Modeling of Membrane Sound Absorbers

Transcription

1 Purdue e-pubs Publications of the Ray W. School of Mechanical Engineering 8-- Modeling of Membrane Sound Absorbers J Stuart Bolton, bolton@purdue.edu Jinho Song Follow this and additional works at: Bolton, J Stuart and Song, Jinho, "Modeling of Membrane Sound Absorbers" (). Publications of the Ray W.. Paper 7. This document has been made available through Purdue e-pubs, a service of the Libraries. Please contact epubs@purdue.edu for additional information.

2 Dearborn, Michigan, USA August 9-, Modeling of Membrane Sound Absorbers August 9th, Jinho Song and J. Stuart Bolton Ray W.

3 Introduction - Background Conventional Sound Absorbing Material Open cell Foams/Glass fibers/polymeric fibers Sound energy dissipation by thermal and viscous interaction of sound field and material fibers Visco-thermal boundary layer Oscillatory flow Sketch of Fibrous Material

4 Introduction - Background Conventional Nonfibrous Material Usages Some environmental needs - Healthy Surroundings/Ease of Maintenance - Recycling - Moisture-Resistance Fibrous materials can be used with impermeable membranes to tune their performance m s α Fibrous Material Fibrous material Membrane Fibrous Material + Membrane Frequency - Conventionally, membrane does not dissipate any energy

5 Introduction Motivation & Objectives Recently, it has been observed that stacked sheets of accordion-folded, impermeable membranes (e.g., mylar sheets) offer substantial levels of low frequency absorption even though such arrays feature no obvious dissipative elements. Alternating layers of folded mylar OBJECTIVES: Identify the origin of the sound absorption and behavior capacity of such a treatment - Develop models of those processes - Use those models to optimize the acoustical performance - How does this sound absorption arise? - How do you model this effect? HYPOTHESIS: Dissipation results from losses due to local flexing of membranes stiffened by curvature (i.e., by folding) or tension

6 Theoretical Model Theoretical Approach System Model I REFLECTED WAVE II TRANSMITTED WAVE INCIDENT WAVE TENSIONED MEMBRANE Assumed Solutions (Modal Sums) jkz Region I : P ( r, z, t) e B J ( k r) e Region II : P I II Membrane Displacement : y ( r, t ) An Jo ( k r ) N n N ( r, z, t) C J ( k r) e N o n rn o rn jk z zn n jkz n z Boundary Conditions Continuity of Velocity : P P y P I PII y j z t j z t o z Membrane Equation of Motion : y k f PI P y T II o z T ( c k f f, c ) f s - Dissipation introduced by modeling T as complex : T To ( j) ( : Loss Factor)

7 Theoretical Model Solution Method Solution Method Apply two velocity continuity boundary conditions o and membrane e equation of motion on a point-bypoint basis across the membrane System Response (Membrane Displacement) Membrane Displacement Particle Position I II r Frequency[Hz] Radius[m] y 6 Membrane Displacement (Top View) A ~... A N B Coefficien t... B N Matrix C... C N Number of Point at which B.C. s applied forcing Vector Frequ uency[hz] Radius[m]

8 Model Verification Power Comparison Power Comparison Compare power calculated by using Acoustical Solution with power calculated using Membrane-based based solution - Acoustical Solution a WI Re{ ( *)( ) } PI ui r dr WII Re{ ( *)( ) } PII uii r dr Power Dissipated : - Membrane Based Solution a W d, a W I a W * * Wd, m Re{ ( )( ) } PI ui PIIuII r dr a * Re{ ( )( ) ( ) } T y k f y iy r dr Losses introduced by using T To ( j) II Should be equal if model works properly

9 Model Verification Power Comparison 8 x -7 Using 4 modes 7 Acoustical Solution Membrane Based Solution Using 4 membrane modes x -7 Using 3 modes 7 6 Acoustical Solution Membrane Based Solution 3 8 x -7 Using modes Acoustical Solution Membrane Based Solution Using membrane modes Using 3 membrane modes

10 Model Verification Velocity Measurement Power Amplifier Pre- Amplifier Signal Analyzer Microphone Membrane Amplifier Sound Source Finite i Backing Laser Sensor

11 Model Verification Vibrational Modes Theory Experiment Absolute velocity of membrane - Experiment Phase st v/p / v/p max y x Magnitude v/p / v/p max y x Phase.5 y y x x Absolute velocity of membrane - Experiment Phase nd v/p / v/p max y x Magnitude v/p / v/p max y Phase x.5 y y x x

12 Model Verification Experimental Set-up Power Amplifier Pre- Amplifier Signal Analyzer Sound Source Microphone Test Sample Anechoic Termination B&K Pulse System Speaker Amplifier Microphones Computer B&K Standing Wave Tube

13 Model Verification Experimental Results Sound Pressure from microphones : P to P4 Normal Incident Pressure Transmission Coeff. & Reflection Coeff.: T, R Acoustic Impedance: Z=(+R)/(-R) Transmission Loss: TL= log (/ T ) Sound Absorption: =- R Sound Dissipation: a quick and convenient method for determining i the fundamental d =- R - T acoustical properties

14 Model Verification Model Optimization Given experimental results as input, Find appropriate material properties (T o, ρ s, η ) TL I II Membrane Surface Density ρ s Membrane Tension T=T o (+i) Tensioned Membrane T 8 Pa kg s m Note : Most absorption results from transmission through membrane in anechoic termination case

15 Absorption Mechanism Frequencies of Peak Absorption Membrane with finite-depth backing space Imaginary part of Impedance Many resonances because of membrane dynamics Absorption Coefficient Z n Z n, Membrane Z n, Backing Frequency [Hz] T 75Pa Absorption peaks when Im{ Z n } =.9 kg s m Frequency [Hz] l. 9m - Significant sound absorption in narrow frequency regions produced by dissipation in the membrane

16 Absorption Mechanism Frequencies of Peak Absorption Limp Membrane with finite depth backing space (Conventional Picture) m (Z) I T Pa Membranes with various Tensions Absorption Coefficient To=5 To=75 To= - -5 Duct only Membrane Only Sum Im(Z)=.5.4 Absorption Coefficient Frequency[Hz] No loss mechanism from the membrane Only dissipation comes from wall losses Theory Frequency[Hz] Frequency[Hz] Different kinds of sound absorption characteristics with various tension values.63.9 kg s m l. 9m

17 Conclusions & Future Work Theoretical models for the various membrane systems were developed, which can reproduce the acoustic behavior of stiffened membrane systems A sound absorption mechanism for a tensioned membrane was suggested and verified by using the relation between the acoustic impedance and sound absorption The effects of various parameters in the sound absorption performance were presented, which can provide guidelines for designing a membrane system to enhance its sound dissipation Effect of membrane e permeability, eab stiffening by curvature u and use of light weight dissipative material in backing space will be considered in future