‘ ,1 w _-'I1..ur— 1...: ‘. ‘ w 1155f. "',“'~ ‘ 3"“. "‘1'4'1’ "f: ‘ n . L',;3113\'"'3 7531?“? :1" 7,1 5111‘} {313- 31kg! '117, , 5331" '1 ‘Iu'm', ' {3"5'1' '3 1:31.43 ' 1 '4" '4'?" 1" :5“ '1 15"1' '344'1 13" 17773 "1' 491l_“ :7‘ ‘ 13h," '41""'4'1'r"'1"'1“"'1'1 ' . '- 47 5.3‘ 3., -~~7. , 4 ., 7 ‘ '7'777777'1433'4 ‘ 73.5. . m: ..'7‘ ' " 47.4.. 4 3'01““ '77-'4-4347' 1"5 :11" '- "1 71".. 4 “£7 j- , “734‘ .7H49‘7754W _, '57‘3533,';i‘,';3i,;11‘11«471353113 f'ii‘.’ @112'1331319511 11'.,~:',131>'..-35 . ! {3:3'1' ~ 1 31"] 1 1,, 17’ L} [$147. 5 ..,. '7‘ .555II;" .«_ Z .' . 37‘, 4::.,1};{3‘:'4"111 III 4 "‘z'ut' $541,971,"; -—<3I' 2,5'} “,5, ”5:51 I 4 5; 171% 7 1 {9,53, ’)=;‘('7:1’ ,4_ 13.17 377(11{L“3,5T~f— ,‘3 ~ "134-1” """'.'1'4"3'11'" . ‘ ' '3 .‘h’. I7 1""':" :‘fl "" *" 3i 'i" 'l 5'" ”131:5;3““,163'5‘5‘J'II'115 "" 9- ' sit—2: 4"4"1j""""'"i""'3113'2'31‘133' 3W“, '111~""§1"13"‘1.1f1:54, :,,3,111'k1,.‘ 375.. "1' l 31' ‘1. , $5143; 313'3 - 7:3'355'5'171'1' . .. ‘33 ' "'1‘1‘1'177' '1'" 47771143” 11111155113331.“ 41.714111'7541143 314 "'4‘ 419-4“ 1341314- 311317 '31 30" 8I'm-.- , g. 7'1'7‘13 5411'1311'.1'3"11"14,1'-’" 37'3 11"131E1'E'41'J. 1: ,11'5. 35,,311. 41-1'7 13 5,5513: '1331 1:31-11 1,117.7" ,7! 3111'7'4' '3, ,51'3 . .' ,5. 53?? T4153 733,11 -"74".113f~"744534.-7 1311‘ 10717 4311' . L1;‘74‘~ 1145' ‘44 11-1" 5711.7 :2 7753141.. ,743 1- '7"411"‘1i""1j - ‘- 13777-4‘1 4"7-7-,“-'11331.'-1174; "7111134 11171171 ~7‘—.'.’4' ‘73 7"7;‘7‘ 3 ' '1'1"'1'"1';"' '1'13'33113'11113'1" L 11172'4'13'1'111111"11 " 113313 \4'71 1‘1 31"1'1 7 V"3.3;1"' . . ' r: ' ,"§,‘-- 1. 3" 35(31'15'53533IE, 4 "' 1,511” ""1 1'3 '1'" 73'; 13, 153,31“; '11" 1 3"}:""'5.'5:'.,1' {'3 ""€ 31 1" ' '51' ' '371157» 3E 133%: '1155'317'1 33113 “'1; #‘IHJLEL' 5 J 1315 7 179,315; '1'1'535F11'."1 ,154 17‘31 ‘6 f“fi1Ir?1I m4"_2!3""1L"1"""14,:3:r" _ 3 $119,355 13171 1' 1'1" 1:731" 333511 ""11'34; 7311,,5'.1,1' 5,1513; 2.4-: x "- 3:" . ..::,., 4 _ . ,7. {24344; 341.4417 312-14144 744‘ ,4- 447 7 1 1‘ 7,, €4,147“- 7. 7' 334“ 1417 4 .4. 4 371411175311 71,. 4-" 17‘1“ {1,1,1} 3 3-7 "7133177531., - , I ‘5' 7 ’32: (15‘ Ffilrfif'w ' 1" '3 5 5' 4 o “ii-51133?” 11"! '1'? u "'111' {can} 9 c 4 7,1'37 '."-1w"4" . 3 ' ' A, _' ' 9 ~ 4 3371,,7"""7 1 “,,,,,,;,x,,,,,,,l5, 5,, 357,113] ”“3311911711131331"J'h' a 35%,, 514,5 ; ,E i .'4 ‘ ‘ “4.3411337114437477, 134: .2241. “3331735533773: .. ~- .7 14.111 ' ~~ ~ . ‘ " . . '. "1 " "'7 “ " '4'4'17'1'." 311'11' 77111143171‘7 1,217 1731313,,1 '1 .' ' 4 ,‘1‘ " 0’ 3 5 [1774 7 7 P 1'13 £173 '13 l0 . o ‘ 4 ‘ ,4774‘7‘ 3’;‘7‘734" 14433344114 ,. . 35,3," '. 5,5,1“; 13’": 5‘. :4 ., "1".',J,4'.'3 ‘1, ""1'f1"'3"""'1fi"311""""11'"'-7'3'EI"$151'.";":,' ‘ 11344‘." '; . '33-'31 ' "- 7 . ;, :‘7 "‘74- _‘;?§.~77|'4‘4,'32'7,443.'~Ii '2. 7471117143}; {33 ; ' “ 3331-5195 . , 1"”hfl'1"1'-"'31'1"7j'11313"H'L1'1" 'u M = a 1:157", 7.5. . A .4 17-; ~47- 44,. “13.4 .4 7:? ~74. .- L 4'4 4 73%, 1-7 4 - 43‘ 4‘4 ,. 43' 4.; :7 1'1"",5331'1759335'1' 3,32,, 55353.34, 11.515 :1“ 5 3 .753: .154: 113,33”. "‘4’ “34!";i' ' 4 ‘7‘7‘,. {HI-q .4 " 'hz “1'3 n 31,215,315 L37 ‘ 51555315 [5* ,1, 7 ,2 44-4423 1,35% 77 H ‘ 7‘ ‘: .,, ,, ,, ‘0 472777,, , . . ,-=.7§,7._4 547,5 .7: ,& 711113531"; 7.11 ““134, ' ,"E;,fi :x . . "L51! ‘1'.”43 . 211' 1"”, 1" 71,. 7' "' .1333 0" '5111135114I"153"4 71,: . "%'11""’1"'4"""L" ,5 . 7 7"11 314;"‘1L‘ 7;“‘1"1'11'7‘7'4' ' '3'1111'4":1 '37'437'311‘ 7 ~ ~‘1 '14 ,,1|,,(,.q3,, 1' 1 "' ' 1'1' '1111 131' '51" 1-35 I 1 55113L'fi 7. ';3_ "'11 ',‘ '77 ,5,- 544 5,] 137711,} '73 ,4331'953133513 555331134 37.1,"; "1 '31," . "'53, 57:55 ’ 35': 1 1' 1 15"1'1. 11,3." '3,""13'1'1"','1"1"" "'§"1"5","‘11;§"""§ ,. ~35 1., 7'; 57:5}1’ 'kg’m‘ I ~, .i'sisr. 73 1771-_ , 51,7 347:” 'f '.. "'"1' Si, 5",", 711" 3,375., ..11"3: 3::53' 33" “41%; inf,“ 55:; 14577145735357. 3"_"|J1"‘.5{‘ 35.1113 , ".555. 7753. 717'451‘4'1 iI,;.~f~T1',“ 1171-" 7 14,: 39,1“?"1'5 £11313 ""1"?“ "1.1‘417. '13" .144" 1"» 45:: 4 3444 .471“1444‘4747 '. 414 72.7741 4474444441773'1L".f'1" -.. A 77 33 1‘ 4"""""" 311'1'11'1'.':"1"""""13. - 7,, ,, "11-,é""1'i4 1'4'1 5 . .,,‘,, '7 1'33“ ,120'147 3171375, ,7':'r, "11"'1l1'7Jd-15‘ “.55, .77 1'3437 "1'3"""1"'I1,l)93,'""1',"".13"fi11"'1' (,11 3'11“ ‘ 5,81,; [55,51 I1,"'I:1:‘1 ' 12.7343 '8 3? {13'11‘1 13"; .1. 1 - 117""!1111'5 ., 74 5;, 57 .47 ,7, 7. 743371, 4; ','4.~,;J i'i""'1"1'11*1' 11’3'15113'" ,n, ,,4 - $53,. at 4,7547 , Q 7, [‘4' '11 , “44‘4 ’7-4..7, ..,3--'-ET~1T‘.11.-..§‘§714.$1454.45” . i137 '1",‘1~‘7-11“‘1 4,7, '4-13‘1‘4'741,7413474747,“,2.7,"31"',7" 344477; , . - g“ '53: 147 ‘-7!'~ 77,574.- 4: 7 JAN: 4,; .'-_.""..'7144“,"41'4E "','.-‘ ' " ' .. 7 774-7 17 '4. 44-47-174. :7 4‘47- 77... :77 474,: 71‘"- 431417-174 1331341471 - "‘ ‘ ' ‘ "- 7': 7.4: 444 4.- ' ' " - ‘1 "7' “344'. ,3?" .. 7‘ '4', "I 7 1' ""';"t":' 91W" ’7 1“” ‘1'.- 1' ‘44 5447444 144447- “"34“ 4333- 4, 4 44.44.44 474: 2... "744’7 431313111 . 4.471171. 4.3731". 11‘ 42111-17317“. ‘77” 7744777 1‘4 1 :‘i7'1"‘~'4777 7- 4 7 1'“ 17‘ “7147717417414 4-744? ,7433 4734,7374 ‘1 “7'3 "‘1' 7771131"! 4473'- : $1" 7 I. '" 2331:9317." lg! 1 '55, ' 1'": "RL fiz’ii'm 1:5;1" 1‘ ‘r 1"" .131", 3351411 ,11'1 Hf? 55,3913367'11 ‘1‘11,,13'1“:’,,“551,5 7, '31'1‘11931153'311q'7'1 5'1' 17t|7751255575 ' 1" I-TI'v'I 74 53'5’ 17 "_1.',,' 94,7 1151331 3’ 17"473'31‘1'r-5'hn'1 1'31771133' '31:.17',"7'3'.1'1"1 1513534713," '1'? 9".' 117'77'113'wi'"11 "35,11'11' 31". '33,,31'153'113 4 '1ng 335111-111; 1'61' "33' ' 47.7175 9,573,777 3"'1"""r' ,9 , 4374441,; 3., ”,2 4,3'4‘1'1324' ‘- 1’ W1 71 1,1 ‘=~4~~ ~ 47 4‘44. ,7, 74747 7574737» 7 ,4, 4- 47" 77";‘4 477774" .7 .. 7‘" 4 4.- 47 ~4-444 7.. ~44... .. .747" 444“ 44747477344 44 447744444 4714444 1‘ 37 3‘1":3,1,:fi.,"%'i:,“11,153‘7” ‘h' 5;‘F"1'wl$ivfif1fl'{ 355 3‘13;'}' 75-0 I73. ’19,.44 ~‘..747 '31""'1’3 "-1’3': ~47 "'.':' :51}.4474,.., -,..’3357 ,559 1143(3'513434 724-77447 477 .747413 41=~347,1'4411<47' 1,, 111‘ . ' 7417'11'11'7‘” 1} . $2.14 11447331111191.1133 35,7: '13, ,5”: 5,1313! "'11 1411'14 7’ '3 1 111474.374 " 3313111 11"'1" 3" 1"1':" "1""11'11' fl .7 *' 7 77 7474 7 7744.474 : . 4v, ,5» 5,131,7731113113'3 5,33”, 531113311131 5"811'3: _ 111533145573 ' '1' " "'11" ""7 34" '1' 77731. 771-1.. Q ,.7-' 1.4177- 41514537"? 74 me Ill llllllllllllllllllilllllHlllllllllllllllllllll "W , 3 1293 02058 This is to certify that the dissertation entitled A Semi-Active Helmholtz Resonator presented by i Charles Birdsong has been accepted towards fulfillment of the requirements for _Eh_._D_.___ degree in Menhanicallngineering ajor profess 77 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 F..- fil_. _ v- —-. "—~ LEBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record.‘ TO AVOID FIND return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 553:? (132% 11m m.m14 A Semi-Active Helmholtz Resonator By Charles Birdsong A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Mechanical Engineering 1999 ABSTRACT A Semi-Active Helmholtz Resonator By Charles Birdsong Helmholtz resonators are commonly used to reduce the sound transmitted through acoustic systems such as industrial processes, vehicle exhaust, industrial ducting systems and more. They function by reflecting sound in a narrow, tuned frequency band back to the source, where the tuned frequency is a function of the dimensions of the resonator cavity. If the frequency of the unwanted noise changes from the tuned value, the noise reduction is diminished. Past efforts at active tuning of Helmholtz resonators resulted in limited tuning capability and significant mechanical complexity. This work considers a system that actively modifies the acoustic response of a Helmholtz resonator continuously, on-line, allowing optimum performance over a range of operating conditions. The system consists of a static Helmholtz resonator designed to enforce a nominal resonance and a feedback control system that provides a variable acoustic impedance. The combination of the nominal impedance of the resonator and the differential impedance of the controller results in a semi-active controlled, variable frequency Helmholtz resonator. A model is presented and analysis, numerical simulations, and experimental results are used to demonstrate the effectiveness of the device. An analysis of the power flow in an acoustic duct with the device is presented to answer the question, “Where does the sound go?” A comparison of actuator implementation techniques is presented that concludes that the actuator dynamics must be included in the controller design, and that different configurations produce results with competing advantages and disadvantages. Finally, a closed-loop adaptive control strategy is presented that uses a gain-scheduled controller to tune the device on-line and track a disturbance signal with slow time- varying frequency. Cepyright by Charles Birdsong 1 999 This work is dedicated to my beautiful wife, Bernadette, and son, Indigo A special thanks to all my friends and family who supported me during this time: Lenny Monterrosa Mark Minor Brooks Byam Joga Setiawan Joe Derose Gary Gosciak Tuhin Das Brennan Sicks Nancy Albright Carol Bishop Aida Rodriguez Roy Bailiff Dave and Sara Hunter Martha Quant Lisa Saltman Augie Hernandez Pete Jewett Mike Jewett The MSU Karate Club Sally Star and the whole Subway crew ACKNOWLEDGMENTS I would like to thank Lakhi Goenka of Visteon Automotive and the members of the Manufacturing Research Consortium at Michigan State University for funding this work vi TABLE OF CONTENTS List of Tables ............................................................................................................... ix List of Figures ................................................................................................................ x List of Abbreviations .................................................................................................... xiv Chapter 1 Introduction ................................................................................................................ 1 Chapter 2 A Semi-Active Helmholtz Resonator ............................................................................... 5 Analytical Resonator Model ................................................................................ 5 Resonator Model with Actuator Dynamics ............................................. 17 Experimental Validation .................................................................................... 24 Speaker Compensation ........................................................................... 26 PI Controller Design .............................................................................. 28 Conclusions ....................................................................................................... 32 Chapter 3 Sound Reduction and Power Flow of the Semi-Active Helmholtz Resonator in an Acoustic Duct .............................................................................................................. 34 Model Development .......................................................................................... 34 Power Flow Model ................................................................................. 34 Impedance Control of SHR .................................................................... 39 Experimental Verification .................................................................................. 49 Conclusions ....................................................................................................... 60 Chapter 4 A Comparison of Acoustic Actuators for the Semi-Active Helmholtz Resonator Analytical Model Development ......................................................................... 64 Resonator ............................................................................................... 65 Controller ............................................................................................... 66 Speaker .................................................................................................. 67 Compensator .......................................................................................... 68 Coupled System Simulation ............................................................................... 7O Resonator and Controller with Ideal Actuator ......................................... 71 Resonator and Speaker ........................................................................... 73 Resonator, Speaker, and Compensator .................................................... 75 Resonator, Speaker, Compensator and Controller ................................... 78 Resonator, Speaker, and Controller (No Compensation) ......................... 83 Experimental Validation .................................................................................... 88 Compensated Actuator Results ............................................................... 9O Uncompensated Actuator Results ........................................................... 94 Conclusions ....................................................................................................... 97 vii Chapter 5 Adaptive Control of a Semi-Active Helmholtz Resonator .............................................. 98 Controller Design .............................................................................................. 99 Analytical Controller Design ................................................................ 105 Gain Scheduled Adaptive Control ........................................................ 106 Gain Scheduled Controller Simulation ................................................. 108 Actuator Dynamics .............................................................................. 11 1 Experimental Validation .................................................................................. 120 Speaker Compensation ......................................................................... 121 P1 Controller Design ............................................................................ 123 Noise Reduction of a Time Varying Disturbance Tone in an Acoustic Duct ...................................................................................... 126 Conclul3lsions ..................................................................................................... Chapter 6 Conclusions ............................................................................................................ 133 References ............................................................................................................ 1 37 viii Table 2.1 Table 2.2 Table 3.1 Table 3.2 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 4.5 Table 4.6 Table 5.1 LIST OF TABLES SHR Model Parameter Values ...................................................................... 20 Controller Gains Identified by Model Based Empirical Design .................... 23 Controller Gains K p and K I used to Move the Resonant Peak to Various Frequencies, a)" , While maintaining Constant Peak TL .............. 46 Controller Gains used to Generate Figure 3.11 ............................................. 56 SHR Model Parameter Values ...................................................................... 71 Controller Gains used to Create Figure 4.8 ................................................... 73 Compensator and Controller Gains used in Figures 4.14 and 4.15 ................ 81 Controller Gains used in Figures 4.17 and 4.18 ............................................ 85 Controller Gains used in Figure 4.22 ............................................................ 92 Controller Gains used in Figure 4.25 ............................................................ 95 Acoustic Parameters used in Simulation ..................................................... 105 ix Figure 1.1 Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.11 Figure 2.12 Figure 2.13 Figure 2.14 Figure 2.15 Figure 2.16 Figure 2.17 Figure 3.1 LIST OF FIGURES Helmholtz resonator connected to a primary acoustic system ................... 3 Ideal Helmholtz resonator ........................................................................ 6 Frequency response of a Helmholtz resonator .......................................... 8 Helmholtz resonator with complex impedance boundary condition .................................................................................................. 9 Plot of the real and imaginary parts of the controllable acoustic impedance for x0 = 21, 31, and 100 vs. normalized command frequency ............................................................................... 11 Frequency response of SHR system with variable controllable cavity impedance for 22 defined by (ac = 0.8, 1.0, and 1.2, and x0 = 12, 31, and 100 with arrows indicating direction of increasing x0 ....................................................................... 12 Plot of controller gains K P and K I vs. command frequency for x0 = 22, 31, and 10 with arrow indicating direction of ‘ increasing x0 .......................................................................................... 14 Block diagram of speaker model coupled with resonator model through the PI and Q2 signals ................................................................ 18 Block diagram of resonator and speaker model with compensator and PI controller ................................................................ 20 Closed-loop frequency response of the Q1/ P1 transfer function for model including actuator dynamics ..................................... 22 Closed-loop SHR frequency response for PZ/Dl transfer function for model including actuator dynamics ..................................... 23 Photograph of SHR connected to an acoustic duct with a second audio speaker to inject noise ....................................................... 25 Schematic diagram of experimental SHR apparatus ................................ 25 Block diagram of dual voice coil speaker compensation used in SHR actuator .............................................................................. 28 Graph of K p and K , vs. a)” determined using an experimental empirical technique ........................................................... 30 Experimental frequency response PZ/D; transfer fimction with (0,, = 80, 110, 140, and 170 Hz and with gains set to zero .............. 31 Schematic of SHR applied to acoustic duct ............................................ 32 Time response of pressure 2 inches from the duct end with a 140 Hz disturbance as the controller is activated showing a 16 dB noise reduction ............................................................ 32 Schematic diagram of SHR and acoustic duct showing incident, reflected, absorbed, and transmitted power towards open end ................................................................................................ 34 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 3.12 Figure 3.13 Figure 3.14 Figure 3.15 Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 Schematic diagram of SHR showing inertia effect in neck and the movable surface in the cavity interior ......................................... 4O Closed-loop SHR system block diagram ................................................. 40 Transmission loss vs. normalized frequency with varying acoustic damping 0.2, 0.5, 2.0, and 4.0 * Rag .......................................... 43 Plot of transmitted, reflected, and absorbed power ratios vs. frequency as acoustic loss term is reduced .............................................. 44 Transmission loss vs. normalized frequency for closed-loop SHR, placing resonance at 1.0, 1.1, 1.2, 1.3, and 1.4, showing that the center frequency and damping can be changed by varying K p and K, .............................................................................. 45 Transmitted, reflected, and absorbed power for gain settings used in Figure 3.6 ................................................................................... 46 Plot of reflection coefficient vs. frequency for open end duct, 3 trials with various controller gains ....................................................... 48 Schematic of experimental apparatus used to measure the reflection coefficient in the duct and SHR system ................................... 49 Reflection coefficient of open end duct with SHR removed showing the duct end can be modeled as a purely reflective boundary ................................................................................ 51 Experimental plots of the reflection coefficient upstream of the SHR with the controller in closed-loop to change the frequency of the reflection added to the duct by the device .................... 53 Plot of transmission coefficient for data in Figure 3.11.D ....................... 57 Parametric polar plot of reflection coefficient magnitude and phase vs. frequency with controller turned on showing vectors for minimum transmission coefficient, T and associated SR ....... 58 Schematic of SHR applied to acoustic duct ............................................ 59 Time response of pressure at duct end with pure tone disturbance as controller is activated showing 10 dB noise reduction ....................................................................................... 59 Schematic of a semi-active Helmholtz resonator connected to a primary acoustic system .................................................................. 62 Local actuator feedback compensation used to boost actuator authority, minimize actuator dynamics, and simplify controller design ..................................................................................... 63 Schematic diagram of SHR showing inertia effect in neck and the movable surface in the cavity interior ............................................... 66 Closed-loop positive feedback SHR system block diagram with disturbance through P] ........................................................................... 66 Dual voice-coil speaker diagram ............................................................ 68 Block diagram of speaker and compensator ............................................ 69 Block diagram of simple coupled system model including acoustic resonator, closed-loop feedback controller, and ideal actuator model ............................................................................... 71 xi Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.11 Figure 4.12 Figure 4.13 Figure 4.14 Figure 4.15 Figure 4.16 Figure 4.17 Figure 4.18 Figure 4.19 Figure 4.20 Figure 4.21 Figure 4.22 Figure 4.23 Figure 4.24 Figure 4.25 Figure 4.26 Figure 5.1 Frequency response simulation of resonator and closed-loop feedback controller with ideal actuator showing that the resonant frequency and damping can be changed by varying the controller gains ................................................................................. 72 Block diagram of resonator and speaker models showing coupling through Q2 and P2 ................................................................... 74 Frequency response simulation of the Q2/ ep transfer function for the resonator and speaker model with the controller removed ........... 75 Block diagram of resonator and speaker with local feedback compensation, and controller removed ................................................... 76 Frequency response simulation of the Q2 /D, transfer fimction with the resonator and speaker with local feedback compensation added and controller removed ................................................................ 77 Block diagram of resonator, compensated speaker, and feedback controller with disturbance D2 ................................................................ 78 Frequency response of the Q1 / P1 transfer function with the resonator, compensated speaker and feedback controller for four cases with gains shown in Table 4.3 ................................................ 79 Frequency response of the compensated Pz/Dl transfer function .................................................................................................. 80 Frequency response of P2 to current sensor disturbance for resonator, compensated speaker and feedback controller model with gains from Table 4.3 ....................................................................... 82 Frequency response simulation of the Q1 / P1 transfer function with the resonator, uncompensated speaker, and feedback controller coupled model with controller gains from Table 4.4 ............................... 84 Frequency response simulation for the PZ/D; transfer function with the resonator, uncompensated speaker, and feedback controller coupled model with controller gains from Table 4.4 ............................... 85 Photograph of SHR connected to an acoustic duct with a second audio speaker to inject noise ....................................................... 89 Schematic diagram of experimental SHR apparatus ................................ 89 Primary coil current sensing circuit ........................................................ 91 Experimental closed-loop frequency response of coupled system with compensated actuator .......................................................... 92 Schematic of experimental setup ............................................................ 94 Sound pressure level in acoustic duct with SHR used to reduce pure tone disturbance ............................................................................. 94 Experimental frequency response of closed-loop SHR with uncompensated actuator ......................................................................... 95 Sound pressure level spectrum with pure tone disturbance at 185 Hz with open and closed-loop SHR and uncompensated actuator ......................................................................... 96 Schematic diagram of SHR showing inertia effect in neck and the movable surface in the cavity interior ............................................. 100 xii Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 Figure 5.18 Closed-loop SHR system block diagram with disturbance through P, ........................................................................................... 100 Plot of stable gain space for controller gains K P and K, ..................... 104 Plot of pole locations for gain scheduled controller algorithm for command frequency ranging fi'om 560 to 820 ................................. 107 Adaptive gain scheduling control block diagram .................................. 109 Numerical simulation of SHR to disturbance tone with time varying frequency ......................................................................... 111 Block diagram of resonator, controller, and compensated speaker with disturbance through D, .................................................... 1 l4 Pole locations of SHR with actuator dynamics and ideal SHR gain scheduling algorithm .................................................................... 1 15 Plot of K p and K, vs. are determined using model based empirical controller design technique ................................................... 1 17 Plot of closed-loop pole locations for 7 controller settings derived from the model based empirical controller design technique ................................................................................... l 18 Photograph of SHR connected to an acoustic duct with a disturbance speaker to inject noise ..................................................... 121 Schematic diagram of experimental SHR apparatus .............................. 121 Block diagram of dual voice coil speaker compensation used in SHR actuator ............................................................................ 123 Graph of K p and K, vs. arc determined using an experimental empirical technique ......................................................... 125 Experimental frequency response Pz/Dl for gain scheduling controller .............................................................................................. 126 Schematic diagram of SHR connected to acoustic duct for noise quieting experiment ..................................................................... 127 Pressure near duct opening, P3 and pressure in SHR cavity, P, vs. time with 130 Hz disturbance noise as controller is activated ........................................................................... 129 Time varying disturbance results .......................................................... 131 xiii LIST OF ABBREVIATIONS Uppercase AD ................. duct cross sectional area (m2) Ca .................. acoustic compliance (ms/N) D ................... disturbance signal Ge .................. controller transfer function H b, ................ velocity sensor transfer function for secondary coil H p ................. velocity sensor transfer function for primary coil I a ................... acoustic inertia (mszlms) Ic ................... coil inductance (H) Im .................. imaginary part K amp .............. amplifier gain K, .................. integral gain K P ................. proportional gain L ................... length of duct (m) M c ................. mutual inductance (H) P, ................... pressure on opening to resonator neck (N /m2) P2 .................. pressure in resonator cavity (N/mz) PA .................. absorbed power P, .................. incident power PR .................. reflected power P, .................. transmitted power Q, .................. volume velocity in resonator neck (m3/s) Q2 .................. volume velocity in resonator cavity (m3/s) QA ................. volume velocity in resonator neck (m3/s) Q, .................. volume velocity in duct upstream of resonator (m3/s) QT ................. volume velocity in duct downstream of resonator (m3/s) R .................... acoustic radiation loss (N s/m5 ) RC .................. speaker coil resistance (ohms) Re .................. real part Rm ................. current sensing resistor resistance ohms S .................... area (m2) 5,, .................. speaker face area (m2) SHR ............... semi-active Helmholtz resonator Smic ............... microphone sensitivity (volts/Pa) TL .................. transmission loss V .................... volume (m3) Z, .................. acoustic impedance of resonator (N s/m5 ) Z, .................. acoustic impedance of cavity surface (N s/rn5 ) Z A ................. acoustic impedance in resonator neck (N s/m5 ) xiv Z, .................. acoustic impedance of duct cross section upstream of resonator(Ns/m5) Z, ................. acoustic impedance of duct cross section downstream of resonator (N s/m5 ) Lowercase bl ................... electromechanical coupling factor (N/amp) c0 ................... speed of sound in air (m/s) ebs ................. secondary coil voltage (volts) e p .................. primary coil voltage (volts) f ................... incident pressure wave amplitude (Pa) i,” .................. secondary coil current (amps) i p ................... primary coil current (amps) k ..................... wave number (rad/m) l ..................... length (m) q 1 ................... volume displacement in resonator neck (m) q 2 ................... volume displacement of actuator s ..................... Laplace variable va .................. speaker face velocity (m/s) x ..................... spatial variable (m) x0 .................. scalar that controls amplitude of 22 Greek Af ................. half power band--width A ................... electromagnetic flux p0 .................. density of air (Kg/m3) we .................. command frequency (rad/s) a) ................... circular frequency (rad/s) C .................... damping ratio XV Chapter 1 INTRODUCTION Many engineering systems create unwanted noise that can be reduced by the careful application of engineering controls. These can be divided into two classes; applications where controls are added to reduce unwanted noise from the environment outside of the system, improving the comfort and safety of human beings, and second, applications where controls are added to change the acoustic response inside of the system, improving performance. The first deals with the attenuation of unwanted environmental noise. The classical example is automobile exhaust noise. The automobile engine creates a pressure pulsation in the engine that transmits down the exhaust system where it is injected into the environment through the tail pipe. Humans perceive this pressure pulsation as undesirable noise (SAE Handbook, 1995). The traditional engineering control consists of adding a muffler to the exhaust system, which clamps the dynamic pressure variation resulting in reduced environmental noise (Beranek, 1971; Hosomi, et al., 1993; Morel et al., 1991). Other examples of this class of problem and their traditional noise controls include treating jet engine noise with acoustic liners (Tang and Sirignano, 1973; Kraft et al. 1994), applying Helmholtz resonators to wind tunnels (Heidelberg and Gordon, 1989; Parrott et al. 1988; Anwar 1991), and applying mufflers to pneumatic exhaust nozzles (ASTM E1265-90). A second class of noise control problem deals with removing acoustic resonances in pressurized systems to improve performance, product quality, or safety. An example of this type of problem is an industrial furnace (Tang, and Sirignano, 1973). Oscillations in pressure in a fumace can grow without bound as increases in pressure magnify the combustion rate. Uncontrolled, this scenario can result in catastrophic failure. In cases where a narrow frequency band of noise exists in an enclosed space (referred to here as the primary acoustic system), a traditional engineering control consists of adding a Helmholtz resonator (Figure 1.1), which is tuned to reduce the unwanted noise by sending it back to the source (Temkin, 1981). The Helmholtz resonator, named in honor of Herman L. F. Helmholtz (1821 -— 1894) is a passive acoustic device, which consists of an enclosed hard-walled cavity that communicates with the primary acoustic system via a narrow short neck. It has a tuning frequency determined by the physical dimensions of the neck and cavity. As long as the frequency of the unwanted noise falls within the tuned resonator frequency range, the device is effective. However, if the frequency of the unwanted sound changes to a frequency that does not match the tuned resonator frequency, the device is no longer effective. Some efforts have been made to vary the resonator dimensions with time, to achieve a variable tuned device (Graham, Graves et. Al., 1992; Garret, 1992; and Bedout, Franchek, et. AL, 1997). Nonetheless, these designs require complex mechanisms and have limited tuning capability. Neck \ Cavity \ Primary acoustic system Figure 1.1. Helmholtz resonator connected to a primary acoustic system This dissertation presents the invention of an electronically tuned semi-active Hehnholtz resonator (SHR). This device can be attached to a primary acoustic system, such as a duct, to reduce the transmission of narrow frequency band noise. It can be adaptively tuned on-line to track a disturbance signal with slowly time-varying frequency. It has several advantages over similar inventions. There are no complex moving parts or mechanisms so it is cheaper and easier to implement. The sensitive components are removed from the primary acoustic system and placed in the resonator cavity, so they are less susceptible to damage from harsh environments. The device is fault tolerant: in the event that the controller is turned off, it continues to provide nominal noise reduction. Also, it requires only one connection to the primary acoustic system. No sensors are required external to the device, so that its operation is not dependent on the structure of the primary acoustic system. The device consists of a classic Helmholtz resonator with a surface of the cavity interior replaced by an acoustic actuator. The actuator is driven by a microphone that senses the pressure in the cavity and a controller that provides the appropriate magnitude and phase between cavity pressure and actuator velocity. This magnitude and phase relationship can be related to an acoustic impedance. The overall SHR impedance is defined by the ratio of the pressure at the resonator inlet to the volume velocity through the inlet. It can be changed by modifying the actuator impedance. With this configuration, the overall acoustic impedance of the device is a function of the resonator’s dimensions, which are fixed, and the controller gains, which can be changed electronically on-line. Each chapter in this dissertation is written to address a separate issue in the performance of the SHR. The chapters are intentionally written to stand alone as separate articles. For this reason, the reader may notice repeated text, equations and figures in separate chapters. This work presents four major issues. Chapter 2 presents a physical analytical model of the SHR, based on first principles of physics and a control strategy. Chapter 3 presents a model of the power flow and answers the question “where does the sound go?” Chapter 4 compares methods of implementing the SHR. And Chapter 5 presents an adaptive control strategy for tuning the device on—line and tracking a disturbance noise with a slowly time-varying frequency. Concluding remarks and suggestions for future work are presented in Chapter 6. Each article includes experimental results which demonstrate the capability of the device and are compared with the modeled results. Chapter 2 A Semi-A ctive Helmholtz Resonator This chapter presents a new invention; the semi-active Helmholtz resonator (SHR). The SHR consists of a Helmholtz resonator (Ingard, 1953; Selamet and Dickey, 1995) with the addition of a microphone and controller driven, compensated acoustic actuator on one surface of the cavity. An analytical model is presented and used to show that an acoustic impedance on the resonator cavity interior can be controlled to modify the overall acoustic response of the system. This changes the apparent resonant frequency and peak amplitude of the SHR. It demonstrates that a simple proportional- integral controller can be used in the feedback control of the device. It presents an analytical controller design based on an ideal actuator model. A compensated actuator is included in the model-to illustrate that the analytical controller design is sensitive to actuator dynamics. This motivates a model-based, empirical controller design which is shown to successfully re-tune the resonator. Experimental lab measurements are presented which demonstrated the tuning ability of the controller and its ability to quiet noise in a duct. ANALYTICAL RESONATOR MODEL The Helmholtz resonator is a classic acoustic device which consists of a rigid-wall acoustic cavity with at least one short and narrow orifice, or “neck” through which the fluid filling it communicates with the external medium (Figure 2.1). Temkin, (1936) developed a model to obtain the impedance of an acoustic resonator. He studied the action of a monochromatic wave on the device, under the assumption the lateral dimensions of the cavity were small compared with the wavelength of the incident wave. The cavity creates an acoustic compliance which can be computed from the physical dimensions of the resonator as Ca _—. V 2 (ms/N) (2.1) 00 where V is the cavity volume, p0 is the density of the medium, and co is the speed of sound in the medium. Mass of air in resonator Q1 neck Cavity , volume, V _.> Neck cross I e section area, S Figure 2.1: Ideal Helmholtz resonator The mass of air in the neck will oscillate in response to the wave as a solid body with effective inertia [a = Pg’e (st/ms) (2.2) where le and S are the effective length and cross sectional area of the neck. When dissipation is small, resonance occurs at a frequency a), = ycw (rad/s) (2.3) Summing the forces on the inertia produces a second order differential equation relating the pressure, P, (N/mz), at the entrance of the neck to the volumetric flow rate, or “volume velocity,” Q, (m3/s). Temkin’s model is extended here by converting the differential equation into the transfer function model 91. = .1. S (2.4) Pl Ia 32+59—s+ 1 [a Ca Ia _ where Ra is the resistance due to radiation losses and viscous damping of the medium. Figure 2.2 shows a frequency response of (2.4) with typical values for the acoustic parameters. The magnitude attains a peak at 194 Hz and the phase crosses zero degrees at the same frequency. The peak in magnitude and zero phase are the key characteristics that produce the pressure release boundary that makes the device reflect sound back to the source. This is discussed in detail in Chapter 3. Magnitude (dB) Phase(deg) 50 100 150 200 250 300 350 Frequency (Hz) Figure 2.2. Frequency response of a Helmholtz resonator The complex acoustic impedance of the Helmholtz resonator, 2,, relates P, and Q1 as P1 = ZlQl (2'5) The ideal Helmholtz resonator model can be modified by adding a boundary condition to the cavity interior surface, relating the surface volume velocity, Q2, to the pressure acting on the surface, 1’2 (Radcliffe and Gogate, 1994) as P2 — = 2.6 Q2 Z2(5) ( ) where Z, is an arbitrary acoustic impedance. Figure 2.3 shows a Helmholtz resonator with this boundary condition added. Q2 Mo ' Mass of suerZeg air in resonator Q1 neck \ _:w , _[— P1 —> ® 2 i Z? > . // //// Figure 2.3. Hehnholtz resonator with complex impedance boundary condition State equations can be written for the system by summing the forces on the mass and summing the volume velocities into the cavity. Taking Q, and the volume displacement, q I, as the states gives 2R, -1 i 1 [91]: ’a 61,10 914;}, (2.7) 41 1 _‘11 0 _ Z2932- The transfer function that relates P, to Q, can then be found as 1 s- Qri Z232 P, 1,, 2 [Ra 1 J 1 [ Ra] s + —— s+ l-— _ Ia ZZCa Cola Z2 _ The dynamic response of the system can be modified by specifying Z,. In (2.8) particular, a specific value of Z, can be found which produces a resonance in the Q,/P, transfer firnction at an arbitrary frequency and with arbitrary peak magnitude. Solving (2.8) for Z, and replacing Q,/ P, with a constant, x0, which specifies the height of the resonant peak, and letting s = jarc gives (2.9) where we is a scalar value that represents the frequency of the desired resonant peak. This impedance can be separated into real and imaginary parts by multiplying the complex conjugate of the denominator of (2.9) and collecting real and imaginary parts. This acoustic impedance control technique can be demonstrated by examining the frequency response of (2.8). For simplicity, the acoustic parameters, 1,, and Ca, are set equal to 1, while an arbitrarily small value, Ra = 0.1, was chosen. This creates a system with a nominal resonant peak at a) = 1 with the maximum gain = 1/Ra = 10 (20 dB). Next, x0 was set to a sufficiently large value and Z, was computed for values of (0C between 0.5 and 1.5. Three graphs were computed with x0 = 12, 31, and 100 (Figure 2.4). These values for x0 were chosen to correspond to resonant peak magnitudes of 22, 30, and 40 dB. These graphs give the value of Z, required to produce a resonant peak at a frequency me with amplitude x0. The circles represent nine specific values of Z; for arc = 0.8, 1.0, and 1.2 that are used to generate Figure 2.5. Note the three circles nearly overlap on the Re(Zz) graph for (DC = 0.8, and 1.2, and for all values of we on the Im( Z,) graph. This indicates less variation in Z, for these values of x0 . Note that a smaller impedance requires a larger Q, for a given P, from the definition of Z, (2.6). Therefore, the maximum Re( 2,) decreases as x0 increases. Also, note the sign of Im( 2,) changes from positive below the nominal resonant frequency, w=1 , to negative above a) = 1. 10 40- ....... 3 .......... g .......... g .......... 3 ......... ,3 .......... g .......... g .......... 3 ......... 3 ‘ i Z ‘ :1 i i l I I . . . . 'Z‘ I - 30 u— ........ .......... ......... .......... ........ “2‘ ......... 3. . Increasing x0 . . . 3| . 20,. ........ ........ ........ ......... ll, ......... .......... 30 N O a o O Imaginary(Z) _L O 1'» o _30 i 1 i i i i i i i i 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Command Frequency Figure 2.4. Plot of the real and imaginary parts 'of the controllable acoustic impedance for x0 = 21, 31, and 100 vs. normalized command frequency Next, the controllable acoustic impedance was applied to the SHR to show how the various values of Z, affect the frequency response of the system. The impedance Z, was computed from (DC and x0, substituted into (2.8) and its frequency response was plotted. Values of we used were 0.8, 1.0 and 1.2. Values of x0 used were 12, 31, and 100. The nine combinations are shown in Figure 2.5. The resonant peaks are located at a) = 0.8, 1.0, and 1.2, and the arrows indicate the direction of increasing x0. As x0 is increased, the height of the resonant peaks is also increased. The maximum magnitudes correspond to the dB values of 12, 31, and 100 (22, 30, and 40 dB). Note that increasing the value of x0 has the affect of reducing the damping of the system as noted by the increased Q factor and the increased slope of the phase at the zero degree crossing. The Q factor is a measure of the damping in a resonant system that is computed by measuring 11 the band-width, Af, corresponding to the half power points, -3 dB, from the peak. Q is computed by f/Af, where f,. is the frequency at the peak (resonant frequency) (Hartmann, 1997). The percentage damping, Z: of a second order system can be computed from the Q factor as g = 1/2Q. InCreasin x, : Magnitude (dB) ' 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Frequency (rad/s) Phase (deg) 0.9 1 1.1 Frequency (rad/s) Figure 2.5. Frequency response of SHR system with variable controllable cavity impedance for Z, defined by are = 0.8, 1.0, and 1.2, and x0 = 12, 31, and 100 with arrows indicating direction of increasing x0 Assuming that the disturbance is a narrow frequency band tone at a relatively low frequency, Z, can be implemented with a positive feedback, proportional-integral (PI) controller. A positive feedback arrangement is used to keep the sign conventions consistent with Figure 2.3. A PI controller is ideal because it is insensitive to noise, it is generally stable, and it has characteristics that are well understood. The PI controller is not suitable for higher frequencies because the magnitude of the integral signal falls off 12 with increasing frequency. Under this assumption, (2.9) need not be implemented for all frequencies and the controller need not generate a control law corresponding to Figure 2.4. Instead, the controller can implement a single valued complex impedance associated with a single point on the curve in Figure 2.4 which corresponds to the disturbance frequency. Each point on this curve represents an impedance with a real part which can be implemented with a proportional controller, and an imaginary part which can be implemented with an integral controller. The gain, K ,, must be increased with frequency to compensate for the attenuation that occurs with frequency in an integral controller. Note the controller outputs a Q, for an input 1’, therefore the controller transfer function is the inverse of Z,. The control law is given by G(s)=Q-3-=Kp+-I-<—’- (2.10) P, s where the gains K P and K, are given by KP = lie/(71;) (2.11) K, = to * 111171;) (2.12) This produces an analytical controller design based on computing the controller gains K P and K, from the relation for Z, (2.9) for desired values of we and x0. Figure 2.6 shows plots of K p and K, vs. we computed for all valrfes between 0.5 and 1.5 and x0 = 22, 31, and 100. The circles represent the values for (ac = 0.8, 1.0, and 1.2 as these controller gains correspond with the response curves in Figure 2.5. The arrow indicates 13 the direction of increasing x0. Note larger values of K P are required for larger values of XO. A negative K, is required when we < 1, and K, changes sign when we increases above 1. It should also be noted that the shape of the graphs in both Figures 2.5 and 2.6 will vary depending on the values of the acoustic parameters Ia, Ra, and Ca. 0.35: ....... .' ......... . ........ , .......... ' ..... .......... - ........ . ......... . .......... . ........ ... \ .' i J L I Z I I I ‘ 0.3L..\ ..... ......... .......... ‘ 0,5 _ _ ‘ .. Increasmg x0 ................................... 0.2). ..... \. .\....\\ ......... ......... 3 ........ , .......... I ......... ......... 3 § a 2 3 2 2 2 2 : a 0.15 , .l .......... .......... ....... .. 0.1 . 4, ::_~g‘~‘ .......... ............................ 0.05 .................. ,.....‘...‘.. .... :.:8: 2.2.3.: : _.,_.,_.._._. .... G ~~~~~~ > __ - —————— 1 1 1 1 f -1_" $‘-—1—-—-1———1 05 06 07 08 0.9 1 11 12 13 14 15 1.5.. ........................ , ......... . .......... ................ ‘ .......... . .......... . ,_ .......................... ......... .......... .................... .......... ........ .293. ..... 3 05_ ....................... ......... ......... ................. Ara/...; ................... '32 E f 3 ,’:” 3 O- ........................... .......... ..... _‘,.‘»Q"" ................ ............................. ,-e—"2 ‘ s -0.5_ ........ ‘Hé.b’fi ....... ...................................... ............................. -"— I _1 1 1 1 1 1 1 1 L 1 1 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 CommandFrequency Figure 2.6. Plot of controller gains K P and K, vs. command frequency for x0 = 22, 31, and 100 with arrow indicating direction of increasing x0 The closed loop transfer function in terms of controller gains K p and K, can be computed by replacing 22 with the inverse of G0 in (2.8) and simplifying which gives s —— Kp+—s _QI=_1_ R K Ca K3 H H (2.13) P I 3 __q___£ 2 1 __i_ a P _ a I ' a 3 + 1,, a S + lac, C, laC 1,0,, 14 This form of the transfer function shows how the controller gains affect the system response. The system is obviously not a simple second-order resonator. The denominator of the transfer function is third order polynomial in ‘s’ and the numerator is second order. However, Figure 2.5 shows that it exhibits resonator characteristics. More insight can be drawn from (2.13) by making some simplifying assumptions. For example, letting Ra = K P = 0 and rearranging yields a. 1 sz—Kz I), 810 32+(1-Klla) L Cola (2.14) This equation shows how K, affects the poles of the transfer fimction. The term in brackets in the denominator of (2.14) represents the effective resonant frequency (squared) of the system. A positive K, decreases the effective resonant frequency, and a negative K, increases the effective resonant frequency. Recall that the control system uses positive feedback. This trend is the reverse of negative feedback systems where positive gain increases the effective stiffness of the system. Also, note the system response is volume velocity relative to pressure. In mechanical resonator systems, the displacement relative to force impedance becomes stiffer as the proportional gain is increased. Displacement is the integral of velocity, explaining why an integral controller gain changes the apparent stiffness in this system. Equation (2.14) also identifies a limit on K ,. If the product of K, and 1,, is made larger than 1 then the system will become unstable as noted by a change in the sign of the so coefficient in the characteristic polynomial (denominator) of (2.14). This occurs when K, = 1/ 1,, which makes the 15 resonant frequency equal to zero. However, K, can be made more negative without bound. This suggests that the resonant frequency can be decreased to zero and increased without bound. Similarly, by setting K, = 0 gives -51 -Q—1=—1— 3 Ca (2.15) P1 Ia s2+(§f—§—:)s+,lc:(1—R0KP) This equations shows that K P appears as a term that is subtracted from the acoustic loss term Ra. Increasing K P reduces the apparent acoustic loss and hence reduces the system damping. The apparent acoustic loss can be either increased by a negative K P, adding additional damping to the system, or, decreased with a positive K P. Reducing system damping increases the peak magnitude at resonance. Equation (2.15) identifies limits on K P. If the value of K P/ Ca becomes less then Ra/ 1,, then the system will become unstable, as can be noted from the change in the sign of the s1 coefficient. Therefore, the largest K p = Ra“ Ca/ Ia, which produces zero damping and marginal stability. This analysis illustrates the ability of the SHR to change the dynamic response of the system. Each gain K P and K, gives an input to change the apparent resonant frequency and peak amplitude of the Helmholtz resonator. The advantage of this affect is that the response of the Helmholtz resonator is modified without changing the physical dimensions, i.e., cavity volume, neck length or cross section area. The change in 16 response is caused only by the interaction of the controllable cavity acoustic impedance, 2,, with the Helmholtz resonator. This design has two important implementation benefits. First, it has the benefit that when the boundary condition is removed, i.e., the controller is turned off, the system reverts to the nominal resonance defined by the physical dimensions of the Helmholtz resonator. Since these dimensions can be designed to meet nominal performance requirements, turning off the controller will only remove the variable tuning leaving the nominal tuning intact. Second, it has the benefit that the sensitive moving parts- microphone and actuator- are not directly in the path of the fluid flow. Instead, they are located inside of the Helmholtz resonator. This provides the advantage that debris carried by the fluid in the system will not come in direct contact with the microphone and actuator. RESONATOR MODEL WITH ACTUATOR DYNAMICS An actuator model with finite dynamics can be introduced to study the effects on the system performance and the controller design. The previous analysis assumed that the actuator transfer function was a pure gain. Unfortunately, most commercially available acoustic actuators do not have such ideal response characteristics. Birdsong and Radcliffe (1999) presented an actuator which uses feedback compensation to improve the response of a dual voice coil speaker by compensating for the internal dynamics and the pressure interaction with the acoustic system. While this actuator does not have a transfer function that is a pure gain, its response is a significant improvement over uncompensated speakers. This actuator will be used here. Figure 2.7 shows a block 17 diagram that illustrates how the resonator model is coupled with the speaker model (without the compensator). The speaker has two identical intertwined coils that produce the electromechanical force that drives the speaker. It can be modeled as a three port device with a voltage input, ep, and current output, ip, on the primary coil; a current input, ibs, and voltage output, ebs, on the secondary coil; and a pressure input, P,, and the volume velocity, Q, output on the speaker face. The speaker is driven by e p, and ib$ is modeled as an open circuit. The other speaker outputs, ip and ebs, are used as the inputs to the compensator, which is not included in Figure 2.7. The pressure in the resonator cavity impinges on the speaker face producing the input to the speaker model, resulting in a tightly coupled dynamic system. This combined model will be considered a single block in subsequent models. Resonator model ._> ,___> Speaker model P] Q] ,__fl —> P —> —> 2 Q2 92 P, l" —->- e i L—> P P "" ibs elm“ Figure 2.7. Block diagram of speaker model coupled with resonator model through the P, and Q, signals The state equations of the Helmholtz resonator and the uncompensated dual voice coil speaker can be assembled by coupling the SHR state equations (2.7) with the model from Birdsong and Radcliffe (1999) through the P, and Q, signals. These coupled system state equations are given by 18 P—Ra _l . P 0 0 in . o 0 0 _ - FQI 1,, Cal“ Q1 la 4, 1 0 1 o o q, 0 0 0 7b, (2.16) i Q _ o ”5‘12 “Rs 1 2134. Q + $de 1-9—[9] 0 0 e d: 2 15C“ 1, C4, 1,], 2 1. Ic ” (12 o o 1 o o 42 0 0 0 _P1 11-00;“0M.A.Ma+_a,0 Sd lc c I. . _ , . The output equation is given by 1 o 0 0 0 1 Qt“ o o 0, bl M —M Q1 0 — 1——C— C +R M. +R M ,_ er 5,, I, If (R‘ ) 41 R...+ ((1 m) ,4 0 ‘bl (2.17) S ,- = 0 0 0 o i 92 + -M,c c e, p 1 0 0 P P l C 42 IC _ 2 2 0 — 0 o o 1 0 0 o . Ca .. d — where the states are: the acoustic volume velocity and displacement from the neck Q, and q], the volume velocity and displacement from the speaker Q, and 42 , and the electromagnetic flux in the speaker coil 21. The inputs are the primary coil current, ip, the secondary coil voltage, ebs, and the input pressure to the Helmholtz resonator neck, P, . The outputs are Q, , the voltage in the secondary coil, 6b,, the current in the primary coil, 1p, and the pressure in the cavity, P,. Radcliffe and Gogate (1996) discuss the derivation of the parameters of the speaker model which include: the speaker face area, Sd, speaker inertia, 1,, speaker compliance, C5, speaker friction, Rs, speaker coil resistance, R, speaker coil inductance, 1,, speaker coil mutual inductance, MC, speaker electromechanical coupling factor, bl, and the primary coil current sensing resistance, Rm. The closed-loop response of the system can be computed by adding the compensator, PI controller, a microphone, and the amplifier models to the speaker and resonator model. Figure 2.8 shows the interconnection of these components with the resonator and speaker models combined into a single block. The pressure in the 19 resonator cavity, P,, is measured by the microphone with sensitivity, Smic. This signal is fed to the controller which generates a control signal, ec. The speaker outputs, i and ebs, are combined in the compensator and produce the speaker velocity estimate, veg. This signal is subtracted from ec and a disturbance signal D, is added. The resulting signal is amplified by K amp and the amplified signal becomes the input to the actuator, e p. Combined resonator and speaker model Speaker compensator —> P, Q, 1—-> ,— ———————— —, l > Hp 1 1p I —> ep ,+ D, Hbs ebs I_+ _ _L__ _1 _ P2 ——->‘ SMIC PI controller kamp Figure 2.8. Block diagram of resonator and speaker model with compensator and PI controller Table 2.1. SHR Model Parameter Values bl 2.45 N/A R, 4.875 ohm ca 343 m/s 5,, 50 ohm C, 0.000868 m/N R, 3.745 N sec/m I, 0.002 H S 0.000254 m2 I, 0.0076 Kg S, 0.0133 m2 le 1 cm V 0.0102 m3 M, 0.001 H p, 1.18 Kg/m3 Sm. 4 mv/ PA I, 160 Ns,/m5 Ra 7.5e4 Ns/m5 Ca 7.20e-9m5/N 20 These results show that the analytical controller design is not effective in the presence of unmodeled actuator dynamics. The feedback controller, resonator, and compensated speaker model did not exhibit the desired resonant frequencies and amplitudes when the controller gains, predicted by the analytical controller design, (2.9) — (2.12), were used. The analytical mapping of the gains to the resonant frequency and peak amplitude was no longer effective. Even with the compensator, the speaker response exhibited excessive deviation from the ideal actuator model response. This leads to the conclusion the analytical controller design was useful in motivating the use of a PI controller, but is too sensitive to actuator dynamics and is not effective in choosing the controller gains. It should be noted that this does not mean the SHR can not be implemented. It only means the relatively simple mapping between controller gains and system response is not sufficient. The mapping must include more factors, in particular, the actuator model must be included in the controller design. Unfortunately, the actuator model adds considerable complexity to the mapping, making a closed-form solution intractable. This motivates the use of a model-based empirical controller design. An empirical technique, motivated by the above formulation was applied to the model to produce the controller design. The model response was simulated for various gains to determine the mapping between the controller gains and the system response. The strategy was to find gains, K P and K ,, that placed the resonance at various frequencies, while maintaining the same peak amplitude. It was found that K P modified the amplitude and K, modified the resonant frequency as predicted by the analytical controller design, but the functional relationship was different than predicted. Figure 2.9 shows the frequency response of the Q,/ P, transfer function for three sets of controller 21 gains. Figure 2.10 shows the frequency response for the P,/D, transfer function with the same gains. Figure 2.10 will be compared to laboratory measurements. The controller places the resonant frequency between 110 and 150 Hz while maintaining a maximum peak of equal magnitude for all cases. Table 2.2 shows the gains K ,1 and K, vs. a)" used to generate Figures 2.9 and 2.10. Chapter 4 examines the relationship between the PI controller gains and system response in more detail. A B C _70_ ........ , ........... ............ . ............ .' ........... . ........... . ....... .. .2 mt ............................................. s ........... I, ........... ,j ....................... a ' E 90 ,. ................................... 5 .................... » ...................... 8 f E : .3 81-100 ~ ........................................................................................ (U 2 41° _ ............................................................................................... 420 1 1 1 1 1 1 1 1 80 90 100 110 120 130 140 150 160 150,. ........... . ............ ...... . ..... . .......... _. ......... ‘ ............ 100 a .8 50 o m 2 o a. -50 we 1 1 1 1 1 1 1 1 80 90 100 110 120 130 140 150 160 Frequency(Hz) Figure 2.9. Closed-loop frequency response of the Q,/P, transfer function for model including actuator dynamics 22 on O ‘1 0'1 ‘1 O Magnitude (dB) 0) 0'1 60.. ................................................................................................ 55 -. ..... g ............ g ............. g ............ g ............ ............ g ............ g ............ 5 so I l l 1 l J l l 80 90 100 110 120 130 140 150 160 100 50 Phase (deg) o -50 _,00 1 1 1 1 1 1 1 —_1 80 90 100 110 120 130 140 150 160 Frequency (Hz) Figure 2.10. Closed-loop SHR frequency response for P,/D, transfer function for model including actuator dynamics Table 2.2. Controller gains identified by model-based empirical controller design Graph Resonant K P Gain K, Gain Fregrency Hz A 112 0.99 -100 B 1 30 0.99 0 C 145 0.99 100 Figures 2.9 and 2.10 show that the controller successfully achieves the goal of re- tuning the resonator with the actuator. Figure 2.9 clearly shows that varying K, and K ,1 results in changing the system resonant frequency and peak amplitude. This result indicates that the compensated acoustic actuator will perform the task of the complex boundary condition as hoped. 23 EXPERIMENTAL VALIDATION An experimental apparatus was constructed to validate the theoretical model and to demonstrate the noise reduction capability of the device. In this section the Helmholtz resonator and actuator implementation will be discussed, the PI controller design will be demonstrated and finally, the SHR will be applied to an acoustic duct to demonstrate the control algorithm and noise reduction capability. The experimental SHR setup consisted of two components: a Helmholtz resonator cavity and a microphone-compensated actuator system. Figure 2.11 shows a photograph of the SHR connected to an acoustic duct and Figure 2.12 shows a schematic of the setup. A cylindrical Helmholtz resonator cavity was constructed from PVC with dimensions 0.075 m in diameter and 0.15 m in length. A cylindrical neck with dimensions 0.018 m diameter and 0.01 m in length, was fitted on one face of the cavity. The microphone- compensated actuator system consisted of a half inch B&K type 4155 microphone sealed through the wall of the cavity. A D-Space Model #1102 floating point, digital signal processor (DSP) was used to implement the speaker compensation, and an acoustic actuator was sealed in the opposite face of the cavity. A DSP sampling rate of 5 kHz was used for all experiments. 24 Microphones Acoustic duct Disturbance ‘ speaker \ Speaker enclosure Figure 2.11. Photograph of SHR connected to an acoustic duct with a second audio speaker to inject noise D 1 V Digital signal processor I IE, P2 l lebs l1P Amplifier Dynamic signal analyzer e \ P Fl _ Speaker neck 1‘: enclosure 1 Microphone “ Dual voice coil speaker cavity Figure 2.12. Schematic diagram of experimental SHR apparatus In all phases of the system design, the device was separated from any primary acoustic system. This was not done arbitrarily for convenience, but because traditionally, mechanical and acoustic resonators are designed independently from the primary system. The usefulness of the device would be limited if the resonator response was dependent on the structure of the primary system. Fortunately, this is not the case; resonators can be 25 designed with a tuning fiequency and then applied to any suitable primary system. In the absence of a disturbance pressure, P, , the system was disturbed electrically by the signal D1 injected via the actuator input voltage (Figure 2.8). Although it would be useful to measure the Q,/P, transfer function directly since it is key in the interaction between the resonator and the primary acoustic system, this was not done. The quantity Q, is difficult to measure experimentally since it is a zero mean, oscillating air velocity. Although a laser velocity anemometer is a device that can be used for such measurements, it is extremely costly, and experimentally complex. Consequently, the transfer function Q,/ P, is difficult to measure directly. Alternatively, the signal, P,, and therefore, the transfer function P,/D1 can be measured easily with a microphone. The system can be disturbed by either inputs P, or Q, since the characteristic polynomial which defines the resonant frequencies is the same regardless of the input. Therefore, the model was validated by comparing the model response for P,/D1 with experimental measurements. Speaker Compensation The goal of the compensation is to produce a constant magnitude and phase relationship between the desired velocity and the actual speaker face velocity. The actuator consisted of a 6 inch dual voice coil speaker with local compensation (Birdsong and Radcliffe, 1999). It was compensated for the mechanical dynamics associated with the mass and compliance of the speaker assembly. It was also compensated for the pressure impedance on the speaker face which becomes critical when applying a control input to an acoustic cavity near an acoustic resonant frequency, as in this experiment. 26 The actuator was compensated to improve the speaker performance, but finite gain and phase errors in the actuator response affected the system. A speaker velocity estimator (Birdsong and Radcliffe, 1999; Radcliffe and Gogate, 1996) was implemented by combining the voltage in the secondary coil with the current in the primary coil. A 10 ohm resistor was placed in series with the primary speaker coil to measure the current. The velocity estimate was then used to close the loop on the speaker velocity with a proportional controller (Figure 2.13). A value of 30 was used for Km], for all trials. This value for K amp is somewhat smaller than values used by Birdsong and Radcliffe (1999) where gains of as much as 100 were used. The gain K W was chosen to increase the robustness of the system, which comes at the expense of performance. The speaker velocity was measured directly to confirm that the transfer function of the actual speaker velocity relative to the desired velocity was acceptable. This was done by directing a laser velocimeter through the SHR neck onto reflective tape on the speaker face. With the relatively low value of Kamp, there was less than 5 dB and 100 degrees of magnitude and phase between desired and actual speaker velocity in the frequency range of 20 to 200 Hz in all experiments. While this represents a 15 dB and 80 degree improvement over uncompensated audio speakers, it clearly does not approach the response of the ideal actuator model. After the speaker compensator was implemented, the closed-loop speaker compensation was considered a single block in the SHR, and all subsequent open and closed-loop SHR experiments included closed-loop speaker compensation. 27 ——————————————————— I Actual ° I ligated 1 l velocity v 1 K amp Controller I + - | current secondary I; 1pm : I Velocity estimate “’1ng . | l Velocrty I estimator I | Figure 2.13. Block diagram of dual voice coil speaker compensation used in SHR actuator PI Controller Design The PI controller was implemented and closed-loop SHR response was recorded with controller gains K P = K, = 0. In this configuration, the compensator attempts to hold the speaker face fixed in the presence of the disturbance. White noise was input as the disturbance to the system and the transfer function of P,/D, was measured using a Hewlett Packard dynamic signal analyzer model #35660A. With the SHR disconnected from a primary acoustic system, resonance was observed as a peak in the frequency response of the P,/D, transfer fimction, The results indicated that a resonant peak occurred at 120 Hz, however there was significant damping in the system which reduced the peak amplitude. This damping was attributed to mechanical damping in the form of friction and electrical power dissipation in the current sensing resistor, Rm, in the speaker compensator. This damping is large compared to the acoustic damping expected in a passive Helmholtz resonator. Other experiments indicated that damping was significantly reduced when current was not allowed to flow through the sensing resistor. 28 The closed-loop response was then measured with various non-zero controller gains. As predicted by the model, the analytical mapping between K P and K, and the resonant frequency and peak amplitude, (2.9) — (2.12), did not produce the desired results. This was attributed to the deviation of the actuator from the ideal model. Even with the compensator, the effects of the speaker dynamics were not sufficiently minimized. The empirical technique was used in place of the analytical mapping to tune the system in the presence of significant actuator dynamics. The PI controller design was based on qualitative information learned from the model. The objective was to find gains, K P and K ,, that placed the resonance at various frequencies, while maintaining the same level of damping. The data was collected by fixing K ,, searching for a K P that produced the desired peak amplitude, and recording the resonant frequency. The gains K P and K, are plotted against resonant frequency in Figure 2.14. Note that although there is a difference in the magnitude, the overall trends of these graphs agree with the data derived from the model in Table 2.2. The K P gain is negative for all values of 60c with the most negative value at the nominal resonant frequency (130 Hz). Only a 10% change in K P is required for the entire range of w". The model predicted that K P had no change in the range of w" = 110 to 145 Hz. The K, gain ranges from —100 to 200 and passes through zero at the nominal resonant frequency. 29 Kp Vs Resonant Frequency 80 100 120 140 160 180 Ki Vs Resonant Frequency 200 1 1 , a A c 100- ....................................... .............................. ..1 .3 . 3 0,. ..................................... . ................................ .. >4 a 4001...... . ................................. - 1 1 1 1 1 1 80 100 120 140 160 180 Frequency Hz Figure 2.14. Graph of K P and K, vs. (0,. determined using an experimental empirical technique The timing capabilities of the device are illustrated by the closed-loop frequency response measurements (Figure 2.15). The five graphs show the resonant peak for five separate experiments with w,, = 80, 110, 140, and 170 Hz and one graph for the controller gains set to zero. Note that the SHR with the gains set to zero is over-damped with approximately 45% damping. With the controller turned on, a resonance is exhibited and positioned at the desired frequency with approximately 5% damping for all cases. This result demonstrates the ability of the controller to tune the SHR to arbitrary frequencies. 30 Ch <3 1 1 1 1 1 1 20 40 60 80 100 120 140 160 180 200 Frequency (Hz) 1‘.) <3 \ Figure 2.15. Experimental frequency response P,/D1 transfer function with w" = 80, 110, 140, and 170 Hz and with gains set to zero An additional experiment was performed to demonstrate the application of the SHR in an acoustic duct system. The SHR was connected to a 3 inch diameter and 32 inch long duct (Figure 2.16). A separate audio speaker was mounted on one end of the duct to inject noise and the other end was left open. A pure 140 Hz tone was injected into the duct and the sound level P, was measured in the duct 2 inches from the end with a 1/2 inch B&K microphone. The gains K P and K, were set to tune the SHR to 140 Hz and the time response of the microphone signal was recorded as the controller was applied. Figure 2.17 shows P3 vs. time in open-loop for 0.05 seconds, then the controller is activated. There is a short transient then the P3 signal is reduced by a factor of approximately 6.7 (-16 dB). Similar results were obtained with the disturbance tones with frequencies ranging from 80 to 180 Hz. 31 Oscilloscope Em Pure tone disturbance Acoustic duct Microphone Figure 2.16. Schematic of SHR applied to acoustic duct 40 I I l 1 1 r 1 l 1 Controller off 30_ .. ..,. .. .1. .. . .. . '. .’ .. _ Centroller on 20— 3 - 10— '1 m 0. g 0 12 2 CL 10 ........ 1 -20 4 ~30 ‘ 1 ‘ ’40 l I l I l l l o 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.10 0.2 Tlmesec Figure 2.17. Time response of pressure 2 inches from the duct end with a 140 Hz disturbance as the controller is activated, showing a 16 dB noise reduction CONCLUSIONS This work presented an electronically tuned semi-active Helmholtz resonator which can be used to quiet noise in enclosed acoustic systems. An analytical model was presented and used to show that an acoustic impedance on the resonator cavity interior can be controlled to modify the overall acoustic response of the system. This changes the 32 apparent resonant frequency and peak amplitude of the acoustic response. It was shown that a simple proportional-integral controller can be used in the feedback control of the device. An analytical mapping solution between the actuator impedance and the controller gains was developed for an ideal actuator, and an empirical mapping technique was developed for an actuator with significant dynamics. A compensated audio speaker was included in the model to demonstrate the effects of actuator dynamics. Experimental lab measurements were presented which demonstrated the tuning ability of the controller and its ability to quiet noise in a duct. The SHR system represents a powerful new tool in tuning acoustic systems. It adds a nominal resonance to an acoustic system and allows the resonance frequency and peak magnitude to be changed on-line continuously over a range of frequencies. It has advantages over other technologies in that it does not add significant mechanical complexity, it is fault tolerant, and the design places the sensitive sensor and actuator away from the direct path of the process. 33 Chapter 3 Sound Reduction and Power Flow of the Semi-Active Helmholtz Resonator in an Acoustic Duct Examination of the acoustic power flow for the semi-active Helmholtz resonator (SHR) and its role in noise reduction in an acoustic duct is considered here. A theoretical model is presented and analyzed to determine the optimum conditions for reducing the transmitted noise in a primary acoustic system with the SHR. These results are used to demonstrate the effectiveness of the SHR in changing the tuned frequency. Experimental results are given and compared with the theoretical model. MODEL DEVELOPMENT Power Flow Model The system of interest is a rigid, long straight duct with a constant cross sectional area, AD, a SHR attached to the wall at x = 0, and a source generating sound at one end of the duct. Figure 3.1 shows a schematic of the system with arrows indicating the directions of incident, reflected, absorbed, and transmitted power. Disturbance Acoustic source /11101 9C @¢\ Figure 3.1. Schematic diagram of SHR and acoustic duct showing incident, reflected, absorbed, and transmitted power towards open end 34 The objective of this study is to develop a relationship for the various power quantities in terms of the SHR properties. A lumped-parameter model (Pierce, 1981) is used which neglects dissipation in the duct, incorporates one-dimensional acoustic theory, assumes continuous pressure, and assumes that the volume velocity is conserved at the junction giving QI=QA+QT (3.1) where Q,, Q,,, and Q, are the volume velocities (m3/s) of the incident, absorbed and transmitted paths. The pressure in the duct is continuous, so the pressure in each path at the intersection is equal g=azg GE where P,, P,,, and P T are the pressures (N/mz) in the incident, absorbed, and transmitted paths. The relationship between P and Q across a section is named acoustic impedance. Dividing (3.1) by (3.2) gives a relationship between the acoustic impedances for each _ | C where Z ,, Z A, and Z, are the acoustic impedances (N s/m5 ) of the incident, absorbed and transmitted paths. The acoustic impedance of the incident path is analyzed by considering the forward traveling and reflected pressure wave in the incident path. The total incident pressure is given by the sum of the forward and reflected wave amplitudes, 35 P, = f(e“"“ +916“) (3.4) where f is the pressure amplitude, k is the wave constant (w/c), x is the spatial variable, and 91 is the pressure-amplitude reflection coefficient that gives the ratio of the reflected to incident pressure wave amplitude. The volume-velocity in the incident path is given by Q: = -%f(e“’“ - 918’“ ) (3.5) where p is the density (kg/m3) of the medium, and c is the speed (m/s) of sound. The quantity pc/AD can be defined as Z; and represents the acoustic impedance of air in a free field. The acoustic impedance in the incident path is given by the ratio of the pressure to the volume-velocity, Z, E — (3-6) Zr _ = 3.7 20 1_ 9, < 1 Solving SR gives, <3 z £10. (3.8) 2, + 20 Substituting (3.3) for Z, in (3.8) gives 36 Z‘1+ "—1—2 (A Zr) 0 ER = —1 (2,,"1 + 2,“) + 20 (3.9) These results are valid for general values, Z A, 20, and Z,. However, by considering an idealized boundary condition at the duct end at x = L, further analysis can be developed. Consider “anechoic termination” at the duct end so that no reflections occur. While it is difficult to realize such a boundary condition on an acoustic duct, it simplifies the model sufficiently and deviations from this ideal case can be accounted for in experimental results. Under this assumption, Z, = 20 and (3.9) can be simplified to give the reflection coefficient, 91 in terms of Z A and 20 as The transmission coefficient, T gives the amplitude of the transmitted pressure relative to the incident pressure and is given by T = 1 + SR (3.1 1) The fractions of reflected, transmitted and absorbed power to incident power are giveh by (Pierce, 1981) P __ 2 9‘, -|9i| (3.12) P _ 2 %, _|r| (3.13) 37 P94] = 1 - |Sii|2 — |T|2 (3.14) These equations, (3.12), (3.13), and (3.14), are ratios of reflected, transmitted, and absorbed power relative to incident power. The ratios PR/P1 and PW] can range from zero to one, and it can be shown that P,,/P, attains a maximum of one half when the ratio of Z A/ Z, is one half. The transmission of power through the intersection depends on the impedance Z A, which can theoretically vary from 0 to 00. When Z A = co, the SHR acts as a rigid wall. In this case, SR = 0, and T = 1, i.e., there is no reflection (PR/P, = 0) and all the power is transmitted (Pp’P; = 1). This scenario is equivalent to no SHR present and the system consists of simply a rigid duct which transmits sound perfectly. When Z A = 0 the intersection behaves as an ideal “pressure release” boundary. In this case 91 = —1 and T = 0, i.e., all of the incident power is reflected (PR/P1 = l) and none is transmitted (Pp/P, = 0). The pressure release is the boundary condition used to model a duct with the end open to atmospheric pressure; the pressure at the open end equals zero regardless of the flow. Theoretically, a duct with an open end will be purely reflective with the reflected wave inverted. A pressure release boundary is also used to model the boundary between fluids of significantly varying density such as water and air (Pierce, 1981). In reality the SHR impedance will fall between these extremes, resulting in finite values of Z A. Transmission Loss (TL) is a measure of the power flow which is commonly used in acoustics (Blaser and Chung, 1978). It is the ratio of transmitted to incident power and is given by 38 TL=(P%I)-l =|Ifili|§ (3.15) Note that TL is the inverse of 1’7”] and increases as the transmitted power is reduced relative to the incident power. For example, the pressure release condition (Z A: 0, ‘31 = —1) gives TL = 00. Impedance Control of SHR The purpose of the SHR is to create a variable Z A to control the power flow in the system, therefore, a careful analysis of the relationship between QA and PA is needed. A theoretical model of the SHR (Chapter 2) will be described briefly here so that it can be examined in terms of power. The SHR consists of a Helmholtz resonator with one surface of the cavity replaced by a moving surface (Figure 3.2). The system can be represented by the linear time invariant state equations of the form QA _ i :1— QA Tla' ()[PA] [V]_[Ila 115.][V]+[0 lch (3.16) Q 1 0 Q 1sz0 2.1-ll 1‘] <3” where the states are Q, , the volumetric flow rate or “volume velocity” from the neck (m3/s) and V, the sum of the volumes introduced through the neck and the inner surface of the cavity (m3). The inputs are P, , the pressure at the neck inlet to the cavity (N/mz), and Q,, the volume velocity from the movable surface in the cavity (m3/s). The outputs are Q, and P,, the pressure in the cavity (N/mz). The other parameters are Ra, the 39 acoustic loss that represents viscous and radiation losses (NS/m5), la, the acoustic inertia of the mass of air in the resonator neck (st/ms), and Ca, the acoustic compliance of the cushion of air in the resonator cavity (ms/N). An acoustic impedance can be generated on the moving inner surface of the cavity by enforcing a control law between Q, and P,. This closed-loop, positive feedback configuration is shown in Figure 3.3. $1 Mass of . Q, Movrng air in resonator surface neck : » Cavity / Figure 3.2. Schematic diagram of SHR showing inertia effect in neck and the movable surface in the cavity interior SHR state space model P1 (Jr—>91 —h COHtI‘OIlCI' H Q2 P2 Figure 3.3. Closed-loop SHR system block diagram Once a control law is chosen, the closed-loop transfer function relating Q,, to PA and the power flow can be computed. A proportional-integral (PI) controller, Gc(s)=-Q—C=Kp+—I—(i (3.18) P s C 40 produces the desired response (Chapter 2), where K P and K, are the proportional and integral gains respectively. The closed-loop transfer function for QA/ P,, can be computed by combining (3.13) with (3.11) and (3.12) and converting the state equations into the transfer function (Phillips and Harbor, 1996) 2 1 ( K,) _— +— _QA =L=_1_ 5 Ca K” 1 S (3.19) P Z I 3 Bl- KP 2 1 _£L __ RaKP _ RaKl A A a s + I, C, S 1' lac, Ca lac, 1,0,, This equation gives Q,,/PA (the reciprocal of Z A) in terms of acoustic parameters which are fixed, and controller gains which can be changed on—line. The above transfer function shows how the acoustic impedance can be changed on line by varying the control gains. The power flow of the duct and SHR system can be modified on-line by varying Z A through the controller gains K ,1 and K ,. The simple case where K ,1 and K, are zero corresponds to the condition where the movable boundary in the cavity is held fixed. In this case (3.19) reduces to a simple second order transfer function, consistent with classic Helmholtz resonator theory (Pierce, 1981; Selamet et at., 1995; Tang and Sirignano, 1973), giving 9¢=i=i - s (3.20) The denominator of (3.15) is a second order polynomial and resonance will occur when the inertia effects balance the compliance effects leaving only the resistive forces. This occurs at the resonance frequency 41 w" = (3.21) The resonance frequency is important because at this frequency, for the classic Helmholtz resonator resulting from no control action (3.19), Z A attains the maximum value Z =R, (w=w,K =K =0) (3.22) A a n p I Substituting (3.19) into (3.10) and computing TL (3.15) gives the maximum TL at the resonance frequency, TL=Rfi-+1, (w=wn,KP=K,=O) (3.23) a This introduces the somewhat surprising result that the TL is inversely proportional to Ra, that is, reducing the dissipation in the system will decrease the transmitted power. This highlights the fact that the device does not actually absorb energy, but rather, it reflects energy thus reducing the transmitted energy. The frequency dependence of the SHR transfer function (3.20) indicates how the SHR (with gains, K P and K ,, set to zero), selectively absorbs sound in a narrow frequency range. Figure 3.4 shows the frequency dependence of the TL, computed by substituting s = jw, and plotting TL in dB vs. normalized frequency w/wn as Ra is decreased. The parameters used in the simulation were identified experimentally for the SHR used in the model validation. They represent the nominal response of the SHR with the gains set to zero. The parameters are A D = 4.40e-3 m2, C, = 7.20e-09 mS/N, Ia = 162 42 st/ms, and R. = 7.54 e 4 Ns/ms. Although these parameters seem somewhat arbitrary, the effect of reducing the acoustic loss term is dramatic. The acoustic loss is decreased from 4.0, 2.0, 1.0, 0.5, and 0.2 times the measured loss coefficient, Ra. As R0 is decreased, the TL increases and the bandwidth decreases. For a fixed value of Ra this system will operate most effectively at the center frequency, w/wn = l, and less effectively above and below w’ar, = 1. It should be noted that Ra is determined by material properties and the geometry of the Helmholtz resonator. This analysis shows the SHR should be designed to minimize R. to attenuate noise most effectively at a single frequency. If the unwanted noise has a wider bandwidth, an alternative objective might be to increase R, to increase the absorption bandwidth, however this will come at the expense of reduced TL at the center frequency. Ideal HR with variable Ra Transmission Loss dB 1.5 2 0 0.5 1 Normalized Frequency Figure 3.4. Transmission loss vs. normalized frequency with varying acoustic damping 0.2, 0.5, 2.0, and 4.0 * Rao 43 The frequency dependence of the transmitted, reflected and absorbed power can also be computed using (3.12) — (3.14). The same parameters used in the previous result were used in Figure 3.5 showing the effect of reducing the acoustic loss term in the model. At (1),, = 1 PHP; approaches 0, PR/P1 approaches 1 and P,,/P) approaches 0 as R. is reduced. 1 \:\ T T Decreasing Ra' . Transmitted Power P U! I O . P as m Reflected Power 0 O O .4 .2 0 0 0'8 l l l l l l l l l 3 : ' '- R: : : : : : EMS-"m“. """ .ecrfiasmg qm" """" """" : """ i 30.4-.......g ....... g ..... ,: ................... ' .................. T g 1 : —~.( ”0_2_ ....... ,,/-’ ........................................................... .1 o 1 1 1 1 1 1 1 0 02 0.4 06 1 12 1.4 1.6 1.8 2 0 . 8 . Normalized Frequency Figure 3.5. Plot of transmitted, reflected, and absorbed power ratios vs. frequency as acoustic loss term is reduced The unique feature of the SHR is the ability to tune the bandwidth and center frequency of the TL by changing the gains K P and K ,. The controller gains were selected to move the center frequency, (0,. while maintaining a constant peak, TL (Figure 3.6), by choosing the appropriate value of K P and K, (Table 3.1). The gains in Table 3.1 define the controller impedance, Q,/P,. The microphone sensitivity (SW = 0.05 volts/Pa) and actuator area (S; = 0.01 m2) are not included in the model. When these are 44 included, the gains that must be implemented on the controller are increased by 1/(S,,.,-C*Sd) = 2000. Transmission Loss dB A 01 l l (A) l 0.8 1. 1.4 1 .6 1.8 2 Normalized Frequency Figure 3.6. Transmission loss vs. normalized frequency for closed-loop SHR, placing resonance at 1.0, 1.1, 1.2, 1.3, and 1.4, showing that the center frequency and damping can be changed by varying K ,1 and K, 45 Transmitted Power .0 01 l Reflected Power 0 N I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Absorbed Power 0 J) r L 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Normalized Frequency Figure 3.7. Transmitted, reflected, and absorbed power for gain settings used in Figure 3.6 Table 3.1. Controller gains K P and K, used to move the resonant peak to various while constant peak TL 1 _ 2.6e—6 1.5e- Thus far, the analysis has been presented to explain the power flow in the acoustic duct and SHR system. The model was simplified by the anechoic duct termination 46 assumption and results were presented to illustrate that the firnction of the SHR is to reflect sound, reducing the transmitted sound from propagating in the duct. When the duct end is not perfectly anechoic, this analysis must be revised. An open duct end, for example, will reflect sound, resulting in two reflecting regions in the duct: one at the duct end and one at the SHR. The reflection coefficient upstream of the SHR can be computed from (3.10), but (3.11) through (3.15) require the anechoic duct assumption and are no longer valid. Therefore, the transmitted and absorbed power and transmission loss can not be determined from the reflection coefficient alone. A more complex model could include more details of the duct to show the SHR reduces sound transmitted from the duct end. Nonetheless, this is beyond the scope of this paper. Furthermore, this would obscure the concept of the SHR acting to reflect sound in the duct and reduce the transmission of sound. However, it is necessary to analyze this scenario since the experimental tests used to verify the model employ an open duct end. A duct open end can be modeled as a perfectly reflective boundary with 91 (Hull and Radcliffe, 1991; Speakerman and Radcliffe, 1988; Seto, 1971; Levine and Schwinger, 1946) given by l a-iw) 91,, = e( 1 (3.24) where I, is twice the distance from the point of measurement to the duct end. This results in a reflection coefficient of —1 with a time delay from the wave propagating to the boundary and back. The duct end impedance can be computed by 47 _1+%D l-flb 20 (3.25) This expression is substituted into (3.9) for Z, and the reflection coefficient upstream of the SHR, 91 vs. frequency is plotted in Figure 3.8 with various controller gain settings. The interpretation of these results is difficult. Clearly, the dynamics that created a peak in the magnitude of ER in the anechoic duct termination model create a dip when the duct end is open. These results are compared with experimental results obtained in the next section to validate the model. Reflection Coeffiecient Vs Frequency .L N l Magnitude linear 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Normalized Frequency Figure 3.8. Plot of reflection coefficient vs. frequency for open end duct, 3 trials with various controller gains 48 EXPERIMENTAL VERIFICATION An experimental apparatus (Figure 3.9) was constructed to verify the analytical model. The setup consisted of a 40 inch long and 3/4 inch diameter acoustic duct with a disturbance noise generating audio speaker at x = O, a SHR at x = 20 inches, and the duct end at x = 40 inches lefi open. Two microphones, spaced a distance 0.0254 m apart were placed in the duct upstream of the SHR to measure the reflection coefficient 9?. Disturbance Acoustic DSP | SHR Controller [8“ /uct Dynamic signal analyzer and digital computer Microphones f Duct end Figure 3.9. Schematic of experimental apparatus used to measure the reflection coefficient in the duct and SHR system The experimental SHR setup, consisted of two components: a Hehnholtz resonator cavity and a microphone-compensated actuator system. A cylindrical Helmholtz resonator cavity was constructed from PVC with dimensions 0.075 m in diameter and 0.15 m in length. A cylindrical neck with dimensions 0.018 m in diameter and 0.01 m in length was fitted on one face of the cavity. The microphone-compensated actuator system consisted of a half inch B&K type 4155 microphone sealed through the wall of the cavity, a D-Space Model #1102 floating point, digital signal processor (DSP) 49 used to implement the controller, and a 6 inch dual voice coil speaker. The actuator was used in the uncompensated configuration (Chapter 4) because the compensated configuration introduced excessive noise which corrupted the experimental measurement of the reflection coefficient. A DSP sampling rate of 5 kHz was used for all experiments. The reflection coefficient 5R, was measured using a transfer function technique for determining the acoustic characteristics of duct systems (ASTM Designation: E 1050 —90; Blaser and Chung, 1978; Chung and Blaser, 1980) The transfer function H12, between the two pressure signals P1 and [’2 was measured using a Hewlett Packard 35660A dynamic signal analyzer. This data was then used to compute 9? as 9? _ H12 ' H12r _ (3.26) H121—H12 where H12, and H12, are the transfer functions (pure time delays) associated with the right- and left-running pressure waves and can be expressed as Hm = e‘”6 (3.27) H121: e+iks (3.28) where sis the distance between the two microphones, and k is the wave number, (m/c). White noise was injected into the duct by the disturbance speaker and H12 was measured using 100 time averages. A digital computer was then used to compute 9? using (3.26), (3.27) and (3.28). The first experiment examined the reflection coefficient of the open duct end. The SHR was removed from the duct and the two microphones were placed 1 and 2 inches from the open end. Figure 10 shows SR vs. frequency with the top graph showing linear magnitude, and the bottom showing phase in degrees. These results show that the 50 reflection coefficient of the duct open end agrees well with (3.24). The magnitude is a constant = 1 and the phase increases with frequency at a rate that corresponds to the pure time delay between the microphone, the duct end and back. These results agree with the theory of Levine and Schwinger (1946). 1-5 F ! 1 f F (D 1 ..J e—T A A. A W: ‘O - . 3 . .9 I c . O) I (U T 20.5,.“ .................................................................................... _l 0 l l l l l 50 100 150 200 250 300 Frequency (Hz) 180 r l f T r -185 .. ...... t ....................................................................................... . A490 _ ....................................................................................... _. 8’ 3.195 ... .............................................................................................. _. 0 8 .200 ,_ .............................................................................................. _ .C a 205 t. ....................................................................................... .. 210 r. ......................................................................................... .. 215 l I l l I 50 100 150 200 250 300 Frequency (Hz) Figure 3.10. Reflection coefficient of open end duct with SHR removed showing the duct end can be modeled as a purely reflective boundary Next the SHR was inserted in the duct at x = 20 inches, the controller was turned on and SR was measured for various controller settings. Graphs of 9i with four different controller settings (Table 3.2) are shown in Figure 3.11. The top graph shows the magnitude of SR vs. frequency and the bottom shows phase vs. frequency. These results show additional dynamics superimposed over a response similar to Figure 3.10, i.e., the SHR adds to the reflection in a narrow frequency band in addition to the reflection of the open end duct. For example, in Figure 3.11.D a dip in the magnitude and a decrease then 51 increase in phase is observed around 240 Hz. The frequency at which this dip occurs changes for each controller setting, demonstrating the ability of the SHR to tune the system and move the frequency of the maximum reflection. The overall trends of the graphs in Figure 3.11 agree with the theoretical graphs of Figure 3.8 verifying that the model agrees with the experimental results. It should be noted that the controller gains for the theory and experiment are not equivalent. This is not surprising since the theory models the actuator as a pure gain with no dynamics and the actuator used in the experiment exhibits significant dynamics. In this article, the controller and actuator models are kept simple intentionally, to focus the attention on the acoustics of the system. A thorough examination of the controller and actuator dynamics is given in Chapter 5. 52 .0 on 8 3 0.6 .Q’ E g o 4 2 . 0.2 . o l l I l l 50 100 150 200 250 300 Frequency (Hz) 150 § 3 E O . B 100 f 8 3 cu . .c . n. : 50 ._ ......... : ......................................................................................... ..l o I l I l l 50 100 150 200 250 300 Frequency (Hz) Figure 3.11.A. Experimental plot of the reflection coefficient upstream of the SHR with the controller in closed-loop to change the frequency of the reflection added to the duct by the device R, at 150 Hz 53 0.8 3 30.6 .9 C 304 2 . 0.2 0 l l I l l 50 100 150 200 250 300 Frequency (Hz) l l l I T 150.. .................................................................................... .. ’6‘: 0 3100... ........................................... f ..................................................... _. 3 : (5 J: I 0- : 50... ..................................................... _ 0 1 l I L l 50 100 150 200 250 300 Frequency (Hz) Figure 3.11.B. Experimental plot of the reflection coefficient upstream of the SHR with the controller in closed-loop to change the frequency of the reflection added to the duct by the device R1 at 165 Hz 54 53 m 8 E 3 0.6 : .9 I c . so. 5 2 ' t 0.2 I O I I I I I 50 100 150 200 250 300 Frequency (Hz) 150 6 O 3 100 3 (U S o. 50 ,_ ........................................................................................... —1 o I I I I I 50 100 150 200 250 300 Frequency (Hz) Figure 3.11.C. Experimental plot of the reflection coefficient upstream of the SHR with the controller in closed-loop to change the frequency of the reflection added to the duct by the device R1 at 210 Hz 55 1.5 r 7 l l I Magnigude o I I I I L 50 1 00 1 50 200 250 300 Phase (deg) 0 50 100 150 200 250 300 Frequency (Hz) Figure 3.11.D. Experimental plot of the reflection coefficient upstream of the SHR with the controller in closed-loop to change the frequency of the reflection added to the duct by the device R1 at 240 Hz Table 3.2 Controllegains used to generate Figure 3.11 Figure Minl T | Kp K1 Frequency Q12) 11.A 150 2245 -1.0 11.B 165 2430 -0.3 11.C 210 1220 1.0 11.D 240 2520 2.5 While the relation for the transmission coefficient (3.15) does not strictly hold for the reflective duct end, it is instructive to consider the transmission coefficient for these results. Figure 3.12 shows a plot of the magnitude of T, computed from T = 1 + 9?, where SR is defined by the data in Figure 3.11.D. The minimum IT] occurs at 240 Hz. 56 Note that this does not correspond to the frequency of the minimum in |9i|; both the magnitude and phase of SR must be considered to determine the minimum IT]. 1.5 f T 1 l l l 1 .. .............................................................................................. _. a) 1: 3 .9 C or w 2 0.5r ............. 'i .............. .: .............. 3 ............... ............... ...... .. 0 I i i i i r O 50 100 150 200 250 300 Frequency (Hz) Figure 3.12. Plot of transmission coefficient for data in Figure 3.11.D Another view of the relationship between 9i and T is made by plotting a parametric polar plot of 98 vs. frequency (Figure 3.13). At low frequencies, 9? is real and equal to —1. As frequency increases the angle increases. However, near 240 Hz a loop occurs and SR approaches a negative real number indicating more reflection. At each point on the graph a vector can be drawn to represent 9i(a)), and an associated vector T(a)) by subtracting 1- 58(0)). A vector diagram (Figure 3.12) locates the magnitude (and frequency) for the minimum IT]. For this experiment the minimum l7] was 0.45 and the 57 associated 9i was 0.6 which occurred at 240 Hz. Similar analysis of the other data in Figure 3.11 results in the minimum l7] located at 150, 165, 210 and 240 Hz for the four different controller settings. 90 150 RI éo-é;.1.60i°°t.. ' 180 . ..... 1 Figure 3.13. Parametric polar plot of reflection coefficient magnitude and phase vs. frequency with controller turned on showing vectors for minimum transmission coefficient, T and associated 9i An additional experiment was performed to demonstrate the application of the SHR in an acoustic duct system (ASTM Designation: E 477 -96). The SHR was connected to a 3/4 inch diameter, 20 inch long duct (Figure 3.14). A separate audio speaker was mounted on one end of the duct to inject noise and the other end was left open. A pure tone of 185 Hz frequency was injected into the duct and the sound level P3 was measured near the duct end with a 1/2 inch B&K microphone. The gains K P and K I were set to tune the SHR to 185 Hz and the time response of the microphone signal was recorded as the controller was activated. Figure 3.15 shows P3 vs. time in open-loop for 0.055 seconds; then the controller was activated. There was a short transient then the P3 signal was reduced by a factor of approximately 3 (10 dB). Similar results were obtained 58 with disturbance tones with frequencies ranging from 180 to 300 Hz. This demonstrates the ability of the device to create a reflective region in the duct, which increases the reflected power and reduces the transmitted power. DSP - _ Microphones dlsturbance source, SHR speaker compensation and controller Oscilloscope P2 P3 E] U Disturbance 10 in I _ 3/4 inch 20 1n speaker diameter acoustic duct Figure 3.14. Schematic of SHR applied to acoustic duct Iv I l I I l I F T l 8 Controller off 3 1 Controller on 6 _ 4 4 2 ....... I (U D. 2 3 o .— 22 2 O. .2 — _ .4 - - -6 — _ -8 — _ 40 i . i r i l i i 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.00 0.09 0.1 Tlme see Figure 3.15. Time response of pressure at duct end with pure tone disturbance as controller is activated showing 10 dB noise reduction 59 CONCLUSIONS In this article a controllable resonator was presented which can be tuned to move the resonant frequency and change the damping characteristics via a closed-loop controller. It was shown that by attaching the resonator as a side branch in a duct, the resonator impedance creates a narrow frequency band pressure release region in the duct. This increases the sound reflection and reduces the sound transmission through the duct when the resonator is tuned to match a disturbance noise with a narrow band frequency. Although the duct was modeled with an open end to compare the experimental and model results, the conclusions are valid regardless of the duct termination. It was shown that the optimum resonator impedance to reflect sound is a large, positive real impedance. Theoretically, an infinite impedance will result in all of the incident sound reflected and none transmitted. Experimental measurements of the reflection coefficient in a duct were made with the closed-loop control system and compared with theory. Finally, experimental results demonstrated that the closed-loop resonator provided as much as 10 dB of noise reduction to a disturbance tone in a duct. 60 Chapter 4 A Comparison of Acoustic Actuators for the Semi—Active Helmholtz Resonator This paper examines the implementation of an acoustic actuator in a tunable semi- active Helmholtz resonator (SHR). The advantages and limitations of a compensated speaker are studied and compared with an uncompensated speaker. It is shown that the speaker compensator reduces the effects of internal dynamics and boosts the actuator control authority but inadvertently introduces noise into the system. Alternatively, the uncompensated speaker has weaker control authority, but does not introduce noise into the system. These results suggest that both compensated and uncompensated actuators can be useful for different applications. The semi-active Helmholtz resonator (Temkin, 1981) is an acoustic device with behavior that can be used to selectively quiet narrow band noise in acoustic systems. It consists of a static Helmholtz resonator with a sensor, controller, and actuator added to the interior of the resonator cavity (Figure 4.1). The nominal resonant frequency and damping of the device was determined by the dimensions of the resonator neck and cavity but can be modified by the closed loop feedback system. When driven by a pressure from a primary acoustic system, such as an acoustic duct, the resonator responds with a large magnitude volume velocity through the resonator neck which is in phase with the pressure. This creates a “pressure release” boundary condition, which inverts and reflects the incident pressure wave back up the duct, thus reducing the transmitted pressure wave and reducing the transmitted sound (Pierce, 1981). 61 Resonator Primary Resonator cavity acoustic neck system _ j) :1 Actuator ‘__. I ._~ Controller l—— Sensor Figure 4.1. Schematic of a semi-active Helmholtz resonator connected to a primary acoustic system The actuator is a critical component in the implementation of the SHR, and strongly affects the closed-loop performance of the system. Internal actuator dynamics will affect the closed-loop response of the system since the actuator and acoustic resonator become tightly coupled by the pressure interaction between the two systems. Electromechanical audio speakers are often used for acoustic actuators because of their low cost and commercial availability. However, audio speakers are not ideal actuators. They typically have a resonance frequency between 50-150 Hz (for bass speakers) resulting in large magnitude and phase variation in their operating frequency range. F urthermore, the speaker velocity response is strongly affected by the pressure interaction with the acoustic system. A resonance in the acoustic system will impede the speaker velocity, resulting in weak control authority. The closed-100p feedback control design for the SHR is also affected by the actuator performance. A simple proportional-integral (PI) controller is used in the SHR, and an analytical solution can be found that maps the controller gains to the acoustic resonant frequency and damping ratio (Chapter 2). However this assumes that the 62 actuator has no dynamics, and the transfer function is a pure gain. Actuator dynamics complicate this mapping, resulting in the need of a higher order controller. This sensitivity of the system response and the desire to avoid using a higher order controller motivates the use of local feedback compensation of the actuator. This technique adds a local feedback loop to the actuator (Figure 4.2), which drives the actuator output to the input voltage making the response approach a pure gain of one, as the loop gain is increased. The goal of the compensator is to boost the control authority. It also simplifies the controller design, since actuator response approaches the ideal response. Compensation for audio speakers has been proposed in many forms (Harvvood, 1974; Klaassen and de Koning, 1968; Holdaway, 1963; Tanner, 1951). Birdsong and Radcliffe (1999) proposed a technique using a dual voice coil speaker with local feedback compensation that resulted in a compensated acoustic actuator with minimal magnitude and phase error below 400 Hz. This design compensated the internal speaker dynamics and the pressure interaction with the acoustic system. The compensated acoustic actuator was chosen as the actuator for the SHR because of these strengths. + . »? Controller Actuator Plant F» + — . Local actuator feedback compensation loop Closed-loop feedback Figure 4.2. Local actuator feedback compensation used to boost actuator authority, minimize actuator dynamics, and simplify controller design 63 This paper discusses three major topics: analytical model development, coupled system simulations, and experimental results. In the first section, separate analytical models for each component are presented, including the acoustic resonator, controller, speaker, and compensator. In the second section, the models are coupled and various configurations are examined. First, the closed-loop resonator model with an ideal actuator is presented. Then, the ideal actuator is replaced by the uncompensated speaker model. Next, the speaker compensator is added, and the compensated speaker model is applied to the closed-loop control system model. Finally the uncompensated speaker model is applied to the closed-loop control system demonstrating the advantages and disadvantages of the compensation technique. The last section presents experimental results which are compared with the analytical model and which demonstrate the effectiveness of the actuator implementation in the SHR. ANALYTICAL MODEL DEVELOPMENT The closed-loop compensated SHR consists of several interconnected components: an acoustic resonator, a feedback controller, an audio speaker, and a compensator. Analytical models for each have been developed in other works, and will be presented here briefly. The reader is referred to the references for complete descriptions of the components. These component models will be assembled into coupled system models in the next section. Resonator The central component of the SHR is a Helmholtz resonator with one surface of the cavity replaced by a moving surface (Figure 4.3). The system can be represented by linear time invariant state equations (Chapter 2) Q1 = ‘15“ 7:11—0[Q1] i OFT] [V] [I 0 v + 01Q2 (4.1) Q1 _ 1 0 Q1 [cl—i0 allvl <42) where the states are Q, the volumetric flow rate or “volume velocity” from the neck (m3/s) and V, the sum of the volumes introduced through the neck and the inner surface of the cavity (m3). The inputs are P1 , the pressure at the neck inlet to the cavity (N/mz), and Q, the volume velocity from the movable surface in the cavity (m3/s). The outputs are Q] and 1’2, the pressure in the cavity (N/mz). The other parameters are Ra, the acoustic loss that represents viscous and radiation losses (N s/m5 ), Ia, the acoustic inertia of the mass of air in the resonator neck (N szlms), and Ca, the acoustic compliance of the cushion of air in the resonator cavity (ms/N). With the movable surface held fixed, the system is a second order oscillator (Temkin, 1981) with resonant frequency and damping given by (0,, = inaIa (4.3) 65 (4.4) 92 Mass of . air in resonator Q1 Movrng neck \ surface > P1 —'=\\\\ (a a 2 la Figure 4.3. Schematic diagram of SHR showing inertia effect in neck and the movable surface in the cavity interior Controller A proportional-integral controller can be used to generate an acoustic impedance between Q2 and P2, on the moving inner surface of the SHR cavity. This creates a closed-loop, positive feedback configuration (Figure 4.4). Resonator model —> P1 Ql—p Controller r—p Q 2 P 2 Figure 4.4. Closed-loop positive feedback SHR system block diagram with disturbance through Pl A PI controller can be modeled by the transfer function, G(s)=—Q—2—=K,p+fl (4.5) P s 2 66 where K P, and K I are the proportional and integral gains respectively. Speaker The actuator is modeled as a dual voice coil speaker (Figure 4.5). The dual voice coil speaker has certain characteristics that make it ideal foruse as an acoustic actuator. It has 2 independent wire coils intertwined and wrapped around a bobbin which is allowed to slide over a permanent magnet. The state equations for a dual voice coil speaker can be represented by the linear time invariant state equations (Radcliffe and Gogate, 1996; Birdsong and Radcliffe, 1999) —R, —1 (2154 Q2 Is Cs]: 1c]: Q2 0 .95' ep (46) — q2 = I 0 0 ‘12 ‘I' O 4 dt A i —R,,,- A 1. P2 Sd c l 0 where the states are the volume velocity and volume displacement from the speaker Q2, and q2, and the electromagnetic flux in the speaker coil 11. The inputs are the primary coil voltage, e,,, and pressure on the speaker face, P2. The output equation is given by —M bl[ we] 0 ——C—(RC+R,,,) M. e,, ’c ’3 1 IQ: 7; 0 e (4.7) [p = O O ]— q2 + 0 0 Pp] Q2 1 0 5 _A o 0 2 where the outputs are the voltage in the secondary coil, e)”, the current in the primary coil, in and Q2. The parameters in (4.6) and (4.7) are the speaker face area, Sd, speaker inertia, 1,, speaker compliance, Cs, speaker friction, Rs, speaker coil resistance, RC, 67 speaker coil inductance, 1,, speaker coil mutual inductance, M 6, speaker electromechanical coupling factor, bl, and the primary coil current sensing resistance, Rm. Primary Coil Bobbin Terminal Permanent Magnet \ '1 ,_ Air Cone Secondary Coil Teminal Figure 4.5. Dual voice-coil speaker diagram Compensator The velocity frequency response of the speaker can be improved with local feedback compensation. The volume velocity of the speaker, Q2, is strongly affected by the dynamics of the speaker and the pressure input, Pz. These effects will combine to create magnitude and phase variations in the primary coil voltage to speaker velocity response, Q / e . One method of reducing these unwanted effects is to apply a 2 p proportional feedback controller (Figure 4.6) resulting in the closed system, Vspkr(s) = Kamstpkr(s) Vd(s) 1+ KWGspk,(s)H(s) Two) = (4.8) where Vspk, is the speaker velocity, Va is the desired velocity, Gspk, is the transfer function that relates the input voltage to speaker velocity, Kamp is an amplifier gain, and H(s) is a velocity sensor. If the sensor transfer function is a real constant, k, over the controller 68 bandwidth, then the closed loop transfer fimction, Twas), will approach a constant, 1/k with zero phase. This compensation forces the speaker cone velocity vspk, to accurately follow the desired velocity input. The speaker volume velocity, Q, is equal to the speaker area, Sd, multiplied by the speaker velocity, vsph. The result is independent of the speaker dynamics and the input pressure provided that the sensor has a constant transfer function over the controller bandwidth. V Amplifier L, Speaker I > V des + Kamp Gspkr I spkr Velocity Sensor H Figure 4.6. Block diagram of speaker and compensator ASK amp is increased, the transfer function approaches 1/H(s) and the magnitude and phase variation approach zero This approach requires that the velocity of the speaker face be measured. A speaker velocity sensor is therefore needed which accurately predicts the speaker velocity in the presence of speaker and plant dynamics. The relation between the speaker velocity and the two other measurable outputs (the secondary coil voltage, em, and the primary coil current, i,,) can be computed from (4.6) and (4.7) in terms of ebs and ip yielding ”SP/«(3) = Hbsebs(s) — H p (s)ip(s) (4.9) where Hbs = 1/ bl and Hp(s)= sM/ bl . 69 The secondary coil voltage, ebs, can be measured directly from the speaker coil. The primary coil current, ip, can be determined from the voltage across a resistor, Rm, placed in series with the primary coil, while Hbs is a pure gain (1/bl). The mathematically improper, differentiating transfer function, H p, cannot be strictly realized exactly, but an approximation . M P13 H =-—C -———- 4.10 ”(5) bl [S+p1) ( ) can be used, where p, is a pole location selected such that HAS) approximates H p(s) over the controller bandwidth. Feedback compensation can now be implemented using the signal from the velocity sensor to compute the error between the desired velocity and the sensor velocity and a proportional controller to drive the speaker velocity to the desired velocity. COUPLED SYSTEM SIMULATION Assembly of the component models allows the investigation of the coupled system dynamics using numerical simulation. The simulation was performed using Matlab and Simulink software on a digital computer. This sofiware allows state space and transfer function models to be interconnected in a single model to compute coupled system, frequency response graphs. The numerical values for the acoustic resonator and speaker parameters used in the simulation are given in Table 4.1. These values were measured from the physical devices used in the experimental results section and have been shown to be accurate (Birdsong and Radcliffe, 1999). The speaker was a 6 inch 70 dual voice coil speaker and the resonator cavity was constructed from PVC pipe fittings with dimensions given in Table 4.1 Table 4.1. SHR Model Parameter Values bl 2.45 N/A RL 5.7 ohm c, 343 m/s Rm 10 ohm C, 0.000260 m/N RL 3.745 N sec/m 1,. 0.002 H S 0.000254 m2 I, 0.0076 Kg 5,, 0.0133 m2 L 0.010 m V 0.002 m3 M, 0.001 H p, 1.18 Kg/m3 SW 4 mv/ PA Resonator and Controller with Ideal Actuator The first coupled system model that will be considered is the acoustic resonator with a closed-loop feedback controller and an ideal actuator (Figure 4.7). This is a simple model which assumes that the actuator is ideal, i.e., it has a transfer function that is a pure gain of one. The cavity pressure, P2, is fed to the controller and the controller output is fed into the resonator cavity volume velocity input, Q2. The system can either be disturbed by the input, P1 , or by the disturbance signal, D1, which is also added to the controller output. Resonator model Ideal actuator . P1 Q1 . model Controller I Figure 4.7. Block diagram of simple coupled system model including acoustic resonator, closed-loop feedback controller, and ideal actuator model 71 The eigenstructure of the system can be modified with the positive feedback controller. With the controller gains set to zero (open-loop), the system resonates at the nominal resonant frequency and damping (4.3) and (4.4). The numeric values for the resonator model nominal, resonant frequency and damping are fl, = 205 Hz and g = 0.025. By varying the controller gains K p and K ,, the resonant frequency and damping can be varied. Figure 4.8 shows the Pl/ Q, transfer fimction for this model for various values of K P and K, (Table 4.2). Magnitude (dB) Phase (deg) -70- ............................. A r : . . E B 3 C -80' .......................................... j ................. . ....... .90.. ....... . .................... -100- .............................. .................................................... 410,. ........................................... .................................................. _120 I I I I I I I I 80 90 100 110 120 130 140 150 160 150.. .......... ‘ ............ ............ ' ........... ....... ....... 100 5 50 Q o 5 -50 f _100 I I I I I I I I 80 90 100 110 120 130 140 150 160 Frequency (Hz) Figure 4.8. Frequency response simulation of resonator and closed-loop feedback controller with ideal actuator showing that the resonant frequency and damping can be changed by varying the controller gains 72 Table 4.2. Controller gains used to create Figure 4.8 Graph K P Gain K, Gain Resonant Percent Frequency Damping (HZ) A 0.99 -100 112 10 B 0.99 0 130 10 C 0.99 100 145 10 The feedback controller makes the system response appear identical to the response of three different passive Helmholtz resonators with different tuned frequencies. In each curve the magnitude attains a maximum at the same frequency that the phase crosses zero. This is identical to the response of a passive resonator. The important feature here is that the change in frequencies was created by electronic tuning, not by changing the physical dimensions of the resonator. This system, with the ideal actuator model, can be used to compute an analytical solution that maps the PI controller gains, K, and K p, to the closed-loop frequency response values of 00,, and g . This is the basis for an adaptive control algorithm that changes the gains on line to tune the system to track a disturbance signal with slow time varying frequency (Chapter 5). However, without the ideal actuator assumption, this mapping is not valid, and a different, more complicated controller design is required. Resonator and Speaker The ideal actuator will be replaced by the dual voice coil speaker model and the coupled system will be simulated with the controller removed. The speaker must provide a controlled Q2 to the resonator for the response to be similar to the previous model. Figure 4.9 shows the speaker model coupled with the resonator model. The speaker output, Q2, is connected to the resonator input, and the resonator output, P2, is connected 73 to the speaker input. The speaker is driven by a voltage applied to the primary coil, and the secondary coil is lefi in open circuit. Resonator model —> P Q __.> Speaker model 1 1 Q2L> Q2 P2 P2? Figure 4.9. Block diagram of resonator and speaker models showing coupling through Q2 and P2 The frequency response of the output, Q2, to input, e p, reveals that the speaker does not behave as an ideal actuator with pure gain of one. The open-loop frequency response of the Q2/ e p transfer function was simulated (Figure 4.10). The response shows two resonance peaks, one at 80 Hz and one at 290 Hz. There is 20 dB of magnitude variation and 180 degrees of phase variation between 50 and 300 Hz. Clearly, the speaker does not respond as an ideal actuator. This motivates the application of the local feedback compensation. 74 Magnitude (dB) 0 50 100 150 200 250 300 350 Phase(deg) O _1 00 i . i . r i i 0 50 100 150 200 250 300 350 Frequency (Hz) Figure 4.10. Frequency response simulation of the Q2/ ep transfer function for the resonator and speaker model with the controller removed Resonator, Speaker, and Compensator A local feedback compensator model is added to the speaker model (Figure 4.11) to improve Qz/ep response. The velocity estimator generates a signal, veg, which is subtracted from the input D1. The error is then amplified by a proportional gain, Kamp, and the resulting signal becomes the input to the speaker, e The local feedback p. compensation closes the loop around the speaker velocity, driving the actual velocity to the desired velocity. The compensated speaker model can be represented in subsequent models with a single block with inputs, D1, and P2, and output, Q2, as indicated by the dashed box. 75 R o t d l Compensated speaker model es na or mo e I ___________ —> P Speaker model I 1 Q1 b —:——> Q2 P2 b I I $ l I : Controller 1% Figure 4.11. Block diagram of resonator and speaker with local feedback compensation, and controller removed The compensator improves the low frequency response somewhat, but degrades the high frequency response. The effect of the compensator on the speaker can be illustrated by examining the frequency response of the Qz/DI transfer function. Figure 4.12 shows the response with Kamp = 30, 50, 100, and 200. As Kamp is increased, the low frequency response is improved. The magnitude and phase variation is reduced from 15 dB and 125 degrees between 20 and 100 Hz without the compensation (Figure 4.10) to 7 dB and 70 degrees with the compensation and Kamp = 200 (Figure 4.12). However, above 100 Hz the response is worse than with the uncompensated actuator. The compensator adds phase above 100 Hz increasing the maximum phase from 75 degrees at 150 Hz for the uncompensated actuator to 150 degrees for the compensated actuator with Kamp = 200. The compensator is not effective in driving the speaker velocity to the 76 desired velocity at the acoustic resonant frequency (205 Hz). The increases in K W have only marginal effect at increasing the magnitude at 205 Hz. These results indicate that the compensator does not provide adequate speaker performance to make the speaker model approach the ideal actuator model. This does not imply that the actuator is not useful in the confioller. It only implies that the speaker dynamics must be considered in the controller design as discussed in the introduction. A more detailed discussion of the controller design is given in Chapter 2. Magnitude (dB) Pha3e[deg) -150 ' ' I i l ' 0 50 100 150 200 250 300 350 Frequency (Hz) Figure 4.12. Frequency response simulation of the Qz/D, transfer function with the resonator and speaker with local feedback compensation added and controller removed, arrows indicate direction of increasing Kamp = 30, 50, 100, 200 77 Resonator, Speaker, Compensator and Controller. Finally, with the compensated speaker model assembled, the closed loop control of the resonator can be modeled. Figure 4.13 shows the block diagram of the resonator and compensated speaker and controller with a disturbance D2. The compensated speaker block contains the blocks in the dashed box in Figure 4.11. Resonator model Compensated —-> P] Q] —> W1 Q 2—> Q 2 P2 —-—>— Dz D1 + + P 2 < Controller ‘— Figure 4.13. Block diagram of resonator, compensated speaker, and feedback controller with disturbance D2 The closed-loop, compensated system response can now be simulated to verify that the acoustic resonance of the system can be modified by the feedback controller. A model based, empirical controller design (Chapter 2) was used to find gains that produced the desired response. Figure 4.14 shows the frequency response of the Q,/ P, transfer function for four cases with the controller gains given in Table 4.3. These results show that the compensated actuator successfully implements the closed-loop control. The controller moves the frequency of the peak and zero phase to 106, 123, and 139 Hz. Note that the maximum amplitude of each resonant peak decreases with frequency. Also, the magnitude of graph C falls below the open-loop graph A at 45 Hz. This shows that if the 78 SHR is mistune then the closed-loop response can be worse than the open-loop response. These results show that with proper tuning, the compensated actuator and SHR behave as a tunable acoustic resonator. MagnfludemB) i i i i r 0 50 100 150 200 250 300 350 Phase[deg) r l l i l 50 100 150 200 250 300 350 Frequency (Hz) Figure 4.14. Frequency response of the Q, / P, transfer function with the resonator, compensated speaker and feedback controller for four cases with gains shown in Table 4.3 Another transfer function Pz/Dl, is of interest in this system because it is used to compare the model and experimental results. Although the Q,/ P, transfer function is the key to the effectiveness of the device for noise control, it is difficult to measure experimentally. The volume velocity flow, Q,, is a zero mean oscillating air velocity. A laser velocity anemometer can be used to make such a measurement, but this is an expensive and complex device. Instead, the Pz/D, transfer function can be examined to 79 observe the resonant frequency and damping. Figure 4.15 shows the closed-loop 1’2/D1 transfer function with the gains in Table 4.3. The model based empirical controller design finds gains, K p and K ,, that produce resonant peaks with constant magnitude in the Pz/DI transfer function. Note this results in resonant peaks that decrease in amplitude with increasing frequency in the Q,/P, transfer function (Figure 4.14). As before, the magnitude attains a peak and the phase crosses zero at the resonant frequency. Magnitude (dB) a. O (a) O 20 . i i l i i i 0 50 100 150 200 250 300 350 200 100 B d.) E o 0 8 .C D. -100 200 l i I I l 50 100 150 200 250 300 350 Frequency (Hz) Figure 4.15. Frequency response of the compensated 1’2/D1 transfer function: A: open- loop, B, C, and D closed-loop with gains selected to increase resonant peak and move resonant frequency 80 Table 4.3. Compensator and Controller gains used in Figures 4.14 and 4.15 Curves A B C D Compensator In or Out In In In In Kamp 30 30 30 30 K P 0 -0.81 -0.90 -0.90 K, 0 -250 O 250 Resonant Frequency (Hz) 102 106 123 139 Percent Damping 50 10 10 10 One undesirable feature of the speaker compensator is that it introduces noise into the actuator output. This is because it uses the voltage from the speaker secondary coil to estimate the secondary coil current. The secondary coil voltage is a low-level signal with a low signal-to-noise ratio. The noise is amplified by the compensator gain Kamp. This can be analyzed by modeling a disturbance D3 input to the secondary coil current. The frequency response of the transfer function for Pz/D3 is shown in Figure 4.16 for the compensator and controller settings in Table 4.3. These results indicate that random noise in the frequency range of 50 — 400 Hz will be injected into the actuator output. Although the signal-to-noise ratio could be increased by increasing the number of windings on the secondary coil, this was not done in this work. 81 Magnitude [dB] 50 100 150 200 250 300 350 400 Figure 4.16. Frequency response of P2 to current sensor disturbance for resonator, compensated speaker and feedback controller model with gains fiom Table 4.3 In summary, these simulations were presented to show that the compensated speaker is effective in implementing the control of the SHR. It was shown that the uncompensated actuator performance had excessive magnitude and phase variations and did not have unity gain. These actuator dynamics complicate the closed-loop controller design because the analytical mapping of the controller gains to the acoustic resonant frequency and damping assumes the actuator has unit gain. The speaker compensation was added as an attempt to minimize the actuator dynamics and simplify the controller design. It was shown that the compensation improved the speaker response below 100 Hz, but degraded the response above 100 Hz and that the coupled resonator and compensated speaker response did not converge to the resonator and ideal actuator model. The inability to remove the actuator dynamics from the system motivated the development of a model based, empirical controller design which was used to find controller gains that successfully re-tuned the acoustic system. These results bring into question the value of the speaker compensation and whether the compensation is worth the expense of the added complexity and introduction of noise. 82 Resonator, Speaker, and Controller (No Compensation) A final model will be examined which includes the resonator, uncompensated speaker, and feedback controller, to compare the compensated system with the uncompensated system. The block diagram for the speaker compensation, Figure 4.11 includes a switch that removes the local feedback compensation from the loop. The model was assembled with the same components as the previous model, but with the local feedback compensation removed. The frequency response of the P,/ Q, transfer function for the resonator, uncompensated speaker, and feedback controller was then simulated and shown in Figure 4.17. It was found that the PI controller was not able to amplify and move the low frequency resonance (80 Hz). Instead the high frequency resonance at 290 Hz was amplified and moved by the application of the controller. Note that a peak in magnitude is attained and a zero phase occurs at the different resonant frequency. This verifies that the uncompensated actuator can be used in the SHR system. 83 Magnitude [dB] Phase[deg) 50 100 150 200 250 300 350 Frequency (Hz) Figure 4.17. Frequency response simulation of the Q,/ P, transfer function with the resonator, uncompensated speaker, and feedback controller coupled model with controller gains from Table 4.4 The frequency response of the Pz/D, transfer function was also simulated, shown in Figure 4.18 for comparison with experimental results. 84 Magnitude (dB) Phase[deg] 0 . 50 100 150 200 250 300 350 Frequency (Hz) Figure 4.18. Frequency response simulation for the Pz/Dl transfer function with the resonator, uncompensated speaker, and feedback controller coupled model with controller gains from Table 4.4 Table 4.4. Controller gains used in Figures 4.17 and 4.18 Curves A B 0 -10 10 0 298 322 Percent 50 10 10 The controller gains were selected to increase and decrease the resonance approximately 10% from the nominal value while maintaining a damping ratio of 10%. Note considerably different gains are needed to obtain these results as compared with the compensated system. The integral gain, K ,, is much larger than before, 1,600 to 12,800 compared to —100 to 100 for the uncompensated system. There are several explanations for this. A value of Kama = 1 was used in the uncompensated system compared with 85 Kamp = 30 in the compensated system. Also, the integral of the pressure signal decreases with frequency, requiring a gain three times as large to affect the resonance at 300 Hz as one at 100 Hz. Note, too, that the trends of the gains are very different from those of the uncompensated system. A more complete discussion of the mapping of the controller gains to the resonant frequency and damping is beyond the scope of this article and is given in Chapter 5. Finally, note that the peak magnitudes of graphs B, C and D in Figure 4.17 are approximately 5 dB less than those with the compensated speaker model, Figure 4.14. Increasing the peak magnitude further would require reducing the system damping, which would lead to reducing the stability margin of the system. These results indicate that the SHR with the uncompensated actuator is capable of producing an electronically tuned acoustic resonator. No significant noise is introduced into the system because the compensator is not present. Also the nominal resonant frequency is increased significantly to 300 Hz compared to the compensated system (130 Hz). However, the maximum magnitude of the uncompensated system is less than that of the compensated system. A comparison of the compensated and uncompensated actuators suggest that each have use for different applications. The larger magnitude response of the compensated actuator suggests that it will be more effective in controlling noise in a narrow frequency band. That is, the compensated SHR will reflect more narrow frequency sound in a duct than the uncompensated SHR. It, therefore, may be more effective when the objective is to minimize narrow band pressure oscillations. However, the compensated SHR adds broadband noise to the system, thus degrading some of the noise reduction that is sought. 86 The uncompensated SHR does not reflect as much noise in a duct, but does not add as much noise to the system. It, therefore, may be more effective when the objective is to improve overall sound quality. 87 EXPERIMENTAL VALIDATION An experimental apparatus was constructed to validate the theoretical model and to demonstrate the noise reduction capability of the device. A Helmholtz resonator and actuator are presented. The compensated, then uncompensated actuator was implemented, and the results are compared with theory. The experimental SHR setup consisted of two components: a Hehnholtz resonator cavity and a microphone-compensated actuator system. Figure 4.19 shows a photograph of the SHR connected to an acoustic duct and Figure 4.20 shows a schematic of the setup. A cylindrical Helmholtz resonator cavity was constructed from PVC with dimensions 0.075 m in diameter and 0.15 m in length. A cylindrical neck with dimensions 0.018 m in diameter and 0.01 m in length was fitted on one face of the cavity. The microphone- compensated actuator system consisted of a half-inch B&K type 4155 microphone sealed through the wall of the cavity. A D-Space Model #1102 floating point digital signal processor (DSP) was used to implement the speaker compensation, and an acoustic actuator was sealed in the opposite face of the cavity. A DSP sampling rate of 5 kHz was used for all experiments. The actuator consisted of a 6-inch dual voice coil speaker with local compensation (Birdsong and Radcliffe, 1999) to improve the speaker velocity response. 88 Microphones Acoustic duct ijll a“, Distceurban . speaker . l. \ Speaker enclosure Figure 4.19. Photograph of SHR connected to an acoustic duct with a second audio speaker to inject noise D 1 V Digital signal processor I I l l l - El, P 2 ebs 1[7 Amplifier Dynamic signal I analyzer e N P \ _ Speaker neck l‘ enclosure 1‘ \\ /I-\ cavity Microphone Dual voice coil speaker Figure 4.20. Schematic diagram of experimental SHR apparatus In all phases of the system design, the device was separated from any primary acoustic system. This was not done arbitrarily for convenience, but because traditionally, mechanical and acoustic resonators are designed independently fiom the primary system. The usefulness of the device would be limited if the resonator response was dependent on the structure of the primary system. Fortunately, this is not the case; resonators can be 89 designed with a tuning frequency and then applied to any suitable primary system. In the absence of a disturbance pressure, P, , the system was disturbed electrically by the signal D1 injected via the actuator input voltage (Figure 4.20). Although it would be useful to measure the Q,/P, transfer function directly since it is key in the interaction between the resonator and the primary acoustic system, this was not done. The quantity Q, is difficult to measure experimentally since it is a zero mean, oscillating air velocity. A laser velocity anemometer is a device that can be used for such measurements, however it was not used here because it is extremely costly, and experimentally complex. Consequently, the transfer function Q, / P, is difficult to measure directly. Alternatively, the quantity P2 and therefore, the transfer function Pz/D, can be measured easily with a microphone. The system can be disturbed by either inputs P, or Q, since the characteristic polynomial which defines the resonant frequencies is the same regardless of the input. Therefore, the model was validated by comparing the model response for Pz/D; with experimental measurements. Compensated Actuator Results A speaker velocity estimator (Birdsong and Radcliffe, 1999; Radcliffe and Gogate, 1996) was created by combining the voltage in the secondary coil with the current in the primary coil. A 10 ohm resistor was placed in series with the primary speaker coil to measure the current (Figure 4.21). The velocity estimate was then used to close the loop on the speaker velocity with a proportional controller (Figure 4.11). A value of K W = 30 was used in all trials. This value for Kamp is somewhat smaller than values used by Birdsong and Radcliffe (1999), where gains of as much as 100 were used. 90 The gain K amp was chosen to increase the robustness of the system, which comes at the expense of performance. Secondary coil Primary coil R c MC R c e b S e p Ic Ic Figure 4.21. Primary coil current sensing circuit Next, the controller‘was added to the system and gains were found to amplify the resonant peak and shift the resonant frequency. The gains were found using a model- based empirical technique (Chapter 5) that produced a response with different resonant frequencies and constant peak amplitude. Figure 4.22 shows the results for 4 experiments, curves labeled A, B, C, and D, which were generated using the controller gains in Table 4.5. These gains reduced the percent damping from 50% with K, = K p = 0 to 5% and shifted the peak from 130 Hz to 100 Hz and 170 Hz. Some discrepancies were found between the experimental and model gains, however the overall trends agreed with the gains in Table 4.3. The reader is refered to Chapter 4 for a more complete discussion of the mapping of the controller gains to the resonant frequency and peak amplitude. 91 0 I I l l I 20 40 60 80 100 120 140 160 180 200 Frequency (Hz) Phase deg i i i n i 1 i i j i "20 4o 60 80 100 120 140 160 180 200 Figure 4.22. Experimental closed-loop frequency response of coupled system with compensated actuator showing that controller amplifies the peak magnitude and moves the resonant frequency Table 4.5. Controller gains used in Figure 4.22 0 —0.34 -0.24 30 30 30 30 1 5 5 The next experiment demonstrates noise reduction in a duct and the introduction of random noise into the system by the actuator compensator. In this experiment the SHR was attached to an acoustic duct and a pure tone of 130 Hz was injected into one end of the duct by a second audio speaker (Figure 4.23). The sound pressure level (SPL) was then recorded at the duct end with the SHR in two configurations: first, with the 92 uncompensated open-loop system, then with the compensated, closed-loop system with the controller gains selected to tune the system to 130 Hz. Figure 4.24 shows both spectra. With the uncompensated, open-loop system, the pure tone appears as a 117 dB spike in the spectrum at 130 Hz (dashed line). Other harmonics at 60 and 180 are attributed to distortion in the disturbance speaker. With the compensated, closed-loop system, the spectrum (solid line), shows the tone at 130 Hz is reduced dramatically to 85 dB, representing a 32 dB noise reduction. Nonetheless, significant background noise is introduced by the closed-loop actuator in a broad band between 60 and 200 Hz. This sound is below 80 dB, but becomes significant since the disturbance has been reduced to 85 dB. The shape of the broadband noise is similar to the Pz/Dz transfer function result predicted by the model in Figure 4.10. It is attributed to random electrical noise introduced in the speaker secondary coil voltage and magnified by the compensation amplifier K amp. The overall sound pressure level was also recorded at the open duct end using a B&K sound level meter. With the uncompensated, open-loop system, the overall sound pressure level was 118 dB, and with the compensated closed-loop system, the overall sound pressure level was 100 dB, representing 18 dB of overall SPL noise reduction. In summary, the narrow band noise at 130 Hz was reduced by 30 dB, but the overall SPL reduction was only 18 dB because of the addition of the broadband noise by the compensated actuator. Similar results were obtained with a disturbance frequency ranging from 100 to 150 Hz. 93 Dynamic signal analyzer Em Pure tone disturbance SHR I: Disturbance Acoustic duct Microphone speaker Figure 4.23. Schematic of experimental setup 12° F I l f ! .' l l .' 4 ' I I l I O en-loo 110,_ ....................................................... V‘— p p 100- ...... ........ ........ ........ ......... ........ ...... ......... 3 ....... _ __ : : : ; r 1 3 Closed-loop m . E . _l 90.. ......................................................... _. Q. 0’ . . : . 80— ...... ........ Z ........ Q ...... .' ...... 70_ ....... ......... ........ ..... 60 i i [i Z '. i i i 0 2o 40 60 120 140 160 180 200 80 100 Frequency (Hz) Figure 4.24. Sound pressure level in acoustic duct with SHR used to reduce pure tone disturbance, dashed line: compensator and controller out of the loop, solid line compensator and controller in the loop with controller gains set to tune SHR to the disturbance tone frequency of 130 Hz Uncompensated Actuator Results The previous experiments were repeated with the actuator in the uncompensated configuration. Figure 4.25 shows the uncompensated closed-loop Pz/D, response Without the duct with gains from Table 4.6. As predicted by the model, the uncompensated actuator does not resonate at the lower frequency. Instead the only resonance occurs at a higher frequency near 240 Hz. The controller was able to amplify 94 the magnitude by 20 dB and shifi the resonant frequency from 240 Hz with gains K P = K, = 0 higher and lower in frequency by 25 Hz. The resonant frequency for K P = K, = 0 is lower than the value predicted by the model, but otherwise, there is good agreement between these results and those derived from the model (Figure 4.12). 20 l I r - r B C D § 3 10 _ .............................................. I ................................... B 8 3 0 - .............................................................................. .9 : : e a: = E _10 ._ .............. If, ....... ' ......... A ........ ‘.\ .............. 50 100 150 200 250 300 35 400 Frequency (Hz) 1 00 I I l T I l \‘m—C‘“ “9 WI B C D o _ ...................... x. . . . . ........... _ 6 . I) U ‘5' -100 _ ...................... . ....................................................... _, 9 : g . _200 _ ....................... ................. A ............ r. ...... ", 50 100 150 200 250 300 350 400 Frequency (Hz) Figure 4.25. Experimental frequency response of closed-loop SHR with uncompensated actuator A C D 1.0 2.0 2520 1740 l 1 l Resonant 250 Percent The last experiment shows that the SHR with the uncompensated actuator reduces noise in a duct without introducing significant random noise into the system. The SHR 95 and duct setup (Figure 4.23) was repeated with the uncompensated actuator and SHR. A pure tone of 185 Hz was injected into the duct end by the second audio speaker. Figure 4.26 shows the SPL spectrum recorded at the duct end with 2 configurations: first open- loop (dashed line), then closed-loop with gains set to tune the system to match the noise frequency (solid line). With the open-loop system, the peak SPL is 107 dB at 185 Hz. With the closed-loop system, the noise level is reduced to 98 dB, representing a 9 dB noise reduction. The background noise level is below 60 dB indicating that the uncompensated actuator does not introduce significant noise to the system. The overall SPL measured with a sound level meter showed identical results (9 dB noise reduction) indicating that in both open and closed-loop settings the noise is dominated by the narrow band tone at 185 Hz. 110 I l 1 i 514—— Open-1001) 100..., ............ . ..... ............. d fl ‘ Closed-loop 90L ................... g ............. .................... g ................... I a : : : : 80p ................... ', ............ .. ... ................... .., Q I I I (n 70 ........... ................... . 60 ......... ,5, ......... ................... _ 100 150 200 250 300 Frequency (Hz) Figure 4.26. Sound pressure level spectrum with pure tone disturbance at 185 Hz with open and closed-loop SHR and uncompensated actuator 96 CONCLUSIONS The actuator is a critical component in the implementation of the SHR. Compensated and uncompensated actuators were presented. Both were shown by analytical model and experimental results to be effective in the SHR. Both uncompensated and compensated actuators introduced significant dynamics into the system, requiring modification of the feedback controller design. However the compensated actuator was found to introduce random noise, degrading the overall SPL noise reduction. The uncompensated actuator did not introduce noise into the system, but could not generate as strong a resonance as the compensated actuator. It also had a higher resonant frequency. These conclusions lead to a criterion for choosing which actuator is optimal for different applications. For applications where a narrow band disturbance must be minimized without concern for the sound quality, the compensated actuator is superior. Alternatively, if sound quality is of concern, then the random noise introduced by the compensation may be objectionable even though the overall SPL is higher. In this case, the uncompensated actuator may be a better choice. 97 CHAPTER 5 Adaptive Control of a Semi-Active Helmholtz Resonator A closed-loop adaptive control strategy is presented in this article for the SHR. Previously, the device was presented in a configuration that could be tuned to reflect sound back to the source at a single command frequency, selected to match the fiequency of the unwanted noise. If this frequency changed, the command frequency could be modified by redesigning the control gains. This involved the operator turning potentiometers to vary the controller gains until the peak in the frequency response of the SHR coincided with the unwanted noise frequency. In this article, control strategies are considered which eliminate the human tuning process and replace it with an automated algorithm. The objective is to develop a self-tuning acoustic resonator that will track a pure tone, noise source with a slow time-varying frequency. A slow time-varying signal will consist of a pure tone with a frequency that changes at a rate on the order of 1 Hz/sec, which is intended to be typical of a rotating machine’s performance. A theoretical model of the system with an ideal actuator is presented and an analytical controller design is developed. An actuator model with internal dynamics is then added and shown to degrade the performance of the analytical controller design. This motivates the use of a model-based empirical controller design based on qualitative information learned from the model. A gain-scheduled adaptive controller is developed using this design technique. Numerical simulations and experimental results are given to demonstrate the effectiveness of the system at reducing noise in a duct and tracking a disturbance tone with slow time-varying frequency. 98 CONTROLLER DESIGN A model of the system is first presented for the controller design. The central component of the SHR is a Helmholtz resonator with one surface of the cavity replaced by a moving surface (Figure 5.1). The system can be represented by linear time invariant state equations (Chapter 2) Q1 = 7:" la—Cla [Q1] i ()[Pl] [V]|:l 0 V+01Q2 (5'1) Q1 _ l 0 Q1 [p.l-[o all] where the states are Q,, the volumetric flow rate or “volume velocity” from the neck (m3/s) and V, the sum of the volumes introduced through the neck and the inner surface of the cavity (m3). The inputs are P, , the pressure at the neck inlet to the cavity (N/mz), and Q2, the volume velocity from the movable surface in the cavity (m3/s). The outputs are Q, and Pz, the pressure in the cavity (N/mz). The other parameters are K,,, the acoustic loss that represents viscous and radiation losses (N s/ms), 1,, the acoustic inertia of the mass of air in the resonator neck (NsZ/ms), and C0, the acoustic compliance of the cushion of air in the resonator cavity (ms/N). 99 Q Mass of . air in resonator QI Movrng neck surface Cavity / Figure 5.1. Schematic diagram of SHR showing inertia effect in neck and the movable surface in the cavity interior An acoustic impedance can be generated on the moving inner surface of the cavity by enforcing a control law between P, and Q2. This closed-loop, positive feedback configuration is shown in Figure 5.2. The actuator is first modeled as a transfer function with a pure gain of 1. The system can be disturbed either by P, or by D1. Resonator model Ideal actuator . P, Q, . model . Controller Figure 5.2. Closed-loop SHR system block diagram with disturbance through P, A proportional-integral (PI) controller, G(s) = Q2— : K,, + 3 (5.3) P2 S 100 can be used to generate the acoustic impedance on the cavity surface, where K P and K, are the proportional and integral gains respectively. The closed-loop transfer function for Q,/ P, can be computed by substituting (5.3) into (5.1) and (5.2) converting the state equation into a transfer function (Phillips and Harbor, 1996), and simplifying which gives 2 1( K1) s —— KP+—s Q1 =—1 R K C, Ks RK RK (5'4) P l 53+(_a___PsZ+ 1 __r_ a P) _ a I ‘ a I. a 1.6.. C. 1.6.. 1.6.. This form of the transfer function shows how the controller gains affect the system response. The system is obviously not a simple second order resonator. The denominator of the transfer function is third order and the numerator is second order. More insight can be drawn from (5.4) by making some simplifying assumptions. For example, letting Ra = K P = 0 and rearranging yields 2 —Q—1=i S ’K’ (5.5) Pr s1. 32+[1—CKlflaj dd J n— This equation shows how K, affects the poles of the transfer function. The term in brackets in the denominator of (5.5) represents the effective resonant frequency (squared) of the system. A positive K, decreases the effective resonant frequency, and a negative K, increases it. Recall that the control system uses positive feedback. This trend is the reverse of negative feedback systems where positive gain increases the effective stiffness of the system. Also note that the system response is volume velocity relative to pressure. In mechanical resonators systems, the displacement relative to force impedance becomes stiffer as the proportional gain is increased. Displacement is the 101 integral of velocity, explaining why an integral controller gain changes the apparent stiffness in this system. Similarly, setting K, = 0 gives _& 21:1. 5 c, (5.6) Pl Ia 52+(f£—%)S+ITIC:(1-RaKP) a This equations shows that K P appears as a term that is subtracted from the acoustic loss term Ra. Increasing K P reduces the apparent acoustic loss and hence reduces the system damping. The apparent acoustic loss can be either increased with a negative K p, adding additional damping to the system, or decreased with a positive K P, reducing system damping and increasing the peak magnitude at resonance. This analysis illustrates the ability of the SHR to change the dynamic response of the system. Each gain K P and K, gives an input to change the apparent resonant frequency and peak amplitude of the Helmholtz resonator. The advantage of this effect is that the response of the Helmholtz resonator is modified without changing the physical dimensions, i.e., cavity volume, neck length or cross section area. The change in response is caused only by the interaction of the controller with the Helmholtz resonator. Finally, setting K p = K, = 0 produces a transfer function that gives the nominal resonant frequency of the device. Note that K P = K, = 0 implies that the actuator is held fixed and becomes a rigid wall of the cavity. 102 A range of gains K, and K P that produce a stable system can be found by examining the characteristic polynomial (denominator of (5.4)). The system is stable provided that there are no sign changes in the coefficients of the characteristic polynomial (Phillips and Harbor 1996). This produces the following ranges for the gains: Ra C0 0 K,. < (5.7) K, < O (5.8) These bounds on the gains determine the limitation of the controller to tune the system, defining a design space for the controller gains. Acoustic loss is affected by K P (5.6). The limit (5.7) translates into a maximum amount of damping that can be removed from the system before marginal stability is reached. The resonant frequency is affected by K, (5.5). The limit (5.8) translates into a limit on the direction the resonance frequency can be moved from the nominal value. In this case a negative K, increases the value of the resonance frequency, and (5.8) indicates that the resonance frequency can not be decreased from the nominal value. A slightly modified controller increases the limits on the gains and thereby increases the tunable range of the SHR. The integrator in 5.3 has infinite DC gain. By adding a low frequency pole, p], to the integrator G(S)=%‘=Kp+ K1 2 “Pl (5.9) the DC gain is reduced to a finite value. Substituting (5.9) into (5.1) and (5.2) produces the closed loop system. The coefficients of the characteristic polynomial can be cOrnputed in terms of the acoustic and controller parameters. The space of stable gains 103 can be visualized by setting each coefficient equal to 0 and plotting the resulting functions in the K P- K, space. Each curve gives a bound defined by a coefficient in the characteristic polynomial. Figure 5.3 shows the stable gain space indicated by the shaded area, with p, = 100 and acoustic parameters in Table 5.1. The gain K P has an upper limit of K P = 1.1 e -6 (positive) (curve A), and no lower limit. The gain K, has an upper limit of K, = 6.0e -3 (positive) (curve B) and no lower limit. The curve labeled C is a redundant constraint on K ,. The dashed lines indicate the negative direction of each coefficient. These results indicate that K, can have both positive and negative values. The resonant frequency is affected by K ,," therefore the apparent resonant frequency of the system can be both increased and decreased. X106 KP 0. 01 -0.01 -0.005 0 0.005 0.01 Ki Figure 5.3. Plot of stable gain space for controller gains K p and K, 104 Table 5.1. Acoustic parameters used in simulation Parameter Value I, 162 st/m5 Ca 7.2e-9 mS/N Ra 7.5e4 Ns/m Analytical Controller Design An analytical controller design is developed by computing an analytical mapping from the controller gains to the resonant frequency and damping of the closed-loop system. The design space defined by the range of stable gains can be examined using pole placement. The desired denominator of the transfer function can be represented by Gena... = (s2 + 2§w.s + w.2)(s + p) (5.10) The values of the damping ratio, C and the natural frequency, (on can be chosen. The pole p; can not be chosen, but becomes an output of the calculation. The ideal integrator (5.3) is used to simplify the model. Setting (5.10) equal to the denominator of (5.4) and matching coefficients in s, the values of K P, K, and p; can be computed from the resulting linear system of equations. The solution is given by 2 K = 5,, - 20¢,sz — 2carfmn3§ + 4CaIaRawn2§ (5.11) p Ra2 + Iazwn2 — 2IaRawnC (0,2(1, — can,2 - Cargo,2 + 2CaIaRawnC) R,2 + 102(1),} - 21,1200)"; 1" (5.12) 105 Rama,2 + Cargo,2 - I, - ZIaCaRawné') 6,1,,(R,2 + 1020),,2 — ZIaRamnC) P2 = (5-13) where it is assumed that the acoustic parameters are known. The values for (0,. and g are replaced by command variables we and {C which are chosen to produce the desired system response. Gain-Scheduled Adaptive Control An adaptive gain-scheduled control algorithm can be constructed from the above formulations. The gain scheduling variable 60c is updated at a rate much slower than the time constant of the resonant system and must be estimated from the disturbance signal on line. The objective is to tune the system so that a resonance with a large peak occurs at an arbitrary command frequency, (0c, which is set to equal the estimated value of a)". The height of the peak can be controlled by setting the value of CC. The strategy used here is to place the complex poles at a constant distance from the imaginary axis for all values of (0C. This can be accomplished by applying the constraint {C =k/wc (5.14) where k is a scalar constant that defines the distance of the complex poles from the imaginary axis assuming that CC is small. The parameter, k is equivalent to the inverse of the time constant of the complex poles (Phillips and Harbor, 1996). Figure 5.4 shows a plot of the closed-loop system pole locations indicated by an ‘x’ computed with gains from (5.11) - (5.13) with the parameters in Table 5.1 and (DC ranging from 560 to 820 rad/s. The distance of the real pole closest to the imaginary axis for the smallest value of 106 (0c depends on the value of k. Ideally this pole is aligned with the complex poles so that the slowest settling time is minimized. This optimum value of k can be determined from the model by trial and error. For the case shown here, a value of k = 4 (time constant = 0.25 sec) aligned the smallest real pole with the complex poles. The choice of k has little effect on the real poles for larger values of we, where the real pole is far to the left of the complex poles, indicating a fast settling time. 1000- ............. I. ............ , ........... ................ , worm.“ x 400_... L , ,' ..... ; Arrows indicate direction : ; E of increasing command 200,. ........... .............. frequency g . . . g, 01. ........... x ..... x....x ...... x ...... x ....... x ......... x ......... g .......... x .............. x ...... E 3 4 f ? : 1 2 _200.. .............. , ....... 1 ....... ........, ................ 4000 i 1 i '1 "I j -60 -50 -40 -30 -20 -10 0 Real Figure 5.4. Plot of pole locations for gain-scheduled controller algorithm for command frequency ranging from 560 to 820, k = 4 aligns the smallest real pole with the complex poles for the smallest value of (0c 107 Gain-Scheduled Controller Simulation For simplicity, the control will be implemented with the resonator neck open to atmospheric pressure instead of connected to a primary acoustic system. The normal operation is with a primary acoustic system producing P, at the resonator neck and having this input become the disturbance to the system. From the control viewpoint, the system can be disturbed by either input, P, or Q2. The motivation for disturbing the system through Q, is that this approach eliminates the dynamics associated with the primary acoustic system and focuses the analysis on the dynamics of the SHR alone. Also, this is the traditional approach used in mechanical and acoustic resonator design. In the absence of a driving pressure, P,, the system will be disturbed electronically. Figure 5.2 shows the block diagram of the SHR with a disturbance D1, added to the controller output. The other input, P, , is then set to zero, representing atmospheric pressure at the SHR neck, and Q, is a free boundary condition. The closed-loop transfer function relating the system output P2 to the input D1 is computed from (5.1) and (5.2) and given by (5.15). Note the denominator of (5.15) which characterized the system dynamics is identical to the previous implementation (5.4) where the disturbance was through P,. This confirms that the previous stability and gain scheduling analysis applies when the disturbance is through eitherP, or Q2. It should be noted that the numerators of (5.4) and (5.15) are not identical. This will result in different magnitudes and phase characteristics between the two transfer functions. However the resonant frequency and damping is a function of the characteristic polynomial which is invariant to the choice of inputs and outputs. 108 é. SIM?) (5.15) D =3 £11912 in m, Rain 5 +(1 c 5 + lac, C, lac, S‘Iaca The preceding gain-scheduled controller was applied to the plant model and the time response was simulated for various inputs. A block diagram of the system is shown in Figure 5.5. Note that the frequency of P2 must be determined on-line. This can be done by various methods. For the simulation, the instantaneous frequency was known to the controller, i.e., the frequency estimator was perfectly accurate and responded instantaneously. This was a significant assumption used to simplify the model and simulation. 1—> Q 1 SHR P r» 2 Controller Ge 7 v Frequency estimator Figure 5.5. Adaptive gain scheduling control block diagram The numerical simulation demonstrates that the system tracks a disturbance with slow time-varying frequency. The disturbance frequency starts at 565 rad/s (90 Hz) and increases to 595 rad/s (95 Hz) in 2 seconds. Figure 5.6 shows the time response of the System. The t0p graph shows the frequency estimate output. The input D1, is a sinusoidal input with amplitude l N/mz. The middle graph shows P2 with the controller turned off. The amplitude is initially small and decreases further as the disturbance frequency moves 109 away from the nominal resonant frequency (86 Hz). The bottom graph shows P, with the controller turned on. The controller re-tunes the system so that the resonant frequency tracks the changing input frequency. The system starts from rest and the amplitude grows until steady state is achieved, afier about 0.5 seconds. At steady state, the amplitude of P2 is 15 times (23 dB) larger than the open-loop response. As the input frequency changes, the amplitude remains constant compared to the open-loop case, where the amplitude diminished as the driving frequency moves away from the nominal resonant frequency. This simulation demonstrates that the system acts as a resonator with variable tuning which tracks the input signal frequency. 110 wd radls Time sec x 10° P2 Controller Off L I’ I r I I I I T I a 1 ~ ' - 9 3 a, v U) 9 o. _1 — _ _2 i 1 1 i i 1 i i i 0 0.2 0.4 0.6 0.8 1 1.2 1 4 1.6 1 8 2 x 10° P2 Controller On 2 l I I I I . 1 . ‘ ,H , !|,ii1:l‘i|,il ,1[,ii,l,,.iii‘:11 ,l,‘ilii‘l“i,|‘i‘i ,‘ ‘Hiill,,‘ WI”), ,ii‘i.‘i,,i.i,i,,“l,',,il ‘ ‘1‘; i . M ‘M‘ 1 . . 1 . 11mm. Pressure Pa “ a :11 111111.11, 1 ~12 Figure 5.6. Numerical simulation of SHR to disturbance tone with time-varying frequency showing that system resonates with constant magnitude as the disturbance frequency changes Actuator Dynamics Analysis was performed to investigate the effects of actuator dynamics on the closed-loop response of the system. The previous analysis assumed an ideal actuator, i.e., the actuator velocity, and Q, is proportional to the voltage input to the actuator. This is a reasonable assumption for analysis of the fundamental response of the SHR since further detail depends on the structure of the actuator. However, pole placement is known to be sensitive to modeling errors and a gain scheduling control strategy must account for actuator dynamics. lll A velocity compensated dual voice coil speaker will be used in the experimental verification of the SHR. An accurate model of this device was presented by Birdsong and Radcliffe (1999). This actuator compensates the velocity for internal dynamics and pressure interactions on the face of the actuator. The result is an actuator with reduced magnitude and phase variation between command signal and actuator output within a limited frequency range of 20 to 400 Hz. The dual voice coil speaker can be represented by the linear time invariant state equations (Radcliffe and Gogate, 1996; Birdsong and Radclifi‘e, 1999). —R, —1 blSd d 92 I. 6.1. 1.1. Q2 0 .2 .p (5.16) -— q2 = 1 o o q2 + 0 d’ A :95 o ———5£'Rm‘ A. 1 ’5 P2 Sd lc where the states are the volume velocity and volume displacement from the speaker, Q2, and q;, and the electromagnetic flux in the speaker coil, 2.. The inputs are the primary coil voltage, e,,, and pressure on the speaker face, P2. The output equation is given by 1 M -M bl[l--—-C-) 0 ———C-(RC+R,,,)P M e... ’c ’3 1 Q2 7f 0 e (5.17) Q2 1 0 6 _). o o 2 where the outputs are the voltage in the secondary coil, ebs, the current in the primary coil, ip, and Q2. The parameters in (5.16) and (5.17) are the speaker face area, Sd, speaker inertia, 1,, speaker compliance, Cs, speaker friction, Rs, speaker coil resistance, RC, Speaker coil inductance, Ic, speaker coil mutual inductance, M c, speaker electromechanical coupling factor, bl, and the primary coil current sensing resistance, Rm. 112 A velocity estimator can be generated fiom two measurable speaker outputs (the secondary coil voltage, em, and the primary coil current, i,,) vspk,(s) = Hbsebs(3) — H p (s)ip(s) (5.18) where Hbs = 1/ bl and Hp(s)= 5 Mo/ bl . The secondary coil voltage, ebs, can be measured directly from the speaker coil. The primary coil current, ip, can be determined from the voltage across a resistor, Rm, placed in series with the primary coil. The mathematically improper, differentiating transfer function, H p, cannot be realized exactly, but an approximation MC " __ _£1_S_ Hp(s)— bl (“1%) (5.19) can be used, where p3 is a pole location selected such that flp(s) approximates H p(s) over the controller bandwidth. Feedback compensation can now be implemented using the signal from the velocity sensor to compute the error between the desired velocity and the sensor velocity and a proportional controller to drive the speaker velocity to the desired velocity. The compensated speaker model can now be coupled with the resonator and controller model (Figure 5.7) and the closed loop response can be simulated. 113 Resonator model Compensated speaker model | ___________ —> P Q .._> Speaker model 1 I I Q2-—* 92 1”2 > Controller * Figure 5.7. Block diagram of resonator, controller, and compensated speaker with disturbance through D; The effect of the actuator dynamics on the controller design can be determined by examining the closed-loop pole locations. The gain schedule control algorithm used in the results presented in Figure 5.4 is applied to the SHR and speaker model. The pole locations of the closed-loop system are calculated for various values of we and plotted in Figure 5.8. The actuator model adds additional dynamics and therefore there are more poles. Figure 5.8 shows one real pole and 4 complex poles as we is varied from 560 to 820 rad/s. In addition, there are 2 higher frequency complex poles not shown in Figure 5.9 because they are located an order of magnitude to the left of the poles of interest and the different scales would make it difficult to compare with Figures 5.4. The gain schedule objective is to provide varying resonant frequencies with the same time constant. The additional actuator dynamics prevents this from occurring. The complex 114 poles are no longer aligned and the imaginary part of the poles do not correspond with the values of we. Pole-zero map ‘ X ; : : . mom. .. Arrowsindicatedirection of increasing we, g E g i .5001- , \>t;x . . . . . 4000,. .............. ......... . ..... ............ ....... .............. ..... , .:... .1500_,..I ...... ...... ......... ............... .......... , ........ , ...... :. .. . : : xx : E : i .2000_.... ...... . ....... 3 ............. xxx ............... : .............. 1 1 1 1 n 1 i 600 -500 -400 -300 -200 -100 0 Realeis Figure 5.8. Pole locations of SHR with actuator dynamics and ideal SHR gain scheduling algorithm showing that the root locus no longer achieves the desired pole placement strategy when finite actuator gains are included in the model The analytical control design is not effective in the presence of fmite unmodeled actuator dynamics and a different approach must be taken. The ideal actuator assumption was unrealistic. The analytical controller design is too sensitive to changes in the pole locations from unmodeled actuator dynamics. A new control design is needed that captures these effects. The closed-loop response of the system is a complex function of the acoustic and actuator parameters. The SHR (5.4) is a third order model, and the actuator model is a third order model (5.16), so the combination is sixth order. The closed form solution given by (5.11) through (5.13) relating K P and K , to (0,, and C 115 does not account for actuator dynamics. Furthermore, a new solution is not easily found for the sixth order model that maps the controller gains K p and K, to (0,. and C . A different approach that can be used in place of the analytical controller design is a model-based empirical controller design. The assembled model, including the SHR (5.1) and (5.2), the controller (5.9), and the actuator (5.16) and (5.17) can be used to search for values of K P and K, that produce complex poles that meet the same criterion as before i.e., move the resonant frequency while maintaining equal time constants. The model eigenvalues can be computed for different gains using numerical simulation. Although there is no closed-form solution that guides the search for the gains, a good starting place is to use the same trends found in the analytical controller design. The SHR and ideal actuator model indicated that K, affects the apparent resonant frequency (5.5) and K P affects the apparent damping (5.6). The search technique used here fixes a value of K ,, then searches K P to produce the desired damping. Alternatively, instead of maintaining constant damping, the technique could align the real parts of the complex poles, thus maintaining an equal time constant for different resonant frequencies. The distance from the imaginary axis to a pole gives the inverse of the time constant for the pole (Phillips and Harbor, 1996). This criterion was used for K P. This was repeated for various values of K, until an empirical mapping was found between frequency and damping. The empirical controller design successfully produced a mapping between the controller gains and the resonant frequency and damping. Figure 5.9 shows the values of K P and K, on the vertical axis, vs. the resulting resonant frequency on the horizontal 116 axis. The gains were selected to maintain equal time constants for each setting. It was found that the resonant frequency was strongly correlated with K ,, and both negative and positive values of K, produced stable systems (all poles in the left half plane). Negative values of K P increased the time constant (reduced the distance between the imaginary axis and the complex poles). An extremely flat, parabolic-shaped trend was observed between K P and the resonant frequency, with the minimum located at the nominal resonant frequency (130 Hz). Slight variations in K P were required to obtain equal time constants with less than approximately 10% variation in K P for most settings. 0 fl l I I l 1 -05 _ .......... g ........... g .............................................. g ........... n. : : : x G\:o_ : : .1- .......... we, 6 O Q ..... Sure ._ 4.5 1 i m i 1 i 60 80 100 120 140 160 180 200 50° ! l l I v .' _500 i l I l l 1 60 80 100 120 140 160 180 200 Command Frequency Hz Figure 5.9. Plot of K p and K, vs. (DC determined using model-based empirical controller design technique Figure 5.10 shows the pole locations of the closed-loop system for the gains from Figure 5.9. Higher frequency poles are not shown to focus on the complex poles that produce the acoustic resonance. The complex poles are aligned at —30 resulting in a time 117 constant of 1/30 = .033 seconds. The resonant frequency can be computed from the distance of each pole to the origin and range from 62 to 191 Hz. Pole-zero map 1500_.i ....... ....... ....... 1 ....... ....... .‘ ....... ....... ....... ........ . X . : : : : : : : : : X' 1000-.. ....... I ....... ....... . ....... :....x .......... . . . x: . n 1 u c 1 1 1 u x: . 500...: ....... ....... ....... ....... ....... ....... ..... x. .1......._. ' ' ' ‘ : : : : ' x: : 3 i a 0L ....... x .......................... x ....... x..0.x.m ........... ‘. ........ E f : : : : : : : : : x: : .5001... ....... ....... -.......1..4 ....... -...............: ....... ................ . . . . . . . . . xi 1 xi ' x: I I i I t I i I l x: -1000--1 """"" : ------- ------- ------- :--"x-.‘ """"" ' ‘ ' ' ‘ ' ' I I X: "z .1500_..: ....... ....... 1. ....... _1 ....... ....... .1 1 1 1 1 1 1 1 1 1 1 1 -900 '800 -700 -600 '500 '400 '300 '200 .100 0 1 00 Realeis Figure 5.10. Plot of closed-loop pole locations for 7 controller settings derived from the model-based empirical controller design technique Stability is guaranteed provided all closed-loop poles lie in the left half plane. This can be confirmed by examining the eigenvalues of the closed-loop system model for the scheduled gains. The controller design process is based on placing the complex poles. in the left half plane, but other poles appear near the origin in Figure 5.10. These poles are associated with the integrator and choosing the value of p, (5 .9) moves them away from the imaginary axis. Kahlil (1996) presents a stability proof for slow time-varying systems that applies to the gain-scheduled control of the SHR. The proof is simplified by the additional condition that the system with the scheduling variable “frozen” is linear, as 118 is the SHR model. This proof concludes that the system is stable given a limit to the rate of change of the scheduling variables, which can be related to parameters in the model. It is assumed in this control design that the scheduling variable (0c changes much slower than the dynamics of the system. In summary, the SHR gain-scheduled controller was designed using a model- based empirical technique. The scheduling variable is the disturbance noise frequency that must be estimated on line. The details of frequency estimator techniques are not discussed here. The gain-scheduled controller then schedules 2 gains: K P and K, in a PI controller. The model predicts the trends of K P and K, vs. a)" and identifies that (0,, and g are very sensitive to small changes in K ,1. This sensitivity identified the need for experimental verification of the empirical gain scheduling, especially in scheduling K P, to achieve the desired response. Finally, it should be noted that the analysis was performed on the SHR with no primary acoustic system. This is common practice with classical Helmholtz resonator design, where the device is designed to resonate at a frequency while separate from any primary acoustic system. The SHR control algorithm was designed this way, and, in the experimental validation, an acoustic duct will be added to verify the performance of the device on a primary acoustic system. 119 EXPERIMENTAL VALIDATION An experimental apparatus was constructed to validate the theoretical model and demonstrate the effectiveness of the SHR at reducing noise in a duct. In this section the Helmholtz resonator and actuator implementation will be presented, the gain scheduling controller will be implemented and finally, the SHR will be applied to an acoustic duct to demonstrate the control algorithm’s tracking and noise reduction capability. The experimental SHR setup consisted of two components: a Helmholtz resonator cavity and a microphone-compensated actuator system. Figure 5.11 shows a photograph of the SHR connected to an acoustic duct and Figure 5.12 shows a schematic of the setup without the duct. A cylindrical Helmholtz resonator cavity was constructed from PVC with dimensions 0.075 m in diameter and 0.15 m in length. A cylindrical neck with dimensions 0.018 m diameter and 0.01 m in length, was fitted through one face of the cavity. The control system consisted of a microphone, controller and actuator. A half- inch B&K type 4155 microphone was sealed through the wall of the cavity, a D-Space Model #1102 floating point, digital signal processor (DSP) was used to implement the speaker compensation, the controller, and to generate a disturbance tone, and a compensated 6-inch dual voice coil speaker was used as the actuator. A DSP sampling rate of 5 kHz was used for all experiments. 120 Acoustic Disturbance speaker Speaker enclosure Figure 5.11. Photograph of SHR connected to acoustic duct wth a disturbance speaker to inject noise D1 Digital signal processor l l l - D1 .2 , . _ Amplifier Dynamrc Signal analyzer e \ P \ _ Speaker 1‘ enclosure neck I ‘ \ . / I 1 \ Mlcrophone Dual voice coil speaker cavity Figure 5.12. Schematic diagram of experimental SHR apparatus Speaker Compensation The speaker was compensated to improve its performance, but finite gain and phase errors in the actuator response affected the system. A speaker velocity estimator (Birdsong and Radcliffe, 1999; Radcliffe and Gogate, 1996) was implemented by combining the voltage in the secondary coil with the current in the primary coil. A 10 121 ohm resistor was placed in series with the primary speaker coil to measure the current. The velocity estimate was then used to close the loop on the speaker velocity with a proportional controller (Figure 2.13). A value of Kamp = 30 was used for all trials. This value for K amp is somewhat smaller than values used by Birdsong and Radcliffe (1999) where gains of as much as 100 were used. The gain Kamp was chosen to increase the robustness of the system, which comes at the expense of performance. The speaker velocity was measured directly to confirm that the transfer function of the actual speaker velocity relative to the desired velocity was acceptable. This was done by directing a laser velocimeter through the SHR neck onto reflective tape on the speaker face. With the relatively low value of K , there was less than 5 dB and 100 degrees of magnitude and phase between desired and actual speaker velocity in the frequency range of 20 to 200 Hz in all experiments. While this represents a 15 dB and 80 degree improvement over uncompensated audio speakers, it clearly does not approach the ideal actuator model of a transfer function with a pure gain of l. Afier the compensator was implemented, the closed-loop speaker compensation was considered a single block in the SHR, and all subsequent open and closed-loop SHR experiments included closed-loop speaker compensation. 122 ——————————————————— ' Actual I Desired I velocity Controller pnmary: secondary current voltagel Velocity : estimator | Figure 5.13. Block diagram of dual voice coil speaker compensation used in SHR actuator PI Controller Design The PI controller was implemented and the closed-loop SHR response was recorded with controller gains K P = K, = 0. In this configuration, the compensator attempts to hold the speaker face fixed in the presence of the disturbance. White noise was input as the disturbance to the system and the transfer function of P2/D1 was measured using a Hewlett Packard dynamic signal analyzer model #35660A. With the SHR disconnected from a primary acoustic system, resonance was observed as a peak in the frequency response of the Pz/Dl transfer function. The results indicated that a resonant peak occurred at 120 Hz, however there was significant damping in the system, which reduced the peak amplitude. This damping was attributed to mechanical damping in the form of friction and electrical power dissipation in the current sensing resistor, Rm, in the speaker compensator. This damping is large compared to the acoustic damping expected in a passive Helmholtz resonator (Tang, and Sirignano, 1973). Other experiments indicated that damping was significantly reduced when current was not allowed to flow through the sensing resistor. 123 The closed-loop response was then measured with various non-zero controller gains. As predicted by the model, the analytical mapping between K P and K, and the resonant frequency and peak amplitude (2.11) — (2.13) did not produce the desired results. This was attributed to the deviation of the actuator from the ideal model. Even with the compensator, the effects of the speaker dynamics were not sufficiently minimized. The empirical technique was used in place of the analytical mapping to tune the system in the presence of significant actuator dynamics. The PI controller design was based on qualitative information learned fi'om the model. The objective was to find gains, K ,1 and K ,, that placed the resonance at various frequencies, while maintaining the same level of damping. The data was collected by fixing K ,, searching for a K P that produced the desired peak amplitude, and recording the resonant frequency. The gains K P and K, are plotted against resonant frequency in Figure 2.14, which compares the experimentally derived controller gains with the model derived controller gains (Figure 5.9). Note that although there is a difference in the magnitude of the K ,1 gains, the overall trends of these graphs agree. The difference in the K ,1 value is attributed to the difficulty in estimating the damping parameters in the model. The K ,1 gain is negative for all values of (DC with the most negative value at the nominal resonant frequency (130 Hz). Only a 10% change in K p is required for the entire range of (0C. The K, gain ranges from —300 to 400 and passes through zero at the nominal resonant frequency. It should be noted that although there is good agreement between experimental and model results, the controller should not rely on the gains derived from the model, but should be determined experimentally. 124 -O.5,. .......... ........... ........... 2 ........... ........... ........... .......... _, Q . . . I . x N: : : -1- .......... U0 0 O '3 ...... aim—:9 . -1.5 1 . - 1 . i 60 80 100 120 140 160 180 200 60 80 100 120 140 160 180 200 Command Frequency Hz Figure 5.14. Graph of K P and K, vs. a), determined using an experimental empirical technique (+) and model-based empirical technique (0) The tuning capabilities of the device are illustrated by the closed-loop frequency response measurements (Figure 2.15). The curves show the resonant peak for separate experiments with me = 80, 110, 140, and 170 Hz and one curve for the controller gains set to zero. Note that the SHR with the gains set to zero is over-damped with approximately 45% damping. With nonzero controller gains, a resonance is exhibited and positioned at the desired frequency with approximately 5% damping for all cases. This result demonstrates the ability of the controller to tune the SHR to arbitrary frequencies. 125 60 l I I I T GundB 20111111111 I l 20 40 60 80 100 120 140 160 180 200 Frequency(Hz) Figure 5.15. Experimental frequency response P2/01 for gain scheduling controller with we = 80, 110, 140, and 170 Hz and with gains set to zero Noise Reduction of a Time-varying Disturbance Tone in an Acoustic Duct The final demonstration illustrates the SHR’s ability to quiet noise in a primary acoustic system and to track a time-varying disturbance tone. Previous experiments examined the performance with no primary acoustic system to focus on the dynamics of the SHR alone. In this experiment, the SHR was connected to an acoustic duct and a pure tone noise was injected into one end and sound pressure levels were measured at the open end of the duct (Figure 5.16). The setup consisted of a 0.78 m long and 0.076 m diameter PVC pipe with the disturbance signal generated by a 6 inch audio speaker at x = O m, the SHR at x = 0.58 m, and a 1/2 inch B&K microphone type #4155 at x = 0.76 m to 126 record P3. The pure tone disturbance signal was generated by the DSP and the fiequency, we was known exactly by the controller. DSP . Microphones disturbance source, SHR speaker compensation - and controller . ‘_ Oscrlloscope P 2 P3 _I U .__u Disturbance 0.58 m I 0 78 3 inch speaker ‘ ‘ m diameter acoustic duct Figure 5.16. Schematic diagram of SHR connected to acoustic duct for noise quieting experiment, DSP generates disturbance tone and controls SHR, pressure is recorded in cavity and at duct opening on an oscilloscope First, the system was tested to examine the effect of the closed-loop controller with a stationary disturbance tone. The noise reduction was measured by comparing P3 with the controller (and compensator) deactivated, and activated. Other more common measures of noise reduction, such as insertion loss (ASTM Designation: C 634 — 96) compare the sound pressure level with the silencer inserted and removed from the system. This was not used here since the objective was to isolate the effect of the controller on the system. Figure 5.18 shows R, and P3 with wc = 130 Hz as the controller is turned on at time to = 0.05 sec. Before time to, the amplitude of P3 is approximately 30 N/m2 (124 dB), and the amplitude of 1’2 is approximately 40 N/m2 (126 dB). Afier to there is a transient for approximately 0.02 sec, and the amplitude of P3 reduces to 5 N/m2 (107 dB) and P2 increases to 200 N/m2 (140 dB). The decrease in P3 represents 17 dB of noise 127 reduction in the unwanted sound transmitted from the duct to the room. The increase in P2 represents the amplified resonance (reduction in damping) of the SHR. High frequency noise is also exhibited in the closed-loop P3 signal due to noise introduced into the system by the local speaker compensation. This noise is unavoidable with the current speaker compensation algorithm and degrades the noise reduction results somewhat by generating relatively low level, approximately (85 dB overall SPL) broadband noise in closed-loop. This effect does not contribute to the overall SPL significantly since it is 22 dB below the 107 dB dominant noise signal. However, it is easily perceived by a human listener and produces the perception that the controller adds noise, reducing the overall effectiveness. 128 300 200 100 0 -100 -200 -300 1 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Cavrty Pressure P2 Pa 4° Controller off 1 . 1 1 1 1 Duct Pressure P3 Pa An 1 J 1 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Tlmesec Figure 5.17. Pressure near duct opening, P3 and pressure in SHR cavity, Pz versus time with a 130 Hz disturbance noise as controller is activated, showing 17 dB reduction Finally, the system was tested with a time-varying disturbance tone frequency to demonstrate the controller’s tracking ability and the noise reduction at various frequencies. It was found that the range of tunable frequencies was sensitive to the K W, and that by reducing it from 30 to 28, a wider range could be achieved at the expense of reducing the magnitude of the noise reduction. The experiment was run twice: first in open-loop then again in closed-loop. In each trial P3 versus time was recorded as we was ramped from 80 to 180 Hz in 9 seconds. Figure 5.18 shows open- and closed-loop RMS P3 versus time (top), (Dc versus time (middle), and noise reduction versus time (bottom), 129 computed by subtracting closed-loop from open-loop RMS P3. The open-loop SPL increases with we due to the presence of an acoustic duct resonance at 190 Hz. This illustrates the difficulty in separating the SHR dynamics from the primary acoustic system dynamics and explains why the gain scheduling design was performed without a primary acoustic system. In other experiments not shown here, other duct lengths were used, and it was shown that there was little change in the system response. The noise reduction is defined by the difference between closed-loop and open-loop (bottom). The bottom graph shows that the noise reduction varies from 5 dB to as much as 15 dB. The effective range of frequencies for the device, defined as 6 dB or greater noise reduction, is 90 - 170 Hz. The low frequency (< 90 Hz) limitation is attributed to finite actuator travel. As the frequency is reduced, the controller requires larger displacements to generate a constant magnitude velocity. The speaker face can only generate displacements on the order of 1 cm. The high frequency (> 170 Hz) limitation is attributed to the control design. The algorithm found gains that produced peaks with equal magnitude in the frequency response of the P2/D1 transfer function. The model reveals that this will produce peaks with magnitudes that decrease with frequency in the response of the Q,/ P, transfer function. This transfer function, not Pz/Dl interacts with the duct and creates the pressure release boundary condition. An attempt to compensate for this by changing the control strategy would require reducing the stability margin of the system. This experiment demonstrates that the SHR gain-scheduled adaptive control algorithm can track time-varying disturbance tone frequencies and provide significant noise reduction in acoustic systems over a range of frequencies. 130 Figure 5.18. Time-varying disturbance results showing open and closed-loop RMS pressure P3 near duct opening (top), disturbance tone, we (middle), and noise reduction versus time computed from closed-loop subtracted from open-loop RMS P3 (bottom) showing between 6 and 15 dB of noise reduction between 90 and 170 Hz CONCLUSIONS In this article, a model for the SHR was analyzed for closed-loop adaptive control. A stability analysis was presented and an analytical controller design was developed that produced a gain scheduling controller algorithm. This allows the system to track a disturbance tone with slow time-varying frequency while maintaining optimum amplitude and damping characteristics. It was shown that the analytical controller design was not effective in the presence of unmodeled actuator dynamics, and a model-based 131 empirical controller design was presented which included actuator dynamics. The model was used to identify key factors in determining the gain scheduling algorithm, such as the effect of actuator dynamics, and sensitivity. Finally, experiments showed that the device was able to provide at 6 to 15 dB of noise reduction as the frequency of a disturbance tone varied with time from 90 to 170 Hz. The control algorithm presented here has several beneficial features. It is composed of simple, inexpensive components that are commercially available. It requires a low order controller and can be implemented with a PI controller with variable gains. The tuning algorithm can run at a relatively slow speed and does not require a high speed digital signal processor. No sensor is required outside of the SHR cavity. This means that the only connection to the primary acoustic system is the resonator neck and the design is independent of the primary system. Finally, the sensitive components in the controller are all located in the resonator cavity out of the direct path of the acoustic duct. This reduces the likelihood of damage from debris and harsh environmental conditions. The controllable SHR is a powerful and adaptive tool in reducing unwanted noise in acoustic systems. 132 Chapter 6 Conclusions This dissertation presents the invention of an electronically tuned semi-active Helmholtz resonator. This device can be attached to a primary acoustic system, such as a duct, to reduce the transmission of narrow frequency band noise. It can be adaptively tuned on-line to track a disturbance signal with slow time-varying frequency. It has several advantages over similar inventions. There are no complex moving parts or mechanisms so it is cheaper and easier to implement. The sensitive components are removed from the primary acoustic system and placed in the resonator cavity, so they are less susceptible to damage from harsh environments. The device is fault tolerant: in the event that the controller is turned off, it continues to provide nominal acoustic filtering. Also, it requires only one connection to the primary acoustic system. No sensors are required external to the device, so that its operation is not dependent on the structure of the primary acoustic system. The control algorithm is relatively simple and easy to implement. This work focused on four major topics related to the device: the overall invention and development of the SHR, an analysis of the power flow in the device, a study of implementation issues, and finally, the development of an adaptive control algorithm. An analytical model was first developed which included the resonator, controller and an ideal actuator. It was shown that the acoustic impedance of the SHR could be modified by controlling the acoustic impedance on the actuator face. An analytical relationship between the actuator impedance and the SHR impedance was presented and 133 used to derive an analytical controller design. This design produced controller gains for a simple proportional integral controller from the desired resonant frequency and peak amplitude. Numerical simulation was used to show that this controller successfully re- tuned the SHR. Next, the ideal actuator model was replaced with a compensated speaker model and it was revealed that the control design was no longer effective in the presence of unmodeled actuator dynamics. A model-based empirical control design was presented that included the effects of actuator dynamics but required a trial-and-error search technique. Experimental results were presented that demonstrated the control design and the effectiveness of the SHR in quieting noise in a duct. Future topics that relate to the overall design of the SHR include investigating other resonator configurations and generalizing the technique to other vibration reduction problems. The overall concept of this invention does not rely on the use of a Helmholtz resonator. Other classes of acoustic resonators are available and may have advantages in different applications, such as higher frequency noise control, where the use of Helmholtz resonators is limited. The concept may also be applied to mechanical vibration problems. The second major topic in this work concerned the power flow in the SHR and an acoustic duct. This topic answered the question, “Where does the sound go?” It was shown that the SHR creates a region in the duct that reflects the incident sound back to the source, thus reducing the transmitted sound. Expressions were developed for the incident, reflected and transmitted acoustic power, reflection coefficient, and transmission loss as a function of the acoustic parameters and control gains. It was concluded that the device is most effective when there is no dissipation in the resonator. This highlights the point that the commonly used term, “sound absorber” is a misnomer 134 in this case since there is little absorption. Instead, the mode of operation is sound reflection. The model and experimental results demonstrated that the closed-loop control of the SHR moved the center frequency of the reflection coefficient to desired values. The third topic in this work concerned implementation issues regarding the choice of actuator. A local feedback compensator was added to a speaker which was used to implement the control input to the SHR. It was discovered that the compensator improved the low frequency response of the speaker moderately but degraded the high frequency response. It also added broadband noise to the system. The closed-loop SHR controller together with the compensated speaker provided as much as 32 dB of noise reduction in a narrow fi'equency band, yet the overall sound pressure level reduction was only 18 dB due added broadband noise. The speaker was then applied to the SHR without the compensation and it was discovered that the control authority was reduced but no random noise was introduced into the system. The narrow band noise was reduced by 9 dB and the overall sound pressure level was reduced by 9 dB since no noise was added by the controller. This led to the conclusion that the compensated speaker is more desirable when the objective is to reduce narrow band sound and overall sound pressure level, while the uncompensated speaker is more desirable when the objective is to improve the sound quality. Future work in the implementation of the SHR includes improving the actuator. A common problem in active acoustic noise control is lack of effective actuators. While this problem has received much attention, it is clear from this work that more work must be done in this area. The fourth topic in this work concerned a closed-loop adaptive control strategy for the SHR. A gain-scheduled controller was deve10ped from the empirical control 135 design. Numerical simulations and experimental results showed that the device was able to track a disturbance noise with slow time-varying frequency. Between 10 and 15 dB of noise reduction was demonstrated with a disturbance frequency that varied from 100 to 150 Hz. Future work in this topic includes optimal, adaptive control design. The gain scheduling controller is the least complex of the adaptive control algorithms. An algorithm could be developed to start from the controller gains prescribed by this work and search for more optimal gains while minimizing an error function. In conclusion, the SHR developed in this work has many beneficial features that make it a useful tool to quiet noise in enclosed acoustic systems. It is an improvement over passive resonators because it can track a disturbance noise when the frequency changes from the nominal tuned frequency. It is also a relatively simple device that requires only one connection to the primary acoustic system. Furthermore, it can be implemented with a low order controller and the design is independent of the primary acoustic system. 136 References ASTM Designation: C 634 - 96, "Standard Terminology Relating to Environmental Acoustics" ' ASTM Designation: E 1050-90, "Standard Test Method for Impedance and Absorption of Acoustical Materials Using a Tube, Two Microphones, and a Digital Frequency Analysis System" ASTM Designation: E 477-96, "Standard Test Method for Measuring Acoustical and Airflow Performance of Duct Liner Materials and Prefabricated Silencers" ASTM E 1265-90, “Standard Test Method for Measuring Insertion Loss of Pneumatic Exhaust Silencers,” ASTM E 1265-90, 1995. Bedout, Francheck, et. A1. 1997, “Adaptive-Passive Noise Control with Self-Tuning Helmholtz Resonators,” Journal of Sound and Vibration, v 202, p109-123. Beranek L. L., Ver 1. L., 1992, “Noise and Vibration Control Engineering,” John Wiley & Sons, New York. Birdsong, C. B., and Radcliffe, C. J., 1999, “A Compensated Acoustic Actuator for Systems with Strong Dynamic Pressure Coupling,” Journal of Vibrations and Acoustics, Vol. 121, pp. 89-94. Birdsong, C., 1996, “A Compensated Actuator for an Acoustic Duct,” Masters Thesis, Michigan State University. Blaser, D., and Chung, J ., 1978, “A Transfer Function Technique for Determining the Acoustic Characteristics of Duct Systems with Flow,” Inter-Noise, San Francisco, May. Chung, J ., and Blaser D., 1980, "A Transfer function method of measuring in-duct acoustic properties. 1. Theory," J. Acoust. Soc. Am. 68(3), Sept. Furstoss, M., Thenail, D., and Galland, M. A., 1996, “Surface Impedance Control for Sound Absorption: Direct and Hybrid Passive/Active Strategies,” Journal Sound and Vibrations, vol. 203, p 219-236. Garrett, K., 1992 , “Inter-cylinder charge Robbing: Key factor in Induction-Tract Tuning.” Automotive Engineer v 17 n 4 Aug-Sept. Graham, C. R., and Graves, M. C., et. al., 1992, “General Motors High Performance 4.3L V6 Engine,” SAE Transaction, vol. 101, Sect 3, p 1305-1320. Hartmann, W., 1997, “Signals, Sound, and Sensation,” American Institute of Physics, Woodbury, New York. 137 Harwood, H.D., 1974, "Motional Feedback in Loudspeakers," Wireless World, 80, pp. 51-52. Heidelberg, L. J ., and Gordon, E. B., 1989, “Acoustic Evaluation of the Helmholtz Resonator Treatment in the NASA Lewis 8- by 6-Foot Supersonic Wind Tunnel,” NASA technical memorandum ; 101407. Holdaway, H.W., 1963, "Design of Velocity Feedback Transducer Systems for Stable Low-Frequency Behavior," IEEE Transactions, AU-ll, pp. 155-173. Hsomi, M., Goawao, S., Imagawa, T., and Hokazono, Y., 1993, "Development of exhaust manifold muffler," SAE Special Publications New Engine Design and Engine Component Technology International Congress and Exposition Mar 1-5 n 972 Detroit Hull, A., and Radcliffe, C., 1991, "An Eigenvalue Acoustic Measurement Technique," Journal of Vibrations and Acoustics, April 1991, Vol. 113, pp. 250-254 IEEE 1975, "IEEE Recommended Practice for Loudspeaker Measurements," IEEE std. 219-1975. Jameson, R. T., and Hodgins, P. A., 1990, “Improvement of the Torque Characteristics of a Small, High-Speed Engine Through the Design of Helmholtz-Tuned Manifolding,” SAE International Congress and Exposition Detroit, Michigan, February. Khalil, H. K., 1996, “Nonlinear Systems,” Prentice-Hall, Inc., New Jersey Klaassen, J .A., de Koning, SH, 1968, "Motional Feedback with Loudspeakers," Philips Technical Review, 29, No. 5, pp. 148-157. Kong, H. , and Woods, R., 1992, “Tuning of Intake Manifold of an Internal Combustion Engine Using Fluid Transmission Line Dynamics,” International Congress & Exposition, SAE, February. Krafi, R., Janardan, B., Kontos, G., and Gliebe, P., 1994, "Active Control of Fan Noise- Feasibility Study," NASA Contractor Report 195392 Levine, H. and Schewinger, J., 1946, “On the Radiation of Sound from an Unflanged Circular Pipe,” Physical Review, Vol. 73, No. 4, pp. 383-406, February 15. Morel, T., Morel, J., and Blaser D., 1991, "Fluid Dynamic and Acoustic Modeling of Concentric-Tube Resonators/Silencers," SAE Transactions Journal of Engines Section 3 Phillips, C., and Harbor, R., “Feedback Control Systems,” Prentice-Hall, New Jersey. Pierce, Allan D., 1981, “Acoustics : an introduction to its physical principles and applications,” New York : McGraw-Hill Book Co. 138 Radcliffe C. J., and Gogate, S. D., 1996, “Velocity Feedback Compensation of Electromechanical Speakers for Acoustic Applications,” International Federation of Automatic Control, Triennial World Congress, July. Radcliffe C.J., and Gogate S.D., Hall 0., 1994, “Development of an Active Acoustic Sink (AAS) for Noise Control Applications,” Active Control of Vibrations and Noise, ASME. Radcliffe, C. J., and Gogate, S. D., 1992, “Identification and Modeling Speaker Dynamics for Acoustic Control Applications,” ASME Symposium on Active Control of Noise and Vibration. SAE Handbook, 1995, "Measurement of Light vehicle exhaust sound level under stationary conditions," SAE J l 169 Mar92 Selamet, A., Dickey, N., and Novak,J., 1995, "Theoretical, computational and experimental investigation of Helmholtz resonators with fixed volume: lumped versus distributed analysis," Journal of Sound and Vibration, Vol. 187(2), 358-367. Seto, W., 1971, "Theory and Problems of Acoustics, McGraw-Hill Book Company, New York. Speakerman, C., and Radcliffe, C., 1988, "Decomposing One-dimensional Acoustic Response into Propagating and Standing Wave Components," Journal of the Acoustical society of America, Vol. 84, No. 4, pp. 1542-1548 Tang, P. K,, and Sirignano, W. A., 1973, “Theory of a Generalized Helmholtz Resonator,” Journal of Sound and Vibration, vol. 26 (2), pp. 247-262. Tanner, R.L., 1951, "Improving Loudspeaker Response with Motional F eedback," Electronics, 24, No. 3. Temkin, 1981, “Elements of Acoustics,” John Wiley & Sons, Inc. 139