absorb the noise energy by increasing the levels of damping. Hy-brid noise control systems utilize both active and passive elements to achieve soundreduction. Adaptive-passive noise control utilizes passive devices whose parameterscan be varied in order to achieve optimal noise attenuation over a band of operatingfrequencies. Although all four types of control will be discussed, the major emphasisof this chapter will be on adaptive-passive systems, since these are the primary focusof this thesis.2.1 Passive Noise ControlPassive methods of attenuating undesired noise almost outdate recorded history.For example, Helmholtz resonators were used in ancient Greek amphitheaters to im-prove reverberation. In modern times, passive noise control systems are widely uti-lized in a variety of applications. Helmholtz resonators and expansion chambers canbe found in virtually every automobile mu�er in the world. Acoustic linings andporous membranes are commonly used to diminish the transmission of noise through

4household walls, ventilation systems, automotive and airline passenger compartments,etc. A detailed survey of passive noise control can be found in the literature [ABA93],[Ing94], [KFCS82], [Pie94].The principal advantage of passive noise control systems lies in their simplicity.External energy is not required to achieve the noise reduction e�ect. Furthermore,they are usually easy to manufacture and install, and are often maintenance free. Insituations where the frequency of the noise is relatively high or con�ned to narrowbands, these systems can economically produce substantial reductions in the soundpressure.Unfortunately, the disadvantages of passive noise control schemes also stem fromtheir simplicity; their inability to adapt to changing environments and excitationfrequencies precludes their use in many noise control applications. For example,Helmholtz resonators attenuate noise over a narrow bandwidth, and therefore mustbe precisely tuned to achieve the maximum noise reduction. Additionally, at lowfrequencies passive systems typically need to be large in size, which makes themunattractive for applications where space is limited.2.2 Active Noise ControlDue to the limitations of conventional passive devices such as Helmholtz res-onators, the interest in active noise control methods has become widespread. Usually,these schemes achieve noise reduction in a reactive manner by loading or unloadingthe primary source, not by absorbing the energy or introducing damping [BHJ92].The main advantage of active noise control over passive control schemes is the implicitadaptability of the control system to changing environments and excitations.With the advent of modern computers, the implementation of active noise andvibration control has utilized primarily adaptive digital �lters. There are severaladvantages to this type of control scheme over conventional noise control methods.Among these advantages is the fact that complex signals, including broadband noise,

5can be signi�cantly attenuated with this type of control algorithm if a reference signalcan be utilized [KM96].Some of the disadvantages of active noise control are the complexity of the algo-rithms and the large actuator e�ort often required to achieve signi�cant attenuation.Additionally, since these control schemes add energy into the system, there is the pos-sibility of creating more noise if the system does not adapt successfully. Active noisecontrol has been deemed to be a cost e�ective solution for some applications up tofrequencies of about 500 Hz [ABEG88]. Active control of noise for frequencies above500 Hz tends to become progressively more expensive as higher sampling frequenciesand faster microprocessors are needed to implement the control algorithms.2.3 Hybrid Noise ControlHybrid noise control utilizes both active and passive elements to achieve soundattenuation. The motivation for utilizing hybrid control is to combine the capabilitiesof active control in the low frequency region with the capabilities of passive control inthe mid to high frequency region. Additionally, hybrid systems can be used to aug-ment an active control system with additional passive dynamics to minimize actuatore�ort levels. The main advantages of utilizing hybrid systems are that a wide oper-ating bandwidth can be achieved with less actuator e�ort than required by strictlyactive control systems, and higher performance can be achieved compared to strictlypassive systems. Although hybrid noise control requires less external energy thanstrictly active noise control schemes, this dependence on external energy can make itunattractive in some applications.Recently, several investigations have been performed concerning the use of activeHelmholtz and quarter wavelength resonators to suppress duct noise and noise gener-ated from ow excited cavities. Active resonators utilize a loudspeaker or other activeacoustic actuators in the resonator cavity. For example, Matsuhisa et al. [MTS93]have used an active Helmholtz resonator to suppress tonal noise in duct systems. Aloudspeaker in the resonator cavity was used to decrease resonator damping, which

6increased the amount of attenuation provided by the resonator. Similarly, Okamotoet al. [OBA94] have investigated the use of active quarter wavelength resonators toattenuate duct noise. The investigation showed that the required source strengthfor the active quarter wavelength resonator was less than that required for a strictlyactive controlled loudspeaker mounted at the same location to produce similar noisereductions. Billoud et al. [BGH91] have attenuated the noise generated by a ow-excited Helmholtz resonator by using adaptive active control techniques to tune aloudspeaker in the resonator cavity.2.4 Adaptive-Passive Noise ControlAn emerging class of noise and vibration solutions are adaptive-passive systems.Passive devices such as Helmholtz resonators and vibration absorbers can be inte-grated with on-board intelligence to tune the passive parameters in an e�ort to guar-antee robust performance in changing environments. The control algorithms that tuneadaptive-passive devices should be robust to uncertainty in the system parameters andenvironmental conditions to guarantee optimal performance. The main advantages ofusing adaptive-passive devices over strictly active schemes is that additional energyis not required to achieve noise reduction, and the control algorithms are simple. Theprincipal advantage of adaptive-passive devices over purely passive solutions is thatthe devices remain tuned under changing environments and time-varying excitationfrequencies.The investigation of adaptive-passive devices has been growing over the past fewyears. A subclass of adaptive-passive systems, the semi-active systems, have started tomake appearances in the automotive industry. An example of this are the dual-modemu�ers [KWH92], [KWH93], used by the Chrysler Corporation Dodge Stealth, inwhich one tailpipe of a twin-tailpipe silencer is closed for low engine speeds and bothare opened for higher engine speeds. Noise control schemes such as these, however,are not optimal, since only two possible con�gurations are available, as opposed to acontinuously tunable adaptive device.

7Another adaptive-passive device which has drawn widespread interest is the adap-tive vibration absorber. The natural frequency of an adaptive vibration absorber canbe controlled by altering the absorber mass, sti�ness, or both. Adaptive vibrationabsorbers can be used to attenuate sinusoidal mechanical vibrations over a largebandwidth of excitation frequencies, as opposed to the narrow frequency bandwidthattenuated by conventional vibration absorbers. Adaptive vibration absorbers havebeen used to reduce vibrations caused by the blade passage frequency in helicopters[vFBB94], to suppress vibrations caused by unbalanced rotating machinery [vFBB94],[WL92], and to attenuate tonal vibrations in simple structures [Rya94].The acoustic equivalent of the adaptive vibration absorber is the adaptive Helmholtzresonator. The natural frequency of a Helmholtz resonator can be controlled by ad-justing the resonator neck dimensions, cavity volume, or both. Therefore, adaptiveHelmholtz resonators (and quarter wavelength resonators) can be used to attenuatenoise over a large bandwidth of excitation frequencies. In recent years, several inves-tigations have been performed to study the use of adaptive Helmholtz resonators fornoise control applications in ducts and ventilation systems.Due to the similarities between adaptive vibration absorbers and Helmholtz res-onators, the tuning algorithms for both should be interchangeable. A review of pre-vious work pertaining to the design and control of adaptive vibration absorbers andHelmholtz resonators is described in detail in the following section.2.4.1 Previous Work in Adaptive-Passive Noise and Vibration ControlKoopman and Neise [KN80], [KN82] investigated the use of adjustable quarter-wavelength resonators to attenuate the blade passing frequency tone in centrifugalfans. Tuning of the resonators was achieved by changing the cavity length via amovable Te on piston. Although no suggestions were given concerning the imple-mentation of a controller to achieve the optimal tuning of the resonators, it wasreported that the use of the adjustable resonators provided reductions in the bladepassing frequency tones of up to 29 dB with no adverse e�ects on the fan performance.

8Little et al. [LKKM94] proposed a uidic Helmholtz resonator for use as anadaptive engine mount. Tuning of the resonator was to be achieved by changing theneck cross-sectional area, thereby varying the neck inertance of the device. A novelelectrorheological uid valve was suggested to provide continuous tuning, as opposedto the discrete nature of the valves commonly used. However, a control scheme fortuning the resonator was not provided.Walsh and Lamancusa [WL92] proposed the use of a variable sti�ness leaf springvibration absorber to attenuate vibrations caused by rotating machinery such aspumps and engines. The proposed tuning scheme was an open loop algorithm thatwould adjust the absorber sti�ness based on a theoretically determined optimal sti�-ness variation rate calculated from previous knowledge of the startup and steady stateoperating speeds of the machines. This type of scheme assumes no uncertainty in ei-ther the absorber model or the machine operating speeds, and is constrained to workonly for speci�c startup conditions. Additionally, the e�ects of changing environ-ments, wear and tear associated with aging, and signal noise on system performanceare not taken into consideration with this type of tuning algorithm. Another openloop control algorithm for vibration absorbers was reported by [Mia92].Lamancusa [Lam87] has suggested the use of variable volumeHelmholtz resonatorsas an alternative to expansion chamber mu�ers in automobiles. Two variable vol-ume resonator con�gurations were proposed. For the �rst design, the volume wascontinuously varied by displacing a piston inside the cavity. For the second con�g-uration, several discrete volumes were achieved by manipulating closeable partitionswithin the cavity. The suggested tuning algorithm consisted of varying the resonatorvolume according to an engine rpm signal. Although an experimental veri�cation ofthis open-loop control law was not provided, it was found that manual tuning of thecontinuously variable volume resonator could provide insertion losses of more than 30dB. Krause et al. [KWH92], [KWH93] have also experimented with a variable-volumeand variable-neck Helmholtz resonator to suppress noise in automotive tailpipes.

9On-line optimization, which consists of using a global search algorithm to deter-mine the optimal adaptive-passive device con�guration, has also been used to tuneadaptive vibration absorbers [HS94]. This type of control algorithm is wasteful andine�cient, since it sweeps the entire absorber natural frequency range to identify thecon�guration where vibration attenuation is a maximum. This method of tuninggives no consideration to the e�ects of sweeping the system through resonances, andwill perform poorly under conditions in which the excitation frequency is changing.Matsuhisa et al. [MS90], [MRS92] developed a resonator in which the volumewas changed by displacing a piston within the cavity. Tuning of the resonator, whichwas used as a sidebranch in a duct, was achieved by comparing the phase of thesound pressure in the duct with that in the resonator cavity. The resonator cavitywas adjusted such that the phase di�erence was 90 degrees, thereby achieving theantiresonance of the duct and resonator system. This implementation required threesensors: one microphone to measure the excitation frequency, one to measure thepressure in the duct, and another to measure the pressure in the cavity. This in-vestigation reported decreases in sound pressure levels of 30 dB for a speaker drivensystem, and 20 dB for a fan driven system.McDonald et al. [MHSC] have been granted an international patent on an adaptiveHelmholtz resonator for tonal noise control in duct systems that is tuned in a similarmanner to Matsuhisa's [MS90], [MRS92]. The adjustable resonator is adapted bysimultaneously tuning the cavity volume and neck length according to the phaserelationship between the pressure in the resonator cavity and the duct system.Ryan et al. [Rya94], [RFB94] proposed the use of a variable sti�ness helical springvibration absorber and a feedback tuning algorithm to minimize structural vibrations.The tuning algorithm adjusted the absorber sti�ness in an e�ort to minimize theamplitude from an accelerometer signal mounted at the location in which vibrationwas to be suppressed. The controller e�ort for this tuning scheme was proportionalto the amplitude of the accelerometer signal. A tuning direction was provided by thealgorithm as long as the di�erence between the estimated absorber natural frequency

10and the estimated excitation frequency was outside of a region contingent on theuncertainty of these estimated parameters. The algorithm will tune an adaptive-passive device which can completely suppress the amplitude of vibration or diminishit to within a prede�ned acceptable level. If a prede�ned acceptable level of vibrationcan't be set, however, the algorithm will produce chatter around the optimal absorbernatural frequency.2.5 SummaryAdaptive-passive systems o�er a technological alternative to strictly passive oractive noise control solutions. Compared to strictly active noise control schemes,adaptive-passive systems can provide a simple and cost e�ective solution for tonalnoise control applications. Additionally, adaptive-passive systems have the majoradvantage over passive systems of being able to adjust their passive parameters toalways deliver optimal performance in the presence of changing environments andtime-varying excitation frequencies.Although adaptive-passive systems have been used to attenuate tonal noise andvibrations, in most cases they have been actuated either mechanically or throughopen loop schemes, with little or no attention given to robust tuning strategies. Twotypical techniques for tuning adaptive-passive devices are open loop control and on-line optimization. Open loop control involves tuning the devices to the theoreticaloptimal con�guration, determined either through system models or a detailed systemcalibration. This approach lacks robustness, because of the uncertainty in both thesystem parameters and the excitation. On-line optimization, on the other hand, �ndsthe optimal system con�guration by implementing a global search routine to identifythe maximum performance. This method of tuning disregards the e�ects of sweepingthe system through resonances, and is ine�cient as far as required tuning times dueto the fact that the whole frequency range must be examined before a solution isachieved.

11Closed loop control implementations for tuning adaptive-passive devices have beendeveloped by Matsuhisa et al. [MS90], [MRS92] and Ryan et al. [RFB94], [Rya94].The tuning algorithm developed by Matsuhisa et al. adjusts the adaptive-passivedevice to where a phase di�erence between the excitation and the forced oscillation is90 degrees, indicating the antiresonance of the combined system. However, this tuningmethod utilizes more than one sound pressure sensor, which can be undesirable insystems where cost is a major concern. The tuning algorithm developed by Ryan et al.will tune an adaptive-passive system which can completely suppress the amplitudeof vibration or diminish it to within a prede�ned acceptable level. However, thealgorithm will produce tuning chatter if an acceptable level of vibration is not de�ned.To date, control algorithms for adaptive-passive devices have either lacked robust-ness in achieving the optimal device con�guration or have required costly sensors todo so. The main objective of this thesis is to develop a control algorithm that willtune an adaptive Helmholtz resonator to the optimal con�guration in which noise at-tenuation is maximized, despite large levels of uncertainty in the system models andthe excitation. An experimental veri�cation of the control algorithm will be providedto test the robustness of the algorithm to model uncertainty and changing excitationfrequencies.

123. THEORETICAL ANALYSIS OF CONVENTIONAL HELMHOLTZRESONATORSHelmholtz resonators are high performance reactive devices that suppress tonalacoustic excitations. The design of a conventional Helmholtz resonator is contingentupon the frequency of noise that is to be eliminated, and the available space for thedevice. A thorough treatment of the design of Helmholtz resonators was given byIngard [Ing53].3.1 Conventional Helmholtz Resonator ModelA Helmholtz resonator is an acoustic bandstop �lter comprised of a rigid cavitywith a protruding neck that connects the cavity to the system of interest. The behav-ior of a Helmholtz resonator is analogous to that of a vibration absorber (See Figure3.1). The volume of air in the neck of the Helmholtz resonator behaves much like avibration absorber mass and the volume of air in the cavity acts like a compliance(reciprocal of sti�ness). The excitation is provided by tonal pressure uctuationsacting over the opening of the neck, resulting in oscillations of the volume of air inthe neck. The pressure increase within the cavity provides a reacting force analogousto that of a spring. Damping appears in the form of radiation losses at the neck ends,and viscous losses due to friction of the oscillating air in the neck.It can be shown that the e�ective mass of a Helmholtz resonator is approximatelygiven by m = �SLeff ; (3.1)

13ω

ω

Figure 3.1 Helmholtz Resonator and Vibration Absorber

14where S is the cross sectional area of the neck, � is the density of the uid and Leffis the e�ective neck length, which includes a correction factor for mass-loading dueto air entrainment near the neck extremities [KFCS82]. For a neck that is anged onboth ends, the e�ective length is approximatelyLeff = L+ 1:7a; (3.2)where a is the radius of the neck, and L is the actual neck length [KFCS82].The sti�ness of the resonator is de�ned as the reciprocal of the compliance. Let� denote the displacement in the positive direction pointing inwards along the neckaxis (See Figure 3.1). The sti�ness � is de�ned asdFd� = �; (3.3)where F is the force applied in the +� direction at the resonator neck entrance. Theforce can also be written as F = PS; (3.4)where P is the pressure at the neck entrance.To develop the resonator sti�ness as a function of the resonator dimensions, con-sider the thermodynamic analysis of the resonator cavity. Assuming the system isadiabatic, and the air is an ideal gas with constant speci�c heats, the polytropicprocess equation for the resonator isPV = constant; (3.5)

15where is the ratio of speci�c heats, and V is the cavity volume of the Helmholtzresonator. Di�erentiating (3.5) givesV dP + P V �1dV = 0: (3.6)From (3.4), the di�erential change in pressure dP is given bydP = dFS : (3.7)The change in cavity volume, dV , isdV = �Sd�; (3.8)where the change in volume is negative since the air in the cavity is compressed.Substituting (3.7) and (3.8) into (3.6) givesV dFS � P V �1Sd� = 0; (3.9)or equivalently dFd� = P S2V = �: (3.10)Further simpli�cation of (3.10) can be achieved by substituting in the speed of soundand the ideal gas law, to give the form� = �c2S2V ; (3.11)

16where c is the speed of sound, and � is the density of the medium.Two sources of damping in the Helmholtz resonator were considered. The �rstsource of damping is due to sound radiation from the neck. This radiation resistanceis a function of the outside neck geometry. For a anged pipe, the radiation resistancefor free-�eld radiation has been theoretically determined to be [KFCS82]R = �ck2S22� ; (3.12)where k is the wave number, k = !c : (3.13)Although (3.12) was proposed for free-�eld radiation, and might already be accountedfor in the impedance of the sidebranch connecting ducts, it will be assumed here thatit can be applied to duct cases [Mun87].Another source of resonator damping are viscous losses in the neck. The mechan-ical resistance due to viscous losses was quanti�ed by Ingard [Ing53] asRvisc = 2SRs (L + a)�ca ; (3.14)where L is the neck length, a is the neck radius, and Rs for a su�ciently large neckdiameter, is Rs = 0:83 � 10�3 � !2�� 12 ; (3.15)where ! is the excitation frequency. In many cases, the viscous losses can be neglectedcompared with the radiation losses [NHD81].

17The mechanical impedance of a driven system is de�ned as the ratio of the drivingforce and the velocity of the system. The mechanical impedance of a driven mass-spring-damper system [KFCS82] isZmsdm = � + i�!� � s!� ; (3.16)where � denotes viscous damping, � is the mass element, and s is the sti�ness of thespring element. Using this form, the overall mechanical impedance of a Helmholtzresonator is Zm = Rvisc + �ck2S22� !+ i !�LeffS � �c2S2!V ! : (3.17)From (3.17), the natural frequency of a Helmholtz resonator, denoted as !o, is thefrequency for which the reactance is zero. Thus,!o = c SLeffV ! 12 : (3.18)3.2 Frequency Response of Duct Systems with Sidebranch Helmholtz ResonatorsThe impedance of a Helmholtz resonator, given by equation (3.17), can be used torepresent the behavior of the device in an acoustical system. In general, an acousticalsystem can be represented using electro-acoustic analogies. In the impedance anal-ogy, a sidebranch resonator can be modeled as an electrical element with a complexacoustic impedance given by Zres = ZmS2 : (3.19)

18Shown in Figure 3.2 [Pie94] is a simple open-ended duct system with a side-branch Helmholtz resonator and an analogous electrical circuit of the system near theHelmholtz resonator junction. In the impedance analogy, acoustic pressure is equiv-alent to voltage, and volume velocity is equivalent to current. The impedance Z(0+)is the impedance of the duct immediately after the junction with the Helmholtz res-onator. At resonance, the impedance of an undamped resonator is zero, and thereforethe Helmholtz resonator becomes a short to ground in the analogous electrical circuit.Consequently, no current ows through the branch to the right of the Helmholtz res-onator. By analogy, the undamped Helmholtz resonator at resonance causes a totalre ection of acoustic waves in the duct system back to the noise source. The e�ectsof damping in the Helmholtz resonator can be derived from the electrical analogy;some current will ow through the branch to the right of the Helmholtz resonator inthe analogous electrical circuit, thus indicating that total re ection of the acousticwaves is not achieved in the duct system.Figure 3.2 Simple Duct / Helmholtz resonator System and Electrical Circuit Analogyof System near Resonator JunctionConsider the duct system with the sidebranch resonator shown in Figure 3.2. As-sume the duct is comprised of 5 cm diameter tubing, and the length of the duct fromthe input to the Helmholtz resonator junction is 62 cm, while the length of the ductfrom the resonator junction to the output is 34 cm. Using the Helmholtz resonatorimpedance given by equation 3.19 and transfer matrices (which will be explainedin the following chapter), the output acoustic pressure Pout can be calculated as afunction of the input acoustic pressure Pin for any excitation frequency. Shown in

19Figure 3.3 is the frequency response of the duct system without the Helmholtz res-onator, and with two di�erent volume Helmholtz resonators with the same naturalfrequency. The excitation frequency axis is normalized with respect to the funda-mental frequency of the straight pipe system, which was also chosen as the naturalfrequency of the resonators. The maximum attenuation of sound pressure for ductsystems with sidebranch Helmholtz resonators occurs when the natural frequency ofthe resonator is equal to the excitation frequency. The narrow bandwidth in whichthe attenuation of sound pressure occurs is bounded by the resonances of the com-bined duct-resonator system. Helmholtz resonators are typically used to attenuatesingle frequency noise or noise with narrow bands, to avoid noise ampli�cation by theresonant peaks.It is shown in Figure 3.3 that as the Helmholtz resonator volume increases relativeto the duct volume, the distance between the two resonant peaks widens, thereforewidening the bandwidth of attenuation of the resonator. It is therefore bene�cial todesign the volume of a Helmholtz resonator to be as large as possible, while insuringthat none of its dimensions exceed a quarter wavelength of the natural frequency ofthe device to minimize the e�ects of standing waves in the apparatus. However, sizeconstraints are usually dictated by the available space for the application, in whichcase this space should be utilized as e�ciently as possible.Shown in Figure 3.4 are the frequency responses of the duct system with a lightlydamped and a heavily damped Helmholtz resonator. The lightly damped Helmholtzresonator is not robust with respect to changes in the excitation frequency, since thesound pressure in the duct system can be ampli�ed if the noise frequency shifts tothe vicinity of either of the two system resonances. To increase the robustness ofHelmholtz resonators with respect to changes in the excitation frequency, dampingis usually added to the resonators to decrease the magnitude of the resonant peaks.However, this increase in robustness comes at the expense of decreased levels of per-formance, since the maximum levels of attenuation are signi�cantly less for heavily

20

Straight Pipe Vres/Vpipe = 0.08Vres/Vpipe = 0.84

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Figure 3.3 E�ect of Resonator Volume on Attenuation Bandwidth

21damped Helmholtz resonators. The motivation for creating a tunable Helmholtz res-onator stems from this tradeo� between robustness and performance. A tunableHelmholtz resonator, capable of adjusting its natural frequency to match the noisefrequency, would be able to guarantee the high performance of a lightly dampedHelmholtz resonator despite changes in the noise frequency.Straight PipeHeavy DampingLight Damping

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Figure 3.4 E�ect of Resonator Damping on Attenuation Performance3.3 SummaryThe addition of an undamped Helmholtz resonator to a duct system causes thetotal re ection of acoustic waves at the resonator natural frequency back to the noisesource. Damping, resulting from radiation resistance and viscous losses in the neckdecreases the performance of Helmholtz resonators, reducing the magnitude of atten-uation. To increase the e�ective bandwidth of attenuation of a Helmholtz resonator,the device should be made as large as possible while insuring the dimensions do notexceed a quarter wavelength of the resonator natural frequency in order to minimizethe e�ects of standing waves within the device.

224. EXPERIMENTAL FACILITY MODEL DEVELOPMENTThe main objectives for developing the system models in this chapter were toquantify the behavior of the tunable Helmholtz resonator used in this investigation,and to aid in the development and implementation of the tuning algorithm. Thesystem models developed in this chapter were implemented in MATLAB, and can befound in Appendix B.4.1 Adaptive Helmholtz ResonatorThe tunable Helmholtz resonator used in this investigation is a variable volumedevice, which allows the natural frequency to be adjusted. The resonator dimensionswere chosen such that the tunable frequency range is 60 to 180 Hz. The largestdimension of the resonator was designed to be less than one quarter of the smallestwavelength, to minimize the in uence of higher frequency standing waves within thedevice.The variable volume actuation of the Helmholtz resonator is realized by rotatingan internal radial wall inside the resonator cavity with respect to an internal �xedwall (See Figure 4.1). The �xed wall is attached to the side of the cylinder and thetop end plate of the resonator. The movable wall is �xed to the bottom end plate,which is free to rotate against the bottom face of the cylinder. This bottom plate isattached to a DC motor which provides the motion to change the volume. The �xedand movable walls are hinged together at the center of the cylinder.The volume of the �nal design is bounded between 1,491 cm3 and 14,093 cm3.The radius of the cavity is 15 cm, and the cavity length is 24.6 cm. The thickness ofthe radial walls is 0.95 cm. The radius of the resonator neck is 2.5 cm, and the neck

23Figure 4.1 Variable Volume Helmholtz Resonatorlength is 8 cm. Recall that the e�ective neck length of the resonator is contingentupon conditions at the exit of the neck. Since the resonator for this investigation wasused as a side-branch for a duct system, it was assumed that both ends of the neckbehave as anged terminations. Under these conditions, the theoretical frequencyrange for the resonator is 59 through 180 Hz.To minimize the losses within the resonator, consideration was given to reduce theinteraction between the two cavities produced by the two walls. A exible membranewas attached to both walls along the hinge to seal the two cavities. Additionally, highvacuum stopcock grease was spread across the cylinder walls, to seal the small gapsbetween the wall edges and the cylinder.For the purposes of this investigation, it was assumed that for any given wallposition the tunable Helmholtz resonator behaved like a conventional Helmholtz res-onator with the same cavity volume and neck dimensions. Therefore, the acousticalimpedance of the tunable resonator is given by equation (3.19). This assumptiondisregards the e�ects of wall vibration and the interaction of the two cavities on thereactance of the resonator. Additional dissipative e�ects within the resonator due tothe resistance to wall vibration, which would a�ect the resistance of the resonatorimpedance, are also neglected with this assumption.

244.2 Duct and Noise Source ModelsTo understand the e�ects of the variable volume Helmholtz resonator on the ex-perimental system, models of the duct and the acoustic driver were developed. Thee�ects of the driver on the system frequency response will be additional resonancesdue to the driver dynamics. This type of modeling is necessary in the developmentalstage of the control algorithm to understand the di�erent control approaches that arepossible.Shown in Figure 4.2 is a schematic of the experimental setup. Tonal noise isgenerated from a 10.2 cm diameter loudspeaker, enclosed in PVC tubing. The 10.2cm diameter tubing is then reduced to a 5 cm diameter duct. The duct is comprised ofa 62 cm long section that connects the loudspeaker enclosure to the tunable Helmholtzresonator via a T coupling. Finally, a 34 cm long section of duct is extended fromthe T coupling to an open termination. A microphone located downstream from theresonator provides the sound pressure signal required by the controller to achieveoptimal tuning of the resonator.Figure 4.2 Experimental Setup

254.2.1 Loudspeaker ModelA loudspeaker was used to provide the tonal acoustic excitation. To determinethe sound pressure and volume velocity at the microphone location, the pressure andvolume velocity at the source must be known. These parameters were determinedthrough an analytical model of the speaker.A loudspeaker can be modeled through the impedance (mass-inductance) anal-ogy as an electrical-mechanical-acoustical system [Ber86]. Figure 4.3 illustrates theelectrical circuit analogous of the loudspeaker.+

_ Figure 4.3 Equivalent Loudspeaker CircuitThe input signal to this circuit, Vin, is in series with the loudspeaker coil resistance,Re, and the ampli�er resistance, Rg. An ideal gyrator is used to connect the electricaland mechanical subsystems. The gyrator has a ratio of B � l (product of the magnetic ux density and the coil wire length for the speaker).The mechanical system is comprised of a resistance, inductance, and capacitance inseries. The resistance, Rsp, represents the damping of the speaker rubber suspension.The inductance, Msp, represents the lumped mass of the diaphragm and voice coil.The capacitance, Csp, is equivalent to the compliance of the speaker suspension. Theinteraction between the mechanical and acoustical subsystems is represented by anideal transformer with a turns ratio of Sd, which is the surface area of the loudspeakerdiaphragm.

26The �nal subsystem is the acoustical system. It is comprised of a capacitance, aninductance and a complex impedance, Zsys, in series. The capacitance, Cbk, representsthe compliance of the posterior cavity of the speaker enclosure. The inductance,Mbk,represents the acoustic mass on the back of the speaker diaphragm [Ber86]. Thecomplex impedance, Zsys, is the equivalent acoustical impedance of the downstreamduct and resonator. Simplifying the circuit in Figure 4.3, by moving the circuitelements through the gyrator and transformer, gives the circuit shown in Figure4.4. Values for the loudspeaker model variables can be found in Table 4.1, with theexception of Rsp, which is frequency dependent. The equation for Rsp, as well as theimplementation of the loudspeaker models, can be found in Appendix B.+

_Figure 4.4 Simpli�ed Loudspeaker Equivalent Circuit4.2.2 Determination Of System ImpedanceThe pressure and volume velocity at the face of the speaker is a function of thesystem impedance. This impedance can be found provided the impedance at thetermination is known. Speci�c locations in the system will be described by the pointsindicated in Figure 4.2.The acoustic impedance for the system termination, which is an open ended,un anged pipe is approximately

27Re 8 Rg 50 B � l 4.88 WbmMsp 5.42 gCsp 1.4�10�3 mNSd 0.0057 m2Cbk 8.15�10�9 m5NMbk 3.44 kgm4Table 4.1 Loudspeaker Model Variable Values

28Z1 = �cSp ��14� (kap)2 + i(0:6kap)� ; (4.1)where � is the density of the medium (air), c is the speed of sound, Sp is the crosssectional area of the pipe, and ap is the radius of the pipe [KFCS82]. The impedanceat point 2 is [KFCS82] Z2� �cSp� = Z1� �cSp� + i tan(kL1)1 + i Z1� �cSp� tan(kL1) ; (4.2)where k is the wave number and L1 is the length of the pipe separating the terminationfrom the resonator.The resonator acoustic impedance, Zres, is given by equation (3.19). The volumevelocity at the location of the resonator is split into two parts; one part travels intothe resonator and the other part travels into the remaining duct (see Figure 3.2).Therefore, the acoustic impedance at point 3 is given by the parallel addition of theresonator impedance and the impedance Z2 [Pie94]. Thus the impedance at point 3is Z3 = Z2ZresZ2 + Zres : (4.3)The impedance at point 4 can be determined asZ4� �cSp� = Z3� �cSp� + i tan(kL2)1 + i Z3� �cSp� tan(kL2) ; (4.4)where L2 is the length of pipe between the resonator and the speaker enclosure aper-ture.Finally, the impedance at point 5, denoted Zsys, is given by

29Zsys� �cSenc� = Z4( �cSenc ) + i tan(kLenc)1 + i Z4( �cSenc ) tan(kLenc) ; (4.5)where Senc is the cross-sectional area of the speaker enclosure and Lenc is the distanceseparating the enclosure aperture and the speaker face.4.2.3 Calculation of Sound Pressure at a Desired LocationWith the system impedance (4.5), the pressure and volume velocity at the speakerface can be determined. The pressure and volume velocity at any location in the ductsystem downstream from the speaker may be computed from the pressure and volumevelocity at the speaker face via transfer matrices.The �rst transfer matrix is used to relate the pressure and volume velocity at apoint downstream in a straight pipe to the pressure and volume velocity at the originof the pipe [Mun87]. This transfer matrix is0@Pd(l)Ud(l)1A = 0@ cos(kl) �i � �cSp � sin(kl)�i sin(kl)� �cSp� cos(kl) 1A0@Pd(o)Ud(o)1A ; (4.6)where l is the length of the pipe and o denotes the origin of the pipe. The secondtransfer matrix relates the pressure and volume velocity immediately downstream ofa sidebranch, to the pressure and volume velocity immediately before the sidebranch[Mun87]. Particularly, the pressure and volume velocity downstream of a sidebranchresonator is 0@Pd(x+)Ud(x+)1A = 0@ 1 0� 1Zres 11A0@Pd(x�)Ud(x�)1A ; (4.7)where x is the location of the sidebranch resonator.These transfer matrices are used to �nd the pressure occurring in the system atthe location of the microphone (located slightly before the termination). First, (4.6)

30is used to �nd the pressure and volume velocity at point 4, given the pressure andvolume velocity at the speaker face. Then, (4.6) is used again to �nd the pressureand volume velocity at point 3, given the pressure and volume velocity at point 4.Equation (4.7) is used to �nd the pressure and volume velocity at point 2, given thepressure and volume velocity at point 3. Finally, (4.6) is used to �nd the pressureand volume velocity at the location of the microphone, given the pressure and volumevelocity at point 2.4.3 Analytical Model Results and ValidationThe system models developed in the previous section were used to create a threedimensional surface of the sound pressure level at the microphone location, versusexcitation frequency and Helmholtz resonator natural frequency (see Figure 4.5). Theinput voltage amplitude to the speaker was 2 Vrms. The prominent low sound pressurelevel region corresponds to the Helmholtz resonator natural frequency. Figure 4.6 isa top view of the surface shown in Figure 4.5. The low pressure region created by theHelmholtz resonator follows a 45 degree line between the resonator natural frequencyand the excitation frequency, indicating that the resonator is tuned when its naturalfrequency is equal to the excitation frequency.To quantify the accuracy of the system models, the sound pressure spectrumpredicted from the models was �rst compared to the measured sound pressure spec-trum of the experimental duct system with a conventional rigid walled Helmholtzresonator. The rigid walled resonator was designed to have the same neck dimensionsas the tunable Helmholtz resonator, with a cavity volume of 4575 cm3. To determinethe validity of the assumptions made for the tunable Helmholtz resonator model,these sound pressure spectra were then also compared to the measured sound pres-sure spectrum of the tunable Helmholtz resonator, adjusted to have the same cavityvolume as the rigid walled resonator. According to the modeling assumptions madepreviously, the measured sound pressure spectra of the rigid walled resonator and thetunable Helmholtz resonator should be identical for these conditions. To measure the

31

Figure 4.5 Helmholtz Resonator Sound Pressure Spectrum Surface

Figure 4.6 Top View of Helmholtz Resonator Sound Pressure Spectrum Surface

32sound pressure spectrum of the resonators for a �xed natural frequency, white noisewas generated with the speaker and the transfer function between the microphoneoutput and the speaker input was measured.Shown in Figure 4.7 is the comparison between the analytically predicted sys-tem sound pressure spectrum, and the measured sound pressure spectra for the rigidwalled resonator and the tunable resonator. It is shown in Figure 4.7 that the modelsaccurately predict the behavior of the rigid walled resonator. However, the tunableHelmholtz resonator sound pressure spectrum varies signi�cantly from the analyticalmodel. Speci�cally, neither the Helmholtz resonator natural frequency nor the atten-uation performance are accurately predicted, suggesting that the assumptions madefor modeling the tunable Helmholtz resonator were violated.Additional comparisons between analytically predicted sound pressure spectra andmeasured sound pressure spectra for several resonator geometries were performed tofurther quantify the accuracy of the analytical models. Shown in Figures 4.8, 4.9 and4.10 are the comparisons between the theoretical and experimental sound pressurespectra for measured resonator natural frequencies of 179, 124 and 59 Hz.The models predict the resonator natural frequency within 25%. The largest dis-crepancy between predicted and measured natural frequency occurs at 124 Hz. Themagnitude of this discrepancy is 25 Hz. The reactive component of the Helmholtzresonator impedance is roughly captured by the models. The reactive component ofthe resonator impedance is potentially a�ected by the vibration of the cavity walls,and the interaction of the resonator cavity with the second cavity created by the twowalls. This is evident in Figure 4.9, where the discrepancy between the predicted andmeasured sound pressure spectra indicates the possibility of an additional antireso-nance due to either the e�ects of the secondary cavity, the vibration of the movablewalls, or both.The magnitude of the attenuation of the measured sound pressure spectra was notadequately predicted by the system models. In �gures 4.8, 4.9, and 4.10 it is shownthat the Helmholtz resonator is more damped than predicted. This is an indication

33Analytically PredictedTunable Resonator Rigid Walled Resonator

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34TheoreticalMeasured

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35that the resistive component of the impedance should have been larger to accountfor losses in the resonator due to leakage from the cavity seals, damping from thevibrating walls, and other sources. It was observed that tightening the resonatorseals decreases the damping in the device. The di�erence between the theoretical andmeasured level of damping is a nonlinear function of the resonator natural frequency,and thus of the movable wall position. This is attributable to fabrication toleranceson the inside surface of the resonator cavity.The observed discrepancies between the actual and predicted resonator perfor-mance illustrate the complexity of developing accurate models. The accuracy of themodels could be substantially improved by taking into account the e�ects of wallvibration, cavity interaction and additional losses within the resonator cavity dueto seal leakage and other sources. However, these models would still be susceptibleto environmental conditions, since the resonator natural frequency is dependent onthe speed of sound. Therefore, there would always be a certain degree of inaccuracyinvolved with analytical models. The use of such models as the sole means of tuningan adjustable resonator should therefore be discouraged.The models for the rest of the system show good general agreement with measureddata at mid to high frequencies. At low frequencies, the models have less dampingthan the actual system, as shown in Figures 4.9 and 4.10. However, the trendsobserved at these lower frequencies are correct and the discrepancies in the magnitudedon't a�ect the analysis of the resonator performance.4.4 Summary of System ModelingThe system models roughly captured the behavior of the experimental tunableHelmholtz resonator. The reactive component of the tunable resonator impedance ispotentially a�ected by the interaction of the main cavity of the resonator with thesecondary cavity, and the vibration of the movable walls within the cavity. Thesee�ects on the resonator reactance are responsible for the poor estimation of the res-onator natural frequency. The resistive component of the resonator impedance should

36have been larger to account for the e�ects of losses within the cavity due to seal leak-age and damping from the vibrating walls. The e�ect of these additional losses is todecrease the maximum attenuation performance of the Helmholtz resonator.

375. ROBUST TUNING CONTROL LAW DEVELOPMENTThe main objective of the tuning algorithm for an adaptive-passive device is toachieve and maintain the maximum attenuation of tonal noise in the system of in-terest under changing environments and excitation frequencies. To date, controlalgorithms for tuning adaptive-passive devices have either lacked robustness to modeluncertainty, or have required the use of several sensors to achieve the optimal con�gu-ration. Previous tuning schemes for maximizing the performance of adaptive-passivedevices have been presented in Chapter 2.5.1 Tuning Control LawThe resonator tuning algorithm developed in this investigation consists of twotuning strategies: an open loop gross tuning scheme and a feedback gradient basedprecise tuning control law. Collectively, these individual strategies guarantee therobust attenuation performance of the passive device despite variations in the noisesource frequency and environmental changes.To justify this proposed tuning control law approach, consider the sound pressurelevel at the microphone location in Figure 5.2 as a function of the resonator naturalfrequency (see Figure 5.1). The noise source for this �gure is an 80 Hz tone. Afeedback based tuning algorithm could utilize the slope of the sound pressure versusnatural frequency curve to obtain a tuning direction for the resonator, and thusdescend along the gradient of the curve until the minimumsound pressure is achieved.For the frequency range of the resonator, there are two minima of sound pressure level.The �rst minimum occurs at the resonator natural frequency of 80 Hz. This is theglobal minimum and the frequency to which the adaptive Helmholtz resonator should

38be tuned. The second minimum occurs above 180 Hz. This minimum is a localminimum for the sound pressure level. A tuning strategy strictly based on a gradientdescent approach would not have a mechanism to distinguish the di�erence betweenthese two minimums and hence performance optimization would not be guaranteed.However, if an open loop tuning control law is used to �rst preset the resonatornatural frequency so that the gradient based approach is guaranteed to converge tothe global minimum, then optimal performance can be achieved. This is the tuningcontrol law developed.50 100 150 200

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Figure 5.1 Sound Pressure Level vs. Natural Frequency for Constant ExcitationFrequency of 80 Hz.5.1.1 Experimental SetupShown in Figure 5.2 is the experimental facility and controller instrumentation.The microphone feedback sensor is a Br�uel and Kj�r type 4130 1/2 inch condensermicrophone. The microphone voltage is processed through a 4-pole Butterworth low-pass �lter with a 190 Hz cuto� frequency to remove high frequency noise. This �lteredsignal is used directly by the controller to determine the sound pressure magnitude.

39The wall position, which is measured by a potentiometer attached to the back plateof the tunable Helmholtz resonator, is also input to the controller.Figure 5.2 Experimental Setup and Controller InstrumentationThe �nal input to the controller is a DC signal proportional to the excitationfrequency. This signal is generated with a frequency to voltage converter (FTV)and some additional signal conditioning circuitry (see Figure 5.3). The �ltered ACsignal from the microphone is �rst sent through a high-pass �lter with a 0.3 Hz cuto�frequency to remove any DC o�set. The signal is then passed through a high-gainampli�er which clips the waveform, creating a constant amplitude square wave ofthe same frequency as the waveform from the microphone. The magnitude of thesquare wave is reduced by half, to satisfy the input voltage range of the frequencyto voltage converter. This square wave is then sent through a low-pass �lter with acuto� frequency of 190 Hz to smooth the waveform before entering the frequency tovoltage converter. The frequency to voltage converter produces a DC voltage thatis proportional to the frequency of the smoothed square wave, and thus proportionalto the frequency of the �ltered microphone signal. The output of the frequency tovoltage converter is sent through a low-pass �lter with a 30 Hz cuto� frequency to

40remove the ripple voltage superimposed on the DC signal. This signal is then sent tothe controller.Figure 5.3 Conditioning Circuitry for FTV Excitation Frequency DeterminationSince the focus of this investigation was on using the adaptive Helmholtz resonatorto attenuate tonal noise, no consideration was given to the e�ect of additional simulta-neous tonal frequencies on the frequency determination scheme. This is a valid courseof action, since Helmholtz resonators should only be applied to single frequency noiseattenuation. Additional tonal noise components simultaneously occurring with thenoise tone of interest will not be �ltered out by the high-pass or low-pass �lters ifthey are within the operating frequency band for the resonator, and will thereforecause problems for the frequency to voltage converter. The e�ect of multiple tonalfrequency inputs on the output of the frequency to voltage converter depends on theamplitude and frequency of the additional components.Other frequency determination schemes could be used in this proposed tuningcontrol law. For example, the frequency could be measured by counting the numberof zero-crossings of the microphone signal for a �xed time duration [Rya94], [RFB94].Calculating the excitation frequency based on source characteristics may also be pos-sible for applications where the noise source has measurable parameters such as rpm.The controller is realized using a Compaq Prolinea 486-66 tower computer andKeithley Metrabyte data acquisition boards. The sampling rate used by the controller

41is 8600 Hz. The controller output is sent to a current ampli�er and then to the DCmotor used to tune the resonator.5.1.2 Open Loop Tuning AlgorithmThe objective of the open loop tuning algorithm is to preset the resonator naturalfrequency within the vicinity of the excitation frequency to insure the convergence ofthe closed loop gradient descent algorithm. To accomplish this, the frequency of thenoise source is estimated from the frequency to voltage converter signal. The openloop controller then drives the DC motor to rotate the interior wall to a position inwhich the theoretical resonator natural frequency is close to the estimated excitationfrequency.The natural frequency of the resonator as a function of the interior wall positioncan be derived from equation (3.18). The relationship between the resonator naturalfrequency and the angle between the two cavity walls is�wall = 360�Rcav2Lcav 0B@ SLeff�2�foc �2 + Vwall1CA ; (5.1)where �wall is the interior wall position in degrees, fo is the desired resonator naturalfrequency in Hz, Rcav is the resonator cavity radius in cm, Lcav is the resonator cavitylength in cm, and Vwall is the volume in cm3 subtracted from the resonator cavityvolume due to the thickness of the walls.Shown in Figure 5.4 is a comparison between equation (5.1) and measured discretewall positions as a function of resonator natural frequency. Shown in Figure 5.5 isthe di�erence between the measured discrete natural frequencies and those used inequation (5.1). As shown in Figure 5.5, the largest discrepancy between the predictedand measured natural frequency is about 25 Hz, at a natural frequency of 124 Hz. Theerror incurred in predicting the Helmholtz resonator natural frequency with equation(5.1) is the result of the simpli�ed models used to represent the Helmholtz resonator.

42PredictedMeasured

50 100 150 2000

50

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250

300

Natural Frequency (Hz)

Mov

able

Wal

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degr

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Figure 5.4 Theoretical and MeasuredWall Angles as a Function of Natural Frequency0 50 100 150 200 250 300

-5

0

5

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20

25

Movable Wall Angle (degrees)

Nat

ural

Fre

quen

cy D

iffer

ence

(Hz)

Figure 5.5 Di�erence Between Predicted and Measured Natural Frequencies as aFunction of Wall Angle

43In general, other approaches to developing a relationship between the tunablecharacteristic of the resonator and the resulting natural frequency can be used asappropriate. For example, a curve �t relating the natural frequency to wall anglecan be generated experimentally. The only criterion to be satis�ed in developing arelationship between the tunable characteristic of the resonator and the resonatornatural frequency is that the calculated natural frequency must be su�ciently closeto the real natural frequency to ensure convergence of the gradient based feedbackcontroller to the global optimum.The transfer between the open loop gross tuning algorithm and the closed loopprecise tuning algorithm occurs whenjf̂o � f̂noisej � �t; (5.2)where f̂o is the predicted resonator natural frequency, f̂noise is the estimated excitationfrequency, and �t denotes a speci�ed tolerance contingent upon model accuracy.To motivate the selection of �t, consider the plot of the Helmholtz resonator andduct system resonance and antiresonance, shown in Figure 5.6. Shown in this plotis the theoretical system resonance and antiresonance in the Helmholtz resonatoroperating range (the same information is shown in Figure 4.6). Also shown arethe actual measured resonance and antiresonance points from the measured transferfunctions. Notice that the resonant ridge exists only between approximately 70 and100 Hz. The minimumseparation between the measured resonance and the theoreticalantiresonance is 6.3 Hz, which occurs at an excitation frequency of 71 Hz. Therefore,if the open loop algorithm is to guarantee convergence of the closed loop gradientbased algorithm, the open loop algorithm must place the resonator natural frequencywithin less than 6.3 Hz of the theoretical natural frequency. The �t value used in thisinvestigation is 2 Hz, to give the controller a margin of safety while accomplishingthis task.

44

Measured Resonance Measured Antiresonance Theoretical Res., Antires.

60 80 100 120 140 160 18060

80

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160

180

Theoretical Excitation Frequency (Hz)

Theo

retic

al N

atur

al F

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(H

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Antiresonance

Upper Resonance

Figure 5.6 Plot of the System Resonance and Antiresonance in the Resonator Oper-ating Range

455.1.3 Gradient Based Feedback Tuning Control LawThe objective of the gradient based tuning control law is to �nd maximum res-onator performance. The open loop tuning algorithm presented in the previous sectionis used to achieve coarse tuning. Environmental changes, variations in the excitation,and the inability to precisely identify the resonator natural frequency through acous-tic models limits the performance potential of an open loop strategy. Therefore, thegradient based tuning control law is employed for precise tuning.The gradient based tuning control law operates on the change in microphone volt-age amplitude due to a change in the interior wall position. The algorithm commencesby making 1 Hz incremental changes in the resonator natural frequency. The result-ing change in the amplitude of the microphone voltage is measured. The gradient iscalculated as r = Mi �Mi�1f̂oi � f̂oi�1 (5.3)where M is the microphone voltage amplitude. When there is a sign change in thegradient, the controller reverses tuning direction and changes the increment quantityin the resonator natural frequency to 0.7 Hz. Each time the minimum is passed,the controller reverses direction, and every time the slope changes from negative topositive, the increment quantity is decreased by 0.3 Hz. The minimum is passed inthis manner a total of six times with a �nal increment quantity of 0.1 Hz. Oncethe �nal sign change is detected, the gradient based tuning is complete. This avoidscontrol chattering of the resonator.The gradient descent algorithm only utilizes the gradient of the sound pressureversus natural frequency curve to obtain a tuning direction for the Helmholtz res-onator. The speed of tuning is held constant throughout the tuning process. On theother hand, the speed of tuning could be made inversely proportional to the gradient

46of the sound pressure versus natural frequency curve. This would provide faster tun-ing when the algorithm is far away from the optimal natural frequency, and slowertuning when the algorithm is near the optimal natural frequency.Once the gradient based tuning is complete, the controller continuously monitorsthe amplitude of the microphone voltage and the excitation frequency to determinewhen retuning is needed. Shown in Figure 5.7 is the ow chart for this tuning algo-rithm. The tuning algorithm code can be found in Appendix A.5.2 Summary of Robust Tuning Control LawMany schemes have been developed to tune adaptive-passive devices for operationunder changing environments and excitations. To date, these strategies have eitherlacked the robustness to system and model uncertainty that is necessary to guaranteeoptimal performance under varying operating conditions, or have required the use ofseveral sensors to achieve the optimal con�guration. The control law presented inthis chapter utilizes a combination of open loop control gross tuning and closed loopcontrol precise tuning to achieve optimal performance of the resonator with only onemicrophone. The open loop strategy is used to preset the resonator natural frequencywithin the vicinity of the excitation frequency. This guarantees the convergence ofthe closed loop gradient descent algorithm to the optimal resonator con�guration.

47Measure Excitation Frequency

Measure resonator natural frequency, Fn.

Move natural frequency to excitation frequency - δT Hz.

I=0

Read Microphone Voltage, Mikeold, and natural frequency, Fnold.

Increase natural frequency, Fn, by Increment.

Read Microphone Voltage, Mike, and natural frequency, Fn.

Is Delta Mike / Delta Fn > 0?

Delta Mike = Mike - Mikeold ; Delta Fn = Fn - Fnold.

Decrease Fn by Increment.

Mikeold = Mike, Fnold = Fn

I=I+1

Is I >2?

Set Increment Quantity, Increment, to 1.

Increment = Increment - 0.3

Is Delta Mike / Delta Fn < 0?

Monitor Excitation Frequency and Microphone Amplitude.

Read Microphone Voltage, Mike, and natural frequency, Fn.

Delta Mike = Mike - Mikeold ; Delta Fn = Fn - Fnold.

Is there a change in excitation or output amplitude?

Mikeold = MikeFnold = Fn

No

Yes

Mikeold = MikeFnold = Fn

No

Yes

Yes

Mikeold = MikeFnold = Fn

No

No YesFigure 5.7 Logic Flow Chart of Control Algorithm

486. RESULTSSteady state and transient response tests were performed on the controlled systemto evaluate the robustness of the proposed tuning algorithm. The resonator naturalfrequency range was restricted in software to be between 65 and 160 Hz to avoidpossible damage to the resonator wall hinge at the extreme locations.6.1 Steady State Response of Controlled SystemThe steady state performance of the controlled system was tested by measuringthe steady state microphone voltage amplitudes for discrete excitation frequenciesover the operating frequency band of the tunable Helmholtz resonator. This test wasused to evaluate the robustness of the tunable Helmholtz resonator over its naturalfrequency range. A comparison of the noise in the duct system with and without thetunable resonator is shown in Figure 6.1. The maximum noise attenuation is 30 dBand occurs at 160 Hz. The minimum noise attenuation is 15 dB and occurs at 80Hz. Shown in Figure 6.2 is a comparison between the noise control realized with theproposed tuning algorithm and the sound pressure spectrum of the acoustic systemfor several preset resonator natural frequencies. The tuning control algorithm ise�ective in adjusting the cavity volume in order to achieve the optimal noise reductionregardless of the excitation frequency.6.2 Transient Response of Controlled SystemTransient testing was used to verify the robustness of the tuning algorithm to tun-ing direction and performance optimization for a time varying excitation frequency.The �rst transient test involved an instantaneous change in excitation frequency from

49With Resonator Without Resonator

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Excitation Frequency (Hz)

Soun

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evel

(dB

)

Figure 6.1 Comparison of Treated and Untreated System Sound Pressure SpectraVariable Resonator Fixed Resonator Volumes

40 60 80 100 120 140 160 180 20090

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Excitation Frequency (Hz)

Soun

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)

Figure 6.2 Comparison of Tuned Performance with Discrete Natural Frequencies

5065 Hz to 160 Hz. These two excitation frequencies are near the extremes of the fre-quency range of the Helmholtz resonator. The change in the excitation frequencywas performed 5 seconds after the beginning of the test. The transient response ofthe sound pressure level is shown in Figure 6.3 for both the proposed tuning schemeand a strictly open loop scheme. Shown in Figure 6.4 is the Helmholtz resonatorwall angle measured by the potentiometer for these transient tests. The gradientbased tuning achieved an additional 1 dB reduction in the sound pressure level overthe strictly open loop scheme. This slight improvement is expected, since the res-onator natural frequency is accurately predicted by equation (5.1) (see Figure 5.4).However, additional performance improvement would be achieved if the temperature,for example, were to change thereby changing the speed of sound. In such a casethe gradient based tuning strategy would be more in uential. The di�erent startingsound pressure levels for the gradient based scheme and the strictly open loop schemeare the steady state sound pressure levels for each of these schemes for an excitationfrequency of 65 Hz.The observation to be made for this transient test is the robustness of the tuningalgorithm, not the time duration required for tuning. This time duration is contingentupon the test hardware, which was not optimized for minimal tuning times. Thesound pressure peaks were generated by wall motion during the natural frequencyadjustments of the gradient based tuning scheme.An additional transient test was performed for an instantaneous excitation fre-quency change from 65 Hz to 100 Hz. The transient response of the sound pressurelevel is shown in Figure 6.5, and the resonator wall angle measured by the potentiome-ter is shown in Figure 6.6. The change in the excitation frequency was performed5 seconds after the beginning of the test. The excitation frequency of 100 Hz waschosen since the open loop control scheme does not accurately predict the requiredwall position as shown in Figure 5.4. A 5.5 dB improvement using the gradient basedtuning algorithm was achieved over the strictly open loop approach. In a similar man-ner, the e�ects of variations in air temperature and other environmental conditions

51Open Loop and Gradient DescentStrictly Open Loop

0 10 20 30 40 50 60 70 80 90

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Figure 6.3 Transient Response of Controlled System to Excitation Change from 65to 160 HzOpen Loop and Gradient DescentStrictly Open Loop

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Wal

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degr

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Figure 6.4 Helmholtz Resonator Wall Angle for Excitation Change from 65 to 160Hz

52are minimized with the gradient based approach. Once again, the di�erent startingsound pressure levels for the gradient based scheme and the strictly open loop schemeare the steady state sound pressure levels for each of these schemes for an excitationfrequency of 65 Hz.Open Loop and Gradient DescentStrictly Open Loop

0 20 40 60 80 100 12090

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Soun

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Figure 6.5 Transient Response of Controlled System to Excitation Change from 65to 100 HzShown in Figure 6.7 is the transient response of the controlled system to twodi�erent instantaneous excitation frequency changes. Shown in Figure 6.8 is theresonator wall angle for this transient test. The system was originally at steady statefor an excitation frequency of 160 Hz. Five seconds after the beginning of the test, theexcitation frequency was suddenly dropped to 100 Hz, and the system was allowed toreach steady state. Then, 100 seconds after the beginning of the test, the excitationfrequency was instantaneously dropped to 65 Hz, and the system was once againallowed to settle.The transient response of the controlled system to external disturbances was alsotested. Shown in Figure 6.9 is the transient response of the system to a signal magni-tude disturbance. Shown in Figure 6.10 is the resonator wall angle for this transienttest. The system was originally at steady state for an excitation frequency of 90 Hz.

53Open Loop and Gradient DescentStrictly Open Loop

0 20 40 60 80 100 12050

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Wal

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Figure 6.6 Helmholtz Resonator Wall Angle for Excitation Change from 65 to 100Hz0 20 40 60 80 100 120 140 160 180

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Time (seconds)

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Figure 6.7 Transient Response of Controlled System to Two Excitation FrequencyChanges

54

0 20 40 60 80 100 120 140 160 1800

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Wal

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Figure 6.8 Helmholtz Resonator Wall Angle for Transient Test with Two ExcitationFrequency Changes

55Two seconds after the beginning of the test, a cap was momentarily placed over theopen ended termination of the duct, and removed approximately 4 seconds later. Thetwo spikes in the sound pressure level that are visible between two and six secondsafter the beginning of the test correspond to the placing and removing of the rigid capon the duct termination. The cap produced an ampli�cation of the sound pressurein the duct, but did not a�ect the frequency of the noise. The controller sensed thechange in the steady state conditions, and retuned the resonator back to the originalwall position, since the excitation frequency did not change.0 20 40 60 80 100 120

96

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Figure 6.9 Transient Response of Controlled System to Noise Magnitude Disturbance6.3 Summary of Experimental ResultsThe steady state experiments performed with the tunable Helmholtz resonatorverify the ability of the controller to achieve the maximum sound pressure attenuationwith the adaptive device. The controller is e�ective at adjusting the resonator cavityvolume to provide optimal performance of the resonator over its entire operatingregion. Sound pressure level reductions of up to 30 dB were observed when comparingthe controlled system to an untreated duct of the same length.

56

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Time (seconds)

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Figure 6.10 Helmholtz Resonator Wall Angle for Noise Magnitude Disturbance Test

57The transient response tests performed on the tunable Helmholtz resonator verifythe robustness of the tuning algorithm in selecting the appropriate tuning directionto attenuate tonal noise. From these tests, the bene�ts of closed loop control can bestbe appreciated in instances when system models are inaccurate. In these instances,the sound pressure has been shown to approach the minimumpossible sound pressurelevel. For cases when the system models are accurate, the closed loop algorithm stillperforms better than a similar open loop algorithm, but this improved performanceis not as drastic as the case when the models are poor. The tuning times were limitedprimarily by the speed of the DC motor that provides the cavity volume actuation.

587. CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORKThe major contributions and conclusions of this investigation are presented inthis chapter. The emphasis of the chapter will be directed toward the robustness ofthe tuning strategy used to achieve optimal performance of the tunable Helmholtzresonator. Suggestions for future work related to this investigation are also provided.7.1 Thesis ContributionsA robust tuning algorithm that guarantees optimal performance of an adaptiveHelmholtz resonator under changing environmental conditions and excitation frequen-cies was developed in this thesis. The tuning strategy ensures optimal tuning of theresonator despite nonmeasurable system uncertainty and errors in the lumped systemmodels. This tuning scheme utilizes only one sound pressure sensor and one positiontransducer to achieve the optimal con�guration.7.2 ConclusionsTypically, adaptive-passive devices have been tuned with open loop algorithmsthat neglect the e�ects of changing environments, system wear, and model inaccu-racies on system performance. Open loop schemes are not well suited for robustperformance guarantees. These limitations motivated the proposed tuning algorithmdeveloped in this thesis, which achieves the optimal system con�guration with aclosed loop gradient descent scheme. Therefore, this algorithm guarantees optimalperformance despite changing environments and model uncertainty.

59Several closed loop algorithms have been previously developed to tune adaptiveHelmholtz resonators and their mechanical counterparts, the tunable vibration ab-sorbers. Usually, these algorithms utilize two or more vibration or sound pressuresensors to identify the antiresonance of the system [MS90], [MRS92], [MHSC]. Incontrast, the proposed tuning algorithm requires one sound pressure sensor, whichresults in an economic advantage over other schemes.Although a single sensor robust tuning scheme has been developed by Ryan et al.for tunable vibration absorbers [Rya94], [RFB94], this tuning scheme is not applicableto the Helmholtz resonator in this investigation. Speci�cally, the large resonatordamping over the natural frequency range prevents complete noise attenuation. Inthis situation, the structure of Ryan's control algorithm would lead to chattering ofthe resonator about the optimal natural frequency.Experimentally, the proposed control algorithm was demonstrated to be e�ectivein tuning the Helmholtz resonator to achieve maximum sound pressure attenuation.The combination of the open loop gross tuning scheme with the closed loop precisetuning strategy successfully achieved system convergence to the minimumsound pres-sure level over the operating range of the resonator. The controller was demonstratedto be robust with respect to large changes in the excitation frequency and externalsystem disturbances.7.3 Suggestions for Future WorkThe e�ects of the secondary cavity and the vibrating radial walls on the Helmholtzresonator impedance should be analyzed in detail. A thorough investigation of thesee�ects would illuminate some of the model discrepancies observed and would be help-ful in �nding a solution to designing a high performance adjustable resonator.The development of a hybrid, variable volume active Helmholtz resonator is thenext logical step from this work. Such a device would be able to provide a much largertonal excitation attenuation than the variable volume Helmholtz resonator used in

60this investigation. The addition of a loudspeaker in the resonator cavity of the presentdesign would be a feasible implementation of such a device.

LIST OF REFERENCES

61LIST OF REFERENCES[ABA93] J. S. Anderson and M. Bratos-Anderson. Noise; Its Measurement, Anal-ysis, Rating and Control. Ashgate Publishing Company, Brook�eld, VM,1993.[ABEG88] M. C. Allie, C. D. Bremigan, L. J. Eriksson, and R. A. Greiner. Hard-ware and software considerations for active noise control. Proceedingsof the IEEE International Conference on Acoustics, Speech, and SignalProcessing, 5:2598{2601, 1988.[Ber86] Leo L. Beranek. Acoustics. American Institute of Physics, Inc., NewYork, Ny, 1986.[BGH91] Guy Billoud, Marie Annick Galland, and Can Huynh Huu. Adaptiveactive control of instabilities. Journal of Intelligent Material Systemsand Structures, 2(4):457{471, October, 1991.[BHJ92] Robert J. Bernhard, Henry R. Hall, and James D. Jones. Adaptive-passive noise control. Proceedings of Inter-Noise 92, pages 427{430, 1992.[HS94] J. J. Hollkamp and T. F. Starchville. A self-tuning piezo-electric vibra-tion absorber, Paper No. AIAA-94-1790. Proceedings of the 35th AIAAStructures, Structural Dynamics, and Materials Conference, 1994.[Ing53] Uno Ingard. On the theory and design of acoustic resonators. The Journalof The Acoustical Society of America, 25(6):1037{1061, 1953.[Ing94] Uno Ingard. Notes on Sound Absorption Technology. Noise Control Foun-dation, Poughkeepsie, NY, 1994.[KFCS82] L. Kinsler, A. Frey, A. Coppens, and J. Sanders. Fundamentals of Acous-tics. John Wiley and Sons, New York, NY, third edition, 1982.[KM96] Sen M. Kuo and Dennis R. Morgan. Adaptive Noise Control Systems -Algorithms and DSP Implementations. John Wiley and Sons, New York,NY, 1996.[KN80] G. Koopmann and W. Neise. Reduction of centrifugal fan noise by use ofresonators. Journal of Sound and Vibration, 73(2):297{308, 1980.

62[KN82] G. Koopmann and W. Neise. The use of resonators to silence centrifugalblowers. Journal of Sound and Vibration, 82(1):17{27, 1982.[KWH92] P. Krause, H. Weltens, and S. Hutchins. Advanced design of automo-tive exhaust silencer systems paper 922088. SAE Technical Paper Series,pages 1{11, 1992.[KWH93] P. Krause, H. Weltens, and S. M. Hutchins. Advanced exhaust silencing.Automotive Engineering, pages 13{16, February, 1993.[Lam87] J. S. Lamancusa. An actively tuned, passive mu�er system for enginesilencing. Proceedings of Noise-Con 87, pages 313{318, 1987.[LKKM94] Eric Little, A. Reza Kashani, J. Kohler, and F. Morrison. Tuning of anelectrorheological uid-based intelligent helmholtz resonator as appliedto hydraulic engine mounts. Proceedings of ASME DSC TransportationSystems, 54:43{51, 1994.[MHSC] A. M. McDonald, S. M. Hutchins, J. Strothers, and P. J. Crowther.Method and apparatus for attenuating acoustic vibrations in a medium.International Patent Number W092/15088.[Mia92] L. A. Mianzo. An adaptable vibration absorber to minimize steady stateand start-up transient vibrations-an analytical and experimental study.Master's thesis, The Pennsylvania State University, 1992.[MRS92] Hiroshi Matsuhisa, Baosheng Ren, and Susumu Sato. Semiactive con-trol of duct noise by a volume-variable resonator. JSME InternationalJournal, 35(2):223{228, 1992.[MS90] Hiroshi Matsuhisa and Susumu Sato. Semi-active noise control by a res-onator with variable parameters. Proceedings of Inter-Noise 90, pages1305{1308, 1990.[MTS93] Hiroshi Matsuhisa, Ikuo Tsujimoto, and Susumu Sato. Control of an-tiresonance by a variable damping active resonator. Transactions of theJapan Society of Mechanical Engineers, Part C, 59(562):1824{1829, June1993.[Mun87] M. L. Munjal. Acoustics of Ducts and Mu�ers with Application to Ex-haust and Ventilation System Design. John Wiley and Sons, New York,NY., 1987.[NHD81] P. A. Nelson, N. A. Halliwell, and P. E. Doak. Fluid dynamics of a owexcited resonance, part 1: Experiment. Journal of Sound and Vibration,78(1):15{38, 1981.

63[OBA94] Yasushi Okamoto, Hans Boden, and Mats Abom. Active noise controlin ducts via side-branch resonators. Journal of the Acoustical Society ofAmerica, 96(3):1533{1538, September 1994.[Pie94] Allan D. Pierce. Acoustics: An Introduction to Its Physical Principlesand Applications. Acoustical Society of America, New York, NY, 1994.[RFB94] Matthew W. Ryan, Matthew A. Franchek, and Robert J. Bernhard.Adaptive-passive vibration control of single frequency excitations appliedto noise control. Proceedings of Noise Con 94, pages 461{466, May 1994.[Rya94] Matthew W. Ryan. Adaptive-passive vibration control. Master's thesis,Purdue University, December 1994.[vFBB94] Andreas H. von Flotow, Andrew Beard, and Don Bailey. Adaptive tunedvibration absorbers: Tuning laws, tracking agility, sizing and physicalimplementations. Proceedings of Noise-Con 94, pages 437{454, May 1994.[WL92] P. L. Walsh and J. S. Lamancusa. A variable sti�ness vibration absorberfor minimization of transient vibrations. Journal of Sound and Vibration,158(2):195{211, 1992.

APPENDICES

64Appendix A: QuickBASIC 4.50 Adaptive-Passive Controller Algorithm10 REM ADAPTIVE-PASSIVE NOISE CONTROL20 REM30 REM PRINCIPAL INVESTIGATORS: MATTHEW A. FRANCHEK40 REM ROBERT J. BERNHARD50 REM LUC MONGEAU60 REM70 REM RESEARCH ASSISTANT: JUAN M. DE BEDOUT80 REM90 REM THIS CONTROLLER VERSION FOR THE THE TUNABLE HELMHOLTZ RESONATOR100 REM WILL FIND THE MINIMUM MICROPHONE VOLTAGE MAGNITUDE BY MOVING THE110 REM RESONATOR WALL IN THE REQUIRED DIRECTION. THIS CHANGES THE DEVICE'S120 REM COMPLIANCE AND THEREFORE ITS NATURAL FREQUENCY. BY TAKING THE130 REM GRADIENT OF THE MICROPHONE VOLTAGE MAGNITUDE VERSUS140 REM RESONATOR NATURAL FREQUENCY CURVE AND THUS DETERMINING THE150 REM SLOPE AT THE OPERATING POINT IN THE MAGNITUDE VS. NATURAL FREQUENCY160 REM CURVE, THE DC MOTOR IS GIVEN A TUNING DIRECTION. IT CAN BE170 REM SHOWN ON A MAGNITUDE VS. NATURAL FREQUENCY PLOT THAT THE CURVE IS180 REM V SHAPED WITH A CLEARLY IDENTIFIABLE MINIMUM.190 REM200 REM INITIALIZE D/A COMMAND210 REM $INCLUDE: 'Q4IFACE.BI'220 REM230 REM ------------------------ DECLARE VARIABLES ------------------------240 REM250 DIM NUMOFBOARDS AS INTEGER260 DIM DERR AS INTEGER270 DIM DEVHANDLE AS LONG280 REM290 REM MIKE AND POT ARE THE CHANNEL NUMBERS FOR THE MICROPHONE AND BACK PLATE300 REM POTENTIOMETER, RESPECTIVELY.310 REM320 DIM MIKE AS INTEGER330 DIM FVC AS INTEGER340 DIM POT AS INTEGER350 DIM ADVALUE AS LONG360 DIM FVCVOLT AS SINGLE370 DIM VOLTIN1 AS SINGLE380 DIM VOLTIN2 AS SINGLE390 DIM MIKEMAG AS SINGLE

65400 DIM POTVOLT AS SINGLE410 DIM INGAIN AS INTEGER420 DIM INGAINPOT AS INTEGER430 DIM INGAINVAL AS SINGLE440 DIM INGAINVALP AS SINGLE450 DIM DABASE AS INTEGER460 REM470 REM DCMOT IS THE D/A CHANNEL FOR THE DC MOTOR.480 REM490 DIM DCMOT AS INTEGER500 REM510 REM --------------------- COMMENCE PROGRAM ------------------------520 CLS530 LOCATE 1, 3:540 PRINT "ADAPTIVE-PASSIVE HELMHOLTZ RESONATOR CONTROLLER V1.00"550 REM560 REM INITIALIZE A/D AND D/A570 REM580 A$ = "DAS1600.cfg" + CHR$(0)590 DERR = DAS1600DEVOPEN%(SSEGADD(A$), NUMOFBOARDS)600 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "OCCURREDDURING '..DEVOPEN'": STOP610 DERR = DAS1600GETDEVHANDLE%(0, DEVHANDLE)620 IF (DERR <> 0) THEN BEEP: PRINT "ERROR "; HEX$(DERR); "OCCURREDDURING '..GETDEVHANDLE'": STOP630 MIKE = 0640 POT = 1650 FVC = 2660 POTIN = 1670 MOTORVOLT = 3680 DCMOT = 0690 DABASE = 784700 INGAIN = 0710 DELFN = 1720 CAVRAD = .1510919#730 LCAV = .2460625#740 PI = 3.14159750 SRES = (PI) * (.0254 ^ 2)760 LEFF = .0802894# + .02159 + .02159770 C = 343.53780 COEFF1 = (PI * LCAV * (CAVRAD ^ 2)) / 360790 COEFF2 = .000177062#800 FRONT = (C / (2 * PI))810 IF INGAIN = 0 THEN INGAINVAL = 1

66820 IF INGAIN = 1 THEN INGAINVAL = 10830 IF INGAIN = 2 THEN INGAINVAL = 100840 IF INGAIN = 3 THEN INGAINVAL = 500850 INGAINPOT = 0860 IF INGAINPOT = 0 THEN INGAINVALP = 1870 IF INGAINPOT = 1 THEN INGAINVALP = 10880 IF INGAINPOT = 2 THEN INGAINVALP = 100890 IF INGAINPOT = 3 THEN INGAINVALP = 500900 REM910 REM ---- FEED A CONSTANT 10 VOLTS TO THE POTENTIOMETER INPUT. --------920 REM930 DP = 4095940 HIGHBP = INT(DP / 256)950 LOWBP = DP - 256 * HIGHBP960 OUT DABASE + 2 * POTIN, LOWBP970 OUT DABASE + 1 + 2 * POTIN, HIGHBP980 REM990 REM ------------------- CALIBRATE POTENTIOMETER ---------------------1000 REM1010 GOSUB 22601020 REM1030 REM ------------- MEASURE THE EXCITATION FREQUENCY. -----------------1040 REM1050 DELFN = 11060 GOSUB 19901070 FREQLAST = FREQUENCY1075 SLEEP 11080 GOSUB 19901090 IF ABS(FREQLAST - FREQUENCY) > 1 THEN GOTO 10701100 REM1110 REM ---- NOW WE HAVE DETERMINED THAT THE SIGNAL IS STATIONARY. ------1120 REM1130 FREQSS = FREQUENCY1140 IF (FREQUENCY < 65) OR (FREQUENCY > 160) GOTO 19301150 MOTORVOLT = 31160 REM1170 REM -- MEASURE THE BACK PLATE POSITION ON THE HELMHOLTZ RESONATOR. --1180 REM1190 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)1200 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP1210 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP1220 THETA = (SLOPE * POTVOLT) + BIAS1230 HELMFN = FRONT * SQR(SRES / (LEFF * (THETA * COEFF1 - COEFF2)))

671240 REM1250 REM -- MOVE THETA TO THEORETICAL BACK PLATE POSITION WITH BETA MARGIN.1260 REM1270 BETA = FREQUENCY - 21280 IF (BETA < 65) THEN BETA = 651290 IF (BETA > 165) THEN BETA = 1651300 IF ABS(BETA - HELMFN) < 1 GOTO 13601310 REM DECREASE HELMFN BY DELTA FN IF HELMFN > BETA1320 IF HELMFN > BETA THEN GOSUB 30801330 REM INCREASE HELMFN BY DELTA FN IF HELMFN < BETA1340 IF HELMFN < BETA THEN GOSUB 27101350 GOTO 13001360 REM1370 PRINT "NATURAL FREQUENCY OF H. R. IS ", HELMFN1380 REM1390 REM ---------- BEGIN CLOSE LOOP PORTION OF ALGORITHM. ----------------1400 REM1410 I = 01420 GOSUB 20801430 MIKEOLD = MIKEPEAK1440 HELMFNOLD = HELMFN1450 REM1460 REM INCREASE HELMFN BY DELTA FN.1470 REM1480 IF HELMFN > 160 GOTO 17801490 GOSUB 27101500 REM1510 REM READ MICROPHONE AT NEW LOCATION1520 REM1530 GOSUB 20801540 DELTAM = MIKEPEAK - MIKEOLD1550 DELTAFN = HELMFN - HELMFNOLD1560 EMM = DELTAM / DELTAFN1570 IF EMM < 0 GOTO 14301580 MOTORVOLT = 11590 DELFN = DELFN - .31600 IF HELMFN < 65 GOTO 17801610 MIKEOLD = MIKEPEAK1620 HELMFNOLD = HELMFN1630 REM1640 REM DECREASE HELMFN BY DELTA HELMFN1650 REM1660 GOSUB 30801670 REM

681680 REM READ MICROPHONE AT NEW LOCATION1690 REM1700 GOSUB 20801710 DELTAM = MIKEPEAK - MIKEOLD1720 DELTAFN = HELMFN - HELMFNOLD1730 EMM = DELTAM / DELTAFN1740 IF EMM > 0 GOTO 16001750 I = I + 11760 IF I > 2 GOTO 17801770 GOTO 14301780 REM1790 REM --- CLOSED LOOP FINISHED. MONITORING EXCITATION AND MAGNITUDE. ----1800 REM1810 REM READ MICROPHONE MAGNITUDE AND COMPARE TO OLDER ONE.1820 REM1830 MIKEOLD = MIKEPEAK1840 GOSUB 20801850 IF (MIKEPEAK > (1.1 * MIKEOLD)) OR (MIKEPEAK < (.7 * MIKEOLD)) GOTO 10301860 REM1870 REM MONITOR EXCITATION FREQUENCY1880 REM1890 GOSUB 19901900 IF ABS(FREQSS - FREQUENCY) > 5 GOTO 10301910 PRINT "FREQSS", FREQSS1920 GOTO 18101930 PRINT "EXCITATION FREQUENCY OUTSIDE RANGE OF HELMHOLTZ RESONATOR"1940 IF (FREQUENCY < 65) THEN GOSUB 34501950 IF (FREQUENCY > 160) THEN GOSUB 38701960 GOSUB 20801970 GOTO 10301980 REM --------------------- SUBROUTINES --------------------------------1990 REM SUBROUTINE TO MEASURE EXCITATION FREQUENCY WITH FREQUENCY TO2000 REM VOLTAGE CONVERTER.2010 REM2015 KOW = 02017 SUM = 02020 DERR = KADRead%(DEVHANDLE, FVC, INGAIN, ADVALUE)2030 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP2040 FVCVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVAL

692050 FRQNCY = 100 * FVCVOLT2052 SUM = SUM + FRQNCY2054 KOW = KOW + 12056 IF KOW < 4000 GOTO 20202058 FREQUENCY = (SUM / 4000)2060 PRINT "THE MEASURED EXCITATION FREQUENCY IS "; FREQUENCY; " HERTZ."2070 RETURN2080 REM SUBROUTINE TO READ MICROPHONE OUTPUT PEAK TO PEAK VALUE2090 SLEEP .552100 TIME1 = TIMER2110 DERR = KADRead%(DEVHANDLE, MIKE, INGAIN, ADVALUE)2120 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP2130 MIKEMAG = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVAL2140 MIKEMAGMAX = MIKEMAG2150 MIKEMAGMIN = MIKEMAG2160 DERR = KADRead%(DEVHANDLE, MIKE, INGAIN, ADVALUE)2170 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP2180 MIKEMAG = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVAL2190 IF MIKEMAG > MIKEMAGMAX THEN MIKEMAGMAX = MIKEMAG2200 IF MIKEMAG < MIKEMAGMIN THEN MIKEMAGMIN = MIKEMAG2210 TIME2 = TIMER2220 IF TIME2 - TIME1 < .5 GOTO 21602230 MIKEPEAK = MIKEMAGMAX - MIKEMAGMIN2240 PRINT "MIKEPEAK IS ", MIKEPEAK, HELMFN2250 RETURN2260 REM SUBROUTINE TO CALIBRATE POTENTIOMETER.2270 PRINT ""2280 PRINT "CONNECT THE FOLLOWING EQUIPMENT TO THESE CHANNELS: "2290 PRINT ""2300 PRINT "EQUIPMENT CHANNEL BOARD"2310 PRINT "------------------------------------------------"2320 PRINT "FILTERED MIKE 0 DAS1600"2330 PRINT "POT OUTPUT (YELLOW) 1 DAS1600"2340 PRINT "SMOOTH FTV OUTPUT 2 DAS1600"2350 PRINT "CURRENT DRIVER INP. 0 DDA06"2360 PRINT "POT INPUT (RED) 1 DDA06"2370 PRINT ""2380 PRINT "DO NOT CONNECT MOTOR TO THE CURRENT DRIVER UNTIL PROMPTED TO."

702390 PRINT ""2400 PRINT "PLEASE MOVE THE RESONATOR BACPLATE TO THE 34 DEGREE REGION."2410 PRINT "THIS POSITION IS MARKED BY THE RED TAPE."2420 INPUT "", WAITER2430 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)2440 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP2450 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP2460 POTVOLT1 = POTVOLT2470 PRINT ""2480 PRINT "PLEASE MOVE THE RESONATOR TO THE 291 DEGREE REGION."2490 PRINT "THIS POSITION IS MARKED BY THE BLACK TAPE."2500 INPUT "", WAITER2510 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)2520 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP2530 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP2540 POTVOLT2 = POTVOLT2550 SLOPE = 257 / (POTVOLT2 - POTVOLT1)2560 BIAS = 34 - (SLOPE * POTVOLT1)2570 PRINT SLOPE, BIAS2580 REM2590 REM OUTPUT ZERO VOLTAGE TO MOTOR BEFORE CONNECTION.2600 REM2610 D = 20482620 HIGHB = INT(D / 256)2630 LOWB = D - 256 * HIGHB2640 OUT DABASE + 2 * DCMOT, LOWB2650 OUT DABASE + 1 + 2 * DCMOT, HIGHB2660 REM2670 PRINT ""2680 PRINT "YOU MAY NOW HOOK THE MOTOR TO THE BOARD"2690 INPUT "", WAITER2700 RETURN2710 REM SUBROUTINE TO INCREASE HELMFN BY DELTA FN2720 REM2730 HELMCOMP = HELMFN2740 IF HELMFN > 160 GOTO 29202750 D = INT((-1 * MOTORVOLT * (4096 / 20) + 2048))2760 IF D > 4095 THEN D = 40952770 IF D < 0 THEN D = 02780 HIGHB = INT(D / 256)

712790 LOWB = D - 256 * HIGHB2800 OUT DABASE + 2 * DCMOT, LOWB2810 OUT DABASE + 1 + 2 * DCMOT, HIGHB2820 REM2830 REM READ BACK PLATE POSITION2840 REM2850 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)2860 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP2870 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP2880 THETA2 = (SLOPE * POTVOLT) + BIAS2890 HELMFN2 = FRONT * SQR(SRES / (LEFF * (THETA2 * COEFF1 - COEFF2)))2900 IF (HELMFN2 > 160) GOTO 29202910 IF HELMFN2 < (HELMFN + DELFN) GOTO 28302920 D = 20482930 HIGHB = INT(D / 256)2940 LOWB = D - 256 * HIGHB2950 OUT DABASE + 2 * DCMOT, LOWB2960 OUT DABASE + 1 + 2 * DCMOT, HIGHB2970 REM2980 REM READ BACK PLATE POSITION2990 REM3000 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)3010 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP3020 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP3030 THETA = (SLOPE * POTVOLT) + BIAS3040 HELMFN = FRONT * SQR(SRES / (LEFF * (THETA * COEFF1 - COEFF2)))3050 IF (HELMFN = HELMCOMP) THEN HELMFN = HELMFN23060 PRINT "HELMHOLTZ RESONATOR NATURAL FREQUENCY IS ", HELMFN3070 RETURN3080 REM SUBROUTINE TO DECREASE HELMFN BY DELTA FN3090 REM3100 HELMCOMP = HELMFN3110 IF HELMFN < 65 GOTO 32903120 D = INT((MOTORVOLT * (4096 / 20) + 2048))3130 IF D > 4095 THEN D = 40953140 IF D < 0 THEN D = 03150 HIGHB = INT(D / 256)3160 LOWB = D - 256 * HIGHB3170 OUT DABASE + 2 * DCMOT, LOWB3180 OUT DABASE + 1 + 2 * DCMOT, HIGHB

723190 REM3200 REM READ BACK PLATE POSITION3210 REM3220 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)3230 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP3240 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP3250 THETA2 = (SLOPE * POTVOLT) + BIAS3260 HELMFN2 = FRONT * SQR(SRES / (LEFF * (THETA2 * COEFF1 - COEFF2)))3270 IF (HELMFN2 < 65) GOTO 32903280 IF HELMFN2 > (HELMFN - DELFN) GOTO 32003290 D = 20483300 HIGHB = INT(D / 256)3310 LOWB = D - 256 * HIGHB3320 OUT DABASE + 2 * DCMOT, LOWB3330 OUT DABASE + 1 + 2 * DCMOT, HIGHB3340 REM3350 REM READ BACK PLATE POSITION3360 REM3370 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)3380 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP3390 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP3400 THETA = (SLOPE * POTVOLT) + BIAS3410 HELMFN = FRONT * SQR(SRES / (LEFF * (THETA * COEFF1 - COEFF2)))3420 IF (HELMFN = HELMCOMP) THEN HELMFN = HELMFN23430 PRINT "HELMHOLTZ RESONATOR NATURAL FREQUENCY IS ", HELMFN3440 RETURN3450 REM SUBROUTINE TO MOVE HELMHOLTZ RESONATOR NATURAL FREQUENCY TO 65 HZ.3460 REM3470 REM READ BACK PLATE POSITION3480 REM3490 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)3500 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP3510 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP3520 THETA = (SLOPE * POTVOLT) + BIAS3530 HELMFN = FRONT * SQR(SRES / (LEFF * (THETA * COEFF1 - COEFF2)))3540 IF HELMFN < 65 GOTO 37203550 D = INT((MOTORVOLT * (4096 / 20) + 2048))3560 IF D > 4095 THEN D = 40953570 IF D < 0 THEN D = 0

733580 HIGHB = INT(D / 256)3590 LOWB = D - 256 * HIGHB3600 OUT DABASE + 2 * DCMOT, LOWB3610 OUT DABASE + 1 + 2 * DCMOT, HIGHB3620 REM3630 REM READ BACK PLATE POSITION3640 REM3650 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)3660 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP3670 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP3680 THETA2 = (SLOPE * POTVOLT) + BIAS3690 HELMFN2 = FRONT * SQR(SRES / (LEFF * (THETA2 * COEFF1 - COEFF2)))3700 IF (HELMFN2 < 65) GOTO 37203710 IF HELMFN2 > 65 GOTO 36303720 D = 20483730 HIGHB = INT(D / 256)3740 LOWB = D - 256 * HIGHB3750 OUT DABASE + 2 * DCMOT, LOWB3760 OUT DABASE + 1 + 2 * DCMOT, HIGHB3770 REM3780 REM READ BACK PLATE POSITION3790 REM3800 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)3810 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP3820 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP3830 THETA = (SLOPE * POTVOLT) + BIAS3840 HELMFN = FRONT * SQR(SRES / (LEFF * (THETA * COEFF1 - COEFF2)))3850 PRINT "HELMHOLTZ RESONATOR NATURAL FREQUENCY IS ", HELMFN3860 RETURN3870 REM SUBROUTINE TO MOVE HELMHOLTZ RESONATOR NATURAL FREQUENCY TO 160 HZ.3880 REM3890 REM READ BACK PLATE POSITION3900 REM3910 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)3920 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP3930 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP3940 THETA = (SLOPE * POTVOLT) + BIAS3950 HELMFN = FRONT * SQR(SRES / (LEFF * (THETA * COEFF1 - COEFF2)))3960 IF HELMFN > 160 GOTO 4140

743970 D = INT((-1 * MOTORVOLT * (4096 / 20) + 2048))3980 IF D > 4095 THEN D = 40953990 IF D < 0 THEN D = 04000 HIGHB = INT(D / 256)4010 LOWB = D - 256 * HIGHB4020 OUT DABASE + 2 * DCMOT, LOWB4030 OUT DABASE + 1 + 2 * DCMOT, HIGHB4040 REM4050 REM READ BACK PLATE POSITION4060 REM4070 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)4080 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP4090 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP4100 THETA2 = (SLOPE * POTVOLT) + BIAS4110 HELMFN2 = FRONT * SQR(SRES / (LEFF * (THETA2 * COEFF1 - COEFF2)))4120 IF (HELMFN2 > 160) GOTO 41404130 IF HELMFN2 < 160 GOTO 40504140 D = 20484150 HIGHB = INT(D / 256)4160 LOWB = D - 256 * HIGHB4170 OUT DABASE + 2 * DCMOT, LOWB4180 OUT DABASE + 1 + 2 * DCMOT, HIGHB4190 REM4200 REM READ BACK PLATE POSITION4210 REM4220 DERR = KADRead%(DEVHANDLE, POT, INGAINPOT, ADVALUE)4230 IF DERR <> 0 THEN BEEP: PRINT "ERROR "; HEX$(DERR); "DURING 'KADRead'": STOP4240 POTVOLT = (20 / 4096) * (CSNG(ADVALUE) / 16 - 2048) / INGAINVALP4250 THETA = (SLOPE * POTVOLT) + BIAS4260 HELMFN = FRONT * SQR(SRES / (LEFF * (THETA * COEFF1 - COEFF2)))4270 PRINT "HELMHOLTZ RESONATOR NATURAL FREQUENCY IS ", HELMFN4280 RETURN4290 END

75Appendix B: MATLAB Duct and Helmholtz Resonator System Modeling Program% Modeler.m M-File to model a speaker driven acoustic system with% a side branch Helmholtz Resonator and an open-ended% termination.% The complex impedances for the tube system and driver% found first. Then, the transfer matrix method is used% to find the sound pressure and particle velocity at a% specific location in the system.%% by: Juan Manuel de Bedout%%% Start by asking for the input voltage conditions.%Amplitude=input('Please enter the input RMS voltage');T=input('Please enter ambient Temperature in degrees Celsius');Theta=input('Please enter Back Plate Angle ');for f=10:1:300;omega=2*pi*f;freq(f-9)=f;% Declare Variables and Set Values.co=331.6; % Speed of sound, air at 0 degrees Celsius (m/sec)c=co*(1+(T/273))^0.5; % Speed of sound, air at ambient temperature.rho=1.21; % Density of air at 20 C. (kg/m^3)ap=0.0254; % Radius of pipe. (m)Sp=pi*(ap^2); % Cross Sectional Area of pipe. (m^2)k=omega/c; % Wave number. (m^-1)Lenc=0.10; % Length of pipe from speaker to pipe. (m)L1=0.34+(0.6*ap); % Length of pipe from open end to H.R. (m)L2=0.62+(0.85*ap); % Length of pipe from H.R. to speaker. (m)Lx=0.254; % Length of pipe from H.R. to microphone. (m)Sres=Sp; % Cross Sectional Area of Resonator Neck. (m^2)Lphys=0.0802894; % Physiscal neck length of Helmholtz Resonator (m)Lin=0.02159; % Internal mass end correction for H.R. (m)Lout=0.02159; % External mass end correction for H.R. (m)

76Lres=Lphys+Lin+Lout; % Effective neck length of H.R. (m)Lcav=0.2460625; % H.R. Cavity length. (m)CAVrad=0.1510919; % H.R. Cavity radius. (m)Vre=(Theta/360)*(pi*CAVrad^2)*Lcav; % H.R. Cavity volume. (m^3)Vres=Vre-0.000177062; % H.R. Cavity Volume with wall volume comp. (m^3)Re=8; % Electrical resistance of voice coil. (ohm)Rg=50; % Internal resistance of voltage source. (ohm)Bl=4.884; % Flux density * length for speaker. (W/m)Msp=(5.415/1000); % Mass of diaphragm and voice coil. (kg)Csp=(1/692); % Compliance of the driver suspension. (m/N)OmegaN=345; % Natural Frequency of the speaker. (rad/sec)Q=0.9259*exp(0.7702*k*0.0425); % Quality factor for the speaker.a=0.0425; % Effective radius of diaphragm. (m)Senc=pi*(a^2); % Cross Sectional Area of diaphragm. (m^2)Mml=2.67*rho*(a^3); % Air load mass on both sides of the driver. (kg)% Only used to calculate speaker diaphragm% resistance.Rsp=OmegaN*(Msp+Mml)/Q; % Speaker diaphragm resistance. (N/m/sec)Gamma=1.4; % Ratio of Specific Heats for air.Vback=(0.9*1.26715e-03); % Back Cavity Volume for Speaker enclosure. (m^3)Po=1e05; % Atmospheric Pressure (N/m^2)Cback=Vback/(Gamma*Po); % Acoustic Compliance of back volume. (m^5/N)b=0.38; % Constant. See Fig. 8.6 BeranekMback=b*rho/(pi*a); % Acoustic mass on back of diaphragm. (kg/m^4)Pref=20e-06; % Reference Pressure for Sound Pressure Level% calculations. (Pa)% Calculating the resistance due to viscous losses. (Ingard pg. 1045).Rs=(0.83e-03)*sqrt(f);Rhes=(1/Sres)*(2*Rs*(Lphys+ap)/(rho*c*ap));% Calculating the equivalent capacitance of the resonator.capres=(Vres)/(rho*(c^2)*(Sres^2));% First, lets calculate the system impedance.Z1=(rho*c*Sp)*(((1/4)*((k*ap)^2)+i*(0.6*k*ap))/(Sp^2));Z2=(rho*c/Sp)*(((Z1/(rho*c/Sp))+i*tan(k*L1))/(1+i*(Z1/(rho*c/Sp))*tan(k*L1)));Zres=Rhes+(1/(Sres^2))*((rho*c*(k^2)*(Sres^2)/(2*pi))+i*((omega*rho*Lres*Sres)-(1/(omega*capres))));

77Z3=Zres*Z2/(Zres+Z2);Z4=(rho*c/Sp)*(((Z3/(rho*c/Sp))+i*tan(k*L2))/(1+i*(Z3/(rho*c/Sp))*tan(k*L2)));Zsys=(rho*c/Senc)*(((Z4/(rho*c/Senc))+i*tan(k*(Lenc)))/(1+i*(Z4/(rho*c/Senc))*tan(k*(Lenc))));% Now, lets calculate the driver impedance.% Lets start with the sum of inductances (acoustic masses).Induct=(Msp/(Senc^2))+Mback;% Followed by the sum of the capacitances:Capact=1/((1/(Csp*Senc^2))+(1/Cback));% And finally, the sum of the resistances:Resist=((Bl^2)/((Senc^2)*(Re+Rg)))+(Rsp/(Senc^2));Drivimp=Resist+(i*omega*Induct)+(1/(i*omega*Capact));% Now, we find the total impedance of the system and driver.Impedance=Zsys+Drivimp;% Now, given the input voltage, we can find the volume velocity.Voltage=Amplitude*Bl/((Re+Rg)*Senc);Velocity=Voltage/Impedance;% The pressure drop seen by the system may now be calculated.Pressure=Velocity*Zsys;% Converting RMS pressure and velocity to Peak pressure and VelocityPressure=Pressure*sqrt(2);Velocity=Velocity*sqrt(2);% Now, given the volume velocity and the pressure going in to the% system, we may calculate the pressure and volume velocity at any% location in the system, via the transfer matrix method.

78TR1 = [Pressure;Velocity];TR15 = [cos(k*(Lenc)) -i*(rho*c/Senc)*sin(k*(Lenc));-i*(sin(k*(Lenc)))/(rho*c/Senc) cos(k*(Lenc))];TR2 = [cos(k*L2) -i*(rho*c/Sp)*sin(k*L2); -i*(sin(k*L2))/(rho*c/Sp) cos(k*L2)];TRres= [1 0; (-1/Zres) 1];TR3 = [cos(k*Lx) -i*(rho*c/Sp)*sin(k*Lx); -i*(sin(k*Lx))/(rho*c/Sp) cos(k*Lx)];Tf=TR3*TRres*TR2*TR15*TR1;Pmike=Tf(1)/(sqrt(2));Vmike=Tf(2);MagP(f-9)=20*log10(Pmike/Pref);end;plot(freq,MagP,'k-.');title('Frequency Response of System with Helmholtz Resonator');xlabel('Excitation Frequency (Hz)')ylabel('Sound Pressure Level (dB)');axis([0 300 55 135]);Natf=(sqrt(1/(rho*Lres*Sres*capres)))/(2*pi);