A Compenated Acoutic Actuator for Sytem with Strong Dynamic Preure Coupling Submitted to ASME Journal of Vibration and Acoutic July.997 Charle Birdong and Clark J. Radcliffe Department of Mechanical Engineering Michigan State Univerity Eat Laning, MI 4884-9 ABSTRACT Audio peaker are commonly ued a acoutic actuator for noie control application. Recent development in the ue of compenated dual-coil peaker have improved the performance of thee acoutic actuator. However, the performance of thee peaker depend on the application. When they are applied in ytem with trong coupling between the plant and the actuator, the velocity enor ued in previou work mut be improved. Thi tudy conider the application of a compenated peaker a an actuator. An acoutic duct i ued an a example of a plant that exhibit trong dynamic preure interaction with the actuator. The peaker dynamic and the acoutic duct dynamic are firt modeled eparately. The two ytem are then coupled, and the reulting ytem i modeled. A velocity enor i developed and ued in feed-back compenation. The reulting peaker ytem behave a an ideal actuator with minimal magnitude and phae variation over a Hz bandwidth. Thee concluion are verified through experimental reult. Thi tudy i important in the overall area of acoutic actuator and active noie control. The actuator developed here will ignificantly aid in the goal of active noie control in an acoutic duct. INTRODUCTION Active noie control i an expanding field in the automotive and aircraft indutrie. Commercial product are currently available to create quiet interior pace [Bradley 995, Warner 995]. Thee ytem ue paive and active control to treat unwanted noie. Paive control conit of applying dampening material to treat high frequency noie. Dampening material mut be of the ame phyical dimenion a the wavelength of the ound wave to be effective [Radcliffe, et al, 994]. Below Hz the wave length in air i approximately.6 m or longer. Thi would require an unrealitic amount of dampening material approximately one quarter of a meter thick for effective paive noie control. For low frequency noie, active control method can be ued. Active control relie on a combination of enor, a controller, and actuator to treat noie in the ytem (plant). One ytem that ha received much attention in thi field i the acoutic duct, which conit of a long, hard walled encloure. Hull howed that the reonance excited by a noie ource in an acoutic duct can be attenuated uing feedback active noie control [Hull 993]. Attempt at wide band noie control were hindered by actuator dynamic that caued the meaured control input to deviate from the deired control. ogate propoed a trategy for eliminating the effect of peaker dynamic through feed-back compenation [Radcliffe et al. 996]. The original deign did not include the effect of the coupled dynamic through the interaction of the plant preure and the actuator. Figure how the peaker face velocity to primary coil voltage frequency repone of the original compenator with two cae: the dahed line repreent the repone of the peaker in free-air, and the olid line repreent the repone when the peaker i coupled with an acoutic duct. In free-air there i little magnitude and phae variation from - Hz. When the peaker i coupled with the acoutic duct, there are large magnitude and phae variation exhibited at the reonance frequencie of the duct. The original compenator fail to eliminate the dynamic aociated with the duct. Magnitude db Phae (Deg) - - - Coupled Sytem Only 5 5 5 3 35 4 Only Coupled Sytem 5 5 5 3 35 4 Figure : Face to Primary Coil Voltage Uing Original Compenator Thi tudy preent an acoutic actuator that compenate for both actuator and plant dynamic. The acoutic duct i preented a a plant in order to demontrate the robutne of the ytem to a plant with trong coupling with the actuator through preure interaction. A model of a dual voice-coil peaker i firt preented, and a velocity enor i developed. It i hown that the peaker dynamic can be eliminated through feed-back compenation. A model of an acoutic duct i preented next, which predict the preure repone due to a velocity input. Finally, the two ytem are coupled, and it i hown that the peaker compenation minimize both the peaker and the acoutic duct dynamic through feed-back compenation. The reult in every tage of the modeling and analyi are verified through experimental teting, and model reult are preented together with experimental reult. Figure how the experimental etup ued to verify the model reult. Figure : Acoutic Duct and Compenated Actuator Setup The work preented here provide a method for creating an ideal acoutic actuator for ytem that include trong plant and actuator coupling and bring the goal of active noie control of ytem uch a the acoutic duct one tep cloer.
ACOUSTIC ACTUATOR SPEAKER MODEL Audio peaker are commonly ued a acoutic actuator in noie control ytem. They are beneficial becaue a mall voltage applied to a peaker can generate a trong control effort. Audio peaker are relatively inexpenive and widely available in commercial ize and model. have the diadvantage that the repone of a peaker can be trongly affected by both the dynamic aociated with the free-air reonance of the peaker and the dynamic of the ytem it i driving. An ideal actuator will have a pure gain over the required bandwidth. When a peaker i affected by dynamic, it can exhibit ignificant magnitude and phae variation limiting it performance. If a peaker i to be ued a an acoutic actuator, thee effect mut be minimized. One method of minimizing magnitude and phae variation i to apply feedback compenation to the peaker. If the peaker repone can be meaured, then the ignal can be applied to a feedback controller, the repone can be driven to the deired output, and the magnitude and phae variation can be reduced. An accurate peaker face velocity enor i therefore required. One variety of peaker named the dual voice-coil peaker ha certain characteritic that make it ideal for ue a an acoutic actuator [Radcliffe et al., 996]. The dual voice-coil peaker ha indepent wire coil intertwined and wrapped around a bobbin which i allowed to lide over a permanent magnet. Thi configuration i hown in Figure 3. Primary Coil Terminal Permanent Magnet e p - i p R m Bobbin x pkr v pkr Face Preure i b Secondary Coil Teminal - e b Figure 3: Dual Voice-Coil Diagram A tranfer function model of the ytem can be developed which relate the input: primary coil voltage, econdary coil current and peaker face preure to the output: econdary coil voltage, primary coil current and peaker face velocity [Birdong,996]. An infinite impedance i applied to the econdary coil forcing the current to zero, eliminating the econdary current a an input. The peaker parameter neceary to define the model are the mechanical inertia of peaker, I pkr ; mechanical compliance of peaker, C pkr ; vicou friction of peaker, R pkr ; electromagnetic coupling factor, bl; peaker coil reitance, R coil ; the current ening reitor reitance, R m,; peaker coil inductance, I coil, mutual inductance, M coil ; and the equivalent peaker area, S D. With the exception of mutual inductance, M coil, thee electrical and mechanical parameter are defined in IEEE tandard 9-975 [IEEE Standard 9-975] for loudpeaker meaurement. The tranfer function model of the ytem i given by eb() eb/ ep eb / P ep ip() () = ip / ep ip / P () P () vpkr() vpkr / ep vpkr / P Each element in the tranfer function matrix () i given by eb / ep bl 3 CpkrMcoil Ipkr Rpkr ( ) Mcoil Cpkr = [( CpkrIpkr ) ( CpkrRpkr ) ] ip / ep = eb / P vpkr / ep ( bl Cpkr ) () (3) = [ ] (4) [ [ ]] bl CpkrSD ( Icoil Mcoil ) Rcoil = ip / P ( bl SDCpkr ) (5) = [ ] (6) [ [ ]] SC D pkr Icoil Rcoil vpkr / P = (7) and the ominator of the () matrix i given a 3 = ( CpkrIpkrIcoil ) ( CpkrIpkrRcoil CpkrIcoilRpkr ) (8) ( bl Cpkr Icoil CpkrRcoilRpkr ) Rcoil Equation (4) and (7) are new and important reult not ued in previou work. All of the peaker model tranfer function equation will be ueful when deigning a velocity enor for the peaker and when modeling the peaker coupled with the acoutic duct. Feedback Compenation of The velocity of the peaker, v pkr, i trongly affected by the dynamic of the peaker and the preure input, P. Thee effect will combine to create magnitude and phae variation in the primary coil voltage to peaker velocity repone. One method of eliminating thee unwanted effect i to apply a proportional feedback controller a hown in Figure 4. The tranfer function for thi ytem i given by (9), where K p i the proportional gain and H() i a velocity enor. If the enor tranfer function i a real contant, k, over the controller bandwidth, then the cloed loop tranfer function, T(), will approach a contant, /k with zero phae. Thi compenation force the peaker cone velocity to accurately follow the deired velocity input. The reult i indepent of the peaker dynamic and the input preure provided that the enor ha a contant tranfer function over the controller bandwidth. v d (t) () - Proportional Controller K p Senor H() e p (t) Dy namic p kr v pkr (t) Figure 4: Proportional Feedback Controller Vpkr() K p pkr() Tpkr() = = (9) Vd () K p pkr() H () A K p i increaed, the tranfer approache /H() and the magnitude and phae variation approache zero. Thi approach aume that the velocity of the peaker face can be meaured. A peaker velocity enor i therefore needed which accurately
predict the peaker velocity in the preence of peaker and plant dynamic. The relation between the peaker velocity and the two other meaurable output (the econdary coil voltage, e b and the primary coil current, i p ) i given in (). The peaker velocity, V pkr can be olved for in term of E b and I p yielding, Vpkr() = HbEb() Hp() Ip() () where H b = /bl and H p ()= M coil /bl. The econdary coil voltage, E b, can be meaured directly from the peaker coil; the primary coil current, I p, can be determined from the voltage acro the reitor, R m ; and H b i a pure gain (/bl). The term H p () i a differentiator tranfer function becaue it contain an in the numerator. Such a tranfer function cannot be realized exactly, but an approximation H ˆ p () can be ued where ˆ M H () coil p = () bl p where p i a pole location elected uch that H ˆ p () approximate H p () over the controller bandwidth. Feedback compenation can now be implemented uing the ignal from the velocity enor to compute the error between the deired velocity and the enor velocity and a proportional controller to drive the peaker velocity to the deired velocity. It hould be noted that the development of the velocity enor did not aume that the preure at the peaker face wa contant, a in previou work. Thi new velocity enor include the effect of preure a an input to the ytem. A a reult, the cloed-loop ytem minimize magnitude and phae variation from not only the peaker dynamic (a in previou work) but in addition, the dynamic aociated with the acoutic ytem, coupled through the preure interaction with the peaker are minimized a well. Thi improvement over the previou velocity enor i eential for the peaker to perform a an ideal actuator in a coupled ytem uch a the acoutic duct. ACOUSTIC DUCT SYSTEM MODEL The acoutic duct i a ytem that exhibit trong dynamic that when coupled with the peaker ytem will caue large magnitude and phae variation in the peaker repone. Thee effect can then be minimized through feed-back compenation. A mathematical model i needed for the acoutic duct before thee effect can be demontrated. In thi ection, a model that accurately repreent the duct preure repone i developed. Sytem equation are firt preented, then they are tranformed into tate pace and tranfer function repreentation. The model i then verified by comparing it with experimental reult obtained from an acoutic duct. Sytem Model An accurate ytem model of the acoutic duct i needed for modeling, analyi. The linear econd order wave equation modeling particle diplacement in a hard-walled, one-dimenional duct i [Seto 97, Doak 973] uxt (, ) uxt (, ) δ( xpt ) ( ) c = x x ρ k M δ x x i() t ( i) [ ] ρs () i= where u(x,t) = particle diplacement, c = wave peed (m/), x = patial location (m), t = time (), ρ = ity of the medium (kg/m 3 ), M I (t) = ma flow input in the domain (kg/), x i = location of ma flow input (m), S = peaker area driving the ma flow input (m ), P(t) = preure excitation at x = (N/m ), and δ(x) = the Dirac delta function. The partially reflective boundary condition at location x = L i the relationhip between the patial gradient and the time gradient of the particle diplacement and i expreed a [Seto 97, Spiekerman 986] u x Lt K u (, ) = ( Lt, ); K i, i, c (3) where K = complex impedance of the termination end (dimenionle). The duct end at x = i modeled a a totally reflective end. Thi boundary condition i u (, t ) = (4) x which correpond to an open duct end. The acoutic preure of the ytem i related to the patial gradient of the particle diplacement by [Seto 97] Pxt (, ) = ρ u c x ( xt, ) (5) The above four equation repreent a mathematical model of the duct. State Space Repreentation To derive the tate equation ued throughout the analyi, eparation of variable i applied to the unforced verion of (), (3) and (4). Solving for the eparation contant and the eigenfunction yield [Spiekerman 99] K nπi λn = log e, n =, ±, ±,... (6) L K L λ x λ x φn x e n n ( )= e (7) where λ n are the natural frequencie and φ n (x) are the eigenfunction of the duct. For a duct with one ma flow rate a the input, the above equation can be manipulated uch that the following tate pace repreentation i produced [Hull,99]: at ( ) = Aductat () Bductmt () (8) where a(t) = the vector of modal wave amplitude A duct = the diagonal matrix [ cλ n ] dφ x B duct = the matrix n( i) 4 cλ L S dx n ρ and m(t) = the ma flow input M The ytem output i the preure at any poition in the duct T Px (, t) = C at ( ) (9) m where Px ( m, t)= the preure in the duct at x = x m, and C duct = the column vector φ ρc d n( x m). dx Equation (8) and (9) repreent the tate pace formulation of the acoutic duct with complex impedance, K, on the termination end. Duct Tranfer Function A velocity to duct preure tranfer function can be computed from the tate pace repreentation of the acoutic duct model for the cae with one ma flow input. The tranfer function repreentation will be ued for the coupled peaker/duct ytem model. The duct tranfer function can be computed numerically from duct 3
Pduct duct() = = Cduct( I Aduct ) Bduct () m where I i the Laplace variable time an itity matrix and duct() i the peaker velocity to duct preure tranfer function. The ma flow rate, m(t) can be replaced by the peaker face velocity, v pkr, by the relation, m() t = Sv D pkr() t, where S D i the peaker area. The tranfer function, duct(), will have a numerator which conit of a polynomial of order *n and a ominator of order *n, where n i the number of mode in the model. COUPLED SPEAKER-DUCT SYSTEM In the previou dicuion both the dynamic of a peaker and a duct were modeled eparately. The model of the peaker aumed that the peaker face wa expoed to atmopheric preure. Thi implied that the peaker velocity wa only affected by the primary peaker voltage. The model of the duct gave the preure at a point in the duct given a velocity input. Thee two ytem can be coupled by allowing the velocity output of the peaker to be the input to the duct and the preure output of the duct to be the input to the peaker. The velocity of the peaker face i then no longer affected only by the primary peaker voltage but alo by the preure generated in the duct, which mut be determined from the coupled dynamic of the two ytem. Thi coupling i illutrated by Figure 5. e p Primary Voltage Face Preure vpkr/ep vpkr/p Σ duct Figure 5: Coupled -Duct Sytem The coupled ytem can be modeled by combining the tranfer function of the peaker and duct model. The reulting tranfer function can be ued to model the open loop repone of the coupled peaker-duct ytem. The peaker velocity, V pkr i given by () a Vpkr() = vpkr / ep() Ep() vpkr / P() P() () The duct preure to peaker velocity tranfer function () i given by Pduct V = duct () duct The preure can be eliminate from () by ubtituting () which give Vpkr() = vpkr / ep() Ep() vpkr / P() duct() Vpkr() (3) which can be olved for the tranfer function of peaker velocity to primary peaker voltage a V pkr vpkr / ep E = (4) p ( vpkr / P duct ) Senor The coupled ytem tranfer function (4) can be ued to model the repone of the velocity enor preented in the previou ection. The velocity enor model will include the effect of etimating the derivative of the primary current [Birdong, 996]. The econdary peaker voltage wa given by (). The preure, P can be eliminated by replacing P with (), giving Eb() = eb / epep eb / PductVpkr (5) The velocity can be eliminated replacing V pkr with (4) giving the econdary peaker voltage to primary peaker voltage tranfer function, eb / P vpkr / ep duct Eb / EP = eb / ep (6) ( vpkr / Pduct ) The primary peaker current, I p i given by (). The preure and velocity can be eliminated a before, giving, ip / P vpkr / ep duct IP / EP = ip / ep (7) ( vpkr / Pduct ) The tranfer function (6) and (7) can be ubtituted into the velocity enor equation () to give the enor velocity to primary peaker voltage tranfer function a, M coil ip / P vpkr / ep duct Venor / Ep = ip / ep bl( p ) ( vpkr / Pduct ) (8) eb / Pvpkr / ep duct eb / ep bl ( vpkr / Pduct ) Equation (8) can be ued to imulate the enor velocity repone of the coupled ytem. The feedback compenation trategy can be applied to the coupled ytem. The enor velocity account for the preure input a well a the primary voltage input, and the cloed-loop ytem compenate for the dynamic aociated with both the peaker and the duct. COUPLED SPEAKER-DUCT MODEL VERIFICATION The coupled peaker/duct ytem model wa verified through experimental teting. The peaker velocity model wa firt compared to experimental reult, then the velocity enor wa hown to accurately predict the meaured velocity. Finally, the velocity enor wa ued in feedback compenation. Model The peaker/duct ytem wa etup a hown in Figure 6. The peaker velocity to primary coil voltage tranfer function wa then meaured uing a Hewlett Packard Signal Analyzer model 3566A from - Hz. The peaker face velocity wa meaured uing a Bruel & Kjaer Laer Doppler -Tranducer Set Type 3544. Duct Sound Preure Level Open End Microphone Laer Tranducer Clear Window Tee Joint Input Signal PVC Pipe Open End Figure 6: Experimental Acoutic Duct Sytem The model given by (4) wa then ued to compute the model repone. Figure 7 how the model repone compared to the 4
meaured repone. The model repone i hown by the olid line and the meaured repone i hown by the dahed line. An end impedance of.5j wa ued in the model. ood agreement wa obtained by the model. There i le than 5 db magnitude difference and degree and phae difference below 4 Hz. Log Magnitude - - -3 Model Magnitude db - -4 ---Senor - 5 5 5 3 35 Phae (degree) -4 5-5 - -5 5 5 5 3 35 4 Model 5 5 5 3 35 4 Figure 7: Comparion of Frequency Repone for Coupled Duct- Sytem with Model The reonance of the duct can clearly be een in the peaker velocity repone. Thee caue a large a 5 db and degree of magnitude and phae variation. The free-air reonance of the peaker i alo uperimpoed on the repone. Clearly, the velocity of the peaker i affected by both the peaker and the duct dynamic. For the application of active noie control of an acoutic duct, the objective i to attenuate the reonance in the duct. Figure 7 how that the peaker repone ha the mot error exactly where the control effort i needed, at the duct reonance frequencie. The repone mut be improved if the peaker i to be an effective acoutic actuator. Senor The velocity enor wa then applied to the coupled peaker/duct ytem a how in Figure 8. A -inch foam plug wa placed in the termination end to add damping to the ytem. An end impedance of.6.j wa ued in the model. Open End Tee Joint Dual Voice-Coil Laer Tranducer Clear Window e p e b Senor PVC Pipe deired velocity enor velocity Foam Plug Figure 8: Diagram of Senor in /Duct Sytem Implementation The enor velocity to deired velocity and meaured velocity to deired velocity tranfer function were then meaured from - Hz uing the ignal analyzer. The modeled enor velocity wa alo computed with (8) uing the value of p = Hz. Figure 9 how good agreement between thee three ignal. Phae (Deg) - ---Senor - 5 5 5 3 35 Figure 9: Comparion of, Modeled and Senor Model for Coupled /Duct Sytem Figure 9 how that the good agreement between the meaured velocity and the velocity enor below approximately Hz. There i le than 4 db magnitude and degree phae difference between the two repone. There i ignificant phae error above Hz. Thi error i attributed to the inductance effect in the peaker which become ignificant at high frequencie and which are not included in the velocity enor. Feedback Compenation of The velocity feedback compenation trategy wa then applied to the coupled peaker/duct ytem. The proportional gain, K p, wa varied from to ; and the meaured velocity to deired velocity tranfer function wa meaured from - Hz uing the ignal analyzer. Figure how that the meaured peaker velocity repone approached the deired velocity a the gain wa increaed. The noticeable deviation between the velocity enor and the meaured velocity may contribute to the 45 degree phae error above Hz in the cloed-loop ytem. Magnitude db Phae (Deg) - - -3-4 - - K= K= open loop 4 6 8 4 6 8 K= open loop K= 4 6 8 4 6 8 Figure : Comparion of Cloed-Loop to Primary Voltage Frequency Repone for Open Loop and Proportional ain of,, and The magnitude and phae variation exhibited in open-loop have been minimized. The effect of the duct reonance and the free-air reonance of the peaker are ignificantly reduced. With a value of 5
K p = there i le than 5 db and 45 degree magnitude and phae variation compared with 3 db and 8 degree in the uncompenated ytem. The compenated peaker velocity i indepent from the peaker and duct dynamic. Thi repone i ideal for an acoutic actuator. CONCLUSIONS Thi paper addree uing a compenated audio peaker a an actuator for ytem with trong dynamic preure coupling. It wa hown that the repone of an actuator i degraded by both the internal dynamic of the actuator and the interaction with the plant. Previou olution are not effective for uch application becaue they only compenate for internal dynamic and not the preure interaction from the plant. A new velocity enor which ue a combination of peaker cone motion induced econdary coil voltage and primary coil current i developed and applied in proportional feed-back controller. An acoutic duct i ued a an example of a ytem with trong dynamic preure interaction. It i demontrated through modeling and experiment that the compenated peaker repone minimize the effect of both internal actuator dynamic and coupling through the preure with the acoutic plant. The work preented here repreent an ideal actuator who deign i indepent of the acoutic plant, and unaffected by the dynamic of the plant. The compenation yield a feaible actuator for acoutic ytem with trong preure coupling. Rough, W. J., 996, Linear Sytem Theory, New Jerey, Prentice-Hall, Inc. Seto, W. W., 97, Theory and Problem of Acoutic, New York, Mcraw-Hill Book Company. Spiekerman, C. E. and Radcliffe, C. J., 986, One- Dimenional Acoutic Repone with Mixed Boundary Condition: Separating Total Repone into Propagating and Standing Wave Component, : Doctoral Diertation, Michigan State Univerity. Warner, J., 995, Active Noie Control in an Off-Road Vehicle Cab Noie and Vibration Worldwide, vol 6 n, 7 July. REFERENCES Birdong C.B., Radcliffe C.J., 996, A Compenated Actuator for an Acoutic Duct, Mater Thei, Michigan State Univerity. Bradley, P., 995, Active Aault on Cabin Noie, Commercial Aviation vol 77, September, 6 pp. Doak, P. E., 973, Excitation, Tranmiion and Radiation of Sound From Source Ditribution in Hard-Walled Duct of Finite Length (I): The Effect of Duct Cro-Section eometry and Source Ditribution Space-Time Pattern, Journal of Sound and Vibration, v 3, n pp-7. Hull A. J., Radcliffe C. J. Southward S. C., 993, lobal Active Noie Control of a One-Dimenional Acoutic Duct Uing a Feedback Controller Journal of Dynamic Sytem, Meaurement, and Control, vol 5, September. Hull A. J., Radcliffe C. J., 99, State Space Repreentation of the Nonelf-Adjoint Acoutic Duct Sytem, Journal of Vibration and Acoutic v, October. IEEE Standard 9-975, IEEE Standard Committee of Acoutic, Speech, and Signal Proceing roup, IEEE Recommended Practice for Loudpeaker Meaurement, IEEE td. 9-975. Karnopp, D. C., D. L. Margoli, and R. C. Roenberg, 99, Sytem Dynamic: A Unified Approach, New York, John Wiley & Son, Inc. Radcliffe C. J., ogate, S. D., 996, Feedback Compenation of Electromechanical for Acoutic Application, International Federation of Automatic Control, Triennial World Congre, July. Radcliffe C.J., ogate S.D., Hall., 994, Development of an Active Acoutic Sink (AAS) for Noie Control Application, Active Control of Vibration and Noie, ASME. Radcliffe, C. J., ogate, S. D., 99, Itification and Modeling Dynamic for Acoutic Control Application, ASME Sympoium on Active Control of Noie and Vibration. 6