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Tunable Flat-Plate Absorber Design for Active Sound Absorption K

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Tunable Flat-Plate Absorber Design for Active Sound Absorption K Tunable Flat-Plate Absorber Design for Active Sound Absorption K. Bjornsson1, R. Boulandet1, H. Lissek1 1. Signal Processing Laboratory LTS2, Ecole Polytechnique Fédérale de Lausanne, Switzerland 1 Introduction to analytical predictions of a lumped model and the per- formance of a prototype is discussed. Low frequency noise (20-200 Hz) is an issue in many workplaces and indoor environments and can negatively 2 Lumped modeling of the system impact individuals’ work performance and well-being [1]. In rooms and enclosed spaces, the annoyance of low 2.1 Passive system frequency noise can be exacerbated by strong room res- onances leading to highly non-diffuse sound fields and A schematic of the lumped system configuration is prolonged decay times [2]. Effective means of damping shown in Figure 1. The system subject to the analy- these modes and dissipating the low frequency sound sis is composed of a suspended plate, an air back cavity energy are then required to equalize the room response. and an electrodynamic inertial exciter. We consider the As opposed to high frequency noise, dissipating low fre- placement of the system in a waveguide excited by a quency sound energy by passive means is problematic. loudspeaker at one end. Porous absorbers are impractical for sound absorption at frequencies towards the lower end of the audible spec- trum and require thicknesses on the scale of meters for effective sound absorption. Helmholz resonators and membrane absorbers offer low frequency sound dissi- pation at a relatively compact size but are limited by their narrow and fixed frequency range. Active sound absorption is an attractive alternative to passive sound absorption at low frequencies. The con- cept of an electroacoustic absorber is one interesting ex- Figure 1: Schematic of lumped system configuration. ample. It consists of a loudspeaker, used as a membrane absorber, the acoustic impedance of which can be mod- The elastic plate is modeled as a rigid piston with mass ified to target optimal absorption in the low-frequency mp and effective area Sd. The piston suspension in- regime. Using feedback control such absorbers have cludes a mechanical compliance Cms and a mechan- been shown to be capable of providing broadband sound ical resistance Rms. The mechanical compliance due absorption in a tunable manner [3]. In this paper the de- to the closed-box air volume Vb is modeled as Cmb = sign of a tunable flat-plate absorber is introduced. The 2 2 Vb/(ρc Sd) where ρ is the density of air and c is the mechanical system is an elastically suspended plate that speed of sound in air. The inertial exciter consists of a is developed to exhibit rigid behavior in the low fre- frame mass mf rigidly attached to the piston, a proof quency range. An electrodynamic inertial exciter is at- mass ma rigidly attached to the rear end of the cabi- tached to this plate and with feedback control, imple- net and a voice coil with a force factor Bl. The total mented in the same manner as in [3], this system is moving mass of the system is then mtot = mp + mf turned into a tunable electroacoustic absorber. For sim- and the total mechanical resistance and total compliance plicity, the absorber’s performance is evaluated by as- of the system are Rtot = Rms + Rma and Ctot = −1 suming incident plane waves, i.e. the system is mod- (1/Cms + 1/Cma + 1/Cmb) respectively. The to- eled in a one-dimensional waveguide. We present the tal mechanical impedance of the moving mass, relating design and modeling of this electro-mechano-acoustic the force on the piston and its velocity, can be derived system, including the feedback control law which al- as, ( ) lows to turn it into a tunable electroacoustic absorber. 1 Zm = Rtot + j !mtot − (1) The simulated performance of the system is compared !Ctot Excerpt from the Proceedings of the 2018 COMSOL Conference in Lausanne 2.2 Active system 2.3 Sound absorption in a waveguide In the case of active absorption a force is exerted on At low frequencies we can assume plane waves within the piston so that its response is modified and its spe- the duct. The pressure within the duct then follows cific acoustic impedance is closer to a target value. The the one-dimensional wave equation. Transforming equation of motion of the moving mass can be derived to the frequency domain yields the Helmholtz equa- from Newton’s second law of motion. In the frequency tion, ( ) d2 domain it writes, + k2 p^(x; !) = 0 (6) dx2 Z v^ =p ^ S − Bl^{ (2) m L L d We consider a loudspeaker located at x = 0, represented which after some manipulation yields, by a velocity boundary condition. The loudspeaker is ( ) a sound source providing low frequency sound energy to be dissipated by the absorber. The absorber itself is ^{ 1 − Zm = Sd (3) located at x = L and is represented by an impedance p^L Bl Zs boundary condition. Given (6) and these boundary con- where Zs(!) =p ^L(!)/^vL(!) is the specific acous- ditions, the expression for the complex pressure within tic impedance of the controlled system. We now de- the duct can be derived as, fine the transfer function Θ(!) = ^{(!)/^p (!). Mea- L +jk(L−x) j j −jk(L−x) suring the total pressure near the piston surface we can e + R e p^(x) = ρcv^0 − (7) use this transfer function to determine the electrical cur- e+jkL − jRje jkL rent required to a achieve some target specific acoustic where the reflection coefficient R is the complex ratio impedance Z (!), st between the reflected and incident pressures [4]. It’s ( ) magnitude is given by, ^{ 1 Zm Θ = = Sd − (4) p^L Bl Zst Z − ρc jRj = s (8) For an electroacoustic absorber placed in a waveguide, Zs + ρc perfect absorption is achieved when the piston’s specific The absorber’s absorption coefficient, the ratio of the acoustic impedance equals the characteristic impedance acoustic power that is absorbed by the absorber and of air, i.e. Zst = ρc. The characteristic impedance of air the acoustic power that is incident to it, can be derived is a real number whereas, however, the specific acoustic as, impedance is generally not due to the presence of mass α = 1 − jRj2 (9) and compliance. In practice, it will be impossible to completely cancel the effects of the mass and the com- We define effective sound absorption as being the sound pliance. To account for this we define the target specific absorption resulting in sound energy in front of the ab- acoustic impedance in the following way, sorber that is less than twice the corresponding sound ( ) energy for a perfect absorber. The minimum absorption !mtot 1 coefficient for effective sound absorption can then be Zst = Rst + j µ1 − µ2 (5) Sd Sd!Ctot shown to be, p − − 2 ≈ where µ1 and µ2 are positive real coefficients that de- α = 1 ( 2 1) 0:83 (10) crease (or increase) the effective mass µ1mtot and the effective stiffness µ2/Ctot. By using different µ1 and µ2 factors it is possible to change the ratio of the effec- 2.4 System design tive mass and the effective stiffness and thus shift the resonance frequency of the active system from the reso- The lumped system parameters are chosen so that the nance of the passive system [3]. In practice, the choice absorber resonates at around f0 = 100 Hz and is capa- of µ1, µ2 and Rst, will determine the stability of the ble of achieving effective sound absorption in the fre- controlled system. quency range of 50–200 Hz by tunable means. The sys- tem is designed to be as compact as possible, which con- strains the closed-box volume. This will in turn increase the total compliance requiring a plate of relatively large mass for obtaining resonance around f0 = 100 Hz. The Excerpt from the Proceedings of the 2018 COMSOL Conference in Lausanne Table 1: Lumped system parameters. Properties of the inertial 3 Numerical modeling exciter correspond to the HIAX25C10-8/HS by HiWave. 3.1 Model setup Parameter Symbol Value Unit Closed-box volume Vb 9.8 L Multiphysics simulations are carried out coupling two 2 Effective piston area Sd 491 cm acoustical domains with a structural domain. The Piston mass mp 114 g COMSOL model is illustrated in Figure 2. This model Proof mass ma 84 g is created as a replica of an experimental setup for Frame mass mf 1 g testing a prototype of the system. A velocity boundary −5 Compliance of air Cmb 2.77·10 m/N −4 condition is applied to one end of the waveguide Compliance of surround Cms 1.18·10 m/N − and serves the role of a loudspeaker providing sound Compliance of exciter C 6.30·10 4 m/N ma energy to be dissipated. The waveguide is modeled as Resistance of surround Rms 2.00 N·s/m a perfectly rigid structure using sound hard boundaries, Resistance of exciter Rma 0.06 N·s/m Force factor Bl 4.28 N/A as is the cabinet of the resonator. The plate and the Density of air ρ 1.21 kg/m3 surround suspension are implemented as shells. Speed of sound in air c 343 m/s Resonance frequency f0 100.8 Hz lumped system parameters are listed in Table 1. An im- portant assumption in the lumped modeling is that the plate behaves like a rigid object. If bending modes of the circular plate are induced the simplistic approach of modeling the system like a rigid object is no longer ade- quate and more sophisticated methods of modeling and control are required.
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