ICV4 Cairns Australia 9- July, 7 ITAL ITAC OPTIMIZATIO OF A HLMHOLTZ OATO I OI COTOL OF MALL CLOU Gangua Yu, Deyu Li and Li Ceng Department of Mecanical ngineering, Te Hong Kong Polytecnic University Hung Home, Kowloon, Hong Kong, A of Cina mail address: mmlceng@polyu.edu. Abstract Tis paper examines te influence of te internal resistance of a Helmoltz resonator on te noise control in a small enclosure. Te absorptive process mainly occurring witin te neck of a Helmoltz resonator provides te resonator wit a damping (internal resistance) property, wic directly dissipates te input energy in te resonator. Te remaining non-dissipated energy is re-radiated back to te enclosure, suc forming an effective secondary sound source, and resulting in acoustic interaction wit te primary source. If te internal resistance of te resonator is low, te acoustic interaction between te enclosure and te resonator sarply splits te targeted resonance peak of te enclosure into two parts, and te peak response is significantly attenuated witin a very narrow frequency band. By appropriately increasing te internal resistance at te resonance of te resonator, te working bandwidt can be enlarged at te expense of sacrificing te control performance due to te decreased amplitude. However, if te resistance is over-increased, te strengt of volume velocity out of te resonator aperture becomes too low; compromising te effective acoustic interaction wit te enclosure and resulting in insignificant control at te targeted resonance peak. In tis paper, a matematical model describing te acoustic interaction of a resonator and a small enclosure is presented. An analytical solution is obtained on te pressure field inside te enclosure and te radiation of te resonator. Based on te analytical solutions, an energy reduction index describing te strengt of acoustic interaction in te enclosure wit a resonator is defined. eries of numerical simulations are conducted to illustrate te influence of te internal resistance on te energy reduction and on te dissipated and re-radiated energy. Finally, te optimal internal resistance is obtained. xperimental results using one resonator are also carried out and compared wit simulation results. I. ITODUCTIO Low frequency noise inside small enclosures always as always been a concern in many engineering problems. Active noise control is a promising approac in low-frequency noise controls. However, its cost of implementation, robustness of te control system and te
ICV4 9- July 7 Cairns Australia requirement of a large space amper its application in small enclosures. Passive tecniques suc as Helmoltz resonators still sow many appealing features in terms of implementation, control quality and cost effectiveness. Te acoustic interaction between an enclosure and wit Helmoltz resonators as been extensively investigated in te past. Fay and cofield [] built an advanced model to explore some underlying pysics of te resonator. Te experimental results on internal resistance [] sowed tat tere exists an optimal damping value wen oter parameters were fixed. Wit a view to broaden te application of resonators, Cummings [] extended Fay and cofield s single resonator and single room mode model to multiple resonator coupled wit multi room-mode. In is multi-mode model, resonators were taken as pseudo point sources. In order to solve te singularity problem caused by point source, te sound pressure of tat pseudo point source at its own location was calculated by te averaged sound pressure at te surface of a small equivalent pulsating spere. Li and Ceng [3] recently proposed a new model to analyze te interaction between acoustic resonator array and a room wit multiple modes. In tat work, te resonator was treated as a point source wit volume velocity influenced by enclosure. Instead of building te linear equation at te aperture of resonators, tey considered te acoustic equilibrium in te volume of room to avoid te singularity problem previously encountered wen te linear equation set was solved. By comparing te teoretical results wit experimental results, te model sows very good approximation to real multi-mode setup. Despite te consistent effort tat was made in Helmoltz resonator design, researcers are still looking into approaces to improve te noise control performance using a Helmoltz resonator array, wic involves eavy experimental measurements on a trial-and-error basis. Tis paper presents a systemic design tool to optimize te internal resistance of a Helmoltz resonator. It is divided into four sections. Te matematical model of acoustic interaction between an enclosure and one single resonator, te teory of energy optimization, and investigation of te internal resistance of te resonator are presented in ection II. ection III gives numerical and experimental results to examine te internal resistance of resonators. Finally, conclusions for optimal design of a resonator are presented in ection IV. II. THOY A general model for describing te acoustic interaction between an enclosure and only one Helmoltz resonator is presented, and ten, te energy optimization teory is given. Formulas of dissipated energy and radiated energy by an acoustic resonator, wic depends on te internal resistance of te resonator, are derived. Trougout te paper, te superscripts and subscripts,, and indicate te variables associated wit nclosure, esonator, and ource, respectively. A. Acoustic interaction between an enclosure and a single Helmoltz resonator Te inomogeneous wave equation governing te pressure field in te enclosure reads:
ICV4 9- July 7 Cairns Australia p(,) r t p(,) r t = ρ q (,) r t, () were p(r, t) is te acoustic pressure, q is te volume velocity source strengt density distribution in te volume and surface of te enclosure. Assuming tat a set of armonic sources wit volume velocity source strengt density q q, q,, q are located at te points, r,, r r form te primary sound field in te enclosure, and te single resonator wit volume velocity source strengt density q centered at te point r (center of te resonator aperture) forms te secondary sound field in te enclosure. otice tat te volume velocity directed out of te resonator as te same sign as tat of te primary sound source, i.e., te positive sign is directed out off te source into te enclosure. Te volume velocity source strengt density out of te acoustic resonator is computed from q (t)= p(r,t)/z, in wic Z is defined as te output acoustic impedance at te aperture of te resonator, and p(r,t) is te sound pressure at r provided tat te largest dimension of te resonator aperture is muc smaller tan te sound wavelengt of interest, equation () becomes p ( r, t) δ ( r r ) (,) (,) () ( r n ), () p r t p r t = ρ qn t δ c + r Z n= were δ(r-r ) is a tree dimensional Dirac delta function.. Acoustic pressure p(r,t) can be decomposed on te basis of eigenfunctions of te enclosure: p(r,t)=σψ (t)φ (r),ψ (t) is te t modal response, and φ (r) is te t eigenfunction. ubstituting tis modal expansion into q. () and applying ortogonality properties of te eigenfunctions yield a discrete acoustic equation ( r ) ( r ) cz ϕ cz ϕ ψ () ϕ ( r ) ψ () ( γ ) ψ () q n() t, (=,, 3,, J) (3) n t t + t = V Z Λ V n= Λ were z =ρ c; z is te caracteristic impedance of te fluid, V is te volume of te enclosure, Λ is te mode normalization factor, given by Λ = ϕ () dv V V r, ϕ ( rn ) is te averaged ϕ ( rn ) over te volume of te nt source (first source), andγ is te t eigenvalue of te enclosures, wic olds on te omogeneous wave equation. Te eigenvalue can be expressed as γ = ω + ic, in wic te real part is te angular frequency and te imaginary part is an equivalent ad oc damping coefficient. Assuming all time dependent variables are armonic, i.e. Ψ (t)=p e iωt and q () t Q e ω n i t = n, ten, in te absence of te resonators, te modal response P can be solved from q. (3) as ( P) iωcz = witout resonator ω γ Λ V n= ( ) ( rn ) ϕ Q, (=,, 3,, J). (4) n 3
ICV4 9- July 7 Cairns Australia Wen a resonator is installed, te modal response of P can be solved from q. (3) as ( ) ( ) ( ) ϕ ϕ ϕ n Q czω r r r n n= ( ) Λ Z V ( ) P = iωcz ϕ ( ) n Q γ ω γ ω Λ r n n= ( γ ) ω V cz ( ) ω ϕ Λ i r Contribution of te first sound field Z V Λ ( γ ) ω Contribution of te acoustic resonator, (=,, 3,, J). (5) Te first term on te rigt and side of q. (5) is te contribution coming from te primary sound field, and te second term is te effect of inserting an acoustic resonator into te enclosure. B. nergy optimization Wen inserting a designed resonator to target a resonance peak of te enclosure, two new coupled frequencies were produced on te eiter side of original resonance frequency []. It is ard to carry out parameters optimizations of resonator by selecting ust a single frequency as control target since te presence of tose two new coupled frequencies. It is necessary to involve maor frequency components around te resonance peak [4] for optimizations of resonator. To tis end, energy in a frequency band is determined as te obective function for te optimization of resonator location and damping. Te teory of te energy optimization is derived as below. By using p(r,t)=ψ (t)φ (r) andψ (t)=p e iωt, te energy witin a bandwidt [ω,ω] is calculated based on spatially averaged mean-square pressure : ω T [ ω, ω] = [ e( p) ] dt dv d e( ) V P ω ω ω ωi V = Δ T Λ ( ), (6) i were Δω is te numerical integration step. Te energy reduction is defined as te ratio of energy level [ω,ω ]wit resonator to tat [ω,ω ] witout resonator. [ ω ω ] [ ω, ω] [ ω, ω ], = log = log e ( Λ ) P ( ωi) i ( Λ ) e P ( ωi) i, (7) wereω i varies in bandwidt[ω,ω ]. If local noise control is expected, a small V around tis region can be selected to calculate [ω,ω ]. However, if global noise control is expected, te wole enclosure volume sould be selected for te calculation of [ω,ω ]. C. Damping optimization for Helmoltz resonators As mentioned before, inserting a ligtly damped resonator into a ligtly damped enclosure can split te controlled armonic peak to two peaks. In order to enlarge te working bandwidt of a resonator, damping material is introduced to abase tose armonic peaks between te two coupled frequencies. However, excessive damping is not effective, as 4
ICV4 9- July 7 Cairns Australia reported in reference []. Terefore, tere exists an optimal damping to implement optimal control performance of a resonator. Te optimization of internal resistance will be carried out based on te energy reduction described in q. (7). Wen te internal resistance of te inserted resonator varies, te modal response P of enclosure varies, correspondingly. Two important types of energy, dissipated energy and radiated energy by resonator, are te key parameters to evaluate te control performance of a resonator, wic are influenced by resonator resistance. Firstly, te relationsip between dissipated energy and resistance is investigated. Te motion of lumped mass in te resonator neck follows te ewton s second law: ρc ( ) ρ L xt () + z Lxt () + xt () = p( r,) t, (8) V were x(t) is te particle displacement in te resonator neck, wic is assumed positive wen it points to te enclosure, and V te body volume of resonator, te cross sectional area of te neck, L te effective neck lengt, and te internal resistance. Assuming te movement of tis lumped mass is a armonic variable, namely x(t)=xe iωt and considering te decomposition of pressure based on mode sape function sown above, te amplitude of displacement is solved from q. (8) as: J X = ϕ ( ) ( ) P ω ω icω r. (9) + ρl = Because energy is defined in a bandwidt [ω,ω], parameters used to calculate energy must be mean square values in te same band. Tus, based on q. (9), te mean square velocity in tis band can be given as: ( ) J ω ( ) ϕ r P ω ω T = [ e( )] ω, ω ω ω ω v = v dt d = Δ T ( ρl ) ω ω cω ( ) + ( ). () By using te definition of mean square velocity in a band in q. (), te averaged dissipated energy of a resonator in te band [ω,ω] is: cω v P d = i = Δω ω, ω ρl J ϕ ( r ) ( ω ), () = ( ω ) ω + ( cω ) were i =z L is te specific acoustic resistance of a resonator. After discussing te dissipated energy by a resonator, te influence of damping effects on te sound field radiated by a resonator will be investigated as te follows. ince sound pressure in te enclosure is te summation of te primary and secondary sources formed by te inserted resonator, te sound pressure radiated from te resonator can be directly analyzed wen te primary field source is turned off. Assuming tat te sound source velocity strengt of primary source is zero, and te lumped mass in te neck of resonator vibrates wit te displacement x(t)=xe iωt [X can be calculated by q. (9)], te modal response is calculated by: 5
( ) ( γ ) ICV4 9- July 7 Cairns Australia ( r ) czω ϕ P = i ϕ ( ) only _ resonator r P, () ω V Z Λ were P is calculated by q. (5). ubstituting (P ) only_resonator into q. (6), te space averaged energy radiated only by a resonator can be calculated. III. UMICAL IMULATIO AD XPIMT umerical simulations on internal resistance are conducted in te following. Te coupling model between te enclosure and a resonator as been previously validated [3]. One room, wit dimensions l x = m, l y =.7 m and l z =. m, was used as enclosure for experiments and simulations. All pysical parameters used are tabulated in TABL I. umerically 6 enclosure modes are used in modal superposition. atural frequencies wit damping are calculated by formulaγ = ω +, in wic te real part ω is te angular frequency of te ic t enclosure mode witout damping and te image part is te t ad oc damping coefficientc ω Q. Q is set to 56 for te rigid mode () and 46 for oter modes. Te = eigenfunctions φ (r ) of te enclosure for termalviscous boundary conditions were presented in reference []. One square source wit dimensions of mm in te x- and y-directions and zero in z-direction was located at (, 55, ) mm to drive te first sound field. One Brüel & Kær Type 489 ½ micropone located at (.84,.3,.6) m was used to measure PL in te enclosure. TABL I. Pysical parameters Pysical parameter value Ambient temperature, T ( o C) 5 peed of sound, c (m/s) 34.3 Density of air, ρ (kg/m 3 ). pecific eat ratio of air, γ.4 Termal conductivity of air, κ [W/(m K)].63 pecific eat at constant pressure of te air, C p [J/(kg K)]. 3 Coefficient of sear viscosity, µ (Pa. s).85-5 For optimization of internal resistance, mode () was selected as targeted mode. Te calculated resonance frequency for tis mode is 9.8Hz and te experimental value is 4Hz. A Helmoltz resonator was designed to target tis mode and fixed at (.,.3, ) m. Te internal diameter of its neck is mm and te pysical neck lengt is 58mm. Te internal diameter and lengt of its body are 74mm and 67mm, respectively. Te optimization frequency band was from Hz to 4Hz. Te Q-factor of te resonator was varied from to, corresponding to a variation range of te resistance from.5 to 5. mks ayls. Te energy reduction was calculated by q. (7) wit different damping values. Te dissipated energy was calculated by q. (). Te radiated energy was calculated by q. (6). FIG.(a) sows te variation of energy reduction versus te internal resistance. Wen te resistance is equal to 4. mks ayls, te energy reduction in te enclosure reaces te maximum value.34 db. Tus tis resistance is te optimal one based on te energy 6
ICV4 9- July 7 Cairns Australia approac. FIG.(b) sows te variation of radiated energy from te resonator and FIG.(c) sows te energy dissipated by resonator. It is obvious tat te variation tendency in FIG.(c) is similar to tat in FIG.(a). Terefore, te influence of resonator resistance on te energy reduction in enclosure is dominated by te dissipation ability of resonator. From FIG. (a, b, c), it is observed tat wen te resistance is muc small, te radiation from te resonator is efficient and te dissipated energy is very small. Terefore, more energy is returned back to te enclosure from te resonator aperture at tis circumstance. Wit te increase of resistance, more sound energy is dissipated by te resonator and te radiated energy is small. However, bot te dissipated energy and radiation energy decreases after te internal resistance of te resonator increases over te optimal value. FIG.. (a) nergy reduction in enclosure measured at (.84,.3,.6) m; (b) adiated energy by resonator; (c) Dissipated energy by resonator. FIG. sows te experimental results in terms of sound pressure level (PL) at te micropone position (.84,.3,.6) m wen resonator resistances are equal to 3.6 and 8. mks ayls. Te resonator resistance 3.6 mks ayls is near te simulated optimal resistance 4. mks ayls, and a large PL reduction up to 6.9dB at 4Hz was observed. A control over te off-target frequency in a relatively broadband is also obtained from FIG. since te resonator is coupled wit all enclosure modes. Te internal resistance 8. mks ayls, muc larger tan te optimal resistance, results in a very low vibrating velocity of te lumped mass inside te resonator neck and subsequently lower energy dissipation inside te enclosure and low radiated energy out of te resonator aperture. Only a.66db PL reduction at 4Hz was 7
ICV4 9- July 7 Cairns Australia obtained wen te internal resistance of te resonator is 8. mks ayls. FIG.. PL curves at (..84,.,.6) m. xperiment: witout resonator ; wit a damped resonator r = 3.6 mks ayls ; wit a damped resonator, r = 8. mks ayls.. IV. COCLUIO A teoretical metod for te optimal design of a Helmoltz resonator internal-resistance is presented based on te maximization of energy reduction. umerical results sow tat te dissipated energy dominates te control performance of te resonator (a good cutoff between te peak reduction and te resonator working bandwidt), wic greatly depends on bot te internal resistance in te resonator neck and te vibrating velocity of te lumped mass in te neck. A large vibrating velocity of te lumped mass (wen te internal resistance is small) can provide a large re-radiation volume velocity, wic reacts to te primary sound field to sarply reduce te targeted peak by means of acoustic interaction. Te measured results sow tat te optimally designed internal resistance can provide a better control performance for te resonator. Using a Helmoltz resonator wit an optimal internal resistance, a PL reduction up to 6.9dB was obtained around te targeted resonance wit 5 Hz bandwidt. Acknowledgements Te autors wis to acknowledge a grant from esearc Grants Council of Hong Kong pecial Administrative egion, Cina (Proect o. PolyU 537/6) and support by te Central esearc Grant of Te Hong Kong Polytecnic University troug Grant G-Y67. FC [] F. J. Fay and C. cofield, A note on te interaction between a Helmoltz resonator and an acoustic mode of an enclosure, Journal of ound and Vibration, Vol. 7, 365-378 (98). [] A. Cummings, Te effects of a resonator array on te sound field in a cavity, Journal of ound and Vibration, Vol. 54, 5-44 (99). [3] Deyu. Li and L. Ceng, Teoretical study of noise control in small enclosures using T-saped acoustic resonators, J. Acoust. oc. Am, to be publised. [4] L. Ceng and. Lapointe, Vibration attenuation of panel structures by optimally saped viscoelastic coating wit added weigt considerations, Tin-walled tru. 37-36 (995). 8