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PAGE 1 1 A TUNABLE ELECTROMECHANI CAL HELMHOLTZ RESONATOR By FEI LIU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 Fei Liu PAGE 3 3 To my beloved wife Hao Hu PAGE 4 4 ACKNOWLEDGMENTS I would like to express my grat itude to Dr. Mark Sheplak, my advisor, for his guidance, patience, encouragement and support during my academic pursuit in the Interdisciplinary Microsystems Group at University of Florida. I would like to thank Dr. Lou Cattafesta, my coadvisor, for his advice, instructive discussion a nd help. I also wish to express thanks to my committee members, Dr. Toshi Nishida and Dr. Na mHo Kim, for their advice and valuable feedback on this dissertation. I would like to acknowledge th e following individuals for their contributions to the completion of my dissertation: Dr. Stephen Ho rowitz, for many productive discussions and his gracious personality, and Dr. Todd Schultz, for his discussion and help with the acoustic impedance measurements. I am very grateful to Philip Cunio, Guiqin Wang, David Martin, Chris Bahr, Benjamin Griffin, Brian Homeijer, Vijay Chandrasekhara n, Matt Williams, Matias Oyarzun, Drew Wetzel, TaiAn Chen and all other group mates, fo r their friendship, discussion, and help. Finally, I would like to thank my parents, my parentsinlaw, and my wife for their love, understanding, and support over the years of my academic pursuit. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............13 CHAPTER 1 INTRODUCTION................................................................................................................. .15 Prologue....................................................................................................................... ...........15 Background..................................................................................................................... ........16 Research Objectives............................................................................................................ ....23 Research Contributions......................................................................................................... ..23 Outline of the Proposal........................................................................................................ ...23 2 THEORETICAL BACKGROUND........................................................................................32 Helmholtz Resonator............................................................................................................ ..32 Lumped Element Modeling.............................................................................................32 Model Parameter Estimation...........................................................................................33 Electromechanical Helmholtz Resonator...............................................................................36 Lumped Element Modeling.............................................................................................36 Tuning Behavior Analysis...............................................................................................39 Capacitive tuning of the EMHR...............................................................................39 Resistive tuning of the EMHR.................................................................................44 Inductive tuning of the EMHR.................................................................................45 3 MODELING AN EMHR USING THE TRANSFER MATRIX METHOD.........................52 Introduction................................................................................................................... ..........52 Characteristics of the Transfer Matrix....................................................................................55 Choice of State Variables................................................................................................55 Properties of the Transfer Matrix Re lated to Special Acoustic Systems.........................58 Reciprocal system....................................................................................................58 Conservative system.................................................................................................59 Basic Types of Elements of Acoustic Transfer Matrix Representation..........................59 Transfer Matrix Representation for EMHR............................................................................62 Area Contraction.............................................................................................................62 Area Expansion...............................................................................................................63 Duct Element................................................................................................................... 64 Clamped Piezoelectric Backplate with Shunt Loads.......................................................66 PAGE 6 6 Transfer Matrix of the EMHR.........................................................................................69 Transfer Matrix of the EMHR with Perforated Facesheet..............................................70 Comparison between TM and LEM.......................................................................................71 4 EXPERIMENTAL TECHNIQUES........................................................................................83 Acoustic Impedance Measurement.........................................................................................83 Introduction................................................................................................................... ..83 Theoretical Basis of the TMM........................................................................................86 Uncertainty Analysis of the TMM..................................................................................89 Parameter Extraction of the Piezoelectric Backplate..............................................................90 Damping Coefficient Measurement................................................................................90 Effective Acoustic Piezoelectric Coefficient Deduction.................................................92 Experimental Setup............................................................................................................. ....93 Acoustic Impedance Measurement Setup.......................................................................93 Damping Measurement Setup.........................................................................................94 Parameter Extraction of the Piezoelectric Backplate......................................................94 EMHR Construction.............................................................................................................. .95 5 EXPERIMENTAL RESULTS AND DISCUSSION...........................................................103 Evaluation of the Tuning Performance of the EMHR..........................................................103 Comparison with LEM and Transfer Matrix........................................................................106 Damping Coefficient Measurement Results.........................................................................107 Parameter Extraction of the Piezoelectric Backplate............................................................108 6 OPTIMAL DESIGN OF AN EMHR...................................................................................122 Introduction................................................................................................................... ........122 Optimizing Single Tuning Range of EMHR with Noninductive Loads.............................124 Theoretical Background................................................................................................124 Optimization Problem Formulation...............................................................................128 Optimization Results.....................................................................................................130 Sensitivity Analysis.......................................................................................................131 Pareto Optimization of the EMHR with Noninductive Loads............................................133 Optimization of the EMHR with Inductive Loads...............................................................135 7 SUMMARY AND FUTURE WORK..................................................................................143 APPENDIX A NOISE LEVELS AND UNITS............................................................................................146 B ACOUSTIC IMPEDANCE PRED ICTION OF AN ORIFICE............................................149 Linear Impedance Model of Orifices....................................................................................149 Crandalls Model...........................................................................................................149 GE Impedance Model....................................................................................................151 PAGE 7 7 End Corrections.............................................................................................................152 Effect of Nonlinearity at High Sound Pressure Levels.........................................................152 C PARAMETERS ESTIMATION FO R LEM OF THE PIEZOELECTRIC DIAPHRAGM...................................................................................................................... 160 D WAVE SCATTERING BY HELM HOLTZ RESONATOR IN A TUBE...........................163 Helmholtz Resonator as a Termination of a Circular Tube..................................................163 Helmholtz Resonator as a Termination of a Rectangular Tube............................................169 E GEOMETRIC DIMENSION OF EMHRS...........................................................................180 F COMPUTER CODES..........................................................................................................184 Acoustic Impedance Prediction using LEM and TM...........................................................184 Optimizing Tuning Range of an EMHR with Capacitive Loads..........................................195 Pareto Optimization Design of an EMHR with Capacitive Loads.......................................200 LIST OF REFERENCES............................................................................................................. 207 BIOGRAPHICAL SKETCH.......................................................................................................215 PAGE 8 8 LIST OF TABLES Table page 11 Characteristics of the airframe noise..................................................................................25 12 Characteristics of pr opulsion noise components................................................................25 13 Comparison between the passive liners and adaptive liners..............................................25 14 Summary of typi cal acoustic liners....................................................................................26 21 End corrections for orifices or slits....................................................................................47 22 Parameter estimation for lumped element modeling of the EMHR..................................48 41 Material properties of the piezoelectric backplate.............................................................97 42 Dimensions of the EMHRs................................................................................................97 43 Selected loads matrix used in the experiment to tune the EMHR.....................................98 51 Determination of the damping coeffici ent of the piezoelect ric plate in air.....................110 52 Determination of the damping coeffici ent of the piezoelectric plate in vacuum chamber........................................................................................................................ ....110 53 Comparison between predicted and deduced LEM parameters of the Piezoelectric backplate...................................................................................................................... ....110 61 Design optimization variables of the EMHR...................................................................137 B1 Summary on some acoustic impedance models of an orifice..........................................157 PAGE 9 9 LIST OF FIGURES Figure page 11 Percentage of highly annoyed persons (%) as a function of LDN for air traffic, road traffic, and rail traffic...................................................................................................... ...27 12 Historical progress of noise reduction...............................................................................27 13 Components of airfra me and propulsion noise..................................................................28 14 Main sources of the airframe noise....................................................................................29 15 Shock and mixing noise compone nts of the jet noise spectrum........................................29 16 Engine noise source and comparison of noise radiation pattern from nopass turbo engine and bypass turbofan engine.................................................................................30 17 Application of the acoustic liner technology in typical turbofan engine..........................30 18 An adaptive HerschelQuincke tube..................................................................................31 19 An electromechanical Helmholtz resonator.......................................................................31 21 A solidwalled HR under excitation of the incident wave.................................................49 22 Equivalent circuit representation of a solidwalled Helmholtz resonator..........................49 23 Schematic illustration of an EMHR...................................................................................50 24 Equivalent circuit representation of the EMHR.................................................................50 25 Simplified equivalent circ uit representation for a EMHR.................................................51 26 EMHR with passive electrical lo ads is analogous to a 2DOF system...............................51 31 A twoport network with reference directi ons for the positive dire ction of the current variables indicated............................................................................................................ .74 32 An acoustic twoport network with refe rence direction for the current variables indicated...................................................................................................................... .......74 33 Three basic types of elements in an equi valent circuit representation for an acoustic network........................................................................................................................ ......75 34 An area contraction........................................................................................................ ....75 35 A HR mounted in the side of one duct...............................................................................76 PAGE 10 10 36 Modeling EMHR using the transfer matrix method..........................................................77 37 Elements for derivation the transfer matrix representation of the EMHR.........................78 38 A piezoelectric backplate and its equivalent circuit representation...................................79 39 An EMHR with perforated facesheet.................................................................................79 311 Comparison of prediction results for th e normalized specific acoustic impedance of a shortand opencircuited EMHR using TM and LEM......................................................81 312 Illustration of contributions to the vari ation in the prediction performance of the LEM and TM..................................................................................................................... 82 41 Illustration of th e two microphone method........................................................................99 42 Pressure measured by a microphone..................................................................................99 43 Free damped vibration of a SDOF system.......................................................................100 44 Acoustic impedance measurement of the EMHR using TMM........................................100 45 The scattering by a HR mounted at the end of a PWT....................................................101 46 The damping measurement for the piezoelectric backplate of an EMHR.......................102 47 Assembly diagram of modul ar EMHR (not to scale)......................................................102 51 Experimental results for the normalized specific acoustic impedance of the EMHR (Case I) as function of the capacitive loads.....................................................................111 52 Experimental results for the reflection co efficient of the EMHR (Case I) as function of the capacitive load.......................................................................................................11 2 53 Experimental results for the normalized specific acoustic impedance of the EMHR (Case I) as function of the resistive loads........................................................................112 54 Experimental results for the normalized specific acoustic impedance of the EMHR (Case I) as function of the inductive loads.......................................................................113 55 Experimental results for the reflection coefficient of the EMHR (Case I) as function of the inductive load.........................................................................................................1 14 56 Experimental results of the normalized acoustic impedance of the EMHR (Case II) for the shortand opencircuit..........................................................................................114 57 Comparison LEM, TR and measurement re sults for a shortand opencircuited EMHR (CASE I)..............................................................................................................115 PAGE 11 11 58 Comparison LEM, TR and measurement re sults for a shortand opencircuited EMHR (CASE I), the damping loss of the backplate is assumed to be acoustic radiation resistance........................................................................................................... 115 59 Comparison LEM, TR and measurement re sults for a shortand opencircuited EMHR (CASE II)............................................................................................................116 510 Comparison LEM, TR and measurement re sults for a shortand opencircuited EMHR (CASE II), the damping loss of the backplate is assumed to be acoustic radiation resistance........................................................................................................... 116 511 Damping coefficient measurement for piezo electric composite backplate (Case I) in air............................................................................................................................ .........117 512 Damping coefficient measurement for piezo electric composite backplate (Case I) in the vacuum chamber........................................................................................................117 513 Determination of damping coefficien t of the piezoelectric plate in air...........................118 514 Curve fitting the measurement data (in air) using a 2ndorder system.............................118 515 Determination of damping coefficient of the piezoelectric plate in the vacuum chamber........................................................................................................................ ....119 516 Curve fitting the measurement data (in the vacuum chamber) using a 2ndorder system......................................................................................................................... .....119 517 Measured transverse displacement of the piezoelectric backplate due to the application of various voltages........................................................................................120 518 Predictions of the LEM and TM fo r shortand opencircuited EMHRs.........................121 61 Resonant frequency of the EMHR versus s and .......................................................138 62 Normalized sensitivity of the design variables at the optima for maximizing the tuning range of 1 f ..........................................................................................................139 63 Illustration of the change in th e optimum solution as a function of 2 R ..........................140 64 Normalized sensitivity of the design variables at the optima for maximizing the tuning range of 2 f ..........................................................................................................141 65 Choice of the starting point for the mu ltiobjective optimization with different 1 ........141 66 Comparison of the Pareto front obtained via the constraint, traditional weighted sum, and adaptive weighted sum methods.......................................................................142 PAGE 12 12 67 Comparison between initial (dash line) a nd optimal (solid line) acoustic impedance of the EMHR with inductive loads..................................................................................142 A1 Aweighting to sound arriving at random incidence........................................................148 D1 A Helmholtz resonator as termination of a circular tube.................................................178 D2 Schematic of a rectangular tube terminated by a Helmholtz resonator...........................178 D3 Schematic of transform between the Cart esian coordinate system and polar system......179 E1 Schematic of the EMHR..................................................................................................180 E2 Engineering draft of the orifice plate...............................................................................180 E3 Engineering draft of the orif ice plate of the cavity plate.................................................181 E4 Draft of the piezoelectric diaphragm bottom plate..........................................................182 E5 Draft of the piezoelectric diaphragm cap plate................................................................183 PAGE 13 13 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A TUNABLE ELECTROMECHANI CAL HELMHOLTZ RESONATOR By Fei Liu August 2007 Chair: Mark Sheplak Cochair: Louis N Cattafesta III Major: Aerospace Engineering Acoustic liners are used in tur bofan engine nacelles for the suppression of engine noise. For a given engine, there are different optimum im pedance distributions asso ciated with takeoff, cutback, and approach flight conditions. The im pedance of conventional acoustic liners is fixed for a given geometry, and conventional active line r approaches are impractical. This project addresses the need for a tunabl e impedance through th e development of an electromechanical Helmholtz resonator (EMHR). The device consists of a Helmholtz resona tor with the standard rigid backplate replaced by a compliant piezoele ctric composite. Analytical models (i.e., a lumped element model (LEM) and a transfer ma trix (TM) representation of the EMHR) are developed to predict the acoustic behavior of the EMHR. The EMHR is experimentally investigated using the standard twomicr ophone method (TMM). The measurement results validate both the LEM and the TM of the EMHR. Good agreement between predicted and measured impedance is obtained. Shortand open circuit loads define the limits of the tuning range using resistive and capacitiv e loads. There is approximat ely a 9% tuning limit under these conditions for the nonoptimized resonator configurat ion studied. Inductive shunt loads result in a 3 degreeoffreedom (DOF) system and an e nhanced tuning range of over 47% that is not restricted by the shortand opencircuit limits. Damping coefficient measurements for a PAGE 14 14 piezoelectric backplate in a vacuum chamber are performed and indicate that the damping is dominated by structural dampi ng losses. A Pareto optimization design based on models of the EMHR is performed with noninduc tive loads. The EMHR with noninductive loads has 2DOF and two resonant frequencies. The tuning ranges of the two resonant frequencies of the EMHR with noninductive loads cannot be optimized simultaneously, so a tradeoff (Pareto solution) must be reached. The Pareto solution shows how de sign tradeoffs can be used to satisfy specific design requirements The goal of the optimization of th e EMHR with inductive loads is to achieve optimal tuning of the three resonant frequencies. The result s indicate that it is possible to keep the acoustic reactance of the resonator ne arly constant within a given frequency range. PAGE 15 15 CHAPTER 1 INTRODUCTION Aircraft noise is an inevitable environmental im pact of aviation. Ai rcraft noise adversely affects the quality of life of peopl e living near an airport. As a consequence, worldwide policies and laws have been enacted to restrict noise emission from aircraft, making noise reduction an important goal for engineers. In th is chapter, the sources of noise from aircraft and the effects of aircraft noise on people are briefly examined. Some existing technologie s to reduce aircraft noise emission from aircraft engines are presen ted (e.g., acoustic liners). A new method using an electromechanical Helmholtz resonator (EMHR) to tune the acoustic impedance of an acoustic liner insitu is introduced. A main goal of this thesis is to study and understand the behavior of the EMHR via modeling and a companion experime ntal investigation. After the introduction, the objectives and approaches are described. Finall y, the outline of the rest of the dissertation is given. Prologue Originating from the operation of military and commercial airplanes, aircraft noise exposure can extend many miles beyond the boundari es of an airport. Aircraft noise is potentially harmful to aircrew and passengers, ground crews and mechanics, and people working or living in the vicinity of the airport. Pr oblems related to aircraft noise pollution include Physiological effects. An increase in blood pressure or heart rate, or even muscular contractions due to exposure to unexpected or elevated noise levels has been observed (Broadbent 1957). Hearing loss. Exposure to intensive noise for long durations can result in temporary or permanent damage to the human ear and decrease s the ability to hear clearly (Chen et al. 1992; Wu and Lai 1995). Disturbance of sleep and communication. As shown in Figure 11 extensive analysis indicates that aircraft noise is a larger annoyance than automobile and railroad noise (Miedema and Vos 1998). PAGE 16 16 Distraction and reduced performance. Long term exposure to noise will impair the ability to control attention and focus on task performance (Griefahn 2000; Kujala et al. 2004). With more and more people suffering adverse e ffects from aircraft noise, public concerns over noise emitted from aircraft have increase d tremendously. As a consequence, worldwide policies and laws have been en acted to restrict the noise em ission from aircraft (FAA 2001). Hence, modern aircraft must reduce noise emi ssions to satisfy new government regulations. Figure 12 illustrates the historical noise level trend in commercial airplanes. This decrease is obtained with the help of technological developmen ts and the use of effec tive acoustic treatments to aircraft. Background Aircraft noise is mainly generated from the airframe and propulsion systems (Smith 1989). Propulsion noise dominates during the takeoff (with or without cutback in thrust) and cruise phases of the flight. Airframe noise, on the othe r hand, is dominant during approach, as shown in Figure 13 Airframe noise is generated by an aircraft in flight due to the air fl ow around the airframe (Smith 1989; Crighton 1995). The fuselage, main wings, tailplane, leading edge slats, trailing edge flaps, and landing gear with wheel bays potentially contribute to the airframe noise signature, as shown in Figure 14 Since airframe components va ry in dimension from a few centimeters (e.g., the cockpit window) to meters (e.g., wings and fuselage), th e scale of turbulence in duced by air flowing over such structures differs. Thus, airframe noise is typically broadband in nature (Smith 1989; Crighton 1995). Audible, lowfrequency tones occu r occasionally. These tones often arise from the vortexshedding mechanism at the trailing edge of the wings, cavities or other discontinuities PAGE 17 17 in otherwise smooth surfaces, such as landingwheel bays (Smith 1989). Some characteristics of the airframe noise are summarized in Table 11 Another significant component of aircraft noi se is generated by the propulsion system. The components of the propulsion noise include turbomachinery noise, combustor(core) noise and jet noise (Smith 1989; Groeneweg 1995). Fans, compressors, and turbines can each generate significant turbomachineryrelated noise. Tu rbomachinery noise exhibits both tonal and broadband characteristics. The broadband noise re sults from sound generated in the proximity of the surfaces of rotating or sta tionary blades as a result of pressure fluctuations caused by turbulent flow (Smith 1989). The discrete tones are generated from the interaction between the airflow perturbations in the path of a rota ting blade row (Smith 1989). Fan blade passage frequency (BPF) tones and harmonics fall into this category. BPF tones are a function of the number of rotating blades and rotation speed (Smith 1989; Groeneweg 1995). Buzz saw noise is another type of discrete tone that has been observed when the fan has a supersonic tip speed. Buzz saw noise is generated at frequencies that are multiples of the rotation speed of the blade and are distinct from the blade passing frequency (Smith 1989). Combustor noise is generated by the combustion processes in the engine and also occurs when the hot flow leaving the combustion chambe r interacts with the turbine and exhaust nozzle downstream. Combustor noise is usually characterized by a broadband spectrum that peaks at a low frequency (Mahan and Karchmer 1995). The operation temperature, pressure, and physical geometry of the hot gas path through the e ngine affect the combustion noise level. Jet noise encompasses subsonic jet mixing noise and shockassociated supersonic jet noise (Smith 1989; Lilley 1995). Subsonic jet mixing noi se results from the turbulent mixing process between the exhaust flow of the engine and the surrounding air. Supersonic jet noise is PAGE 18 18 associated with the expansion shocks (jet core) fr om the nozzle of the engi ne. Jet mixing noise is broadband in nature and centered at a relatively low frequency. Supersonic jet radiates both broadband noise and screech tones. The broadband spectrum of the shockassociated jet noise is centered at a higher frequency than the mixing noise (Smith 1989; Lilley 1995; Shen and Tam 1998), as shown in Figure 15 Figure 16 shows the propulsion noise source radiation patterns of a conventional turbo engine and a modern turbofan e ngine. A summary of the charact eristics of the propulsion noise components is listed in Table 12 There are still other aircraft noise sources not specifically discussed above, such as propeller noise and ro tor noise from a helicopter. They warrant consideration in the de sign of such aircraft. In recognition of increasing av iation noise regulations, a vast number of techniques have been developed or proposed to reduce or control noise emission from aircraft. Noise control in propulsion systems attracted most of the research attention in the early er a of noise reduction. With more demanding standards, airframe noise suppression has become increasingly important. Airframe noise reduction involve s low noise gear design, devel opment of low noise highlift technology, reducing of aircraft cruise drag using passive/act ive flow control, and other technologies (Willshire 2001). The development of propulsion noise reduc tion technologies has followed two routes: noise source control and noise level reduction (Envia 2001). Noise source control suppresses the noise generation mechanism. The methods incl ude designing low noise engine components, such as using cycle optimization design to reduce the tip speed of the fan blades, fan pressure ratio and the velocity of the exhaust flow, and optimizing the number and spacing of the blades to weaken the impinging wake (Smith 1989; Envia 2001). Noise level reduction, however, PAGE 19 19 adopts acoustic treatments to reduce noise emissi ons. There are many different kinds of acoustic treatments. The use of the acoustic liners has been proven effective. Absorbent liners in the inlet of the engine account for about 5 PNdB of noise reduction (Smith 1989). Physically, acoustic liners provide a complex impedance bou ndary condition for the engine duct and thus suppress the propagation of the noise along the duct. Generally, the environment in which the acoustic liners have to operate is very hostile. The typical temperatures range fr om 50C to 500C between the inlet airflow at high altitude and hot gas in the exhaust system of the engine (Smith 1989). High airflow velocity in engine inlets and fan bypass ducts results in intensive aerodynamic turbulen t pressure fluctuations up to 170 dB (ref. 20 Pa ) (Manglarotty 1970). Moreover, as part of the aircraft, acoustic liners must satisfy structural integrity require ments and weight constraints. As a result, the typical acoustic liner in service has a Helmholtz resonator structur e and is comprised of a simple perforated plate facesheet, a cellular structure such as honeycom b, and a solid backplate (usually the engine wall), as shown in Figure 17 This type of acoustic liner is also called a single degree of freedom (SDOF) resonant absorber (Motsinger and Kraft 1995). The SDOF absorber is effective over approximately one octave of the frequenc y range for noise suppression. Thus, the design parameters should be pretuned to the frequency band that contai ns the single tone of most concern (Motsinger and Kraft 1995). The two degree of freedom (2 DOF) acoustic liner shown in Figure 17 is similar to a SDOF acoustic liner. The major difference is th at there are two layers of honeycomb separated by a fibrous or perforated sept um sheet. The useful bandwidth of 2DOF acoustic liner is approximately two octaves. Thus, 2DOF liners can be used to suppress fan noise encompassing two adjacent harmonics of the fan bladepassag e frequency. Theoretically, wider attenuation PAGE 20 20 bandwidth can be acquired using multiple laye rs of honeycomb structures (Multiple DOF system). Such a design, however, may not be rea lizable due to weight and space limitations of the engine. Wider attenuation bandwidth can be also achieved by nonresonant acoustic liners, such as bulk absorbers. Bulk absorbers have a single layer constructi on, similar to the SDOF absorber, but use a fibrous mat instead of a honeyc omb separator. Bulk absorbers are predicted to work well over three octaves (Motsinger and Kraft 1995). However, bulk absorbers have not been used in commercial engine service in the pa st because of structural design difficulties and maintainability concerns due to their tendenc y to absorb fluids (M otsinger and Kraft 1995; Bielak and Premo 1999). Without grazing flow effects, the frequency range of noise suppre ssion is fixed by the geometry for the acoustic liners mentioned abov e. These types of acoustic liners are called passive acoustic liners. These ar e in contrast to the active/adap tive liners, which can adjust the performance of the acoustic liner insitu. Activ e/adaptive liners have attracted more attention from researchers recently because of their pr omise in suppressing the engine noise under different operating conditions. The typical operating conditions of the engine include takeoff/cutback, cruise, and approach. When operated at different conditions, the characteristics of the aircraft noise may be much different. Po tential ways in which active/adaptive liners can modify acoustic liner performance insitu include using bias flow through the liner resistive elements and changing the geometry characterist ics of the liner (e.g.,the orifice dimension and the cavity volume). In 1957, Thurston et al. (1957) indi cated that steady flow through an orifice will change its impedance. Howe (1979) developed a model of the Rayleigh conductiv ity of an aperture, through which a high Reynolds number flow passes. The interaction between the incident sound PAGE 21 21 wave and the mean bias flow results in the periodic shedding of vor ticity. Consequently, a portion of the acoustic energy is diss ipated into heat. Howes work indicates that it is possible to enhance the sound absorption using a bias flow through the orifice of the acoustic liner. Hughes and Dowling further studied the vortex shedding mechanism for slits and circular perforations (Hughes and Dowling 1990; Dowling and H ughes 1992). The perforated liner was experimentally investigated by Sun and his collea gues (1999). They found that a bias flow can enhance sound absorption and extend the eff ective bandwidth of th e perforated liner. Little et al. (1994) tuned thei r resonator by changing the neck sectional area. De Bedout et al. (1997) developed a Helmholtz resonator with a cavity that allows a continuously variable volume. The variable volume actuation of the cav ity is realized by rotating an internal radial wall based on a tuning (control) algorithm. Selamet et al. (2003) presented a Helmholtz resonator with an extended neck. Their work indi cates that it is possible to control the resonance frequency of the resonator by adjusting the leng th of the extended neck without changing the cavity volume. All methods above mechanically modify the geometry of the resonatorlike acoustic liner. Generally, there is an obvious need for an addi tional actuator, controller, and power supply. Thus, it is difficult to apply this type of method to the acoustic liner in an engine. In essence, the techniques described above seek to enhance the noise absorption by modifying the impedance boundary conditions of the liner system. Moreover, it is worth noting that all adaptive/active system s discussed above, whether by modifying the geometry of the acoustic liner or introducing a bias flow to th e liner, may require actuators, sensors, and controllers. Consequently, such a system is of ten complex in comparison with a passive liner system. A more detailed comparison between passive liners and adapti ve liners is found in Table 13 A summary of typical acoustic liners is provided in Table 14 Ideally, an acoustic PAGE 22 22 treatment system should be robust and lightweig ht, have wide noise s uppression bandwidth, and have the ability to modify the pe rformance of the system insitu. Another type of acoustic treatment using a He rschelQuincke tube merits discussion. A HerschelQuincke (HQ) tube is essentially a hol low sidetube that trav els along the axis of a main duct (but not necessarily parallel to) and a ttaches to the mainduct at each of the two ends of the tube (Stewart 1928), as shown in Figure 18 The HQ tube is a simple implementation of the destructive interference princi ple to attenuate the sound. Histor ically, the HQ tube is used to suppress the tones of special freq uencies due to the limitations of constant crosssectional areas (Panigrahi and Munjal 2005). The limitations require that the crosssectional areas of the parallel ducts be equal, and the sum of the crosssectional areas of the branch ducts be equal to one of the entrant and exit duct (Stewart 1928; 1 945). People later found that such limitations are not necessary and re moving these restrictions can enhanc e the attenuation bandwidth of the HQ tube (Selamet et al. 1994; Selamet and Easwaran 1997). The potential of the HQ tube to attenuate tur bofan engine noise was recently investigated by Smith et al. (2002a; 2002b). They found that the HQ tube can be used to attenuate lowfrequency broadband noise, buzzsaw noise, and BPF tones. The HQ tube can be designed to work together with passive liners, which are most effective at atte nuating higherfrequency noise. They also presented an adaptive HQ tube, in which an internal throttleplate flap tunes the resonant frequency, as shown in Figure 18 Again, additional controller, sensor(s), and actuator(s) are needed for practical implementation. A novel method to tune the impedance of the liner system is presented in this dissertation. The primary element of this liner is a Helmho ltz resonator with the standard rigid backing replaced by a compliant piezoelectric composite diaphragm (Sheplak et al. 2004), as shown in PAGE 23 23 Figure 19 The acoustic impedance of the resonato r is adjusted and additional degrees of freedom added via electromechanical coupling of the piezoelectric composite diaphragm to a passive electrical shunt network. Research Objectives The objective of this disserta tion is to investigate a comp act, practical EMHR for noise reduction applications. To complete this objective, an analytical model, i.e. transfer matrix (TM) representation of the EMHR, is developed to pr edict the acoustic behavior of the EMHR. The model is implemented in a MATLAB program and compared with the lumped elements model (LEM) and measurement results. The EMHR is e xperimentally investigated using the standard twomicrophone method (TMM). The measurement results are used to validate analytical models of the EMHR and to demonstrate the po tential noise suppression applications of the EMHR. Parameter extraction, such as for the da mping coefficient and some LEM parameters of the piezoelectric backplate of the EMHR, is im plemented to improve the model performance. Finally, an optimal design routine is developed to provide the aids in the design of the EMHR to achieve a usable impedance range. Research Contributions The expected contributions of th is dissertation are as follows: Development of a transfer matrix model (T M) and refinement of the lumped element model (LEM) for the prediction of th e acoustic impedance of the EMHR. Experimental validation of bot h TM and LEM for the EMHR. Development of an optimal design routine for the EMHR. Outline of the Proposal The organization of this di ssertation is as follows. Chapter 1 introduces the background, motivation, and technical objectiv es of the research. In Chapter 2 lumped element modeling a PAGE 24 24 Helmholtz resonator and an EMHR are presented. Chapter 2 also presents the theoretical analysis of the electromechanical tuning behavior of an EMHR. The development of a model of the EMHR using the transfer matrix method is detailed in Chapter 3 Chapter 4 demonstrates the use of the standard two microphone method (TMM) to experimentally inve stigate the behavior of the EMHR as well as methods (e.g., the l ogarithmic decrement method) implemented to extract the model parameters. The experiment al results and discussi on are presented in Chapter 5 followed by the development of the design methodology using optimization techniques in Chapter 6 The conclusion of this research projec t and comments on future work are given in Chapter 7 PAGE 25 25 Table 11. Characteristics of the airframe noise (Smith 1989). Work condition Noise source Characteristics of noise Cruise Fuselage Main wings Broadband Distributed low frequency peak Fairly strong tones Takeoff Approach Slats Flaps Landing gear Broadband Generally omnidirectional Table 12. Characteristics of propulsion noise components (Sm ith 1989; Hubbard 1995). Noise component Characteristics Fan noise Tones, blade passing frequency related Buzzsaw noise Broadband noise Compressor noise High frequency tones Broadband noise Combustor noise Broadband noise with spectrum centered at low frequency Turbine noise High frequency tones Broadband noise which spectrum is centered at high frequency Jet noise Screech tones Broadband noise Centered at low frequency for the mixing jet noise Centered at higher frequency for the shockassociated jet noise Table 13. Comparison between th e passive liners and adaptive lin ers (Smith 1989; Kraft et al. 1999). Passive liners Adaptive liners Pros Fixed geometry, simple structure Increased noise suppression frequency range or reduced noise over variable frequency range Cons Relative narrow suppression Frequency range fixed by the geometry More degree of freedoms are required to increase the suppression frequency range Complex structure Power, actuator, or controller needed PAGE 26 26 Table 14. Summary of typica l acoustic liners (Nayfeh et al 1975; Motsinger and Kraft 1995; Parrott and Jones 1995; Bielak and Premo 1999). Liner types Structure, Properties and Application Perforatedplate, honeycomb liner Easy to manufacture Narrow noise attenuation bandwidth (SDOF) Increased degrees of freedom needed to widen attenuation bandwidth Acoustic nonlinearity Widely used in engine duct Resistive resonant liner (linear liner) Porous layer can be wovenscreen metal sheets, sintering fibermetal sheets, etc. Narrow noise attenuation bandwidth (SDOF) Acoustic character is independent of sound pressure level (SPL) Usable in engine noise reduction Parallel elements liner Damping provided by the cylindrical channel Circumvents use of the facesheet Ceramic honeycomb is heavy and fragile Not applicable to the engine in the past Locally reactive linersa (Ingard 1999) Perforatedplate liner with bias flow Introduction of airflow, blowing or suction, perpendicular to the acoustic liner Ability to control acoustic impedance of liner insitu via bias flow Additional actuator or controller needed Promising way to reduce engine noise Nonlocally reactive liners Bulk nonresonant liner Broad attenuation bandwidth Poor mechanical properties (rigidity, etc.) and fluid absorption Not applicable to aircraft in the past a. Locally reactive liners are defined as liners th at permit wave propagatio n only in the direction normal to the acoustic liner surface, while nonlocally reactive liners permit propagation in more than one direction. PAGE 27 27 45 50 55 60 65 70 75 80 0 10 20 30 40 50 60 Percentage highly annoyed persons [%]LDN Air traffic Road traffic Rail traffic Figure 11. Percentage of highl y annoyed persons (%) as a function of LDN for air traffic, road traffic, and rail traffic, where LDN is the av erage level of day/night noise measured in decibels (dBA) over an extended period of time. Adapted from Miedema and Vos (1998). Figure 12. Historical progress of noise reduction (sideline no ise level, normalized to 104 lb thrust), where the noise metric is Effec tive Perceived Noise Level (EPNL), measured in units of EPNdB. Adapted from Willshire (2001). PAGE 28 28 0 20 40 60 80 100 120InletAft fanCombustorTurbineJetTotal airframe Total aircraft noise Approach Cutback (a) 120m 2000m 6500m Approach reference Takeoff/Cutback reference 450m Sideline reference (b) Figure 13. Components of airf rame and propulsion noise. (a) Comparison of airframe and propulsion noise for Boeing 747400 with P&W 1992 technology e ngine during the approach and takeoff/cutback operation with (b) the noise reference points illustrated. Adapted from Smith (1989). Noise level ( EPNdB ) PAGE 29 29 Figure 14. Main sources of the airfra me noise. Adapted from Smith (1989). Shock noise Mixing noise Frequency Figure 15. Shock and mixing noise components of the jet noise spectrum. Adapted from Smith (1989). PAGE 30 30 Compressor Jet Turbine & core Fan Turbo engineTurbofan Engine Figure 16. Engine noise source and comparison of noise radiation pattern from nopass turbo engine and bypass turbofan engi ne. Adapted from Smith (1989). Figure 17. Application of the acoustic liner technology in typical turbofan engine. Adapted from Smith (1989). PAGE 31 31 Figure 18. An adaptive HerschelQuincke tube Adapted from Smith and Burdisso (2002). PZTbackplate Loads Cavity Neck Figure 19. An electromechan ical Helmholtz resonator. PAGE 32 32 CHAPTER 2 THEORETICAL BACKGROUND In this chapter, a review of the lumped el ement model (LEM) of a conventional Helmholtz resonator and an EMHR is presented. The tuning behavior of an EMHR with various electrical loads (e.g., the EMHR with capaci tive loads) is then analyzed with the aid of the LEM. Helmholtz Resonator As mentioned in Chapter 1 the passive acoustic liners widely used as acoustic treatments in engine ducts have a structure similar to a Helmholtz resonator, as shown in Figure 21 The name Helmholtz resonator comes from the German physician and physicist Hermmann von Helmholtz (Rayleigh 1945). A conventional solidw alled Helmholtz resonator is comprised of a narrow neck (i.e., a small orifice) and a cavity vo lume. The resonator is a SDOF system and has a resonant frequency determined by its geometry. Lumped Element Modeling The modeling of a HR can be simplified if any characteristic dimensi on of the resonator is small compared to the acoustic wavelength. When the wavelength is greater than all dimensions, a distributed system can be lumped into idealized discrete elements (Rossi 1988). The linearized continuity and Eulers equations are thus replaced by equivalent Kirchoffs laws for volume velocity and pressure drop. Hence, for a Helmho ltz resonator excited by an incident acoustic pressure of low frequencies, shown in Figure 21 the gas trapped in the n eck of the resonator is modeled as a lumped mass element aN M The air isentropically compressed in the cavity is modeled as an acoustic compliance aNC. Furthermore, the open end of the neck radiates sound, and thus provides radiation resistance aNradR, and mass aNradM. Additional resistance, aNvisR, results from the thermoviscous losses at the neck walls. The aNradR and aNvisR comprise the PAGE 33 33 damping loss of the neck, aN R In the notations above, the fi rst subscript de notes the domain (e.g., a for acoustic), and the second subscript describes the element (e.g., N for neck). The equivalent circuit representation for a solid walled Helmholtz resonator is shown in Figure 22 where P and Q represent the incident acoustic pressure s and volumetric flow rate, respectively. Model Parameter Estimation In Figure 22 the acoustic mass, aN M is given by (Blackstock 2000) 0 2 aNtt M r (21) where t and rare the thickness and the radi us of the neck, respectively, t is the sum of the end correction at the inner an d outer neck ends. Note that E q. (21) assumes no viscosity and the velocity profile within the neck is constant (i.e., plug flow). The end correction in Eq. (21) takes into account the entrained air near the flanged or unflanged mouth of the resonator. The end correction, t is given as (Ingard 1953) innerouter0.8511.250.85 r tttrr R (22) where R is the radius of the cavity. More cases of the end correction of the orifice (hole) are listed in Table 21 The cavity acoustic compliance, aCC, is derived assuming a uniform pressure throughout the cavity and an isentropic co mpression process (Blackstock 2000), 2 00,aCC c (23) where 2 R L is the cavity volume, and L is the depth of the cavity. Another method is to treat the cavity as a short closed tube with depth L The specific acoustic impedance of such tube is thus given by (Blackstock 2000) PAGE 34 34 sp_00cotaC Z jckL (24) If 1 kL i.e., 0.3 kL the tangent function cot kLkL (25) Substitution of Eq. (25) into (24) results in 2 0000 sp_ aCcc Z j kLjL (26) The acoustic impedance of the cavity is therefore 2 00 2 2 0011 .aC aCc Z jC jc jLR (27) Both methods yield the same results. The require ment for the first method that the process is quasistatic is closely related to the requirement for the second method that kL be small ( kL<<1 ). However, the first method can be used for a general case, such as an irregular cavity. The neck resistance aN R is given by ( Ingard 1953) 00 00 1 22222 2(2) 21 2aNc Jkr t R rrrrkr (28) where is dynamic viscosity of the air, and 1J is the Bessel function of the first kind. The neck resistance includes several contributions. The first term in Eq. (28) represents the viscous loss in the neck wall boundary layer which is derived assuming the fluc tuating flow through the neck due to the acoustic excita tion is hydrodynamically incompre ssible viscous flow (Crandall 1926). The second term represents th e viscous loss at the neck ends (also refers to resistance end correction). This approximation is adapted from Ingard (1953), which is originally derived by empirical methods. The third term represents the radiation loss at the oute r neck end. At low frequencies, PAGE 35 35 2 0000 1 22(2) 1 22aNcck Jkr R rkr (29) It is notable that Eq. (28) only accounts for the linear resistance when an incident wave with low sound pressure level (SPL) excites the resonator. When the incident wave has a high SPL, an additional nonlinear resist ance term is added into Eq. (28), wh ich is proportional to the particle velocity within the neck of the resonator. Nonl inear behavior of the orif ice due to a high SPL is reviewed in Appendix B Furthermore, notice that the assumptions for the reactance and resistance in Eqs. (21) and (28), respectively, are different. Other advanced impedance model is presented in Appendix B to alleviate these conflicts. The acoustic impedance of the Helm holtz resonator is thus given by 22 0 0000 HRHRHR 22 tt ckc ZRjXj rj (210) for low frequencies, where HR R and HR X are the acoustic resistance and reactance of the resonator, respectively. The resonant freque ncy of the resonator is the frequency at which HR X vanishes, 2 0 02 c r f tt (211) At the resonant frequency, the pressure amplifi cation, defined as the ra tio of amplitude between the cavity pressure and the harmonic incident pres sure, reaches a peak. It is clear that the geometry of the Helmholtz resonator determines th e resonant frequency. The resonant frequency is fixed when the geometry of the resonator fixed.* No grazing flow is involved. PAGE 36 36 Electromechanical Helmholtz Resonator As discussed above, the resonant frequency of a conventional Helmholtz resonator is fixed when the device geometry is fixed. Since Helm holtz resonators work effectively near their resonant frequency, their application to noise su ppression is thus limited (Kinsler 2000). Recall that a tunable resonant frequency is one of the de sired traits of the modern Helmholtz resonatorlike acoustic liner ( Chapter 1 ). The electromechanical Helmho ltz resonator (EMHR) developed in this dissertation is capable of tuning the ac oustic impedance as well as the resonant frequency of the resonator. As shown in Figure 23 an EMHR is a Helmholtz resonator, in which the rigid backplate is replaced by, for example, a piezoelectr ic composite one that is attached to a passive electrical shunt networ k (Sheplak et al. 2004). The acous tic impedance and the resonant frequency of the resonator are adjusted and additional degrees of freedom are added via electromechanical coupling of the piezoelectric composite diaphragm to a passive electrical shunt network. Specifically, in comparison to a conventional SDOF solidwalled Helmholtz resonator, the EMHR has two or three DOF with a variety of resi stive, capacitive, and inductive shunts. Lumped Element Modeling Similar to the conventional HR at low frequencies, the dimensions of the EMHR, shown in Figure 23 are much smaller than the wavelength of interest. The device components are thus lumped into idealized discrete circuit elements (Rossi 1988). In particular, the EMHR is lumped into two parts. The first part is similar to a conventional Helmholtz re sonator described in the previous section, where the neck of the res onator possesses both dissipative and inertial components, and the cavity is modeled as a compliance. The second part includes the compliance, damping loss and mass of the piezoel ectric backplate and the coupling between the acoustical and electri cal energy domains. PAGE 37 37 Figure 24 shows the equivalent circuit representation for the EMHR, where the lumped parameters are defined as follows: P and P represent the incident and diaphragm acoustic pressures, respectively. Q and Q are incident and diaphragm volumetric flow rates, respectively. In the notation below, the first subscript denotes the domain (e.g., a for acoustic), and the second subscript de scribes the element (e.g., D for diaphragm). aN R and aN M are the acoustic resistance and acoustic mass of the fluid in the neck, respectively. aCC is the acoustic compliance of the cavity, while aDC and aD M represent the shortcircuit acoustic compliance and mass of the piezoelectric compliant diaphragm, respectively. aDradM is the acoustic radiation mass of the diaphragm. aD R is the acoustic resistance which includes the acoustic radiation resistance and other stru ctural damping losses. Finally, eBC is the blocked electrical capacitance of the piezoelectric diaphragm (i.e., when the volumetric flow rate is zero), is the impedance transformation factor, and eL Z is the electrical load impedance across the piezoelectric backplate. Analy tical equations to estimate these parameters are listed in Table 22 The parameter deduction for the backplate is presented in Appendix C As shown in Figure 24 a lossless transformer converts energy between th e acoustical and electrical domains and obeys the following transformer relations (Liu et al. 2003, 2007) i Q (212) and PV (213) PAGE 38 38 Furthermore, the transformer transform s the blocked electrical capacitance, eBC to an equivalent acoustic impedance, aEBC and the electrical impedance, eL Z to an acoustic impedance, aL Z by 2eB aEBC C (214) and 2 aLeL Z Z. (215) Accordingly, a simplified equivalent circu it representation is obt ained by converting the electrical impedance to acous tic impedance as shown in Figure 25 The input acoustic impedance ( aIN Z PQ) is then given by (Liu et al. 2003, 2007) 11 1 11 1aL aDradaDaDrad aCaDaEBaL aINaNaN aL aDradaDaDrad aCaDaEBaLZ RsMM sCsCsCZ ZRsM Z RsMM sCsCsCZ (216) where sj Eq. (216) clearly demonstrates that modifying the shunt networkeL Z tunes the acoustic impedance of the EMHR. When the shunt loads are capacitive or resistive, the EMHR is analogous to a 2DOF system with tw o resonant frequencies, as shown in Figure 26a and 6b respectively. The EMHR with inductive loads in Figure 26c has 3DOF, and thus has three resonant frequencies. As indicated by Eq.(216), the resonant freque ncies of the EMHR are tuned via adjusting the shunt load s. The tuning behavior of the EMHR with various electrical loads is analyzed below. In particular, the tuning mechanism of the EMHR with capacitive loads is described in detail. The tuni ng of the EMHR with resistive and inductive loads is analyzed in an approximate manner because the noncapacitive tuning is not mathematic ally tractable. In this case, it is more convenient to dem onstrate the tuning of the EMHR numerically. PAGE 39 39 Tuning Behavior Analysis Capacitive tuning of the EMHR The EMHR with capacitive s hunts is analogous to a 2DOF system with two resonant frequencies, as shown in Figure 26a Referring to Figure 26a the impedance of loop 1 is 1 1 1 11 1111 1LaNaN aC aN aNaNaC aC aNaC aN L wL LL wLLLZRjM C M RjMC C MC R f f j Zff j (217) where the quantity 11 2L aNaCf M C (218) is the resonant frequency of loop 1. The first term in Eq. (217) 1 LaNaNaC R MC (219) is the dissipation factor, which is the ratio of the power dissipated to the power stored (Fischer 1955). The quantity 111 LLL f fff (220) is the tuning factor which measures the devia tion of the operating frequency of loop 1 from its resonant frequency. Finally, the quantity 1 wLaNaC M C (221) is a weighting factor by which th e magnitude of the impedance of a different system which has the same dissipation factor and the resona nt frequency differs from each other. PAGE 40 40 Similarly, the impedance of loop 2 is 2 22222 22aD L LwLwLLL wLLR f f Zjj Zff (222) where the definitions of the quantities 2L f 2wL 2L and 2L are analogous to their counterparts in loop 1 2 21 2L aDaDradf M MC (223) 22 LaDaDaDrad R MMC (224) 222 LLL f fff (225) and 22 wLaDaDrad M MC, (226) where 2 aEBaLaDaC aBaLaDaBaLaDaCCCCC C CCCCCCC (227) Next, for the coupled system 1121aCPZQQ jC, (228) and 2211 0aC Z QQ jC, (229) which leads to 2 1 1211aINL LaCP ZZ QZjC (230) PAGE 41 41 From Eqs. (217), (222) and (230), one has 2 11 11222111aIN LL wLwLwLaCLLZ j ZjCj (231) If the system is undamped or weakly damped, 120LL (232) Eq. (231) can be rewritten as 2 2 112 1 2 11221 2 21111aIN LLL wLwLwLaCLL L LZ fff f jj ZZZjCjfff f f j f f (233) where 2 212aCaCCCC is the coupling factor of the system, and 1 and 2 are the coupling coefficients which define the ratio of the oscillating energy stored in the coupling elements to that stored in the total capacitan ce for each loop. They are given by (Fischer 1955) 2 1 1 1 2 111aC aCQdtC C C QdtC (234) and 2 2 2 2 2 22 aC aCQdtC C C QdtC (235) where iQ is volume velocity, as shown in Figure 26 At the resonant frequencies, Eq. (233) equals zero, and the solution for the resonant frequencies is 2 2 22222 121212 2 1,241 2LLLLLLffffff f (236) Furthermore, Eqs. (218) and (223) yield PAGE 42 42 22 12 2 2 2 2 22111 4 1111 4 11 1 4 1LL aNaCaDaDrad aBaLaD aNaDaDradaCaDaDradaBaLaD aBaLaD aNaCaDaDradaBaLaD HRDff MCMMC CCC MMMCMMCCC CCC MCMMCCC ff (237) where aN aDaDradM MM (238) is the mass ratio between the neck and pi ezoelectric backplate. In addition, 11 2HR aNaCf M C (239) is the resonant frequency of the Helmholtz res onator with solid wall instead of a piezoelectric backplate, and 1 2aEBaLaD D aDaDradaBaLaDCCC f M MCCC (240) is the resonant frequency of the pi ezoelectric backplate. Similarly, 2222 121 L LHRD f fff (241) When the system is weakly coupled, 0 substitution of equations (237) and (241) into (236) results in 2 2222 1,2 2211 2HR HRDHRD D HRf ffff f f f (242) PAGE 43 43 Furthermore, if the mass ratio between the neck a nd the piezoelectric backplate is very small as well, 0 2 2222 1,22HRDHRD H R D ffff f f f (243) In other words, the resonant frequencies of a lightly damped EMHR with capacitive shunts possessing weak coupling are approximately th e resonant frequency of the solidwalled Helmholtz resonator and the piezoelectric backplat e. As indicated by Eq.(240) the resonant frequency of the piezoelectric backplate is adjust ed with the change of capacitive loads. For opencircuited EMHR 0aLC or 1aLaLZsC open1 2aEBaD D aDaDradaEBaDCC f M MCC (244) While for shortcircuited EMHR, aLC or 10aLaLZsC short11 2D aDaDradaDf M MC (245) Moreover, for the EMHR with a capacitive load, 2 aLeLCC, capacitive1 2aEBaLaD D aDaDradaEBaLaDCCC f M MCCC (246) Clearly, shortcapacitiveopen DDDfff (247) or by the use of Eq. (243) 222 shortcapacitiveopenfff (248) PAGE 44 44 Equation (248) demonstrates how the tunable electromechanical Helmholtz resonator works with different capacitive shunts. The short and open loads define the limits of tuning using capacitive loads. The second re sonant frequency shifts toward the short case when the capacitance increases. Resistive tuning of the EMHR As shown in Figure 26b the EMHR with a resistive load is a 2DOF system. The resistive tuning is not mathematically tractable. Thus, an approximate analysis is given as follows. When an EMHR attached to a resistive load, the effective loads converted from the electric domain, as shown in Figure 26b is given by 21 1eL aE eBeLR Z j CR (249) When 0eLR the EMHR is shortcircuited, while the EMHR is opencircuited as eLR Furthermore, in the frequency range (i.e., for small eL R ) 1 2eBeLf CR, (250) Eq. (249) can be further approximated as 21 1aEeLeBeL Z RjCR (251) which indicates that the resistive loads effectiv ely reduce the acoustic mass of the piezoelectric backplate by 2 eBeLjCR (i.e., 2effaDaDeBeLMMjCR where effaDM is the effective acoustic mass of the piezoelectric backplate). Thus, the resonant fre quencies of the weakly coupled EMHR are given by 2 2222 1,22HRDHRD H R D ffff f f f (252) PAGE 45 45 where H R f is given by Eq. (239), and the resonant frequency of the pi ezoelectric backplate D f is 211 2D aDaDradeBeLaDf M MCRC (253) In comparison with the shortcircuited EMHR, D f increases due to the decrease of the acoustic mass. Moreover, short DDff in Eq. (245). With increasing resistive loads, D f moves further away from the shortcircuit resonant frequency short Df toward the opencircuit resonant frequency open Df Note that Eq. (253) doesnt give that D f goes to open Df for the opencircuited EMHR (eLR ). When eL R is large, the assumption in Eq. (250) breaks down, and Eq. (253) fails to predict the resistive tuning. In such a case, it is more convenient to demonstrate the tuning of the EMHR numerically. Inductive tuning of the EMHR As shown in Figure 26c The EMHR with an inductive load is a 3DOF system. To simplify the problem without loss of generality, it is assumed that the EMHR with inductive shunts has negligible damping. The eff ective impedance of loop 3 is given by 22 L3 22 eBeL eBeLjCjM Z jCjM (254) which can be further approximated as 2 L3 eL Z jM (255) in the frequency range Strictly, D f should be called the natural frequency. PAGE 46 46 1 2E eBeL f f CM (256) where E f is the resonant frequency of loop 3. Physically, loop 3 effectively adds a mass 2 eL M to loop 2. Similarly, the resonant frequencie s of the weakly coupled EMHR are given by 2 2222 1,22HRDHRD H R D ffff f f f (257) where H R f is given by Eq. (239), while D f is 211 2D aDaDradeLaDf M MMC (258) Clearly, short DDff in Eq. (245). With increasing inductive loads, D f moves further away from the shortcircuit resonant frequency short Df. The aforementioned analysis focuses on illustra tion of the tuning behavior of the EMHR with weak coupling between the solidwalled Helmholtz resonator and the piezoelectric backplate. However, the first resonant frequency of the system can be tuned insitu as well. More analysis on the tuning behavior is presented in Chapter 6 PAGE 47 47 Table 21. End corrections for orifices or slits. Items Expression Orifice in baffle wall (flanged) (Blackstock 2000; Kinsler 2000) 0.85 tr where t is the end correction of the orifice, r is the radius of the orifice Orifice in free space (unflanged) (Blackstock 2001; Kinsler 2000) 0.6 tr Orifice in tube (Mechel 2002) 2 20.04457280.7283260.177078 0.03395310.008104710.00100762 trxx yyxy where 102log x rb 10log yb is the wavelength and b is the radius of the tube Orifice array, square arrangement (Mechel 2002) 320.7911.470.47 tr where 2 b is the hole separation distance, and 22=4 rbis the porosity Orifice array, hexagonal arrangement (Mechel 2002) 2 20.04457280.7283260.177078 0.03395310.008104710.00100762 trxx yyxy where 102log x rb 10logyb and 2 b is the length of the hexagon Orifice array (Melling 1973) 83 tr where is the Fock function 0n n na with the first eight coefficients are given by 01 23 45 67 811.40925 00.33818 00.06793 0.022870.03015 0.01641 aa aa aa aa a PAGE 48 48 Table 22. Parameter estimation for lumped element modeling of the EMHR. Acoustic impedance Description aN R 00 00 1 22222 2(2) 21 2aNc Jkr t R rrrrkr where 1J is the Bessel function of the first kind Neck (Ingard 1953) aN M 0 2aNtt M r where 0.8511.250.85r trr R Cavity (Blackstock 2000) aCC 2 00cotaCR C ckL aDC 20 02R aD VCrwrdrP where wr is the transverse displacement of the piezoelectric backplate, P is the applied pressure and V is applied voltage on the backplate aD M 22 2 0 02 ()R aDA V M wrrdr where A is the surface density of the piezoelectric backplate aDradM 0 2 28 3aDradM R aD R 2aDaDrad aD aDMM R C where is the experimentally determined damping factor 20 02R aaDaD PdCrwrdrVC Piezoelectric backplate (Prasad et al. 2006) eBC 2 22 0111r eBeFEMEM pR CC h where r is the relative dielectric constant of the piezoelectric material, 0 is the permittivity of free space, and 22 EMaeFaDdCC is the electroacoustic coupling factor PAGE 49 49 2 rt L 2 R Cavity Orifice Figure 21. A solidwalled HR under excitation of the incident wave. The scattered wave is removed for clarity. aN R aN M aCC P I N Z Q Figure 22. Equivalent circ uit representation of a solidwalled Helmholtz resonator. PAGE 50 50 2 rt L 2 R s h p h1 R 2 R Piezoelectric backplate Cavity Orifice eL Z Figure 23. Schematic illustration of an EMHR. aN R aN M aCCaDCaD R aDradaD M M :1 PP eBCeL Z i V aIN Z QQ Figure 24. Equivalent circu it representation of the EMHR. PAGE 51 51 aN R aN M aCCaDCaD R aDradaD M M PaEBC aIN Z aL Z Q Figure 25. Simplified equivalent circuit representation for a EMHR. aN R aN M aCCaDCaD R aDradaD M M P 2 eBC 2 eLC aIN Z 1Q 2Q 2 eBC 2 eL R 2 eBC 2 eL M 2 aLeL Z jC2 aLeL Z R2 aLeL Z jM Figure 26. EMHR with passive electrical loads is analogous to a 2DOF system for (a) capacitive and (b) resistive loads and a 3DOF system with an (c) inductive load. PAGE 52 52 CHAPTER 3 MODELING AN EMHR USING THE TRANSFER MATRIX METHOD The purpose of this chapter is to develop an analytical model of an EMHR using the transfer matrix method and to compare the transf er matrix of the EMHR with the existing LEM. The chapter is organized as follows. The first section briefly reviews the application of the transfer matrix method in acoustics. The second s ection introduces some e ssential characteristics of the transfer matrix. In the second section, basic types of transfer matrix representation are introduced as well. The development of the transfer matrix for different types of Helmholtz resonators are presented in the third section. Finally, a comparison between LEM and TM is provided. Introduction The transfer matrix method (also called the tr ansmission matrix or the fourpole parameter representation) has been widely used in electrical and structur al engineering (Munjal 1987). In both fields, the analysis of comp licated systems has been greatly simplified by the use of this method. Complex circuits or structures can be simply regarded as bl ack boxes with one input port and one output port, as shown in Figure 31 Matrix algebra can then be applied to the general treatment of such networks The interconnection of parallel seriesparallel, and parallelseries network combinations can then be handled by simple linear addition or multiplication of the transfer matrix (Van Valkenburg 1964). In 1930, Stewart and Lindsay first applied th e transfer matrix method to acoustic filter design, although the name of the transfer matrix was not us ed in their book (Stewart and Lindsay 1930). Later, Peterson et al. (1950) presented a dynamic an alysis of the cochlea in the human ear using the transfer matrix method. In their work, fourpole parameters were deduced based on the equivalent Tnetwork representation of the cochlea. PAGE 53 53 After this time, the transfer matrix met hod was more and more commonly used for modeling a variety of acoustic problems such as Muffler/silencer design w ith/without mean flow (Igarashi and Toyama 1958; Al fredson and Davies 1971),wave propagation in a duct with a simple and/or complicated structure (Miles 1981; Munjal and Prasad 1986), and acoustic absorbers (Parrott and Jones 1995). Alfredson et al. (1971) presented fourparameter models of a number of silencer components, such as the s udden area change and an extended outlet and inlet based on onedimensional linear theory with m ean flow. It was hypothesized that any possible pressure discontinuity across the area change was caused by an entropy variation. Comparisons between measurements and numerical prediction s for these components were made, and it was found that onedimensional models provided a cceptable accuracy for silencer design. Miles (1981) modeled a variable area duct, su ch as a nozzle, as a series of exponential tubes with each tube being shor t enough such that the mean flow could be regarded as uniform along it. The transfer matrix for each subtube was derived by solving a constantcoefficient acoustic state variable differential equation, and the overall transfer matrix was then obtained by multiplying the individual duct transfer matrices. Munjal et al. (1986) developed the transfer matrix method for a duct with an axial temperature gradient and mean fl ow and a similar analysis was performed by Peat (1988). Both analyses were limited to ducts with small axial mean temperature gradients. Sujith (1996) extended the transfer matrix method to a uniform duct with an arbitrar ily large axial mean temperature gradient. His analysis neglected mean flow and is thus only valid for Mach numbers up to approximately 0.1. Lung and Doige (1983), Munjal (1987) and Be nade (1988) derived a transfer matrix model for the case of spherical wa ve propagation in conical pipe s without mean flow. Based on PAGE 54 54 a general differential equation for the propagati on of sound in a variab le area duct with low Mach number flow, Easwaran et al. (1991) presented an expressi on for the fourpole parameter model of the duct system. The shap e of the duct was set to hyperbol ic or parabolic. The conical, exponential, catenoidal, sine and cosine ducts were shown to be special cases of a hyperbolic duct. It was shown that the mean flow can cau se a distinct change in the individual four parameters of the matrix and a transmission loss in a variablearea duct. Parrot et al. (Parrott and Jones 1995) used the transfer matrix met hod to predict surface impedance and sound absorption for paralleleleme nt acoustic liners. The transfer matrix, derived from Zwikker and Kostens theory for propagation of sound in small tubes (Zwikker and Kosten 1949), provided excellent agreement betw een predicted and measured normalincidence impedance for ceramichoneycombstructure distributed systems. More recently, Song and Bolton (2000) used a transfermatrix approach to estimate the acoustic properties of rigid por ous materials. In their pape r, the modified twomicrophone method was used to estimate the 22 transfer matrix of porous material samples, and the result was used to determine acoustic properties of th e material, such as the characteristic impedance and complex wave number. Generally, the application of the transfer ma trix method follows one of two paths. The first uses a onedimensional or quasi onedimensional approach while the second is based on multimodal theory. The former is adopted when the propagation of higherorder modes is minimal and/or neglected. The latter is more complicated and often combines the boundary element method (BEM) or finite element method (FEM) in acoustic applic ations where higherorder modes are not neglected (Wan g et al. 1993; Ji et al. 1995). PAGE 55 55 Characteristics of the Transfer Matrix In the twoport network of Figure 31 four variables are identified: two effort variables and two flow vector variables. A transfer matrix that relates the effort a nd flow state variables is of the form (Van Valkenburg 1964) 111122 121222VTTV ITTI (31) where 1V and 1I are the input effort and flow vector variables, respectively; 2V and 2I are the output effort and flow vector variables, resp ectively; and Tij is a matrix entry in the transfer matrix of the twoport network. The negative si gn before the output current vector variables arises from the convention which defines the po wer flow as positive out of the network. In contrast, the twoport network in acoustic applications, shown in Figure 32 defines the output flow vector as positive when it is leaving the twopor t network. In such cases, the transfer matrix is given by 111122 121222TT TTPP QQ (32) Choice of State Variables The definitions of the general state vari ables in different domains are listed in Table 31 In acoustic applications, the effort an d flow variables are acoustic pressure,P, and volume velocity, Q. The volume velocity is chosen as a flow variable because this quantity will remain continuous, even discontinuities present in a duct. However, particle velocity U is often The convention ite is adopted in the context of this dissertation except for specific indication. Furthermore, Instead of p and u the phasor of the acoustic pressure P and the phasor of the acoustic velocity Uare adopted in this chapter. PAGE 56 56 adopted as a flow variable. In such a case, modi fications should be made for the transfer matrix to take the area change of the acoustic network into account. For the system shown in Figure 32 the PQ formulation is given by 12 1112 12 2122QQ QQPP TT QQ TT (33) where 1,2;1,2Q ijTij is an entry of the PQ transfer matrix, where the superscript Q indicates that the flow variable is the volume velocity. Clearly, 22 111 01Q QP TP (34) 22 121 01Q PQ TP (35) 22 211 01Q QP TQ (36) and 22 221 01Q PQ TQ (37) Physically, 111QT is an opencircuit effort (pressure) gain, 121QT is a shortcircuit transfer admittance, 211QT is an opencircuit transfer impedance, and 221QT is a shortcircuit current (volume velocity) gain. The PU formulation for the system shown in Figure 32 is 12 1112 12 2122UU UUPP TT UU TT (38) PAGE 57 57 If the temperature remains constant in the acoustic network, one can also choose 00cU as the flow variable, where00c is the characteristic acoustic impedance at this temperature (Ingard 1999). The transfer matrix of an acoustic circuit when adopting acoustic pressure P and the flow variable00cU as the two state variables is 0000 000012 1112 001002 2122cUcU cUcUPP TT cUcU TT (39) Note that all entries of the 00PcU transfer matrix are dimensi onless. For waves propagating within a duct with a constant crosssection, th e relationship between acoustic particle velocity U and volume velocity Q is QUS (310) where S is the projected crosssectiona l area perpendicular to the pa rticle velocity direction. Hence, from Eqs.(33)(310), one has 21222 1111121222122 11QQ UQUQUUTTS TTTTSTT SS (311) and 0000000012221222 1111122122 11QQQ cUcUcUcU QTScTTS TTTTT cSS (312) All of these three formulations can be used in an acoustic analysis. However, there are some limitations for the application of these tran sfer matrices. For instance, Eq. (39) provides a transfer matrix with dimensionless elements. Th e acoustic analysis is therefore simplified with these dimensionless matrix elements (Ingard 199 9). However, Eq. (39) cannot be used for a duct with a continuously changing crosssect ional area, such as the Webster horn. PAGE 58 58 Properties of the Transfer Matrix Re lated to Special Acoustic Systems Knowledge of properties of the transfer matrix facilitates the derivation of the transfer matrix for some special acoustic systems (i.e., reci procal systems). For a reciprocal system, the transfer matrix can be determined when three independent entr ies are known. Reciprocal system A reciprocal system refers to one in whic h the system response is the same when the position of the input and output terminal are interchanged (Van Valkenburg 1964; Pierce 1989). Mathematically, a reciprocal acoustic system satisfies 2112 21 00PPPP QQ (313) which consequently requires that the determinant of the PQ formulation of an acoustic twoport network equals unity, 112212211QQQQTTTT (314) For the 00PcU formulation, the determinant of the transfer matrix should be 000000001122122112 cUcUcUcUTTTTSS (315) where 1S and 2S are the projected crosssectional area s of port 1 and port 2, respectively. Clearly, Eqs. (314) and (315) reduce the number of independent transfer matrix parameters to three. From a practical standpoint, a twoport ne twork that is reciprocal will thus simplify the analysis and permit determination of the transfer matrix when three independent elements are known. In some cases, an acoustic system, i.e. an ac oustic duct system, may not be reciprocal when mean flow is present or area discontinuities occu r. The presence of mean flow in an acoustic duct system causes the upstream phase speed to be different from the downstream phase speed, PAGE 59 59 and thus breaks down the reciprocity of the sy stem (Munjal 1987). At an area discontinuity, evanescent higher order modes are ex cited, which may result in the violation of the reciprocity of the system (Ingard 1999). Conservative system If a system is conservative, the net power flux into the system is equal to zero. Mathematically, for a conservative acoustic twoport system, it is required that 11QT and 22QT be real, and 12QT and 21QT be purely imaginary (Van Valkenburg 1964). Basic Types of Elements of Acoust ic Transfer Matrix Representation Dividing a complicated acoustic system into several subsystems for which the transfer matrix model can be easily developed facilitates the derivation of the transfer matrix representation. The overall tr ansfer matrix of the system can then obtained by addition or multiplication of the subsystem transfer matrices. There are three basic types of elements in a transfer matrix representation for an acoustic network.§ They are the distributed element, the series lumped element, and the parallel lumped element (Munjal 1987). Their equivalent circuit representations are shown in Figure 33 For the distributed element shown in Figure 33a a simple corresponding case is plane wave propagation along a lossless duct in the x direction. The crosssec tional area of the duct is S The governing wave propagation equa tions are given by (Blackstock 2000) 111 jkxjkxPAeBe (316) 111 00jkxjkxASBS USee ZZ (317) 222jkxjkxPAeBe (318) § An exception is the transfer matr ix representation for a flare tube. PAGE 60 60 222 00jkxjkxASBS USee ZZ (319) where A and B are coefficients, and 0 Z is the characteristic specific acoustic impedance. Letting 20 x and 21lxx as shown in Figure 33a one obtains 2PAB (320) 2 00ASBS US Z Z (321) 1 jkljklPAeBe (322) 1 00jkljklASBS USee ZZ (323) Equations (320) (323) result in 0 12 0 12cossin sincos kljZSkl PP jklZSkl QQ (324) where QuS is the acoustic volume velocity. Note that 2 00cossinsin1kljZSkljklZS (325) the system is a reciprocal system. Moreover, the system is also a conservative system. For a series lumped element shown in Figure 33b one has 211 s PPQZ (326) 21QQ (327) In matrix form, one has 12 121 01sPP Z QQ (328) PAGE 61 61 where s Z is the inline lumped complex impedance. An example of a series lumped element is wave propagation across an area contraction, as shown in Figure 34 The relationships between 11(,) PQ and 22,PQare given by Eq. (328), and s Z is given by (Ingard 1999) 2 00 0 218 2 23sckr j kr Z r (329) which accounts for the viscous loss and radiati on impedance at the area discontinuity. More details on the areaexpansion are presented in Section 3. Clearly, such a system is not reciprocal (112212211QQQQTTTT ) due to the area discontinuity, and not conservative due to a portion of the energy is dissipated at the area discontinuity. For a parallel element as shown in Figure 33c one can write the relationship between input and output para meters directly as 21PP (330) 211pQPZQ (331) or 12 1210 11pPP Z QQ (332) An example of such a case is shown in Figure 35 where a Helmholtz resonator is used as a sidebranch to a uniform tube. The relationships between 11(,) PQ and 22,PQare given by 12 1210 11HRPP QZQ (333) PAGE 62 62 where H R Z is the complex acoustic impedance of the HR. Such a system is not reciprocal (112212211QQQQTTTT ) due to the area change at the juncti on and not conservative due to a portion of the energy is dissipated by the HR. Transfer Matrix Representation for EMHR In this section, the transfer matrix for an EMHR mounted at the end of plane wave tube (PWT) will be developed. As shown in Figure 36 the EMHR mounted at the end of the PWT is comprised of the following four components: area contraction, area expansion, duct element and piezoelectric backplate with shunt network. Hence, it is conve nient to develop the transfer matrix for each of these elements, and then multip ly these matrices sequentially to obtain the overall transfer matrix of the EMHR Area Contraction The area contracti on is shown in Figure 37a Due to abrupt change of the area, even with a planar incident wave, evanescent higher order ac oustic modes will be excited in the vicinity of the discontinuity, thus the acoustic field near the discontinuity will have a transverse component. Such effects can be taken into account in te rms of plane wave vari ables by means of an additional mass and resistance (Karal 1953; Er iksson 1980; T.Y.Lung and Doige 1983; Ingard 1999). Following Karal and Ingard, the relati on between the acoustic variables for the area contraction is given by 54004444PPcAUA, (334) 00550044cUAcUA (335) where 4P and 5P are the plane wave pressure component be fore and after the discontinuity. Eqs. (334) and (335) represent continuity of th e pressure and the volum e velocity across the discontinuity, respectively. The matrix represen tation for the area contr action is thus given by PAGE 63 63 544 005450041 0PP cUAAcU (336) where2 4 A r is the crosssectional area of the neck of the Helmholtz resonator, 2 5 tube A D is the crosssectional ar ea of the tube, and 4 is the normalized specific acoustic impedance given by (Ingard, 1999a) 2 04 40 00018 2 23A jkr cc (337) Thus, the transfer matrix fo r the area contraction is 4 451 0ACT AA (338) Area Expansion Similarly, the relationship between the acousti c variables for the area expansion, shown in Figure 37b is given by (Ingard, 1999a) 32003322PPcAUA, (339) 00330022cUAcUA (340) or in transfer matrix form, 32 233 003002 231 0PP AA cUcU AA (341) where 2 3 A r is the crosssectional area of th e neck of Helmholtz resonator, 2 2 A R is the crosssectional area of the cavity, and 3 is the normalized specific acoustic impedance given by 2 03 30 00018 211.25 23A jkrr ccR (342) Thus, the transfer matrix fo r the area contraction is PAGE 64 64 233 231 0AEAA T AA (343) Duct Element Determination of the acoustic velocity and pr essure fields due to sound wave propagation in a long circular duct is a cl assic acoustics problem. Tijdeman (1974) reviewed this topic and showed that the solution obtai ned by Zwikker and Kosten (19 49) has the widest range of validity. For planewave propa gation in a duct with radius R as shown in Figure 37c the pressure and axial velocity component are gi ven by (Zwikker and Kosten 1949; Tijdeman 1974) 2 00kxkxc PAeBe (344) and 32 0 0 32 01kxkxJjs j UcAeBe Jjs (345) where is the complex propagation coefficient, 0kc is the wave number, rR is the dimensionless radius, sR is the shear wave number or Stokes number, is the ratio of specific heats, and is the dynamic viscosity coefficient of the air. The complex propagation coefficient is given by 32 0 32 2Jjs n Jjs (346) where 1 32 2 32 01 1 Jjs n Jjs (347) PAGE 65 65 PrpC, Pris Prandtl number and iJis the thi order Bessel function of the first kind. From Eq. (345), the mean velocity over crosssectional area of the duct is thus obtained by 2 0 32 00 2 32 0 0 32 2 0 32 0 01 2 12R kxkx R kxkx kxkxUUrdr R cAeBeJjs j rdr R Jjs Jjs jc AeBe Jjs c j AeBe n (348) where the following identities of the Be ssel function are used (Watson 1996) 112nnnn JzJzJz z (349) 01 0 azJzdzaJa (350) Thus, from Eqs. (344) and (348), one can deduce the relationship between 22, PU and 11, PU, shown in Figure 38, as 21 002001coshsinh/ sinhcosh PkLkLGP cUGkLkLcU (351) where Gjn and and n are given by Eqs. (346) and (347). The same result can be obtained for the relationship between 44,PU and 33,PU. Thus, the transfer matrix for the duct element shown is given by coshsinh/ sinhcoshDEkLkLG T GkLkL (352) PAGE 66 66 Note that such a system is reciprocal, but not conservative because a portion of the energy is dissipated at duct. When the wave field in the duct is isentropic (j and 1G), Eq. (351) can be simplified to 21 002001cossin sincos PkLjkLP cUjkLkLcU (353) or in the p Q formulation given by Eq. (324) Clamped Piezoelectric Back plate with Shunt Loads To develop the transfer matrix representa tion for the piezoelectricbackplate (or PZTbackplate), it is assumed that the maximum dime nsion of the PZTbackplate is much less than the bending wavelength of interest, and that only linear behavior of the PZTbackplate needs to be considered. With these assumptions, an e quivalent twoport netw ork including a lossless transformer can be developed for the PZTbackplate, as shown in Figure 38 where 1P is the pressure exerted on the PZTbackplate, 1Q is the incident volume velocity, and V and I are the voltage and current at the electric al port, respectively. Associated with the twoport network are two impedances that are measurable properties of the system. One is the blocked electrical impedance, 01eB Q eBV Z I jC (354) and the other is the shortcircuit acoustic impedance 01aDaDaD aD VP ZRjM QjC (355) PAGE 67 67 where the definition of eBC aD R aD M and aDC are listed in Table 22. Furthermore, as indicated in Chapter 2 the lossless transformer converts energy between the acoustical and electrical domains and obeys the transformer relations 1 I Q (3.56) and PV (3.57) where is the impedance transformation factor. He nce, using network theory (Van Valkenburg 1964), for the twoport network shown in Figure 38 one has 1 eBeBQ I V j CjC (358) and 11 aDPVZQ (359) Substituting Eq. (358) into (359) results in 2 11 aD eBeBI PZQ jCjC (360) where 2 aDeB Z jC defines the opencircuit (0 I ) acoustic impedance of the PZTbackplate. On the other hand, substitution of 1P and Vfor 1Q in Eq. (358) from Eq. (359) results in 1 22 eBaD eBeBaDeBaDjCZ P I V jCjCZjCZ (361) where 2 2 221111 11eB eBaD EM eBeBaDeBaDeBeBeFjC jCZ CjCZCZjCCC (362) PAGE 68 68 eFC is the free electric compliance (10 P ) and EM is the electromechani cal coupling factor of the piezoelectric backplate. Finally, from Eqs. (358) and (359), one has 2 1 eBaDaDjCZZ PVI (363) 11eBjC QVI (364) or in the matrix form of 2 1 1. 1eBaDaD eBjCZZ P V Q I jC (365) Consequently, when the electrical boundary condition eLVIZ is given, the acoustic impedance, 11PQ can be obtained for a given PZTbackplate as 22 1 111eBaDaDeBaDeLaD eBeBeL j CZVIZjCZZZ P QjCVIjCZ (366) The 00PcU formulation of Eq. (365) is given by 2 1 001 0000 eBaDaD eB DDjCZZ P V cU I jcCc AA (367) where 2 213DAR is the effective area of the PZTbackplate due to the nonuniform displacement of the clamped PZTbackplate. PAGE 69 69 Transfer Matrix of the EMHR Next, the transfer matrix of the EMHR mounted at the end of the plane wave tube (PWT) is obtained by multiplication of the transfer matr ices of each subsystem in the order shown in Figure 36 1 54 233 00545 23 Transfer matrix for Transfer matrix f Transfer matrix for neck of the EMHR area contraction1 coshsinh 1 0 sinhcosh 0NNN NNNP ktktG AA cUAA Gktkt AA or area expansion 2 1 0000 Transfer matrix for cavity of the EMHR Transfer matrix for PZTbackplacoshsinh sinhcosheBaDaD CCC CCC eB DDjCZZ kLkLG GkLkL jcCc AA te EMHREMHR 1112 EMHREMHR 2122 Transfer matrix for the EMHR, V I V TT I TT (368) The subscript Nand C are for neck and cavity, respectivel y, and the definition of the other parameters refers to the analysis above. Eq. (368) simplifies the calc ulation of some acoustic characteristics of the EMHR. For example, when the electrical boundary condition is given by eLVIZ the specific acoustic impedance of the EMHR is EMHREMHR 1112 EMHR EMHREMHR 2122 eL eLTZT TZT (369) For LZ (open circuit), EMHR 11 EMHR EMHR 21T T (370) For 0LZ (short circuit), PAGE 70 70 EMHR 12 EMHR EMHR 22T T (371) For a normalincidence plane wave on the EMHR, the reflection coefficient R is EMHREMHREMHREMHR 11122122 EMHR EMHREMHREMHREMHR EMHR 111221221 1eLeL eLeLTZTTZT R TZTTZT (372) For LZ (open circuit), EMHREMHR 1121 EMHREMHR 1121TT R TT (373) For 0LZ (short circuit), EMHREMHR 1222 EMHREMHR 1222TT R TT (374) Transfer Matrix of the EMHR with Perforated Facesheet For an EMHR with a perforated plate instead of a small neck, as shown in Figure 39 the derivation of the transfer matrix is similar to the derivation above Here, it is assumed that the plate is thin compared to the wavelength of inte rest and that the phase difference of the particle velocity between both sides of the plate can be neglected. Thus, the transfer matrix of the perforated plate is given by 32 0030021 01pPP cUcU (375) where p is the normalized acoustic impedance of th e perforated plate gi ven by (Ingard 1953) 22 0 002 112 2220.85 2pcd t jktd cd (376) PAGE 71 71 and 2dD is the porosity of the perforated plat e. The overall transfer matrix of the EMHR with a perforated plate is thus given by 2 1 3 003 00001 coshsinh 01 sinhcosheBaDaD p CCC CCC eB DDjCZZ P V kLkLG cU I GkLkL jcCc AA (377) Comparison between TM and LEM As discussed in Chapter 2 when the dimensions of the acoustic device are much smaller than the wavelength of interest, the device com ponents can be lumped into idealized discrete circuit elements (Rossi 1988). In comparison with the transfer matrix representation of the acoustic system, the LEM decouples the temporal and spatial variables associated with the acoustic field. There is no spatial variation for a lumped element. On the other hand, sometimes the LEM is no more than a reduced version of the transfer matrix representation when the quasistatic assump tion is satisfied. For example, as shown in Figure 310a a duct with a soundsoft termination redu ces to the following using Eq. (324) 000 1 1tan as 1 jckll P jkl QSS (378) where 0lS is the lumped acoustic mass of the short tube which is the same as aN M discussed in Chapter 2**. One more example, as shown in Figure 310b is when the duct is ended by a soundhard termination, simila rly, from Eq. (324), one has 00 1 1 2 00cot 1 as 1 jckl P kl QS j c (379) ** In chapter 2, aN M is the acoustic mass of the neck of the Helmholtz resonator. PAGE 72 72 where Sl is the volume of the tube, and 2 00c is the lumped acoustic compliance aCC However, in some cases, more assumptions must be satisfied for the LEM to be consistent with the transfer matrix representation. Fo r instance, a duct terminated with a complex boundaryT Z shown in Figure 310c reduces to the following using Eq. (324) 2 000000 1 10000cossincot sincoscotTT TT Z kljcSklcZjklScS P QjklZcSklZcjklS (380) When 1 kl Eq. (380) reduces to 22 000000 1 100cot() cot1()TTaC TTaCcZjklScSZjCcS P QZcjklSZjC (381) where 2 00aCCc Furthermore, when 2 00 TaC Z jCcS, (382) one has 1 1() 1()TaC TaCZjC P QZjC (383) which is the LEM of the system. Figure 311 shows the comparison between LEM and transfer matrix prediction results for a shortcircuited EMHR. The geom etry of the EMHR is listed in Table 42 (Case I). Both models predictions match pretty well except fo r near the second resona nt frequency of the EMHR. The second resonant frequency of the EM HR corresponds to the resonant frequency of the piezoelectric backplate. The acoustic impe dance of the piezoelectric backplate becomes relatively small near the res onant frequency, as shown in Figure 312 where Eq. (382) thus breaks down. Consequently, the LEM is not consis tent with the transfer matrix representation even though the quasistatic assumption is satisfied. PAGE 73 73 Table 31. Conventional st ate variable definitions for the twoport network. Electric Domain Mechanical Domain Acoustic Domain Effort Voltage Force/ Moment Acoustic pressure Flow Current Velocity/ Angular velocity Volume velocity PAGE 74 74 1I2I1V2V Figure 31. A twoport network wi th reference directions for the positive direction of the current variables indicated. 1Q2Q1 P 2 P Figure 32. An acoustic twopor t network with reference direc tion for the current variables indicated. PAGE 75 75 1P0 Z kl 1P2P S Z 1P2P P Z 1Q2Q2Q1Q1Q2Q2P Figure 33. Three basic types of elements in an equivalent circ uit representation for an acoustic network. (a) A distributed element. (b) A series element. (c) A parallel element. r Figure 34. Illustration of an area contraction. PAGE 76 76 11, P Q22, P Q Figure 35. A HR mounted in the side of one duct. PAGE 77 77 5 P 4 P 3 P 2 P 1 P V eL Z 5U4U3U2U1UI 1 2 3 4 5 EMHR PWT1 1 P U2 2 P U3 3 P U4 4 P U5 5 P U eL Z VI 1 2 3 4 5 Figure 36. Modeling EMHR using the transfer matrix method, where each subsystem is denoted as: 1area contraction; 2duct elem ent; 3area expansion; 4duct element; 5piezoelectric backplate. PAGE 78 78 555,, A PU444,, A PU (a) 333,, A PU222,, A PU (b) 22, PU11, PU (c) Figure 37. Elements for derivati on the transfer matrix representa tion of the EMHR. (a) An area contraction. (b) An area expansi on. (c) An acoustic duct system. PAGE 79 79 11 P Q, VI eL Z 2R p h1R s h (a) V 1 P 1Q I aDRaD M aDC:1 eBCeL Z P I (b) Figure 38. A piezoelectric ba ckplate and its equivalent ci rcuit representation. (a) A piezoelectric backplate with shunt loads. (b ) The equivalent circuit representation for the piezoelectric backplate with the lu mpedelement twoport network indicated. td L DeL Z 3 3P U2 2P U1 1P U, VI Figure 39. An EMHR with perforated facesheet. PAGE 80 80 11, P U22, P U a M 1 k (a) 220PU 11, P U22, P U aC1 k (b) 22PU 11, PU22, PU aC1 k T Z Loading effect? (c) 22 TPUZ Figure 310. Comparison between the TM and LEM. (a) Schematic of a duct terminated with soundsoft termination. (b) Schematic of a duct wi th soundhard termination. (c) Schematic of a duct with complex termination. PAGE 81 81 1500 2000 2500 3000 0 5 10 15 LEMShort TMShort LEMOpen TMOpen 1500 2000 2500 3000 5 0 5 10 15 Freq.[Hz] Figure 311. Comparison of prediction results fo r the normalized specific acoustic impedance of a shortand opencircuite d EMHR using TM and LEM. PAGE 82 82 1500 2000 2500 3000 106 108 1010 1012 1014 1016 Freq. [Hz] ZaCZaD [ 0c0/( R2)]2 ZaC+aDTM ZaC+aDLEM Figure 312. Illustration of contributions to th e variation in the predic tion performance of the LEM and TM. PAGE 83 83 CHAPTER 4 EXPERIMENTAL TECHNIQUES This chapter presents the experimental setup used to evaluate the performance of the EMHR and to validate the TM and LEM of the EMHR. The chapter is organized as follows. In the first section, a number of methods for meas uring the acoustic properties of an impedance specimen are briefly introduced, followed by a re view of the theoretical basis for the twomicrophone method (TMM). The second section pres ents the method for measuring the damping coefficient of the EMHR as well as the parameter extraction of the piezoelec tric backplate. The experimental setup is detailed in the third section. The fourth section presents the sample construction. Acoustic Impedance Measurement Introduction A number of methods have been developed ove r the last few decades for determining induct acoustic properties such as the reflection and absorption co efficient. Among these methods, four are commonly used: the standing wave ratio method (SWM) using a probe microphone, the twomicrophone method (TMM), the singlemi crophone method (SMM), and the multipoint method (MPM). The SWM has commonly been used to measure the acoustic impedance (Lippert 1953; Beranek 1988). The SWM is implemented by means of a plane wave tube (PWT) and a traversable probe microphone. It is assumed that only plane waves are propagating along the tube, thus the operating freque ncy of the SWM is limited to below the cuton frequency of higherorder propagating modes in the tube. Th e probe microphone is used to trace the acoustic pressure magnitude along the cen ter line of the duct versus the distance from the impedance sample under test. The maximum and minimum pre ssures and their locations (extrema loci) in PAGE 84 84 the duct can thus be determined and the standing wave ratio (SWR) can be calculated. The SWR is defined as the ratio of successi ve maximum and mi nimum pressures max minP SWR P (41) The amplitude of the reflection coefficient is readily determined from the SWR (Blackstock 2000) 1 1 SWR R SWR (42) The phase of the reflection coefficient is calcul ated from the location of extrema loci. The acoustic impedance of the sample can then be calc ulated from the complex reflection coefficient. The SWM is quite accurate with a si ngle tone source over the 0.1 to 10 00c impedance range (Jones and Stiede 1997). However, the SWM is tedious and timeconsuming due to the required physical movement of the trav ersing microphone. Furthermore, the SWM cannot be used to measure induct acoustic pr operties with mean flow. Seybert and Ross (1977) introduced the TMM which is a standardized technique to measure the impedance and absorption of acoustical materials (ASTME105098 1998). The TMM also employs a PWT but uses two stationary, wallmounted microphones to simultaneously measure acoustic pressure at two known positions in the tube. Broadband excitation is used to find the autoand crossspec tral density functions so that the whole standing wave pattern in the tube can be found at once. The TMM was further developed analytically and experimentally by Chung and Blaser (1980a; 198 0b). Their method uses a simple transfer function relationship between two locations on the tube wall to decompose the acoustic wave in a tube into its incident and reflected com ponents. The wave decomposition leads to the determination of the complex reflection coefficien t, from which acoustic properties of the test PAGE 85 85 sample can be found. Chu (1986) extended th e TMM by including the effect of the tube attenuation due to thermoviscous effects, thus allowing more freedom in choosing the positions of the microphones. Chu (1988) also investig ated the choice of microphone positions, and indicated that a fixed choi ce of the microphone locations compromises the measurement accuracy. For an accurate measurement over di fferent frequency ranges, varied microphone positions are preferred, as long as the separati on of two microphones is not close to onehalf wavelength. Compared to the SWM, the TMM does not require traver sing movement of the microphone and thus is considerably more efficient. Furthermore, the effect of the mean flow can be taken into account. Two major disa dvantages of the TMM are the requirement of accurate knowledge of amplitude and phase rela tionships between the two microphones and the singularities associated with onehalf wavelength microphone spacing. The former can be mitigated using a sensorswitching techni que (Chung and Blaser 1980) in which the measurement of the transfer function is made w ith the initial microphone locations, and then the measurement is repeated with microphone locations switched. The final measurement result is obtained from the geometric mean of the original and switched results. Schultz et al. (2007) indicated that there are still other factors that cause bias and precision errors for the TMM, such as temperature uncertainty, micr ophone separation uncertainty, etc. Such factors can be taken into account during uncertainty analysis for TMM. Chu (1986) introduced the socalled single mi crophone method (SMM) in which a single microphone is used to measure the sound at two locations, and then the transfer function is derived from the measured autoand crossspectral densities with respect to the source signal. The SMM is intended to eliminate the elaborate calibration procedure and any error associated with phase mismatching found in the TMM. In essence, the SMM is an alternative PAGE 86 86 implementation of the TMM (Jones and Stiede 1 997). A potential advantage of the SMM is its use for high frequency measurement, where the diameter of the tube is small and space limitations do not allow for the switching ri g for two microphones. Both the SMM and the TMM are susceptible to singularity issues. The multiplepoint method (MPM) employs multiple fixed microphones or a traversing probe microphone to measure acoustic pressure at a minimum of three distances from the surface of the impedance sample (Jones and Parrot 1989). A least square fitting technique is then used to fit a wave propagation model to the point measur ements. The wave propagation model includes the effect of the mean flow and the duct wa ll absorption and permits reconstruction of the standing wave pattern in the duct. The acousti c properties of the impedance sample are then deduced (Jones and Parrot 1989; Hang and Ih 1998). The MPM can achieve very accurate measurement results if a minimum of six pressu re measurements are taken at points evenly spaced over a distance of onehal f wavelength (Jones and Parrot 1989). The physical movement of the hardware is required for the MPM if the traversing pr obe microphone is adopted. The measurement procedure is thus timeconsuming. However, the multifrequency (source) version of the multipoint method (MPMPR) can reduce th e time necessary to complete an acoustic impedance measurement (Jones and Stiede 1997). Theoretical Basis of the TMM As discussed above, the TMM is timesaving ye t accurate. In this study, the TMM is used to measure the acoustic impedance of the EMHR. An illustration of the TMM is shown in Figure 41 Only plane waves are assumed to propagate along the tube, no mean flow exists, and the effects of tube attenuation are negligible. The acoustic field inside the tube is given by (Blackstock 2000) PAGE 87 87 ,jkxjkxPxfPfePfe, (43) where P and P denote the Fourier transfor ms of, respectively, the incident and the reflected acoustic pressures at 0 x The P and P can be resolved by measur ing the complex pressure P at two different locations, Mic.1 and Mic.2, along the PWT. The pressure spectra at two microphone locations are expressed as 2 jkljklPPfePfe, (44) and 1 jklsjklsPPfePfe (45) where s is the distance between the two microphones and l is the distance from the surface of the impedance sample to the nearest microphone. The complex reflection coefficient R at 0 x is thus given by 2112 21 12 jksjks j2klsj2kls jks jksPPeHe P Ree PePP eH (46) where 121211 HEGG is the frequency response func tion between Mic.1 and Mic.2. E is the expectation operator, 12 G is the estimated cross spect rum between two microphones, and 11 G is the estimated autospectrum of Mic. 1 (Bendat and Pierso l 2000). The specific acoustic impedance of the impedance sample can then be then given by (ASTME105098 1998) 001 1 R Zc R (47) where 00c is the characteristic acoustic impedance of the medium in the tube. Eq. (47) is valid only if Eqs. (44) and (45) are linearly independent (i.e. 1,or20,1,2...jksesnn ). PAGE 88 88 As mentioned earlier, a switchingsensor technique can be used to mitigate the elaborate calibration procedure and any error associated with phase mismatching in the TMM. What follows is a brief presentation of the theoretical basis behind this technique. By assuming that both microphone channel systems are linear and tim einvariant and devoid of noise, one has 1,2iimiPfPfHfi (48) as shown in Figure 42 iPf(1,2i ) is the Fourier transform of the actual acoustic pressure at the microphone locations. miHf is the frequency response associ ated with the first and second microphone channels. iPf is the Fourier transform of the measured acoustic pressure via the microphones. When the measurement is taken w ith the initial microphone positions, the transfer function from Mic. 1 to Mic. 2 is given by 2211 1221 12 11 11 1111 mm o mmPHPH GPP HE G PP PHPH (49) where 12oH denotes the original transfer function from Mic. 1 to Mic. 2. Then the measurement is taken with the microphone locations switched. The transfer function from Mic. 1 to Mic.2 is now given by 1221 12 2121mm s mmPHPH H PHPH (410) where 12 s H denotes the switched transfer function from Mic. 1 to Mi c. 2. Thus, the transfer function from Mic. 1 to Mic. 2 can be obtai ned from the geometric mean (Chung and Blaser 1980) 1221 12 12 11 o sHPP H H PP (411) PAGE 89 89 Eq. (411) shows that the complex frequency re sponse characteristics of the microphone system do not affect the measurement results. In other words, the amplitude and phase characteristics of two microphone channel systems need not match perfectly, as required when the TMM is used (Seybert and Ross 1977). Uncertainty Analysis of the TMM Two types of errors are associated with the TMM: bias and ra ndom errors (ASTME105098 1998; Seybert and Soenarko 1981; Bodn and b om 1986; Schultz et al. 2007). The bias error is the fixed, systematic, or constant compone nt of the total error. It is the same for each measurement. The random error is the random co mponent of the total er ror and has a different value for each measurement (Coleman and Stee le 1989). Random errors associated with the TMM can be kept low by ensemble averaging a nd by maintaining a high coherence between the microphones (ASTME105098 1998; Bodn and bom 1986). Several sour ces of bias error exist; for instance, errors in microphone separa tion and distance from th e test surface of the specimen can cause significant bi as errors associated with the TMM (Bodn and bom 1986). Bias error also can arise from the tube attenua tion and computational error in postprocessing (ASTME105098 1998). The tube attenuation sh ifts the loci of pressure minimums asymmetrically in the standing wave pattern as the distance from the specimen increases (ASTME105098 1998). The bias error caused by the tube attenuation can be minimized by placing microphones as close to the speci men as possible (ASTME105098 1998). Theoretically, the quality of the measuremen t results can be evaluated via the relation between the total error and the true value of the measurand. The total error is the sum of bias errors and random errors. However, one does not know the true value of the measurand in most cases. Alternately, one can evaluate the degree of goodness of the measurement by the PAGE 90 90 statement that the true value of the measurand, true X lies within the interval bestX X U with % C confidence, where best X is the mean value of the measurements and XU is the uncertainty in X of the combination of bias and ra ndom errors (Coleman and Steele 1989). Schultz et al. (2007) presented the uncertainty analysis of the TMM for acoustic impedance testing. They employed both multivariate uncerta inty analysis and the Monte Carlo method to provide a systematic framework for computing the uncertainties of the TMM. The multivariate method, which involves the propagation of comp onent uncertainties, assumes small component uncertainties that cause only linear variations in output quantities. The component uncertainties include uncertainty in the fr equency response function (FRF ), uncertainty in microphone location, temperature uncertainty, etc. The mu ltivariate method matches the results from the Monte Carlo method when all comp onent uncertainties are small (0.1% ) or the specimen is sound hard. However, the component uncertainti es are normally large enough to invalidate the linear assumption. Thus, although the Monte Carl o method is more computationally intensive, the Monte Carlo method is recommended by the au thors for accurate uncer tainty estimation, Parameter Extraction of the Piezoelectric Backplate Damping Coefficient Measurement It is difficult to accurately model the dampi ng loss of the piezoelectric backplate of the EMHR. The damping loss may arise from acoustic radiation, thermoelastic dissipation, compliant boundaries, and other intrinsic loss m echanisms (Tilmans et al. 1992). The LEM as well as TM employed in this study only accounts for the acoustic radiation loss. The damping loss measurement thus provides a way to check if the acoustic radiation loss represents the damping loss of the system. The logarithmic d ecrement method is used to measure the amount of damping of the piezoelectric backplate of th e EMHR. Assuming the piezoelectric backplate is PAGE 91 91 a SDOF system, and its free damped vibration is similar to that in Figure 43 then the logarithmic decrement is de fined as (Meirovitch 2001) 1 2 22 ln 1x x (412) where is the damping coefficient of the system, and 1 x and 2 x are two sequential peak displacements corresponding to the times 1t and 1tT where T is the period of the system. Eq. (412) yields 2 22 (413) For small damping, such that 1 Eq. (413) reduces to 2 (414) The accuracy of the damping coefficient estimati on in Eq. (413) can be improved if the peak displacement is measured at two different times separated by a given number of periods. For instance, let 1 x and 1i x be the peak displacements corresponding to the times 1t and 1it, where 111,2,3...ittiTi and Tis the period of the system. Moreover, 221 1 1i ix e x (415) from which one has the logarithmic decrement 1 2 121 ln 1i x ix (416) Rearranging and letting 1 ii gives 1lnln1ixxi (417) PAGE 92 92 The plot lni x versus i should have the form of a straight line with the slope if the measurement is exact. Hence, the accuracy of the estimation for the damping coefficient can be improved using Eq. (417). In particular, N peak displacements (,1,2,3...ixi ) are chosen. lni x and 1iyi are then calculated. Then a straight line of the form iizayb (418) which minimizes 22 11lnlnNN iiii ii x zxayb (419) is determined. Note that a corresponds to in Eq. (417). The damp ing coefficient is thus calculated by using Eq. (413) 222a a (420) Effective Acoustic Piezoelectric Coefficient Deduction Both LEM and TM of the EMHR assume that the piezoelectric backplate is clamped. However, it is difficult to obtai n a perfectly clamped boundary condi tion in practice. Thus, it is necessary to experimentally i nvestigate the impact of the practical boundary condition on the prediction performance of the models. In th is study, a parameter extraction method is implemented to deduce the effective acoustic piezoelectric coefficient, Ad, which is significant in determining the impedance transformer factor, and the blocked electrical capacitance, eBC. As discussed in Chapter 2 as well as Appendix C and eBC affect the prediction performance of the models, especially, when the EMHR is connected with passive loads. The method to experimentally determine Ad is as follows (Prasad et al. 2006). First, the deformation of the clamped piezoelectric backplate is measured at discrete points while applying a voltage to the PAGE 93 93 plate. Second, the deflection mode shape of th e plate is reconstructed using the least square curvefitting method. Finally, Ad is calculated using Eq. (C4). Experimental Setup Acoustic Impedance Measurement Setup The schematic of the experimental setup for the acoustic impedance measurement is shown in Figure 44 The plane wave tube (PWT) has a cr osssection of 25.4 mm by 25.4 mm and is 96.5 mm long, which permits a plane wave acousti c field at frequencies up to 6.7 kHz (Horowitz et al. 2002). Three Brel and Kjr (B&K) type 4138 micr ophones are used simultaneously to measure the acoustic pressure. Two micr ophones, labeled as Mic. 1 and Mi c. 2, are flush mounted in a rotating plug to the side of the impedance tube. The rotating plug is used to remove amplitude and phase mismatches between the microphones as explained in Section 1. The other microphone, labeled as Ref. Mic., is flush mounted to the end face of the impedance tube to measure the total acoustic pressure at the entrance of the resonator. This microphone also serves as a reference to ensure a constant SPL at the neck of the resonator. Furthermore, Mic. 2 is mounted as close to the test sp ecimen as possible. The distan ce between Mic. 2 and the test surface of the specimen is 32.00.8 mm with a 95% confidence interval es timate (Schultz et al. 2007). The distance is about one and onehalf times the duct dimension, and is enough to facilitate the evanescence of higherorder modes generated from the specimen as shown in Figure 45. The microphone separation is 20.71.1 mm with a 95%confidence interval estimate (Schultz et al. 2005). Thus, there is no frequency sin gularities for measurement below the frequency range of interest (6.4 kHz ). Please see Appendix D for more details. PAGE 94 94 All microphones are calibrated with a Bre l and Kjr 4228 Pistonphone while connected to a Brel and Kjr PULSE MultiAnalyzer System Type 3560. The PULSE system serves as the power supply and data acquisition and proces sing system for the microphones as well as the signal source. A pseudorandom waveform genera ted from the PULSE system is fed through a Techron Model 7540 power supply amplifier to drive a BMS H4590P compression driver. The pseudorandom signal, which has uniform spectra l density and random phase, is commonly used to avoid leakage effects of a nonperiodic signal. The driver, which can produce acoustic waves between 200 Hz and 22 kHz, is connected to one end of the PWT via a transition piece. The pseudo random waveform is bandpass filtered from 300 Hz to 6.7 kHz. Meanwhile, a FFT with 1000 ensemble averages is performed on each incoming microphone signal. Damping Measurement Setup The schematic of the experimental setup for the damping coefficient measurement is shown in Figure 46 The piezoelectric backplate is placed in air or within a vacuum chamber capable of producing a low pressure around 10 To rr (1300 Pa). A LKG32 high accuracy laser displacement sensor is used to measure the vibr ation response of the center of the plate. The repeatability of the sensor is 0.05 m, and th e measurement range is 5 mm. A rectangular waveform (1 Hz, 20.4Vpp) generated from the Agilent 33120A Waveform Generator is used to excite the piezoelectric plate. This signal is also used as the external trigger of the Tek TDS5104B oscilloscope which acqui res data from the laser displa cement sensor. In addition, 1024 ensemble averages are performed. Parameter Extraction of th e Piezoelectric Backplate The experimental setup for parameter extraction of the piezoelectric backplate is similar to the damping measurement setup with two exceptions The piezoelectric backplate is not placed PAGE 95 95 within the vacuum chamber and the transverse displacements along the ra dius of the backplate are measured to reconstruct the deforma tion of the plate due to applied voltage. EMHR Construction The EMHR samples are modularly constructed, as shown in Figure 47 The modular design permits the parameter studies of the orif ice diameter and thickness, cavity volume, and piezoelectric backplate geometric parameters. Th e sample is comprised of an orifice plate, cavity plate, piezoelectric diaphr agm cap/bottom plates and piezoelectric diaphragm. Except for piezoelectric diaphragm, all part s are made from aluminum. The piezoelectric diaphragm is a commercially available piezoceramic circular bend er disk (APC International, Ltd, model APC 850), which consists of a circular lead zirconate titanate (PZT) 850 patch bonded to a brass shim. A very thin silver electrode covers the PZT patch. A passive shunt networ k is then connected to the diaphragm. The piezoelectric diaphragm is clamped between the cap and bottom plates, which are bolted to the backside of the cavity plate. The orifice plate is att ached to the other side of the cavity plate when the acoustic impedance of the EMHR is measured. The orifice plate is removed when the damping measurement and para meter extraction are im plemented. Because the piezoelectric diaphragm is not demounted between measurements, the clamped boundary condition of the piezoelectric di aphragm is maintained for the acoustic impedance measurement, damping measurement and other parameter extraction. Table 41 summarizes the material parameters of the piezoelectric diaphragm. The geometric parameters of the EMHR samples are listed in Table 42 Two samples with different cavity dimensions are involved in this study. Th e dimensions of the piezoelectric diaphragm are the same for both samples. In order to evaluate the tuning performance of the EMHR samples, a variety of shunt networks were used to tune the EMHR. Table 43 shows one of the loads PAGE 96 96 matrices used in the acoustic impedance measurem ent. The measurement results are presented in Chapter 5 PAGE 97 97 Table 41. Material properties of the piezoelectric backplate. Properties Value Youngs Modules (2/Nm) 6.3E10 Poisson ratio 0.31 Density ( 3kgm) 7700 Relative dielectric constant 1750 Piezoceramic (APC 850)a Piezoelectric Strain Constant d31 ( p CN) 175 Youngs Modules (2/Nm) 11.0E10 Poisson ratio 0.38 Shim (260 half hard brass)b Density ( 3kgm) 8530 a. The data is adapted from APC International, Ltd http://www.americanpiezo.com/materials/apc_properties.html b. The data is adapted from www.matweb.com Table 42. Dimensions of the EMHRs (unit: mm, resolution 0.01 mm). Name Case I Case II Radius R 2.42 2.42 Neck Length T 3.16 3.16 Radius R 6.34 6.34 Cavity Depth L 16.42 9.38 Radius R1 10.05 10.05 piezoceramic Thickness hp 0.13 0.13 Radius R2 12.41 12.41 Piezoelectric backplate shim Thickness hs 0.19 0.19 PAGE 98 98 Table 43. Selected loads matrix used in the experiment to tune the EMHR. Resistive loads () Capacitive loads (nF) Inductive loads (mH) Nominal Measured Nominal Measured Nominal Measured Short Short Short Short Short Short 200 199.1 10 10.3 100 101.9 (65.4)a 2k 2.0k 47 37.8 300 304.6 (191.8) 7.5k 7.4k 100 89.7 500 507.8 (319.8) Open Open Open Open Open Open a. the value in () represents the resi stance of the inductive loads in PAGE 99 99 0 x P P s Figure 41. Illustration of the two microphone method. Pf mHf Pf Figure 42. Pressure measured by a microphone. PAGE 100 100 0 Time [s]Amplitude x1x2 Figure 43. Free damped vi bration of a SDOF system. B&K Pulse System Techron Amplifier Mic.2 Mic.1 Ref. Mic. Loads Speaker PWT EMHR s Figure 44. Acoustic impedance meas urement of the EMHR using TMM. PAGE 101 101 (a) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1.96 1.965 1.97 1.975 1.98 1.985 1.99 1.995 2 2.005 z/ap/P+/0=0.1 Pcenter Pwall P(0,0) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 z/ap/P+/0=0.5 Pcenter Pwall P(0,0) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 z/ap/P+/0=1 Pcenter Pwall P(0,0) 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 z/ap/P+/0=1.5 Pcenter Pwall P(0,0) (b) Figure 45. The scattering by a HR mounted at th e end of a PWT. (a) FEM simulation results of the pressure distribution in the tube terminated with a HR. (b) Numeric results of pressure distribution in the tube termin ated with the HR under different exciting frequencies. Note: centerp is the acoustic pressure along the centerline of the tube, wall p is the pressure along the line on the wall where microphones are mounted, 0,0 p is the acoustic pressure distribution with assumption that only plane waves propagate within the duct, za is the ratio of distance away from the test surface of the specimen to the duct crosssection dimension, and 0 is the resonant frequency of the HR. The results show that higher order modes contribute to the pressure distribution near the specimen. PAGE 102 102 Figure 46. The damping measurement for the piezoelectric backplate of an EMHR. Figure 47. Assembly diagram of modular EMHR (not to scale). PAGE 103 103 CHAPTER 5 EXPERIMENTAL RESULTS AND DISCUSSION This chapter summarizes the experimental results of the normal incidence acoustic impedance measurement for the EMHRs with diffe rent shunt loads, the damping coefficient measurement and the parameter extraction. A co mparison between the experimental results and models predictions for the acoustic impeda nce of the EMHR is also provided. Evaluation of the Tuning Performance of the EMHR As detailed in Chapter 2 the acoustic impedance of an EM HR is tuned via adjustment of the shunt network attached to the pi ezoelectric diaphragm of the EMHR. Figure 51 shows the acoustic impedance measurement results for the re sonator (Case I) with a variety of capacitive loads, the measured reflection co efficient magnitude is shown in Figure 52 The EMHR with capacitive loads ( 1eLeL Z jC ) has 2DOF and then has two resonant frequencies (1 f and 2 f ), located where the reactance of the EMHR crosses zero with positive slope (Im()0INZ ). At the resonant frequency, the particle velocity within the orifice of the EMHR is at its maximum and there is a sharp dip in the re flection coefficient magnitude ( Figure 52 ). There is also an antiresonant frequency (a f ) where the reactance of the EMHR cro sses the zero with negative slope and the particle velocity in the orifice of the EMHR goes to its minimum. As the capacitance is increased, the second resonant frequency (2 f ) shifts towards the shortcircuit case from the opencircuit case, as indicated by Eq.(248). The first resonant frequency (1 f ) barely changes for the EMHR (Case I). This is because of the weak coupling (0.17s where s indicates the shortcircuit case) between the solidwalled Helmholtz resonator and the PZT backplate for the tested EMHR (Case I). There is an approximately 9% capac itive tuning range of the second resonant frequency (2 ) for the Case I resonator under the conditions and geometry listed in PAGE 104 104 Table 42 where the tuning range is defined as the difference between the opencircuit resonant frequency and its respective shortcircuit counterpart (e.g., 222 os f ff where 2 o f and 2 s f are the second resonant frequency with ope nand shortcircuit, respectively). Figure 53 shows the measured acoustic impedance for different resistive loads across the EMHR (Case I). Similar to the EMHR with capac itive loads, the EMHR with resistive loads has 2DOF, and thus two resonant frequencies. A trend similar to the capacitive tuning can be observed for the resistive tuning, in which the tuning range is also de fined by the shortand opencircuit limits. When the resistance is incr eased, the resonant frequency moves from the shortcircuit case to the opencircuit case. More over, when resistive load s are attached to the EMHR, a portion of the energy is removed from the EMHR and dissi pated by the resistive loads. In other words, resistive loads increase the system damping (Hag wood et al. 1990). As shown in Figure 53 for a small resistive load, only a slight amount of energy is removed from the EMHR, thus the additional damping due to the load is not significant. As the resistance increases, more energy is removed and the system damping incr eases. The amplitude of the impedance peaks thus continue to reduce. At the optimal resistive load, the maximum energy is extracted from the EMHR, and the system damping becomes far larger than one of the shortcircuited EMHR. The amplitude of the impedance peaks thus reaches their minimum. As the load resistance moves away from the optimal value, the system damp ing decreases with the increase of the load resistance due to the amount of energy removed from the EMHR reduced. At high resistance (i.e., eLR ), the system damping goes to one of the opencircuited EMHR. Figure 54 shows the results for inductive tuning of the Case I EMHR, and the measured reflection coefficient ma gnitude is shown in Figure 55 Unlike capacitive and resistive tuning, inductive tuning provides an additional DOF for the EMHR, resulting in a 3DOF EMHR with PAGE 105 105 three resonant frequencies. The third resonant frequency shifts closer to the second resonant frequency opencircuit case as the inductance is increased. However, as a result of the 3DOF, inductive tuning is not restricted to lie between the shortand opencircu it limits. Rather, the second resonant frequency always lies outside the shortto openc ircuit tuning range as shown in Figures 54 and 55 Notice that the second resonant freque ncy of the resonato r shifts to lower frequencies as the inductance is increased. Mo reover, when the inductive load is large enough, the second resonant frequency is smaller than the first resonant frequency. There is approximately a 47% inductive tuning range ( 22 s f f ) of the EMHR (Case I) under the conditions and geometry listed in Table 42 Again, the first res onant frequency only changes slightly due to weak coupling be tween the piezoelectric backplate and the Helmholtz resonator. Furthermore, a real inductor posse sses finite resistance as well ( Table 43 ). Typically, the resistance increases with incr easing inductance, so the larger inductor also has a larger resistance. Figure 56 shows the experimental results of th e EMHR (Case II) with shortand opencircuits. It indicates that the first resonant frequency of the EMHR shifts due to coupling between the piezoelectric backplat e and the Helmholtz resonator is not as weak as EMHR (Case I). There is an approximately 4.6% tuning range of the first resonant frequency, in contrast, the tuning range of the first resonant frequency of th e EMHR(Case I) is approximately 0.9%. This is because the depth of the cavity of the EMHR (Case II) is shorter th an one of the EMHR (Case I). The acoustic compliance of the cavity consequen tly decreases. Hence, the coupling between Helmholtz resonator and piezoelectric com posite backplate becomes stronger (0.22s) and the first resonant frequency of the EMHR is not the resonant frequency of the solidwalled PAGE 106 106 Helmholtz resonator, as given in Eq.(239). Th e first resonant frequency is adjusted with changing of the shunt loads of the EM HR. More details are presented in Chapter 2 Comparison with LEM and Transfer Matrix The comparison between measurement da ta and LEM and TM is shown in Figure 57 to Figure 510 Clearly, with measured damping loss of the PZT backplate, both LEM and TM prediction match the experimental results pretty well for the EMHR with shortcircuit. The TM gives a better prediction for the acoustic resistan ce than LEM does. This is because the TM includes the viscous loss effect on the wave propagation within the cavity. There are some other factors as presented in Chapter 3 which concern when the TM coincides with the LEM. The predictions for the normalized specific acoustic impedance do not match the experimental data very well for the opencircuit case. One possibl e explanation for the observed discrepancy is that, for the shortcircuit case, the piezoelectr ic backplate is electri cally shorted and thus and eBC do not affect the acoustic impedance of the EMHR, as they do for the opencircuit case. Hence, any inaccuracies in either or eBC will impact the predicted results. Some factors do impact the piezoelectric backplate model, such as the bond layer between the piezoceramic patch and the brass shim (the model assumes a ne gligible bond layer), a ny asymmetry in the piezoceramic patch geometry, and imperfect cl amped boundary conditions. The results also reveal that the measured resistance is larger than the predicted results in the low and high frequency ranges. The deviations may be caused by the fact that the reflection coefficient is close to unity in those regions. When the reflec tion coefficient is near unity, the uncertainty in the acoustic impedance measurement using the TMM will become very large (Schultz et al. 2007). The uncertainty in the acoustic impedance arises from the measurement uncertainties in PAGE 107 107 microphone locations (and s in Figure 44 ), the frequency response function between the two measurement microphones, ambien t temperature and pressure. Damping Coefficient Measurement Results Figure 511 and Figure 512 show the damping coefficient measurement results for a piezoelectric composite backplate in air and in a vacuum chamber. Using the logarithm decrement method and choosing the first two peaks, the damping coefficient is calculated to be 0.026air (51) for a piezoelectric backplate in air, and 0.024vacuum (52) for a piezoelectric backplate in a vacuum chamber. Note that the damping coefficient in Eq. (51) and (52) are from the first and second peak displacement of the damping oscillation of the system. The accuracy of the estimation ma y be improved using more measured peak displacements, as discussed in Chapter 4. Table 51 contains the measurement results for the first twelve peak displacements of the piezoelectric plate in air. To minimize Eq.(419), one has 121212 2 111lniiii iiiyaybxy (53) and 1212 1112lnii iiyabx (54) Inserting the values from Table 51 into Eqs. (53) and (54) leads to 5066640.7598ab (55) and 66124.95673 ab (56) PAGE 108 108 The solutions for Eqs. (55) and (56) are 0.094390.1061ab (57) The damping coefficient is then obtained using Eq.(420) and is 0.015air (58) which differs significantly from the estimation in Eq. (51). The illustration of the determination of damping coefficient using the method above a nd the comparison between numeric fittings and measurement results are shown in Figures 513 and 514 Following a similar procedure, the damping co efficient of the piezoelectric plate in the vacuum chamber is 0.01air (59) The data used to determine the coefficient of the piezoelectric plate in the vacuum chamber are listed in Table 52 The damping coefficient found usi ng the method above and the comparison between numeric fittings and measurement results are shown in Figures 515 and 516 Parameter Extraction of the Piezoelectric Backplate Figure 517 shows the measured displacement of the piezoelectric backplate due to the application of voltages. The re sults indicate that, when applyi ng a voltage to the plate, the transverse displacement of the plate is larger than one calculated by the model. This is likely due to nonidealities in the composite plate boundary conditions. A clamped boundary is difficult to achieve in practice and any comp liance in the boundaries or inp lane compressive stress due to mounting will result in an enhanced Ad. Consequently, the deduced parameters such as Ad, and eBC differ from those used in the models, as listed in Table 53 Applying the deduced and eBC to the LEM and TM results in good predicti ons of opencircuited EMHRs, as shown in Figure 518 The model prediction for the normalized specific acoustic impedance matches the PAGE 109 109 experimental data quite well for both shortcircuit and opencircu it cases. The results indicate that the functional form of th e LEM accurately captures the p hysical behavior of the EMHR. PAGE 110 110 Table 51. Determination of the damping co efficient of the piezoel ectric plate in air. i i x lni x iy 2 iy lniiyx 1 1.220 0.199 0 0 0 2 1.035 0.034 1 1 0.034 3 0.886 0.121 2 4 0.241 4 0.812 0.208 3 9 0.625 5 0.725 0.321 4 16 1.285 6 0.663 0.411 5 25 2.057 7 0.631 0.460 6 36 2.759 8 0.549 0.599 7 49 4.192 9 0.535 0.625 8 64 5.000 10 0.491 0.711 9 81 6.400 11 0.434 0.834 10 100 8.342 12 0.407 0.899 11 121 9.891 Table 52. Determination of the damping coef ficient of the piezoelectric plate in vacuum chamber. i i x lni x iy 2iy lniiyx 1 1.246 0.220 0 0 0 2 1.077 0.074 1 1 0.074 3 0.975 0.026 2 4 0.052 4 0.891 0.115 3 9 0.346 5 0.853 0.159 4 16 0.637 6 0.794 0.231 5 25 1.155 7 0.761 0.274 6 36 1.643 8 0.722 0.326 7 49 2.284 9 0.672 0.397 8 64 3.178 10 0.650 0.430 9 81 3.874 11 0.602 0.508 10 100 5.077 12 0.595 0.519 11 121 5.711 Table 53. Comparison between predicted and deduced LEM para meters of the Piezoelectric backplate. Ad eBC Predicted 48.3e12 82.4 45.7e9 Deduced 71.4e12 122 41.2e9 PAGE 111 111 1500 2000 2500 3000 0 10 20 30 Short 10nF 47nF 100nF Open 1500 2000 2500 3000 10 0 10 20 Freq.[Hz] Short Open Short Open f1s f1o f2s f2o fao fas Figure 51. Experimental resu lts for the normalized specific acoustic impedance of the EMHR (Case I) as function of the capacitive loads. PAGE 112 112 1500 2000 2500 3000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Magnitude of reflection coefficientFreq.[Hz] Short 10nF 47nF 100nF Open f2o f2(10) f2(47) f2(100) f2s f1s f1o f1(100) f1(47) f1(10) Figure 52. Experimental results for the reflection coefficient of the EMHR (Case I) as function of the capacitive load. Note that 2(10) f denotes the first resonant frequency under a capacitive loading of 10 nF. 1500 2000 2500 3000 0 10 20 30 Short 200 2k 7.5k Open 1500 2000 2500 3000 10 0 10 20 Freq.[Hz] Short Open Short Open f1s f1o f2s f2o Figure 53. Experimental resu lts for the normalized specific acoustic impedance of the EMHR (Case I) as function of the resistive loads. PAGE 113 113 1000 1500 2000 2500 3000 0 10 20 30 Short 100mH 300mH 500mH Open 1000 1500 2000 2500 3000 10 0 10 20 Freq.[Hz] Figure 54. Experimental resu lts for the normalized specific acoustic impedance of the EMHR (Case I) as function of the inductive loads. PAGE 114 114 1000 1500 2000 2500 3000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Magnitude of reflection coefficientFreq.[Hz] Short 100mH 300mH 500mH Open f3(300) f1(300) f2(300) f2(500) f3(500) f2(100) f1(500) f1(100) f2s f3o f1s f1o Figure 55. Experimental results for the reflection coefficient of the EMHR (Case I) as function of the inductive load. Note that denot es the third resonant frequency under an inductive loading of 300 mH. 1500 2000 2500 3000 0 20 40 60 1500 2000 2500 3000 20 10 0 10 20 Freq.[Hz] f1s f1o f2s f2o Short Open Figure 56. Experimental results of the norma lized acoustic impedance of the EMHR (Case II) for the shortand opencircuit. PAGE 115 115 101 100 101 102 1500 2000 2500 3000 10 0 10 20 Freq.[Hz] LEMShort TMShort DataShort LEMOpen TMOpen DataOpen Figure 57. Comparison LEM, TR and measurement results for a shortand opencircuited EMHR (CASE I), the damping loss of the b ackplate is determin ed using logarithm decrement method (0.015 ). 102 100 102 104 1500 2000 2500 3000 50 0 50 100 150 Freq.[Hz] LEMShort TMShort DataShort LEMOpen TMOpen DataOpen Figure 58. Comparison LEM, TR and measurement results for a shortand opencircuited EMHR (CASE I), the damping loss of the backplate is assumed to be acoustic radiation resistance. PAGE 116 116 101 100 101 102 1500 2000 2500 3000 20 0 20 40 Freq.[Hz] LEMShort TMShort DataShort LEMOpen TMOpen DataOpen Figure 59. Comparison LEM, TR and measurement results for a shortand opencircuited EMHR (CASE II), the damping loss of the b ackplate is determined using logarithm decrement method (0.015 ). 102 100 102 104 1500 2000 2500 3000 400 200 0 200 Freq.[Hz] LEMShort TMShort DataShort LEMOpen TMOpen DataOpen Figure 510. Comparison LEM, TR and measurement results for a shortand opencircuited EMHR (CASE II), the damping loss of the backplate is assumed to be acoustic radiation resistance. PAGE 117 117 1 2 3 4 5 6 7 8 x 103 1.5 1 0.5 0 0.5 1 1.5 Time [s]Transverse displacement [ m] Figure 511. Damping coefficient measurement for piezoelectric composite backplate (Case I) in air. 1 2 3 4 5 6 7 8 x 103 1.5 1 0.5 0 0.5 1 1.5 Time [s]Transverse displacement [ m] Figure 512. Damping coefficient measurement for piezoelectric composite backplate (Case I) in the vacuum chamber. PAGE 118 118 0 2 4 6 8 10 12 1 0.8 0.6 0.4 0.2 0 0.2 0.4 ilnxi, zi Data zi=0.09439yi+0.1061 Figure 513. Determination of damping coeffi cient of the piezoelectric plate in air. 1 2 3 4 5 6 7 8 x 103 1.5 1 0.5 0 0.5 1 1.5 Time [s]Transverse displacement [ m] Data Curve fitting 1 Curve fitting 2 Figure 514. Curve fitting the measurement data (in air) using a 2ndorder system, curve fitting 10.026 curve fitting 20.015 PAGE 119 119 0 2 4 6 8 10 12 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 ilnxi, zi Data zi=0.06347yi+0.1248 Figure 515. Determination of damping coeffici ent of the piezoelectric plate in the vacuum chamber. 1 2 3 4 5 6 7 8 x 103 1.5 1 0.5 0 0.5 1 1.5 Time [s]Transverse displacement [ m] Data Curve fitting 1 Curve fitting 2 Figure 516. Curve fitting the measurement data (in the vacuum chamber) using a 2ndorder system, curve fitting 10.024 curve fitting 20.01 PAGE 120 120 0 0.2 0.4 0.6 0.8 1 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 x 106 r / R2Transverse Displacement [m] Data8 Data10 Data12 Data14 (a) 0 0.2 0.4 0.6 0.8 1 3.5 3 2.5 2 1.5 1 0.5 0 x 107 Normalized Transverse Displacement [m/V]r / R2 Data8 Data10 Data12 Data14 Num1 (b) Figure 517. Measured transverse displacemen t of the piezoelectric backplate due to the application of various voltages. PAGE 121 121 101 100 101 102 1500 2000 2500 3000 10 0 10 20 Freq.[Hz] LEMShort TMShort DataShort LEMOpen TMOpen DataOpen (a) 101 100 101 102 1500 2000 2500 3000 20 0 20 40 Freq.[Hz] LEMShort TMShort DataShort LEMOpen TMOpen DataOpen (b) Figure 518. Predictions of the LEM and TM for shortand opencircuited EMHRs. (a) EMHR (Case I). (b) EMHR (Case II). PAGE 122 122 CHAPTER 6 OPTIMAL DESIGN OF AN EMHR This chapter discusses the optimal design of an EMHR. The synthesis of the optimal design problem for an EMHR aims at providing a usable impedan ce range. The optimal design studies of an EMHR have been undertaken followi ng two paths. The first path is to maximize the tuning range of an EMHR with noninductive load s, which is presented in Sections 2 and 3. The second path aims to optimally tune the three resonant frequencies of the EMHR with inductive shunts. The optimization design of th e EMHR with inductive loads is presented in Section 4. Introduction The acoustic impedance of an EMHR, shown in Figure 25 is modified insitu by adjusting the electrical impeda nce of passive shunt networks attached to the piezoelectric backplate of the EMHR. The frequency range over which the acoustic im pedance of the EMHR can be tuned significantly is calle d the tuning range. Both experi mental data and the theoretical analysis presented in Chapter 5 indicate that the EMHR w ith noninductive shunts has two tuning ranges, defined by 1 f and 2 f corresponding to the tuning ra nge of the first and second resonant frequencies of the EMHR, respectively. The analytical investiga tion indicates that the EMHR with different geometry has different tuning ranges. The material properties of the piezoelectric backplate also affect the tuning ranges of the EMHR. It is thus necessary to employ an optimal design procedure to obtain a useable tuning ranges 1 f and2 f Mathematically, the optimal design deals wi th minimizing or maximizing the objective function under certain co nstraints in the form PAGE 123 123 0()1,2,..., 01,2,..., =01,2,...,ii i jJXiin X gXim hXjn Minimize such that LBUB, (61) where the design variables jj x (1,2,..., j jp ) are collectively represented as the design vector 12,,...,p X xxx The design variables are limited by lower and upper bounds LB, and UB, respectively. iiJX is termed the objective function, and igX and jhX are known as inequality and equality constrai nts, respectively. The problem stated in Eq. (61) is called constrained optimization problem. Some optimi zation problems do not have any constraints and are called unconstrained optimization problems. Moreover, if only one objective function is involved in the problem stated in Eq. (61), th e process is called singl eobjective optimization problem, otherwise the process is denoted by the term multiobjective optimization problem. In general, solving a multiobjective optimization probl em is much more complicated than solving a singleobjective problem. An p dimensional space with each coordinate axis representing a design variable jj x (1,2,...,jjp ) is denoted as the design space. Each point in the design space is called a design point which represents a possi ble or impossible soluti on of the optimization problem. Constraints can be functio nal or geometric in term of th eir characteristics. Constraints limiting the performance of the sy stem are functional constraints, while constraints defining the physical limitation on design variables are geometric constraints. Geometri c constraints are also known as side constraints. A cons traint is satisfied with a margin is known as inactive, while is called active when it is satisfied with no margin. A design point which satisf ies all constraints is feasible or acceptable. The colle ction of all feasible points is named the feasible region. A feasible point where all constraints are inactive is called a free point. Furthermore, a feasible PAGE 124 124 point X is global optimum point if at this point JXJXfor all X in the feasible region. A feasible point X is called local optimum point if JXJX for all sufficiently small positive and negative value of Finally, optimization pr oblems where both objective function(s) and constraints ar e linear functions of the desi gn variables are called linear programming problems. In contrast, optimiza tion problems where the obj ective function(s) or one of the constraints is a nonlinear func tion of design variables are named nonlinear programming. Generally, it is easier to solve a linear programming problem than to solve a nonlinear programming (Haftka and Gurdal 1992; Rao 1996). In this study, the optimal de sign of the EMHR firstly aims at maximizing the tuning range(s) of the EMHR with nonindu ctive loads to satisfy a certain set of specified requirements. Therefore, the tuning range(s) is (are) the objective function(s) of the optimal design, while the set of specified requirements is the constraint The optimization problem is nonlinear because the objective function(s) and some constraints are nonlinear functions of the design variables (Liu et al. 2006). Optimizing Single Tuning Range of EMHR with Noninductive Loads Theoretical Background As described in Chapter 2 and Chapter 5, the shortand opencircuit cases define the tuning range of an EMHR with a noninductive shunt network. The EMHR is a 2DOF system in such cases and possesses two res onant frequencies, denoted by 1,i f ios and 2i f and the subscripts o and s denote openand shortcircuit conditions, respectively. For the shortcircuit case, the two resonant frequencies are given by 2 2 22222 121212 2 141 2LLsLLssLLs sffffff f (62) PAGE 125 125 and 2 2 22222 121212 2 241 2LLsLLssLLs sffffff f (63) where 1L f is the resonant frequency of loop 1 ( Figure 26 ) 11 2L aNaCf M C, (64) and 2Ls f is the resonant frequency of loop 2 under shortcircuit condition 21 2Ls aDaDradaDaC aDaCf M MCC CC (65) Furthermore, 2 s is the electromechanical coupling f actor for the shortcircuited EMHR 2 12aDaCaDaC s s aDaCaCaDaCaCCCCC CCCCCC (66) where 1 and 2 s are the coupling coefficients of loop 1 and loop 2 for the shortcircuit EMHR, respectively. The coupling coefficient defines the ratio of the energy stored in the coupling elements (aCC) to that stored in the total capacitance for each loop. For loop1, the coupling coefficient is 2 1 2 11aC aC aCQdtC C C QdtC (67) where 1aCCC is the total capacitance of loop 1. Similarly, for loop 2, the coupling coefficient is 2aD s aDaCC CC (68) PAGE 126 126 For the opencircuit case, the two resonant frequencies are given by 2 2 22222 121212 2 141 2LLoLLooLLo offffff f (69) and 2 2 22222 121212 2 241 2LLoLLooLLo offffff f (610) where 2Lo f is the resonant frequency of loop 2 under opencircuit condition 2 21 2Lo aDaDradeBaDaC eBaDeBaDaCf MMCCC CCCCC (611) Furthermore, 2 o is the electromechanical coupling f actor of the opencircuit EMHR 2 12 2 eBaD oo eBaDeBaDaCCC CCCCC (612) where 2 o are the coupling coefficients of loop 2 for opencircuit EMHR 2 2 eBaD o eBaDeBaDaCCC CCCCC (613) Thus, the tuning ranges of the EMHR with noninductive loads are defined by 111 os f ff (614) and 222 os f ff (615) Moreover, note that 1 L f is essentially the resonant frequency of a Helmholtz resonator with a rigid wall instead of a piezoelectric bac kplate. Dividing Eqs. (62) and (63) by 2 1 L f leads to PAGE 127 127 2 2222 2 1 2 11141 2ssss s Lf f (616) and 2 2222 2 2 2 11141 2 s sss s Lf f (617) where is the mass ratio between the neck and piezoelectric backplate aN aDaDradM MM (618) Similarly, dividing Eqs. (69) and (610) by 2 1 L f results in 2 2222 2 1 2 11141 2oooo o Lf f (619) and 2 2222 2 2 2 11141 2oooo o Lf f (620) Equations (614) (620) i ndicate that the tuning ranges vary with four parameters, 1 L f (or H R f ), s o and An example of calculated 11 s L f f and 21 s L f f as function of s and is shown in Figure 61 The 11 oL f f and 21 oL f f versus o and are expected to have similar trend. Note that o is less than s because the opencircuit EMHR becomes less compliant than the shortcircuit case. For a given EMHR, the mass ratio is the same for the shortand opencircuit cases. Therefore, the dimensionless tuning range 1 iL f f (1,2i ) can be represented by two points on the isolines map of the 11 s L f f and 21 s L f f as shown in Figure 61 where the points have the same value of and different value of s Clearly, the change of 11 L f f with PAGE 128 128 the change of the coupling factor is significant when s is large, in contrast, the change of 21 L f f with the change of s becomes significant when s is very small. The observation indicates that it is possible to maximize 1 f by designing the EMHR with large s while 2 f may be maximized by designing the EMHR with small s Optimization Problem Formulation The first optimization problem is formulated by choosing one of the tuning ranges in Eqs. (614) and (615) as objective function. As discussed above, bot h tuning ranges are functions of the geometric parameters, listed in Table 61 as well as the material properties of piezoelectric composite diaphragm and air. To simplify the pr oblem, it is assumed that the material properties of a given piezoelectric composite di sc are constants, as listed in Table 41 Thus, the single objective design optimization problem s eeks to maximize either tuning range, 1 f or 2 f (or to minimize 1 f or 2 f ) to satisfy a certain set of constraints. The design variables are the radius of the neck r, the thickness of the neck t the radius of the cavity R the depth of the cavity L, the radius of the shim 1 R the thickness of the shim ph the radius of the shim 2 R and the thickness of the shim s h The constraints are categorized as geometric constraints and functional constraints. The side constraints include the physical bounds for the design variables. The constraints involved in the optimization problem are: 1) Lower bounds (LB) and upper bounds (UB): 12,,,,,,,psLBrtRLRhRhUB where LB and UB of each design variable are listed in Table 61 ; 2) Geometry constraints that im pose physical limitations on the design variables. The constraints are based on the size of the test apparatus, available commercial piezoelectric benders, and size restri ctions on the EMHR; 3) Freque ncy constraint that confines the first shortcircuit resonant frequency 1 s f of the EMHR to a particular range (1200 Hz to 1900 PAGE 129 129 Hz) where noise suppression is preferred, whil e also prescribing an upper limit of 3000 Hz for 2 s f Clearly, both the objective function and the frequency constraint are nonlinear. Thus, the optimal design of the EMHR with noninductive load s is a constrained nonlinear programming. Mathematically, the single objective optimizati on of the EMHR with noninductive loads is the following: 12 212 43 1110;0 100;100 12000;19000;30000 Minimize or such that LBUBpsps sssfXfX X RRRR hhhh fff (621) where 12,,,,,,,ps X rtRLRhRh is the design variable vector. The optimization problem defined in Eq. (621) is first implemented in MATLAB using its fmincon function and then verified using Gene tic Algorithms also provided by MATLAB Genetic Algorithm Direct Search (GADS) toolbox. The fmincon function employs sequential quadratic programming (SQP) for nonlinear constrained optimization. The SQP method focuses on the solution of the KuhnTucker (KT) equations The KT equations are necessary conditions to be satisfied at a local minimum of JX. For convex programming problems, the KT conditions are necessary and sufficient for a gl obal minimum. The optimization problem is called a convex programming problem if the objectiv e function and the constraint functions are convex (Rao 1996; Mathworks 2005). However, it is difficult to determine the optimization problem set up in Eq. (621) is a convex progra mming problem or not. Thus, the optimal results obtained using the SQP method may be local minima Moreover, due to th e high nonlinearity of the optimization problem defined in Eq. (621), the solution process (i.e ., convergence of the optimal solution and optima) usi ng SQP method is highly dependent on the initial values. The PAGE 130 130 Genetic Algorithms (GA) provided by MATLA B Genetic Algorithm Direct Search (GADS) toolbox are thus adopted to explore the possi bility of finding a gl obal optimum. The GA provides a natural search strategy and is simple to implement. At each iteration, a population of points is generated. The objective function de termines best points in the population as the optimal solution. In consecutive iterations, th e genetic algorithm rando mly selects individuals from the current population to be parents a nd uses them produce the children by applying mutation and crossover rules. Over successive generations, the population evolves toward an optimal solution (Mathworks 2005). By choosing suitable parameters for GA, it is possible to ensure the local optimum does not dominate the population (Mat hworks 2005). However, GA is timeconsuming. An alternative approach is to first employ GA with 10 to 20 generations to find a point close to the optimal solution and then use that point as the initial condition for a gradientbased optimization study. The GAoptimal re sult is then improved by setting the GA optimum as the initial condition of the SQP optimization implemented via the fmincon function in MATLAB. Optimization Results One set of the optimization results using SQP method are listed in Table 62 The initial values of the design variables are chosen based on the prototype of the EMHR discussed in Liu et al. (2003, 2007). The initial desi gn vector satisfies all constraint s and thus is a feasible initial condition. Different initial desi gn vectors are explored next. The result of the optimization depends strongly on the initial values. However, the collective optimal results show that any improvement in one tuning range (1 f or 2 f ) can only occur by comp romising the other tuning range. Both tuning ranges 1 f and 2 f can not maximized simultaneously. Table 62 also lists the optimization results by combination GA and SQP methods. The re sults show different PAGE 131 131 optimum. In terms of characteristics of the optimization problem set up in Eq. (621), multiple optimum are expected. This is the objective f unction is based on the LEM of the EMHR. Some LEM parameters are manytoone function. For in stance, different dimensions of the cavity can leads to the same acoustic compliance 22 00 aCCRLc only if 2 R L is the same. Sensitivity Analysis The sensitivity analysis is very important for understanding which design variables are significant drivers for a optimum solution x. It is impossible to undertake the sensitivity analysis before the optimum solution x has been found, thus the se nsitivity analys is is a postprocessing step. In fact, the sensitivity analys is with respect to design variables seeks to determine how the objective function J changes as the design vector x changes. This can be achieved by the computation of the gradient of JX 1 2 pJx Jx J Jx (622) for the single objective function, and 0 0 011211 12222 12 Jacobian Matrix,, ,, ,, Jn n ppnpJxJxJx JxJxJx JxJxJx (623) for the multipleobjective function. The normalized sensitivity is then derived as following using Eqs. (622) and (623) % change in % change in jj jjjjjjjj XXx JX JJJ xxxx JX (624) PAGE 132 132 which is useful to compare sensitivi ties to the different design variables. The sensitivity analysis results for the optima of the tuning range 1 f related to the design variables are shown Figure 62 The result shows that the optimal 1 f is most sensitive to the radius of the shim of the PZTbackplate of the EMHR (2 R ). An increase of 1% in 2 R will lead to a decrease in the tuning range 1 f (the objective function) of 1.6%. Similarly, the second most sensitive factor is the thickness of the ceramic layer of the PZTbackplate of the EMHR (ph), where an increase of one percent in ph will result in an increase of the tuning range 1 f by 0.52%. However, allowable changes may be re stricted by the constraints, especially the frequency constraint and some of the geometric constraints, as shown in Figure 63 This figure illustrates how the optimal solution is affected by the upper constraint on 1 s f When the tuning range 1 f is maximized, there is a relative strong co upling between the piezoelectric backplate and the cavity, the changes in the geometry of th e neck and the cavity thus have some effect on the optima. The sensitivity analysis result for the optimization of the tuning range 2 f is shown in Figure 64 which indicates that 2 R and ph are also the most sensitive design parameters with respect to maximizing 2 f An increase of one percent in 2 R will lead to a change of the tuning range 2 f by 2.27%, while an increase of one percent in ph will result in a change of the tuning range 2 by +0.73%. Moreover, the optimal solution is not sensitive to changes in the geometry of the neck and cavity. Physically, this is because the piezoelec tric backplate weakly couples with the cavity in this case. The 2 f is thus dominated by th e piezoelectric backplate. PAGE 133 133 Pareto Optimization of the EMHR with Noninductive Loads The results of the optimizing single tuni ng range of EMHR w ith noninductive loads indicates that it is impossible to simultaneously maximize both 1 f and 2 f Therefore, a tradeoff approach is pursued. Pareto optimization is thus explored to optimize both tuning rages at the time to achieve a Pareto solution. The Pareto solution (also called a Pareto optimal) is one where any improvement of one objective degrades at least one other obj ective. Three methods are used to obtain the Pareto solution for multipleobjective optimization of the EMHR: the constraints method (Marglin 1967), the traditio nal weighted sum method (Koski 1988), and the adaptive weighted sum (AWS) method (Kim and de Weck 2005). Mathematically, the constraints method is 12 11 212 43 1110;0 100;100 12000;19000;30000Minimize or such that LBUBpsps sssfXfX X f RRRR hhhh fff (625) where 12,,,,,,,ps X rtRLRhRh The tuning range 2 f is chosen to be the primary objective function which is subject to the original constraints and an additional constraint limiting the tuning range 1 f The advantage of the constraint method is that it is able to achieve the optimal solution even in the nonconvex boundary of the Pareto front (i.e., the set of the Pareto solutions). The problem with this method is that it is generally difficult to choose a suitable In order to speed up the op timization, the choice of 1 follows an ascending order: the result of a previous optimization with the constraint factor 1 i is used as the starting point for the optimization with another set of the constraint factor 1 1 i as shown in Figure 65 PAGE 134 134 The traditional weighted sum method is used to convert the multiobjective optimization to a single objective problem by using a weighted sum of the original multiple objectives 1112 212 43 1111 0;0 100;100 12000;19000;30000Minimize such that LBUBpsps sssfXfX X RRRR hhhh fff (626) where 101 is the weighting coefficient. This me thod is easy to implement, but it has two drawbacks. One is that an even distribution of the weighting factors does not always achieve an even distribution of the Pareto front. The ot her is the weighted sum method cannot find the Pareto solution on nonconvex parts of the Pareto front, as shown in Figure 66 Furthermore, the traditional weighted sum method can produce nonPareto points occasi onally. To address this drawback of the conventional weighted sum method: the adaptive weighted sum method (AWS) (Kim and De Weck 2005) is adopted to s earch the Pareto front of the multiobjective optimization of the EMHR. The AWS produces welldistributed solutions, finds the Pareto optimum in nonconvex regions, and excludes nonPa reto optimums. For the region unexplored by the traditional weighted sum method, the AWS involves changing th e weighting factor adaptively rather than by using a priori weight selections and by specifying additional inequality constraints. The Pareto solution obtained using these three methods are shown in Figure 66 Clearly, optimal solutions obtained using the weighted sum method tends to cl ustered together, as mentioned above. A large portion of th e Pareto front is obtained using the constraints method, which is simply ignored by the weighted sum me thod. This is due to the nonconvex properties of the Pareto front. The results indicate that the AWS addresses the limitations of the PAGE 135 135 conventional weighted sum method, and produces the Pareto optim um with a good distribution. The result from the AWS method matches the result using the constraint method very well. As shown in Figure 66 the points that form the Pareto front do not distinguish themselves from each other in the sense that none of them is better than any other with respect to maximizing both 1 and 2 because any improvement in one objective degrades the other one. In Figure 66 the point A and B are corresponding to the maximized single objective tuning ranges of 1 f and 2 f respectively. At each point, one tuning range approaches its maximum, while the other is essentially invarian t. Between these two extrema, other points, such as point C on the Pareto front, represents a tradeoff for optimizing both tuning ranges. Which optimal solution is attractive is determin ed by other design considerations. Specifically, the Pareto solution provides the information for a designer that shows how design tradeoffs can be used to satisfy specific design requirements. Optimization of the EMHR with Inductive Loads The EMHR with inductive loads has 3DOF and thus has three resonant frequencies (1 f 2 f and 3 f ). One goal of the optimal design of the EM HR with inductive loads is to find an optimal EMHR which all resonant frequencies are with in the frequency range of interest and the frequency shift between the maximum and mi nimum resonant frequency is minimized (123123max,,min,, f ffffff ). Thus, within a wide fre quency range, the variation of the reactance of the EMHR is relative small. Mathematically, the task is described as PAGE 136 136 212 43 1130;0 100;100 12000;19000;30000Minimize such that LBUBpsps sssfX X RRRR hhhh fff (627) where 12,,,,,,,,pseL X rtRLRhRhZ is the design variables vector which also includes the inductive shunts. The lower and upper bound of the inductive shunts are 10 mHand 150 mH, respectively. Due to highly nonlinearity of constraints, Gene tic Algorithm provided by MATLAB Genetic Algorithm Direct Search (GADS ) toolbox is thus explored. The comparison of the acoustic impedance between the in itial and optimal EMHR is shown in Figure 67 The results indicate that it is possible to keep the ac oustic reactance of the re sonator nearly constant over a given frequency range. PAGE 137 137 Table 61. Design optimization va riables of the EMHR (Unit: mm). Description Symbol Lower Bound (LB) Upper Bound (UB) Neck radius r 1.0 3.5 Neck thickness t 1.0 4.5 Cavity depth R 5.0 15.0 Cavity radius L 10.0 20.0 Piezoceramic radius 1 R 1.0 25.0 Piezoceramic thickness ph 0.5E1 1.0 Shim radius 2 R 1.0 25.0 Shim thickness s h 0.5E1 1.0 Table 62. Single objective optimization re sults of the EMHR (Dimensions unit: mm). SQP GA and SQP Initial values Maximizing 1 f Maximizing 2 f Maximizing 1 f Maximizing 2 f r 2.4 3.5 3.5 3.5 3.5 t 3.2 1.0 1.0 1.0 1.0 R 6.3 6.6 13.9 7.8 13.5 L 16.4 14.2 18.8 10.1 20.0 1 R 10.1 21.1 16.2 21.3 15.0 ph 1.2E1 7.2E1 6.5E1 7.4E1 5.6E1 2 R 12.4 22.5 17.2 22.7 16.0 s h 1.9E1 2.3E1 2.1E1 2.3E1 1.8E1 1 f 9.6 170.0 0.4 170.0 0.4 11 s f f 0.5% 8.8% 0.0% 8.8% 0.0% 2 f 113.3 14.6 290.5 14.6 290.5 22 s f f 4.3% 0.5% 9.3% 0.5% 9.3% s 1.7E1 2.4E1 5.5E2 2.4E1 5.5E2 5.6E2 3.1E2 2.0E2 3.1E2 2.0E2 PAGE 138 138 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.4 0.5 0.6 0.7 0.8 0.9 1 sf1s/fL1 Increasing A B (a) 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 1.5 2 2.5 3 3.5 4 4.5 sf2s/fL1 Increasing A B (b) Figure 61. Resonant fre quency of the EMHR versus s and (a) 11 s L f f as function of s and (b) 21 s L f f as function of s and PAGE 139 139 r t R L R1 hp R2 hs 2 1.5 1 0.5 0 0.5 1 Normalized sensitivity Figure 62. Normalized sensit ivity of the design variables at the optima for maximizing the tuning range of 1 f PAGE 140 140 0.0215 0.022 0.0225 0.023 0.0235 0.024 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 f1s 0.0215 0.022 0.0225 0.023 0.0235 0.024 170 160 150 140 f1R2 Optimum Figure 63. Illustration of the change in the optimum solution as a function of 2 R PAGE 141 141 r t R L R1 hp R2 hs 2.5 2 1.5 1 0.5 0 0.5 1 Normalized sensitivity Figure 64. Normalized sensit ivity of the design variables at the optima for maximizing the tuning range of 2 f 1J2J Figure 65. Choice of the star ting point for the multiobjective optimization with different 1. PAGE 142 142 200 150 100 50 0 300 250 200 150 100 50 0 f1f2 method AWS method Weighted sum method A B C Figure 66. Comparison of the Pareto front obtained via the constraint, traditional weighted sum, and adaptive weighted sum methods. 1000 1500 2000 2500 3000 3500 0 5 10 15 1000 1500 2000 2500 3000 3500 20 10 0 10 20 Freq. [Hz] Figure 67. Comparison between initial (dash line) and optimal (solid line) acoustic impedance of the EMHR with inductive loads. Nonconvex region PAGE 143 143 CHAPTER 7 SUMMARY AND FUTURE WORK A lumped element model (LEM) and a transfer matrix representation(TM) of the EMHR have been developed to predict the acoustic be havior of the EMHR. The models have been implemented in a MATLAB code and incorporated into the NASA ZKTL code, which converted to MATLAB code from original FOTRAN code In this study, LEM and TM show good agreement within the frequency range of interest The analysis of the tuning behavior of the EMHR based on the LEM is validate d by the experiment observations. Experiment investigation of the EMHR has been implemented using the standard twomicrophone method (TMM). The measurement results verify both the LEM and the TM of the EMHR. Good agreement between predicted and m easured impedance was obtained. Shortand opencircuit loads define the limits of the tuni ng range using resistive and capacitive loads. There is approximately a 9% noninductive tuni ng limit for the second resonant frequency under these conditions for the nonoptimized EMHR configuration studied. Inductive shunt loads result in a 3DOF system and an enhanced tuning range of over 47% that is not restricted by the shortand opencircuit limits. The damping coefficient measurement for the piezoelectric backplate in the vacuum chamber is also performe d. The results show th at the acoustic radiation damping loss is relative small. Based on models of the EMHR, the Pareto optimization design of the EMHR has been performed for the EMHR with noninductive load s. The EMHR with noninductive loads is 2DOF and has two resonant frequencies. Either of them can be optimally tuned by approximately 9%. However, the results show that it is impossible to maximize both tuning ranges simultaneously. The improvement of one t uning range degrades the other. Consequently, a tradeoff must be reached. In other words, a generally accepted Pareto solution should be PAGE 144 144 achieved. Three methods are explored to obtai n the Pareto optimal set of the biobjective optimization design of the EMHR: the constraint method, the traditional weighted sum method, and the adaptive weighted sum method. Both the constraint method and the adaptive weighted sum method obtain the same Pareto front for the op timization problem, while the weighted sum method clusters the Pareto solu tions around the two singleobjective design points of the Pareto front. The Pareto solution provide s the information for a designer that shows how design tradeoffs can be used to satisfy specifi c design requirements. The EMHR with inductiveloads is 3DOF and has three resonant frequencie s. The optimization design of the EMHR with inductive loads aims at optimal tuning of these thr ee resonant frequencies, i.e. to constrain three resonant frequencies within a given range. The resu lts indicate that it is possible to keep the acoustic reactance of the resonator close to a constant over a given frequency range. Future work should include the testing of sa mple EMHR liner designs in NASA Langleys dualincidence and grazingincidence flow facilities to evaluating both the impedance characteristics as well as the energy reclamation ab ilities. Inductive tuning offers the capability of an increased tuning range, but further work is needed regardi ng the design of such liners. Specifically, system goals are required to develop the formulation of the optimal design problem. Moreover, the optimization design of the EMHR s hould take into account the optimal choice of the materials for the piezoelectric backplate to satisfy specific design requirements. Poor electromechanical coupling limits the electrom echanical tuning capabilities of the EMHR. Novel composite material systems, like an interdigitated piezoelectric composite may dramatically improve the electromechanical c oupling (Hong et al. 2006). Such improvements are essential to mimic both resistance and reacta nce of existing double layer liners. Furthermore, PAGE 145 145 enhanced electromechanical tuning will also en able the use of more complex passive ladder circuits for impedance spectra shaping. PAGE 146 146 APPENDIX A NOISE LEVELS AND UNITS Level is taken on mean the quantity of noise. By definition: Level is the logarithm of the ratio of a given quantity to a re ference quantity of the same kind (Harris 1998). The following is the definition of the noise levels and units adopted in this dissertation deciBel: dB, the deciBel si a measure, on a loga rithm scale, of the magnitude of a particular sound intensity by refere nce to a standard quantity that represents the threshold of the hearing (Smith 1989) Sound Pressure Level: SPL (in units of dB), a measure of the root mean square (rms) pressure of a sound 1020logrmsrefSPLpp (A1) where ref p is a reference pressure. For air the reference pressure is 20 Pa For water the standard reference pressure is 1 Pa (Blackstock 2000). Weighted Sound Pressure Level: the sound pressure level we ighted to reflect human interpretation of the loudness of sounds at di fferent frequencies (Smith 1989). The weighting schemes based on loudness are A, B, and Cwe ighting. The corresponding weighted sound pressure levels are Aweighted SPL (dBA), Bweighted SPL (dBB) and Cweighted SPL (dBC). Figure A1 identifies the A weighting. Perceived Noise Level: PNL (in units of PNdB), An fr equencyweighted SPL obtained by a stated procedure that combines the sound pressu re levels in the 24 onethird octave bands with midband frequencies from 50 Hz to 10 kHz (ANSI 1994). Effective Perceived Noise Level: EPNL, (in units of EPNdB), an internationally recognized unit for describing th e noise of a single aircraft operation. The level of the time PAGE 147 147 integral of the antilogarithm of onetenth of tonecorrected pe rceived noise level over the duration of an aircraft flyover, the reference dur ation being 10s (ANSIS1.1 1994), where the tonecorrected perceived noise le vel (PNLT) recognizes the impact of a discrete tone on human annoyance by adding a penalty to the PNL (Smith 1989). Day/night Equivalent Sound Level: DNL or LDN, dBAbased rating, twentyfour hour average sound level but with nighttime (24000700 a nd 22002400) penalty of 10dBA (ANSI 1994). PAGE 148 148 101 100 101 60 50 40 30 20 10 0 10 Freq. [kHz]Aweighting [dB] Figure A1: Aweighting to s ound arriving at random incidence. PAGE 149 149 APPENDIX B ACOUSTIC IMPEDANCE PREDICTION OF AN ORIFICE As discussed in Chapter 1 a common component of resonant liners is the perforated metal plate. A perforated sheet consists of arrays of orifices/holes. Unde rstanding of the acoustic characteristics of the orifice is critical to achieve full potential of the app lication of the resonant liners. Linear Impedance Model of Orifices The term linear here means that the acousti c impedance of the orifice is independent on the incident SPL. Conversely, the term nonlinea r implies that the acoustic impedance of the orifice is dependent on the incident SPL. Two orifice linear impedance models will be briefly introduced here, one is the Cra ndall model (1926), the other is GE impedance model developed by Motsinger and Kraft (1995). Crandalls Model Crandall (1926) treated an orif ice as a narrow tube in which a shear layer was set up near the wall of the tube. The driving force due to pressure gradient was balanced by the inertial force and viscous force. The e quation of the motion is given as 0 inertial force driving force due to viscous force pressure gradient222pu dxrdrjurdrdxrdrdxdr xrr (B1) or 0p jru xrrr (B2) where the particle velocity ,u, is a function of the radius r. In other words, the velocity profile in the tube only has an axial component whic h varies radically and keeps the same along the length of the tube. The solution of Eq. (B2) is PAGE 150 150 0 2 01s ss p xJkr u kJka (B3) where 2 skj and a is the radius of the orifice. Note that Eq. (B3) satisfies the boundary condition of that the velocity is finite when 0r The mean velocity over the crosssection of the tube is thus given as 2 0 1 2 02 2 12 1a s sssuurdr a Jka p kkaJkax (B4) Hence, the specific acoustic impedan ce of the orifice with the thickness t is orifice 0 0 1 01 2 1t s ssp Zdx ux jt Jka kaJka (B5) The specific acoustic impedance of th e perforate sheet with porosity is then given as orificepZ Z (B6) Equation (B5) can be further approximated based on the value of s ka orifice 2 000084 3Z tt j ccac (B7) for 1ska and 2 orifice 2 2 00081 11 32 9 2s ska Z tt j ccac ka (B8) PAGE 151 151 for 110ska and orifice 000022Z tt j ccaca (B9) for 10ska The real portion of Eq.(B7) is known as Poiseuille coefficient of the resistance for laminar flow in the tube, thus Equation (B7) is also called Po iseuille model. Eq. (B8) is also called Helmholtz impedance model because its real part was first determined by Helmholtz. The approximation in Eq. (B9) was developed by M aa (1987). It is worth noting that Crandalls model assumes infinite tube, thus the end co rrection should be added to account for finite thickness of the perforated sheet. GE Impedance Model Motsinger and Kraft (1995) derived the linear resistance of the orif ice assuming Poiseuille flow within the orifice. However, a plug flow was assumed when developing the reactance of the orifice. Furthermore, Mots inger and Kraft assumed a linear re sistance equivalent to a DC flow resistance inside the orifice, thus their linear specific resistance was independent on the frequency. The GE linear impedance model, as Motsinger and Kraft described, is orifice 2 0002 8DDkta Z t j ccCaC (B10) where 0.76DC is the dimensionless orifice discharge coefficient, and 0.8510.7 is the dimensionless end correction factor and is the porosity of the perforated sheet. The specific acoustic impedance of the perforated sheet with porosity is 2 0002 8p DDZ kta t j ccCaC (B11) PAGE 152 152 End Corrections As mentioned above, Crandalls model assume s infinite tube, thus the end correction should be added to account for finite thickness of th e perforated sheet. This is because the flow of air through the orifice of finite length affects the air close to the inner and outer entries. These nearby airs are taken into the flow and thus c ontribute to the total acoustic impedance of the orifice. For a circular hole of radius a, Lord Rayleigh (1945) pres ented an end correction 8 3a (B12) which corresponds to the reactive part of the radi ation impedance of a circ ular piston of radius a in an infinite wall. This end correction can be used as an approximation for a flanged orifice. However, care must be taken if it is used for an orifice has no flange or not baffled, where 0.6133a (B13) Ingard (1953) did an extensive study on th e design of the resonators and developed expressions for the end corrections for some ci rcular and rectangular geometries. Assuming uniform velocity profile in the orifice, he deve loped the end correction of a concentric circular orifice in a circular tube, a circul ar orifice in a square tube, and a square aperture in a square tube as 8 11.25 3a (B14) where is the porosity which defines th e area ratio of the aperture and tube. Eq. (B14) is valid for 0.16 Effect of Nonlinearity at High Sound Pressure Levels Sivian (1935) observed the phenomenon of increasing acoustic resistance with a corresponding increase of the orif ice velocity. Since then, nonlinea r behavior of the orifice at PAGE 153 153 high SPL has attracted more attention from research ers such as Sivian (19 35), Ingard and Labate (1950), Ingard (1953), Bies and Wilson (1957), I ngard and Ising (1967) Zinn (1970), Melling (1973), and Cummings (1983). The effects of high SPL inci dent acoustic wave on the impedance of an orifice are that the acoustic resi stance increases as a function of the SPL and the acoustic reactance stays constant but decreases after a certain SPL. In his paper, Sivian (1935) re ported that the acoustic resistance increased with a increase in particle velocity within the orifice while the reactance was substantia lly independent of the particle velocity. Sivian tested various orifices and found that th e increase in acoustic resistance for all orifices had a similar trend, the resistan ce was constant when particle velocity in the orifice was small (e.g. <0.5m/s) and the resistan ce appeared to be linearly proportional to the velocity when the particle velocity in the orifice was large. Sivian presented that the increase in acoustic resistance was an effect of the increased kinetic energy in the orifice. At high particle velocity in the orifice, the acoustic resistance of the orifice was given by orifice 00 0 11 00 0016 Real 32 22 11o ss ssssV R jtaj cc JkaJka cc kaJkakaJka (B15) where the first term is the acoustic resistance associated with the low particle velocity in the orifice and the second term corresponded to the nonlin ear behavior of the orifice at high particle velocity or SPL, oV is the root mean square (RMS) velocity of the air particle in the orifice, s k is the modified Stokes wave number in the orifice sj k (B16) PAGE 154 154 where 0 is the effective kinematic viscosity under isothermal conditions (e.g., near a highly conducting wall) and is the effective absolute viscosity 21 1 Pr (B17) where is the ratio of specific heats in air and Pr is the Prandtl numb er in air which is a constant equal to 0.706 over a wide range of temperatures. Ingard and Labate (1950) thorough ly studied the effect of an incident acoustic wave on an orifice by examining the flow patterns in and ar ound 25 circular orifices of different thickness and diameters. They observed the large increas es in the acoustic resistance when the incident SPL is high. They also mentioned that the acous tic mass stayed constant independent of the SPL of an incident wave until turbulence was reached in and around the orifice. Ingard (1953) took account of the nonlinear effect of the orifice exposed to a high SPL incident wave using an addi tional resistance end correction orifice 0002R tt cca (B18) where t is the viscous and nonlinea r resistance end correction 22 100n oV taa (B19) where and n are coefficients which may be dependen t on the frequency of the incident sound wave, oV is the RMS particle velocity in the orifi ce. Ingard found that the nonlinear acoustic resistance was not linearly proportio nal to the particle velocity (0.7,1.7n were used in the paper). However, in his paper, the particle velocity in the orifice was less than 0.6m/s. The nonlinear resistance was believed to be linearly propor tional to particle veloci ty in the orifice at PAGE 155 155 higher SPL as indicated by Bies and Wilson (1957), they measured the orifice impedance for the orifice in which the particle velocity was up to 50 m/s. Like Sivian (1935), they found a similar linear dependence of resist ance with particle velocity in an orifice. Ingard and Ising (1967) investig ated the acoustic nonlinearity of an circular orifice in a plate by measuring simultaneously the particle velo city in the orifice using a hot wire and the acoustic pressure. The particle velocity in the orifice was up to 50 m/s. When particle velocity was above 10 m/s, it was found that the acousti c resistance was linearly proportional to the velocity orifice_NL 000oR B V cc, (B20) where B is the constant which lies between 1 and 1. 5 and is dependent on the fluid mechanical behavior of the orifice, and oV is the amplitude of particle velocity in the orifice. Zinn (1970) investigated the in teraction between high SPL in cident wave and a single Helmholtz resonator. The flow in the entrance re gion of the orifice and ca vity was considered in detail using the appropriate cons ervation equations. His analysis indicated that the energy losses at high SPL were due to viscous damping and the dissipation of the kinetic energy of the jets which are periodically formed at both ends of the orifice. Zinn developed the expression for the nonlinear resistance of an orifice orifice_NL 2 0004 3o D R V ccC, (B21) where oV is the amplitude of the orifice particle velocity, and D C is the discharge coefficient which was 0.61 in his paper, then Eq. (B21) results in orifice_NL 0001.16oR V cc (B22) PAGE 156 156 Melling (1973) presented an expression fo r the nonlinear acoustic resistance of a perforated plate 2 p_NL 2 0000.61o DR V cCc (B23) where 0V is the RMS particle velocity in the orifice. Melling found that the discharge coefficient was a function of porosity as well as Re in the orif ice. In his paper, the discharge coefficients were approximately 0.89 and 0.96 for perfor ates with porosities of 7.5% and 22.5%, respectively, to fit the measured data. A similar expression was given by Cummin g and Eversman (1983). They begun with unsteady Bernoullis equation and arrived at an expression for the nonlinear acous tic resistance 22 p_NL 2 0001 0.57o C CR V C cCc (B24) where CC is the contraction coefficient and is defi ned as the ratio between the area associated with a vena contracta (vcA) and the physical orifice area (oA) vc oACA C (B25) The summary on some impedance models whic h include the nonlinearity of the acoustic impedance of an orifice at high SPL is listed in Table B1 PAGE 157 157 Table B1. Summary on some acoustic impedance models of an orifice. Model Assumption Model Description Pros Sivian modified Crandall Model (Sivian 1935) Turbulence is negligible Adiabatic flow Uniform velocity profile No internal dissipation orifice 00 1 0 0 nonlinear resistance w.r.t. high SPL 1 02 1 2 16 3 2 1s ss o s ssZ jt c Jka c kaJka V c aj J ka c kaJka Particle velocity measurement needed Ingard Model (Ingard 1953) orifice 000002 Z ttjtt ccrc t is the mass end correction, 01.711.25 tr for 0.4 Nonlinear behavior prediction highly dependent on wave frequency and other empirical parameters, such as the critical thickness Ingard and Ising Model (Ingard and Ising 1967) orifice_NL 000,oR V cc where is constant which lies between 1 to 1.5. Prediction of nonlinear behavior dependent on empiricism Zinn Model (Zinn 1971) Large amplitude of the particle oscillation compared to the orifice thickness 2 0004 3NLorifice o D R V ccC For 0.61DC the model predicted Ingard(Ingard 1953; Ingard and Ising 1967),Bies and Wilson (Bies and O.B. Wilson 1957)measurement data pretty well Measurement of the orifice discharge coefficient is necessary and important PAGE 158 158 Table B1 (Continue) Model Assumption Model Description Pros Melling modified Crandall Model (Melling 1973) Incompressible flow Poiseuille flow within the orifice Only axial velocity component with radial and axial variation No higher order harmonics produced nonlinearity 00 1 0 0 2 2 0 nonlinear resistance w.r.t. high SPL 1 0 002 1 1.21 2 16 2 3 1p s ss o D s ssZ jt c Jka c kaJka V cC aj J ka c krJka Interaction between orifices included End correction included Empirical parameter dependent Particle velocity measurement needed Cummings and Eversman Model (Cummings and Eversman 1983) Zero means flow Sinusoidal incident pressure signal 22 21 0.45,c NLorifice CVC R c cC where CCis orifice contraction coefficient which defined as the ratio between the area associated with a vena contracta and the physical orifice area. GE Impedance Model (Motsinger and Kraft 1991) Incompressible flow Poiseuille flow within the orifice when developing the linear resistance term Plug flow within the orifice when developing the reactance term 2 00 0 linear resistance 2 nonlinear resistance w.r.t high SPL reactance8 1 2 2p D o D DZ t c cCa V cC kta C Simple and straightforward Assumptions for the resistance and reactance derivation are different Empirical parameters dependent Measurement of the velocity at the orifice or related parameters required Frequency independent PAGE 159 159 Table B1 (Continue) Model Assumption Model Description Pros Kraft et al. modified Crandall Model (Kraft et al. 1999) 00 1 0 0 2 2 0 nonlinear resistance w.r.t. high SPL 1 0 02 1 1 2 16 2 3 1p s D ss o D s D ssZ jt c Jka cC kaJka V cC aj J ka cC kaJka PAGE 160 160 APPENDIX C PARAMETERS ESTIMATION FOR LEM OF THE PIEZOELECTRIC DIAPHRAGM The extraction of the model parameters for the piezoelectric backplate is more complex due to composite plate mechanics. The piezoelec tric backplate consists of an axisymmetric piezoceramic of radius 1 R and thickness ph bonded in the center of a metal shim of radius 2 R and thickness s h. Up to and just beyond the firs t resonant mode, the onedimensional piezoelectric electroacoustic coupling is given by (Prasad et al. 2006) aDa aeFCd P dC qV (C1) where is the volume displacement of the piezoelectri c backplate due to the application of the pressure P and voltage V. Additionally, q is the charge stored on the piezoelectric electrodes, ad is the effective acoustic pi ezoelectric coefficient, and eFC is the electrical free capacitance of the piezoelectric material. The volume displacem ent is calculated by integrating the transverse displacement, wr, over the whole plate 202Rrwrdr. (C2) Thus, the shortcircuit acoustic compliance of the backplate, aDC, is determined by 20 0 01 2aD V R VC P rwrdr P (C3) Similarly, the effective acousti c piezoelectric coefficient, Ad, is determined by application of voltage to the free piezoelectric plate PAGE 161 161 20 0 01 2A P R Pd V rwrdr V (C4) The electroacoustic impeda nce transformer factor, is defined as A aDd C (C5) The effective acoustic mass of the piezoelectric backplate, aD M is found by equating the total distributed kinetic energy stored in the velocity of the plate to a lumped mass as 22 0 02 ()R aDA V M wrrdr V (C6) whereA is the area density of the piezoelectric b ackplate. The blocked electrical capacitance, eBC, is related to the free electrical capac itance of the piezoelectric backplate, eFC, as 2 2 2 011 1eBeFEM r EM pCC R h (C7) where r is the relative dielectric constant of the piezoelectric material, 0 is the permittivity of free space, 1 R is the radius of the piezoceramic, and ph is the thickness of the piezoceramic, and 2EM is the electroacoustic co upling factor and given by 2 2A EM eFaDd CC. (C8) The acoustic resistance, aD R of the piezoelectric backplate models acoustic resistance and structural damping in the backpl ate. The damping may arise from thermoelastic dissipation, compliant boundaries, and other intrinsic loss mech anisms. The acoustic resistance is given by PAGE 162 162 2aDaDrad aD aDMM R C (C9) where is an experimentally determined damping factor determined using, for example, the logarithmic decrement method (Meirovitch 2001). PAGE 163 163 APPENDIX D WAVE SCATTERING BY HELMHO LTZ RESONATOR IN A TUBE When using the TMM to measure the acousti c impedance of the EMHR (or HR), it is assumed that only the plane wave propagates along the tube, and two microphones should be properly mounted certain distance away from the impedance sample. This is because when the PWT terminated by an EMHR or HR, eventhough a plane wave mode is sent from the source, higher order models will be exci ted and scattered at the interface between th e tube and the HR. If the higher order modes can reach one or two microphones, the measurement accuracy is significantly affected. Thus, it is necessary to solve the acoustic field within the tube to determine the proper distance away from the samp le two microphones should be. After such a distance, all higher order modes are far gone in decay. Helmholtz Resonator as a Termination of a Circular Tube A Helmholtz resonator terminates a ci rcular, solidwalled tube with radius0 R as shown in Figure D1 The lossless linear wave equati on in the tube is in the form 2 2 22 01 0ct (D1) where is the velocity potential and 0c is the isentropic smallsign al speed of sound. Moreover, for a harmonic signal ,jtrxe the equation can be simplified as 220k (D2) where 0kc is the wave number, and rz is the complex magnitude of the velocity potential To resolve Eq. (D2), the followi ng assumptions are taken into account (1) The axis of HR coincides with axis of the tube. (2) At the opening of HR, the velocity distribution is uniform and axial symmetrical. PAGE 164 164 (3) The acoustic filed within the tube is independent of Then, using the method of the separation of vari ables, the solution of Eq. (D2) is given by 00zzjkzjkz rr A eBeCJkrDYkr (D3) where A, B C and D are coefficients. zk is the trace wavenumber in the z direction, and 22rzkkk is the trace wavenumber in the r direction. 0J is the zeroth order Bessel function of the first kind, and 0Y is the zeroth order Bessel func tion of the second kind. Note that the acoustic properties in the circul ar tube are finite. The coefficient D in Eq.(D3) thus goes to zero due to 0at 0rYkrr (D4) Moreover, the wall of the circular tube is soli d enough. The normal component of the particle velocity thus vanishes 00at rR r (D5) From Eqs. (D3) and (D5), one has 100rJkR (D6) which yields 00,0,1,2,rnkRn (D7) where 0n is the root of 10 Jx The complete solution for Eq.(D2) is the sum of all the modes which correspond to different roots of 10 Jx 0 0 0 0zzikzikz n nn nAeBeJr R (D8) PAGE 165 165 where 2 2 00 znkkR Furthermore, by defining 2nnnABe (D9) where n is a complex number, Eq. (D8) is rewritten as 0 0 0 0 0 0 0 0 0 0 0 02sinhzz nznzn nikzikz n nn n ikzikz n n n n nzn nAeBeJr R B eeeJr R BeikzJr R (D10) Hence, the particle velocity at the opening of the HR 0 z is given by 0 0 0 0 0 0 0 0 0 02cosh 2coshn nz n znzn n z n znn nur x ikBeikzJr R ikBeJr R (D11) On the other hand, at the opening of the HR ( 0z ) the particle velocity is in the form 00 000 0 Urr uur rrR (D12) Then, the FourierBessel series expression for ur is given by (Blackstock 2000) 0 0 0 0 n n nuraJr R (D13) where na is the coefficient. Comparing (D13) with Eq.(D11) results in 2coshnn n zna B ike (D14) PAGE 166 166 Plugging Eq (D14) into (D10) leads to 00 0 0sinh coshn znn n zna r iikzJ kR (D15) Furthermore, to integrate both si des of Eq. (D13) as follows 00000 000 0 000 00 RR nnn n nurJrrdraJrJrrdr RRR (D16) One has 000 22 0000 02R nn naurJrrdr RJR (D17) where the following orthogonali ty relation is used 000 00 00 0 1 2 00000 0000 0 1 0 0001010010 00000000 0 R nn nn nnnnn nnJrJrrdr RR rrrr RJJd RRRR rrrrrr RJJJJ RRRRRR 1 0 0 1 2 0 01010 00000 0 2 2 0 10 2 2 0 0020 2 2 0 001000 02 1 22 12 22n nn n nnn nnnn nnn nr d R rrrr RJJd RRRR R J R JJ R JJJ R 2 2 0 002nn nnnR J (D18) PAGE 167 167 where nn is the Kronecker delta function. Substitu tion Eq. (D12) into (D17) results in 2 0000aUrR (D19) and 000 10 2 000002 1nn nnUrr aJn JRR (D20) Using the Euler equation, the pressure in the tube is given by 0 00 0 000 0 0 0 000 0 0 2 0 00 000000001 2 000000 Isinh cosh sinh cosh sinh 2 coshnn zn n zn nn zn n nn n nn p t ick ar ckikzJ kR ar cikzJ R ikz rr cUrRcUJ RJR 0 0 1 0 IIsinh coshzn n n nnikz r J R (D21) where nzkk and the first term of Eq (D21) is plane wave contribution while the second term is caused by the higher modes scattered back from the HR. Furthermore, the tube is supposed to be long enough and the exciting frequency is low, so that there is only plane wave propagating through the whole tube, accordingly 01nAn (D22) which results in n then sinh lim cosh z nzn ikz nikz e (D23) Thus, Eq. (D21) can be simlified as PAGE 168 168 2 0 00000 000000010 0 2 1 0000000 H.O.M.sinh 2 cosh z ikz nn n nnnikz rrr p cUrRcUJJe RJRR (D24) Furthermore, the pressure in the tube can also be expressed as 000 00H.O.M 2sinhH.O.MikzikzpPePe Peikz (D25) where 0P and 0P are the incident and reflected (plane wave components) wave amplitude at 0 x respectively, and H.O.M. represents the higher order modes, and the definition of is given by Eq. (D9). Hence, comparing Eq (D25) and (D24) results in 02 00000 0 02 coshcUrR Pe (D26) or 02 00000002coshcUPeRr. (D27) Furthermore, at 0 z the relationship between the pressu re and the velocity is given by 2 0 0000 0000010 2 1 0000000 2 0 000 0 00sinh 2 cosh sinh coshnn n nnn HRHRrr p crRcJJ UJRR crR cj (D28) where H R and H R are the specific acoustic resistance and reactance of Helmholtz resonator, respectively. Thus, fr om Eq (D28), one has 2 000tanh H RHRRrj. (D29) PAGE 169 169 Finally, from Eqs. (D24)(D27), the pressure in the tube when it is terminated by a HR is given by 0 000 2 00000 000010 2 1 0000002sinh 21 2cosh z ikz nn n nnnpPeikz rrr PeRrJJe RJRR (D30) Helmholtz Resonator as a Termination of a Rectangular Tube A HR terminates a rectangular t ube with crosssectional area, ab, shown in Figure D2 The center of the opening of Helmholtz resonator locates at ,,0ccab. To simplify the problem, the following assumptions are taken into account (1) The wall of the duct is rigid. (2) The acoustic filed in the tube is lossless and linear. (3) At the opening of HR, the velocity distribution is uniform; Thus, using the method of the separation of vari ables, the solution of Eq. (D2) is given by cossincossinzzjkzjkz xxyyAkxBkxCkyDkyEeFe (D31) where A B C, D, E and F are coefficients, and xk, yk and zk are the trace wavenumbers in the x y and z directions. Furthermore, a rigidwall of rectangular tube means that the normal component of particle velo city vanishes at each solid wall 0at0, x a x (D32)\ and 0at0,yb z (D33) Substitution of Eq. (D31) into Eqs.(D32) and (D33) results in 00,1,2xBkmam (D34) PAGE 170 170 and 00,1,2yDknbn (D35) Plugging Eqs.(D34) and (D35) into (D31) and collecting th e coefficients results in coscosmnmnjkzjkz mnmnmnmxanybAeBe, (D36) where mnk is the zdirection trace wave number 22 2 mnkkmanb. (D37) Physically, indices m and n represent the number of half wave in the direction of x andy. The combination ,mn names the acoustic mode in the duct. The complete solution for the magnitude of the velocity potential is thus the sum of all the modes 00coscosmnmnjkzjkz mnmn mnmxanybAeBe (D38) Similarly, by defining 2mnmnmnABe (D39) one has 002coscossinhmnmnmnmn mnmxanybBejkz (D40) and 002coscoscoshmnmnmnmnmn mnzjmxanybBekjkz (D41) Moreover, it is assumed that velocity distribution at the opening of HR (0z) is 000 0otherwiseUrr ur (D42) Thus, from Eq. (D41) and (D42), one has PAGE 171 171 0 00 02coscoscosh ,mnmnzmnmn z mnzjmxanybBek UHxy (D43) where ,Hxy is Heaviside unit step function, which equal one at the opening of HR, was zero at other place. Furthermore, to integrat e both sides of Eq. (D43) as follows 0 opening ,0,0 00coscos 2coscoscoscoscoshmnab mnzmnmn mmnnUmxanybdxdy jmxanybmxanybBekdxdy (D44) one has 2coshmnmnmnzmnmnRHSvjBek (D45) where RHS is the right hand side of Eq. (D44), and for 0 2for 0,0 or 0 0 4for 0mnabmn vabmnmn abmn (D46) When 0mn, the LHS of Eq.(D44) is 2 00LHSUr (D47) Thus, the coefficient 00 B is given by 002 00 00 002coshUr Bj abek (D48) For the case of which 0mn or mn but one of them equals zero, one can solve the coefficients mn B using a transformation between Cartesia n coordinate system and polar system for the opening of HR, as shown in Figure D3 PAGE 172 172 0cos sin :02 :0c c x ra yrb rrr (D49) Thus, the LHS of the Eq.(D44) can be related to 00 opening 2 0 00 Icoscos coscoscossinr ccLHSUmxanybdxdy Umraanrbbrdrd (D50) Further manipulation of (I) results in Icoscoscossin coscossin 1 2 coscossin coscoscossin sinsincossin 1 2 ccc cc cc cc ccmraanrbb mraanrbb mraanrbb maanbbmranrb maanbbmranrb oscoscossin sinsincossin coscoscos sinsincos 1 2 coscoscos sincc cc cc cc cc cmaanbbmranrb maanbbmranrb maanbbr maanbbr maanbbr maa sincoscnbbr (D51) where 22tan rmanbr namb (D52) Furthermore, the integration over in Eq.(D51) and mmnn leads to 2 0 0I2coscosccdmaanbbJr (D53) PAGE 173 173 where some identities as follows are used 22 00 2 0 0coscoscoscos coscos coscos 2 rdrd rd rd Jr (D54) 22 00 2 0sincossincos sincos sincos 0 rdrd rd rd (D55) 01 cossinnJrrnd (D56) and cos 0cosn ir ni Jrend (D57) Furthermore, the integration over r in Eq. (D53) results in 0 00 0 0 0 22 0 10 222coscos 2coscos 2coscosr cc r cc ccmaanbbJrrdr maanbbJrrdr maanbbr Jmanbr manb (D58) where the identity for the Bessel function is used PAGE 174 174 01rJrdrrJr (D59) Thus, the LHS of Eq. (D44) is 22 0 010 222coscosccmaanbbr LHSUJmanbr manb (D60) Then, one can arrive at the followi ng expression for the coefficients mn B from Eqs. (D44) ,(D45) and (D60) 22 0 0 10 22coscos coshmncc mn mnzmnmnmaanbbr U B jJmanbr ek manb (D61) for 0 mn or 0,0 mn or 0,0 mn The complex amplitude of the velocity potential is thus given by 2 00 00 00 I 22 010 22 0 IIsinh cosh coscos 2 cosh coscossinhcc mnzmnmn mn mnmnUr jjkz abk rJmanbr maanbb jU k manb mxanybjkz (D62) where; mn is defined by Eq. (D46), and zmnk is given by Eq. (D37) The first part of Eq.(D62) represents the planar in cident wave with its reflected one, while the second part is for the higher order modes excite d around the discontinuity. The pressure in the tube is thus obtained by using Euler equation PAGE 175 175 0 2 0000 00 00 22 010 22 000sinh cosh coscos 2 cosh coscossinhcc mnmnmn mn mnmn p t cUr jkz ab rkJmanbr maanbb cU k manb mxanybjkz (D63) The tube is long enough and the exciting frequency is low, so that there is only plane wave propagating through the whole tube, accordingly 0mnA (D64) and mn (D65) for ,0,0 mn Hence, one has sinh lim coshmn mnmnmnikz mnikz e (D66) Eq. (D63) is then rewritten as 2 0000 00 00 0 22 22 00010 H.O.Msinh cosh coscos 2cos cosmncc mnmn mn ikzcUr pjkz ab rmaanbb k k manb cUJmanbrmxa nybe (D67) PAGE 176 176 Furthermore, the pressure in the tube can also be expressed as 0000 000H.O.M 2sinhH.O.MikzikzpPePe Peikz (D68) where 0P and 0P are the incident and reflected plane wave amplitude at 0 z respectively. The definition of 00 is the same as Eq.(D39). Compari ng Eqs.(D68) and (D67) results in 002 0000 0 002 cosh cUr Pe ab (D69) or 002 00000002cosh cUPeabr (D70) At 0z, the relationship between the pressu re and the velocity is given by 22 2 22 000 000010 0 2 00000 00coscos tanh2cos cos tanhcc mnmn mn HRHRmaanbb k k manb cr p cJmanbrmxa Uab nyb crab cj (D71) Thus, one has 2 000tanh H RHRabrj (D72) Finally, from Eqs. (D67) (D71), one has the pr essure in the tube when it is terminated by a HR PAGE 177 177 00 00000 22 22 2 00000102sinh coscos 4coshcos coszmncc mnzmn mn ikzpPejkz maanbb k k manb PreabrJmanbrmxa nybe (D73) where 00 is given by Eq. (D72). PAGE 178 178 0 R r z 0r Figure D1: A Helmholtz resonator as termination of a circular tube. 0,0,0yz x ba ,,0ccab Figure D2: Schematic of a rectangular t ube terminated by a Helmholtz resonator. PAGE 179 179 x z r0rba0,0 cbca Figure D3: Schematic of transf orm between the Cartesian coordi nate system and polar system. PAGE 180 180 APPENDIX E GEOMETRIC DIMENSION OF EMHRS As discussed in Chapter 4 the EMHRs are modularly constructed, as shown in Figure E1 The engineering draft of the EMHR (Case I) is presented as follows. Figure E1. Schematic of the EMHR which consists of, from left to right, an orifice plate, cavity plate, piezoelectric diaphragm bottom plat e and piezoelectric diaphragm cap plate. Figure E2. Engineering dr aft of the orifice plate. PAGE 181 181 Figure E3. Engineering draft of th e orifice plate of the cavity plate. PAGE 182 182 Figure E4. Draft of the piezo electric diaphragm bottom plate. PAGE 183 183 Figure E5. Draft of the piezo electric diaphragm cap plate. PAGE 184 184 APPENDIX F COMPUTER CODES In this appendix, the MATLAB codes used to predict the acoustic impedance and to implement the optimization design of an EMHR are presented. Acoustic Impedance Prediction using LEM and TM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % EMHR impedance calculation % % Variable Definitions % % % % % Dtube: the diameter of the PWT [m] % r: the radius of the neck (m) % t: the thickness of the neck (m) % R: the radius of the cavity (m) % L: the depth of the cavity (m) % as: the radius of the shim of the PZT backplate (m) % ts: the thickness of the shim of the PZT backplate (m) % ap: the radius of the piezoceramic of the PZT backplate (m) % tp: the thickness of the piezoceramic of the PZT backplate (m) % Es: Youngs modulus of the shim (N/m^2) % vs:Poisson ratio of the shim % rhos: density of the shim [kg/m^3] % Ep: Youngs modulus of the piezoceramic (N/m^2) % vp: Poisson ratio of the piezoceramic % rhop: density of the piezoceramic [kg/m^3] % d31: the piezo strain constant d31 % dp: relative dilectric constant of the piezoceramic % damp: the damping coefficient of the PZT backplate % % Format for Frequency file (freq.txt) % % f1, f2 ... fn % where the frequencies are provided in Hz % % % Format for the geometry and material properties of the EMHR input file % (input.txt) % % Dtube % r % t % R % L % as % ts % ap % tp % Es PAGE 185 185 % vs % rhos % Ep % vp % rhop % d31 % dp % damp %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function EMHR() global rho0 t0 c0 visc Pref global c rho freqs disp (' ') disp (' ') disp ('') disp ('EMHR impedance calculation linear ') disp (' ') disp ('') disp (' ') file = input ('Please enter output filename: ','s'); fid=fopen(file,'w'); fprintf(fid, 'Output File is:%12s\n\n',file); disp (' ') % Date and Time output b=fix(clock); time=[num2str(b(1)),'',num2str(b(2)),'',num2str(b(3)),'',... num2str(b(4)),'',num2str(b(5))]; fprintf(fid, 'Time (yymmddhhmm):%20s\n\n',time); %Initialization of temperature, density of air disp(' ') disp('Initialization of temprature, density of air') disp(' ') tc = input ('Please enter temperature (C): '); fprintf(fid,'Temperature is (C):%5.2f\n',tc); rho = rho0*t0/(t0+tc); c = c0*sqrt(1+tc/t0); fprintf(fid,'Density is(kg/m^3):%5.2f\n',rho); fprintf(fid,'Sound speed is(m/s):%5.2f\n\n',c); disp('') %Frequency initial disp (' ') disp ('Frequency Initialization') disp (' ') ifrev = input('Frequencies evenly spaced? (Yes 1, No2): '); disp(' ') if ifrev == 1 f1 = input ('Please enter beginning frequency (Hz): '); f2 = input ('Please enter stopping frequency (Hz): '); df = input ('Please enter delta frequency (Hz): '); fprintf(fid,'Beginning Freq.(Hz): %12.2f\n',f1); PAGE 186 186 fprintf(fid,'Stopping Freq.(Hz): %12.2f\n',f2); fprintf(fid,'Deta Freq.(Hz): %12.2f\n',df); nf = (f2f1)/df+1; fprintf(fid,'# of computing points: %6.1f\n\n',nf); freqs = f1:df:f2; else filefr = input ('Enter Filename containing frequencies :','s'); freqstemp = importdata(filefr); freqs = freqstemp.data; nf = str2num(freqstemp.textdata{1,1}); if nf ~= length(freqs) error('Frequency input is invalid, program will be terminated' ) end fprintf(fid,'Frequency File is:%12s\n\n',filefr); fprintf(fid,'# of computing points: %6.1f\n\n',nf); end disp('') %Geometry initial disp (' ') disp ('Geometry and Material Properties Initialization') disp (' ') filein = input ('Enter Filename containing geometry and material properties of an EMHR :','s'); Intemp = importdata(filein); Indata = Intemp.data; Dtube = Indata(1); fprintf(fid,'The diameter of the PWT (m): %12.5f\n',Dtube); r = Indata(2); fprintf(fid,'The radius of the neck (m): %12.5f\n',r); t = Indata(3); fprintf(fid,'The thickness of the neck (m): %12.5f\n',t); R = Indata(4); fprintf(fid,'The radius of the cavity (m): %12.5f\n',R); L = Indata(5); fprintf(fid,'The depth of the cavity (m): %12.5f\n',L); as = Indata(6); fprintf(fid,'The radius of the shim of the PZT backplate (m): %12.5f\n',as); ts = Indata(7); fprintf(fid,'The thickness of the shim of the PZT backplate (m): %12.5f\n',ts); ap = Indata(8); fprintf(fid,'The radius of the piezoceramic of the PZT backplate (m): %12.5f\n',ap); tp = Indata(9); PAGE 187 187 fprintf(fid,'The thickness of the piezoceramic of the PZT backplate (m): %12.5f\n',tp); %Material properties Es = Indata(10); fprintf(fid,'The Youngs modulus of the shim (N/m^2): %12.5f\n',Es); vs = Indata(11); fprintf(fid,'The Poisson ratio of the shim : %12.5f\n',vs); rhos =Indata(12); fprintf(fid,'The density of the shim : %12.5f\n',rhos); Ep = Indata(13); fprintf(fid,'The Youngs modulus of the piezoceramic (N/m^2): %12.5f\n',Ep); vp = Indata(14); fprintf(fid,'The Poisson ratio of the piezoceramic : %12.5f\n',vp); rhop = Indata(15); fprintf(fid,'The density of the piezoceramic : %12.5f\n',rhop); d31 = Indata(16); fprintf(fid,'The piezo strain constant : %12.5f\n',d31); dp = Indata(17); fprintf(fid,'The relative dielectric constant of the PZT: %12.5f\n',dp); damp = Indata(18); fprintf(fid,'The damping coefficient of the PZT backplate : %12.5f\n',damp); %Shunts initial disp (' ') disp ('shunts Initialization') flag01 = input('The shunt load is [1]resitive, [2]capacitive or [3]inductive: '); switch flag01 case 1 disp(' ') ZL = input('Please input the value of resitive load [Ohm]: '); fprintf(fid,'The the value of resitive load [Ohm]: %12.5f\n',ZL); ZL = ZL*ones(1,length(freqs)); case 2 disp(' ') ZL = input('Please input the value of capacitive load [F]: '); fprintf(fid,'The the value of capacitive load [F]: %12.5f\n',ZL); ZL = 1./(j*2*freqs*pi*ZL); case 3 disp(' ') ZL = input('Please input the value of inductive load [H]: '); fprintf(fid,'The the value of inductive load [F]: %12.5f\n',ZL); ZL = j*2*freqs*pi*ZL; end % Compute the acoustic impedance of the EMHR PAGE 188 188 disp (' ') disp ('calculation of the acoustic impedance of the EMHR') disp (' ') flag02 = input('Please choose model of the EMHR [1]LEM [2]TM: '); % LEMLumped element model TMTransfer matrix model if flag02 ==1 zetal = LEMEMHR(r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL); else zetal = TMEMHR (r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL); end %Result output %Write data to output file fprintf(fid,'Freq \t Normalized Zeta \t \r'); for inf = 1:nf f(inf) = freqs(inf); fprintf(fid,'%7.2f\t %12.5f\t \t %12.5f\t\t \r',... [f(inf);real(zetal(inf)); imag(zetal(inf))]); end fclose(fid); %Plot gragh for normalized resistance and reactance figure (1) set(gcf,'paperorientation','landscape') set(gcf,'paperposition',[0.25 0.25 10.5 8.0]) set(gcf,'DefaultlineLinewidth',2) h1=subplot(2,1,1); h2=subplot(2,1,2); subplot(2,1,1); plot(f,real(zetal)) subplot(2,1,2); plot(f,imag(zetal)) subplot(h1) ylabel('\theta','fontsize',12) title ('EMHR impedance calculation linear ','fontsize',12) grid on subplot(h2) ylabel('\chi','fontsize',12) xlabel('Freq.[Hz]','fontsize',12) grid on disp(' ') figure (2) set(gcf,'paperorientation','landscape') set(gcf,'paperposition',[0.25 0.25 10.5 8.0]) set(gcf,'DefaultlineLinewidth',2) plot(f,real(zetal)) xlabel('Freq.[Hz]','fontsize',12) ylabel('Normalized Resistance','fontsize',12) title ('EMHR impedance calculation linear ','fontsize',12) grid on PAGE 189 189 figure (3) set(gcf,'paperorientation','landscape') set(gcf,'paperposition',[0.25 0.25 10.5 8.0]) set(gcf,'DefaultlineLinewidth',2) plot(f,imag(zetal)) xlabel('Freq.[Hz]','fontsize',12) ylabel('Normalized Reactance','fontsize',12) title ('EMHR impedance calculation linear ','fontsize',12) grid on disp ('End of calculation of EMHR') %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Subfunction LEMEMHR % % This subfunction is used to compute the acoustic impedance of the EMHR % using LEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function ZinEMHRLEM = LEMEMHR(r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL) global rho0 t0 c0 visc Pref global c rho freqs % Constants definition mu = visc; gama = 1.4; % Frequency information ww = 2*pi*freqs; k = ww/c; % Acoustic impedance of the neck of the EMHR muprime = mu*(1+(gama1)/sqrt(0.71))^2; nu = mu/rho; nuprime = muprime/rho; ks = sqrt(j*ww/nu); ksprime = sqrt(j*ww/nuprime); Fs = 12*besselj(1,ks*r)./(ks*r.*besselj(0,ks*r)); Fsprime = 12*besselj(1,ksprime*r)./(ks*r.*besselj(0,ksprime*r)); ZaN = j*ww*rho/(pi*r^2).*(t./Fsprime+1.7*r./Fs); % Acoustic impedance of the cavity of the EMHR ZaC = j*cot(k*L)*rho*c/(pi*R^2); % Acoustic impedance of the PZT backplate [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp) Rf0 = 1/2/pi/sqrt(MaD*CaD) RaD = 2*damp*sqrt(MaD./CaD); %RaD = (k*ap).^2/2*rho*csound/(pi*ap^2); ZaD = j*ww*MaD+ 1./(j*ww*CaD)+RaD + Phi^2*ZL./(1+j*ww*CeB.*ZL); PAGE 190 190 ZinEMHRLEM = (ZaN + ZaC.*ZaD./(ZaC+ZaD))*(Dtube)^2/rho/c; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Subfunction TMEMHR % This subfunction is used to compute the acoustic impedance of the EMHR % using TM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function ZinEMHRTR = TMEMHR(r,t,R,L,as,ts,ap,tp,Dtube,Es,vs,rhos,Ep,vp,rhop,d31,dp,damp,ZL) global rho0 t0 c0 visc Pref global c rho freqs % Constants definition mu = visc; gama = 1.4; % Frequency information ww = 2*pi*freqs; % Geometry information A5 = (Dtube)^2; %area of the PWT tube A4 = pi*r^2; %area of the neck A3 = A4; A2 = pi*R^2; %crosssectional area of the cavity A1 = 1/3*pi*as^2; % effective are of the piezoelectric backplate [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); % Computing entries of the transfer matrix and acoustic impedance of the % EMHR for n = 1:length(freqs) w = ww(n); s = j*w; k = w/c; zeta4 = 1/(rho*c)*(sqrt(2*mu*rho*w)+rho*w^2*A4/(2*pi*c))+... j*8*k*r/(3*pi); zeta3 = zeta4; GamaN=j; GGN = 1; [GamaC GGC] = GamaTREMHR(R,mu,rho,gama,w); ZaD = s*MaD+1/(s*CaD)+2*damp*sqrt(MaD/CaD); TR = [1 zeta4;0 A4/A5]*... [cosh(GamaN*k*t) sinh(GamaN*k*t)/GGN; GGN*sinh(GamaN*k*t) cosh(GamaN*k*t)]*... [1 A2/A3*zeta3;0 A2/A3]*... [cosh(GamaC*k*L) sinh(GamaC*k*L)/GGC; GGC*sinh(GamaC*k*L) cosh(GamaC*k*L)]*... [(s*CeB*ZaD+Phi^2)/Phi ZaD/Phi; s*rho*c*CeB/(A1*Phi) rho*c/(A1*Phi)]; ZinEMHRTR(n) = (TR(1,1)*ZL(n)+TR(1,2))/(TR(2,1)*ZL(n)+TR(2,2)); PAGE 191 191 end ZinEMHRTR; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Subfunction Center % % This subfunction is used to compute the acoustic impedance of the % piezoelectric backplate, it is modified based on the code developed by % Guiqin Wang % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp) % Calculate the CaD, MaD of the piezoelectric backplate % Pressure loading P = 1; V = 0; Ef = V/tp; [wtotal wwtot wefftot] = defEMHR(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp,P,V,Ef); % The values of CaS, MaD, FreQ and W0 Area = pi*as^2; CaD = abs((wtotal)/(P)); MaD = 2*pi*wwtot/(wefftot^2)/(Area^2); % Calculate the dA % Voltage loading P = 0; V = 1; Ef = V/tp; [wtotal wwtot wefftot] = defEMHR(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp,P,V,Ef); % The value of dA dA = abs(wtotal/1); % Calculate the Phi Phi = dA/CaD; % Calculate CEB PAGE 192 192 epsilon0 = 8.8542E12; %permitivity of free space in F/m dielectricconstant = dp; %relative permitivity of the piezo epsilon=dielectricconstant*epsilon0; %absolute permitivity of the piezo CeF = epsilon*pi*(ap^2)/tp; kk = 1(dA^2)/CeF/CaD; CeB = CeF*(kk); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Subfunction defEMHR% % The subfunction is used to compute the displacement of the piezoelectric % backplate % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [wtotal wwtot wefftot] = defEMHR(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp,P,V,Ef) % Initializing shim Esi = Es; vsi = vs; densitysi = rhos; R2 = as; tsi = ts; % Initializing piezoelectric tp = tp; R1 = ap; R21 = R1/R2; Ep = Ep; %Youngs modulus of the piezo vp = vp; %Poissons ratio of the piezo material d31 = d31; %electromechanical transduction const of the piezo densityp = rhop; %Density of the piezo epsilon0 = 8.8542E12; %permitivity of free space in F/m dielectricconstant = dp; %relative permitivity of the piezo epsilon=dielectricconstant*epsilon0; %absolute permitivity of the piezo % Computing A,B,D for the central and annular paltes % Constitutive Relations for isotropic circular plates Qsi = [1 vsi; vsi 1].*(Esi/(1(vsi^2))); Qp = [1 vp; vp 1].*(Ep/(1(vp^2))); % Taking the original zxis as centre of the shim layer for zin1 = tsi/2 ; %distance of bottom of shim layer from reference zin2 = tsi/2 ; %distance of interface from reference zin3 = tp + tsi/2; zout1 = tsi/2 ; %distance of top of the piezo layer from reference PAGE 193 193 zout2 = tsi/2 ; A_out = Qsi.*(zout2zout1) ; B_out = Qsi.*((zout2^2zout1^2)/2) ; D_out = Qsi.*((zout2^3zout1^3)/3) ; A_in = Qsi.*(zin2zin1) + Qp.*(zin3zin2) ; B_in = Qsi.*((zin2^2zin1^2)/2) + Qp.*((zin3^2zin2^2)/2); D_in = Qsi.*((zin2^3zin1^3)/3) + Qp.*((zin3^3zin2^3)/3); % Computing D Mark(determinant of matrix mapping defined variables y1,y2 to U0 theta Dstar_in = D_in(1,1)(B_in(1,1)^2)/A_in(1,1); Dstar_out = D_out(1,1)(B_out(1,1)^2)/A_out(1,1); % Computing fictitious forces due to piezo Mp_in = Ef* (Ep/(1vp)) d31 (zin3^2zin2^2)/2; Np_in = Ef* (Ep/(1vp)) d31 (zin3zin2); Mp_out = 0; Np_out = 0; %Computing coefficient matrixa_R = R1/R2; Gamma_out = B_out(1,1)/A_out(1,1); Gamma_in = B_in(1,1)/A_in(1,1); AB12_in = P*(R1^2)*(B_in(1,2)Gamma_in*A_in(1,2))/8/Dstar_in; AB12_out = P*(R1^2)*(B_out(1,2)Gamma_out*A_out(1,2))/8/Dstar_out; BD12_in = P*(R1^2)*(3*Dstar_in+D_in(1,2)Gamma_in*B_in(1,2))/8/Dstar_in; BD12_out = P*(R1^2)*(D_out(1,2)Gamma_out*B_out(1,2))/8/Dstar_out; arf = 1/(a_R^4)1; % Compute the matrix A11 = 1; A12 = 0; A13 = (1/(a_R^2))1; A14 = 0; A21 = 0; A22 = 1; A23 = 0; A24 = 1/(a_R^2)1; A31 = B_in(1,1) B_in(1,2); A32 = A_in(1,1)+A_in(1,2); A33 = B_out(1,1)*(1/(a_R^2)+1) + B_out(1,2)*(11/(a_R^2)); A34 = A_out(1,2)*(1/(a_R^2)1) A_out(1,1)*(1+1/(a_R^2)); A41 = D_in(1,1) D_in(1,2); A42 = B_in(1,1) + B_in(1,2) ; A43 = D_out(1,1)*(1/(a_R^2)+1) + D_out(1,2)*(1 1/(a_R^2)); A44 = B_out(1,2)*(1/(a_R^2)1) B_out(1,1)*(1/(a_R^2) + 1); b1 = P*(R1^2)/8/Dstar_in + P*(R1^2)*arf/8/Dstar_out; b2 = Gamma_in*P*(R1^2)/8/Dstar_in + Gamma_out*P*(R1^2)*arf/8/Dstar_out; PAGE 194 194 b3 = AB12_in + 2*Np_in + AB12_out*arf 2*Np_out; b4 = BD12_in + 2*Mp_in + BD12_out*arf 2*Mp_out + P*(R1^2)*(3+1/(a_R^4))/8; A = [A11 A12 A13 A14; A21 A22 A23 A24; A31 A32 A33 A34; A41 A42 A43 A44]; b = [b1 b2 b3 b4]'; % Calculate the costants c1234 = inv(A)*b; c1 = c1234(1); c2 = c1234(2); c3 = c1234(3); c4 = c1234(4); c6 = P*(R2^4)*(0.25log(R2))/16/Dstar_out 0.5*c3*(R2^2)*(0.5log(R2)); c5 = P*(R1^4)/64/Dstar_in 0.25*c1*(R1^2) P*(R1^4)*(0.25(log(R1)/(a_R^4)))/16/Dstar_out ... + 0.5*c3*(R1^2)*(0.5log(R1)/(a_R^2)) + c6; % Composite plates; % Computing deflections and Forces at the interface by superposition % Finding Deflection in the central region and the annular region % Initialize num1 = 1500; r = linspace(0,1,num1+1); jj=floor(R21*num1)+1; for i=1:num1+1 rad1=r(i)*R2; if (i PAGE 195 195 sig = sqrt(0.71); s = r*sqrt(w*rho/mu); nn = (1+(gama1)... /gama*besselj(2,j^(3/2)*sig*s)/besselj(0,j^(3/2)*sig*s))^(1); GamaTR = sqrt(besselj(0,j^(3/2)*s)*gama/besselj(0,j^(3/2)*s)/nn); GG = j*gama/(GamaTR*nn); Optimizing Tuning Range of an EMHR with Capacitive Loads %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % There are eight design variables to optimize the tuning range of EMHR % x: collective variable for all eight design variables % x(1): >r radius of the neck of EMHR % x(2): >t thickness of the neck of EMHR % x(3): >R radius of the cavity of EMHR % x(4): >L depth of the cavity of EMHR % x(5): >ap radius of the pztlayer of pztbackplate of EMHR % x(6): >tp thickness of the pztlayer of pztbackplate of EMHR % x(7): >as radius of the shim of pztbackplate of EMHR % x(8): >ts thickness of the shim of pztbackplate of EMHR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% clear all close all %starting guess x0 =[2.42e3,3.16e3,6.34e3,9.4e3,10.05e3,0.13e3,12.4e3,0.19e3]; %optimal initial set options = optimset('LargeScale','off','Display','iter',... 'MaxFunEvals',2000,'TolX',1e7,'Tolcon',1e7); %lowerbound for the design variables LB =[1e3; 1e3; 5e3;10e3;1e3;0.05e3;1e3;0.05e3]; %upperbound for the design variables UB =[3.5e3; 4.5e3; 15e3;20e3;25e3;1e3;25e3;1e3]; %linear constraints A =[0 0 1 0 0 0 1 0; 0 0 0 0 1 0 1 0; 0 0 0 0 0 1 0 1; 0 0 0 0 0 1 0 1]; B =[1e3;1e3;1e4;1e3]; disp('Please select the optimization goal:') disp('[1]the first resonant freq.') disp('[2]the second resonant freq.') flag = input(''); PAGE 196 196 %optimal design switch flag case 1 [x,fval,exitflag,output,lambda,grad] = fmincon('OPTEMHRFUN01',x0,... A,B,[],[],LB,UB,'OPTEMHRCONNIND',options); case 2 [x,fval,exitflag,output] = fmincon('OPTEMHRFUN02',x0,... A,B,[],[],LB,UB,'OPTEMHRCONIND',options); otherwise 'Catch you later' end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The objective function to maximize f1 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function f = OPTEMHRFUN01(x) %% % Constants definition rho = 1.231; % air density csound = 343; % speed of sound at air % Material properties of the piezo Es =110e9; % Young's modulus of the shim vs = 0.375; % Poisson ratio of the shim rhos = 8530; % density of the shim Ep = 63e9; % Young's modulus of the piezo vp = 0.31; % Poisson ratio of the piezo rhop = 7700; % density of the piezo d31 = 175e12; % d31 of the piezo dp = 1750; % relative dilectric constant % geometry of the EMHR r = x(1); t = x(2); R = x(3); L = x(4); ap = x(5); tp = x(6); as = x(7); ts = x(8); % Computing the acoutic impedance of the EMHR MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity PAGE 197 197 [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); PhiA = Phi; % The first resonant frequency of the shortcircuited EMHR w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the shortcircuited EMHR w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The first resonant frequency of the opencircuited EMHR w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % % The second resonant frequency of the opencircuited EMHR w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; %objective function f = w02sw02o; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The objective function to maximize f2 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function f = OPTEMHRFUN02(x) %% % Constants definition rho = 1.231; % air density csound = 343; % speed of sound at air % Material properties of the piezo Es =110e9; % Young's modulus of the shim vs = 0.375; % Poisson ratio of the shim rhos = 8530; % density of the shim Ep = 63e9; % Young's modulus of the piezo vp = 0.31; % Poisson ratio of the piezo rhop = 7700; % density of the piezo d31 = 175e12; % d31 of the piezo dp = 1750; % relative dilectric constant PAGE 198 198 % geometry of the EMHR r = x(1); t = x(2); R = x(3); L = x(4); ap = x(5); tp = x(6); as = x(7); ts = x(8); % Computing the acoutic impedance of the EMHR MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); PhiA = Phi; % The first resonant frequency of the shortcircuited EMHR w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the shortcircuited EMHR w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The first resonant frequency of the opencircuited EMHR w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % % The second resonant frequency of the opencircuited EMHR w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; %objective function f = w02sw02o; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The constraints of optimizing problem to maximize f1 or f2 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [c, ceq] = OPTEMHRCONNIND(x) % Constants definition rho = 1.231; % air density PAGE 199 199 csound = 343; % speed of sound at air % Material properties of the piezo Es =110e9; % Young's modulus of the shim vs = 0.375; % Poisson ratio of the shim rhos = 8530; % density of the shim Ep = 63e9; % Young's modulus of the piezo vp = 0.31; % Poisson ratio of the piezo rhop = 7700; % density of the piezo d31 = 175e12; % d31 of the piezo dp = 1750; % relative dilectric constant % geometry of the EMHR r = x(1); t = x(2); R = x(3); L = x(4); ap = x(5); tp = x(6); as = x(7); ts = x(8); % Computing the acoutic impedance of the EMHR MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); PhiA = Phi; % The first resonant frequency of the shortcircuited EMHR w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the shortcircuited EMHR w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The first resonant frequency of the opencircuited EMHR w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the opencircuited EMHR w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; %nonlinear inequality constraints c = [1200w01s,w01s1900,w02s3000]; PAGE 200 200 %nonlinear equality constraints ceq = []; Pareto Optimization Design of an EMHR with Capacitive Loads %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The program is used to implement Pareto optimization design of an EMHR % using epsilon method % % There are eight design variables to optimize the tuning range of EMHR % x: collective variable for all eight design variables % x(1): >r radius of the neck of EMHR % x(2): >t thickness of the neck of EMHR % x(3): >R radius of the cavity of EMHR % x(4): >L depth of the cavity of EMHR % x(5): >ap radius of the pztlayer of pztbackplate of EMHR % x(6): >tp thickness of the pztlayer of pztbackplate of EMHR % x(7): >as radius of the shim of pztbackplate of EMHR % x(8): >ts thickness of the shim of pztbackplate of EMHR % x(9): >MeL inductive loads %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% close all clear all global index %starting guess x0 =[2.42e3,3.16e3,6.34e3,16.4e3,10.1e3,0.12e3,12.4e3,0.19e3]; mm = 18:2:174; % set up the contraint for f1 for nn = 1:length(mm) index = mm(nn); %optimal initial set options = optimset('LargeScale','off','Display','iter',... 'MaxFunEvals',2000,'TolX',1e7,'Tolcon',1e7); %lowerbound for the design variables LB =[1e3; 1e3; 5e3;10e3;1e3;0.05e3;1e3;0.05e3]; %upperbound for the design variables UB =[3.5e3; 4.5e3; 15e3;20e3;25e3;1e3;25e3;1e3]; %linear constraints A =[0 0 1 0 0 0 1 0; 0 0 0 0 1 0 1 0; 0 0 0 0 0 1 0 1; 0 0 0 0 0 1 0 1]; B =[1e3;1e3;1e4;1e3]; [x,fval,exitflag,output,lambda,grad,hessian] = fmincon('OPTEMHRFUN02',x0,... PAGE 201 201 A,B,[],[],LB,UB,'OPTEMHRCONNIND2',options); x0 =x; % update initial value for next iteration % Constants definition rho = 1.231; % air density csound = 343; % speed of sound at air % Material properties of the piezo Es =110e9; %89.6e9; % Young's modulus of the shim vs = 0.375; %0.324; % Poisson ratio of the shim rhos = 8530; %8700; % density of the shim Ep = 63e9; % Young's modulus of the piezo vp = 0.31; % Poisson ratio of the piezo rhop = 7700; % density of the piezo d31 = 175e12; % d31 of the piezo dp = 1750; % relative dilectric constant % geometry of the EMHR r = x(1); t = x(2); R = x(3); L = x(4); ap = x(5); tp = x(6); as = x(7); ts = x(8); % Computing the acoutic impedance of the EMHR MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); PhiA = Phi; % The first resonant frequency of the shortcircuited EMHR w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the shortcircuited EMHR w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The first resonant frequency of the opencircuited EMHR w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the opencircuited EMHR w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; PAGE 202 202 dw011 = w01ow01s; % the tuning range of f1 dw022 = w02ow02s; % the tuning range of f2 % save the result of Pareto solution Presult(1,nn) = dw011; Presult(2,nn) = dw022; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The constraints of optimizing problem to maximize f2 with the tuning of the % f1 is limited % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [c, ceq] = OPTEMHRCONNIND2(x) global index % Constants definition rho = 1.231; % air density csound = 343; % speed of sound at air % Material properties of the piezo Es =110e9; % Young's modulus of the shim vs = 0.375; % Poisson ratio of the shim rhos = 8530; % density of the shim Ep = 63e9; % Young's modulus of the piezo vp = 0.31; % Poisson ratio of the piezo rhop = 7700; % density of the piezo d31 = 175e12; % d31 of the piezo dp = 1750; % relative dilectric constant % geometry of the EMHR r = x(1); t = x(2); R = x(3); L = x(4); ap = x(5); tp = x(6); as = x(7); ts = x(8); % Computing the acoutic impedance of the EMHR MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); PhiA = Phi; % HR = 1/sqrt(MaN*CaC)/2/pi; PAGE 203 203 % PD = 1/sqrt(MaD*CaD)/2/pi; w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; dw01 = w01sw01o; %nonlinear inequality constraints c = [dw01(index),1200w01s,w01s1900,w02s3000]; %nonlinear equality constraints ceq = []; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % The program is used to implement Pareto optimization design of an EMHR % using weighting sum method % % There are eight design variables to optimize the tuning range of EMHR % x: collective variable for all eight design variables % x(1): >r radius of the neck of EMHR % x(2): >t thickness of the neck of EMHR % x(3): >R radius of the cavity of EMHR % x(4): >L depth of the cavity of EMHR % x(5): >ap radius of the pztlayer of pztbackplate of EMHR % x(6): >tp thickness of the pztlayer of pztbackplate of EMHR % x(7): >as radius of the shim of pztbackplate of EMHR % x(8): >ts thickness of the shim of pztbackplate of EMHR % x(9): >MeL inductive loads %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% close all clear all global Windex %starting guess x0 =[2.42e3,3.16e3,6.34e3,16.4e3,10.1e3,0.12e3,12.4e3,0.19e3]; PAGE 204 204 mm = rand(100,1); % set up the contraint for f1 for nn = 1:length(mm) Windex = mm(nn); %optimal initial set options = optimset('LargeScale','off','Display','iter',... 'MaxFunEvals',2000,'TolX',1e7,'Tolcon',1e7); %lowerbound for the design variables LB =[1e3; 1e3; 5e3;10e3;1e3;0.05e3;1e3;0.05e3]; %upperbound for the design variables UB =[3.5e3; 4.5e3; 15e3;20e3;25e3;1e3;25e3;1e3]; %linear constraints A =[0 0 1 0 0 0 1 0; 0 0 0 0 1 0 1 0; 0 0 0 0 0 1 0 1; 0 0 0 0 0 1 0 1]; B =[1e3;1e3;1e4;1e3]; [x,fval,exitflag,output,lambda,grad,hessian] = fmincon('OPTEMHRFUN03',x0,... A,B,[],[],LB,UB,'OPTEMHRCONNIND',options); x0 =x; % update initial value for next iteration % Constants definition rho = 1.231; % air density csound = 343; % speed of sound at air % Material properties of the piezo Es =110e9; % Young's modulus of the shim vs = 0.375; % Poisson ratio of the shim rhos = 8530; % density of the shim Ep = 63e9; % Young's modulus of the piezo vp = 0.31; % Poisson ratio of the piezo rhop = 7700; % density of the piezo d31 = 175e12; % d31 of the piezo dp = 1750; % relative dilectric constant % geometry of the EMHR r = x(1); t = x(2); R = x(3); L = x(4); ap = x(5); tp = x(6); as = x(7); ts = x(8); % Computing the acoutic impedance of the EMHR MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity PAGE 205 205 [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); PhiA = Phi; % The first resonant frequency of the shortcircuited EMHR w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the shortcircuited EMHR w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The first resonant frequency of the opencircuited EMHR w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the opencircuited EMHR w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; dw011 = w01ow01s; % the tuning range of f1 dw022 = w02ow02s; % the tuning range of f2 % save the result of Pareto solution Presult(1,nn) = dw011; Presult(2,nn) = dw022; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % The objective function to maximize f1 and f2 using weighting sum % method %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function f = OPTEMHRFUN03(x) global Windex %% % Constants definition rho = 1.231; % air density csound = 343; % speed of sound at air % Material properties of the piezo Es =110e9; % Young's modulus of the shim vs = 0.375; % Poisson ratio of the shim rhos = 8530; % density of the shim Ep = 63e9; % Young's modulus of the piezo PAGE 206 206 vp = 0.31; % Poisson ratio of the piezo rhop = 7700; % density of the piezo d31 = 175e12; % d31 of the piezo dp = 1750; % relative dilectric constant % geometry of the EMHR r = x(1); t = x(2); R = x(3); L = x(4); ap = x(5); tp = x(6); as = x(7); ts = x(8); % Computing the acoutic impedance of the EMHR MaN = rho*(t+2*0.85*r)/(pi*r^2); %acoustic mass of the neck CaC = pi*R^2*L/(rho*csound^2); %acoustic capacitance of the cavity [MaD,CaD,CeB,Phi]=Center(Es,vs,rhos,as,ts,Ep,vp,rhop,d31,dp,ap,tp); PhiA = Phi; % The first resonant frequency of the shortcircuited EMHR w01s = sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The second resonant frequency of the shortcircuited EMHR w02s =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % The first resonant frequency of the opencircuited EMHR w01o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; % % The second resonant frequency of the opencircuited EMHR w02o =sqrt((1/(MaN*CaC)+1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))/2+... sqrt((1/(MaN*CaC)1/MaD*(1/CaC+1/CaD+PhiA^2/CeB))^2+... 4*1/(MaN*MaD*CaC^2))/2)/2/pi; %objective function f = Windex*(w01sw01o)+(1Windex)* (w02sw02o); PAGE 207 207 LIST OF REFERENCES Alfredson, R. 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Sound Absorbing Materials (Amsterdam, NY, Elsevier), Chaps. I and II. PAGE 215 215 BIOGRAPHICAL SKETCH Fei Liu was born in Chongqing, China, in Janua ry 1973. He received the BE degree with an aerospace engineering major from Beijing University of Aeronautics and Astronautics (BUAA), China in 1994. He then worked at th e Department of Jet Pr opulsion of the BUAA as an Assistant Lecturer from 1994 to 1998, and Lect urer and Head of the Engine Control System Laboratory from 1998 to 2001. He came to the Un ited States to pursue his Ph.D. study in 2001, and entered to the Department of Mechanical and Aerospace Engineering of the University of Florida as a graduate assistant. There he re ceived his M.S. degree and Ph.D. degree with an aerospace engineering major in 2003 and 2007, respec tively. His research involved in designing, modeling and testing on an electrom echanical Helmholtz resonator. He was a student member of American Instit ute of Aeronautics a nd Astronautics (AIAA) and Acoustical Society of America. 