Modeling of Membrane Sound Absorbers

8/19/2019 Modeling of Membrane Sound Absorbers

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Purdue University

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Publications of the Ray W. Herrick Laboratories School of Mechanical Engineering

8-1-2002

Modeling of Membrane Sound Absorbers J Stuart Bolton Purdue University , [email protected]

Jinho Song

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Bolton, J Stuart and Song, Jinho, "Modeling of Membrane Sound Absorbers" (2002). Publications of the Ray W. Herrick Laboratories.

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3/17

Introduction - Background

Conventional” Sound Absorbing Material

Sound energy dissipation by thermal and viscousinteraction of sound field and material fibers

-

boundary layer

Sketch of

Fibrous Material

sc a ory ow

Purdue University Herrick Laboratories


4/17

Introduction - Background

Conventional” Nonfibrous Material Usages Some environmental needs

- Healthy Surroundings/Ease of Maintenance- Recycling

- Moisture-Resistance

Fibrous materials can be used with impermeable membranesto “tune” their performance

Fibrous Materialα

s

- Conventionally, membrane does not dissipate any energy

Fibrous material

+ MembraneMembrane Fibrous Material

Frequency



5/17

Introduction –

Motivation & Objectives Recently, it has been observed that

stacked sheets of accordion-folded,

OBJECTIVES:

Identify the origin of the. .,

sheets) offer substantial levels of lowfrequency absorption even thoughsuch arrays feature no obviousdissi ative elements.

behavior capacity of sucha treatment

- Develop models of those

- Use those models to optimizethe acoustical performance

Alternating layers of folded mylar

-

HYPOTHESIS:

Dissipation results fromlosses due to local flexing

- How do you model this effect? by curvature (i.e., byfolding) or tension



6/17

Theoretical Model – Theoretical Approach

REFLECTED WAVE TRANSMITTED WAVE

I II

System Model

INCIDENTWAVE

TENSIONED MEMBRANE

Region I : z jk

r on

jkz

I n

z

ner k J Bet z r P )(),,(

Assumed Solutions (Modal Sums) Boundary Conditions

Continuity of Velocity :

Region II :

Membrane

z jk

r N

on II

n z

ner k J C t z r P

)(),,( t y

z

P

j z

I

o

0 t z j z

II

o

0

Membrane Equation of Motion :

Displacement : , 0n N on T yk y II I

f 22

f f c

, s f c

- Dissipation introduced by modeling T as complex : )1( jT T o ( : Loss Factor)



7/17

Theoretical Model – Solution Method

Apply two velocity continuityboundar conditions and membrane

Solution Method System Response(Membrane Displacement)

Membrane Displacement

equation of motion on a point-by-

point basis across the membrane

Particle

Position

I II

r

Radius[m]Frequency[Hz]

y

~

N A

A

...

1

1000

1200

1400

1600

Membrane Displacement (Top View)

e n c y [ H z ]

Number

of Point

at which

B.C.’s

applied

N

N

C

C

B

B

...

...

0

0

Matrix

t Coefficien

Vector

orcing

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

200

400

600

Radius[m]

F r e q



8/17

Model Verification – Power Comparison

Compare power calculated by using Acoustical Solution with power

Power Comparison

-

- Acoustical Solution

})2*)((Re{1

a

I I I dr r u P W })2*)((Re{1

a

II II II dr r u P W

Power Dissipated :

0 0

II I ad W W W ,

Should be equal- em rane ase o u on

})2)((Re{2

1 **,

a

II II I I md dr r u P u P W

if model

works properly

Losses introduced by using )1( jT T o

})2())((Re{2

1

0

*22

a

f dr r yi yk yT



9/17

Model Verification – Power Comparison

6

7

8x 10

-7 Using 4 modes

Acoustical SolutionMembrane Based Solution

Using 4 membrane modes

2

3

4

5

7

8x 10

-7 Using 30 modes

Acoustical SolutionMembrane Based Solution

102

103

0

1

8x 10

-7

Using 10 modes Acoustical Solution

Membrane Based Solution

3

4

5

6

3

4

5

6

7

102

103

0

1

2

102

103

0

1

2

Usin 10 membrane modes

Using 30 membrane modes



10/17

Model Verification – Velocity Measurement

SignalAnalyzer

Pre-Amplifier

Power

Amplifier

Microphone Membrane

Laser Sensor

Amplifier

n te ac ngSound Source



11/17

Model Verification – Vibrational Modes

Theory Experiment Absolute velocity of membrane - Experiment Phase

-0.050

0.05

-0.05

0

0.050

0.5

1

xy

| v / p

| / | v / p | m a x

-0.050

0.05

-0.05

0

0.050

2

4

xy

| v / p

| / | v / p | m a x

0.05Magnitude

0.05Phase1st

Absolute velocity of membrane - Experiment Phase

-0.05 0 0.05-0.05

0

x

y

-0.05 0 0.05-0.05

0

x

y

-0.050

0.05

-0.05

0

0.050

0.5

1

xy

| v / p | / | v / p | m a x

-0.050

0.05

-0.05

0

0.05-2

0

2

xy

| v / p | / | v / p | m a x

0.05Magnitude

0.05Phase

2nd

-0.05 0 0.05-0.05

0

x

y

-0.05 0 0.05-0.05

0

x

y



12/17

Model Verification – Experimental Set-up

SignalAnalyzer

Pre-Amplifier

Power

Amplifier

SoundSource

Anechoic Termination

Microphone

Test Sample

B&K Pulse Speaker Amplifier

Computer

ys em

B&K Standing

Microphones

Wave Tube



13/17

Model Verif ication – Experimental Results

microphones : P1

to P4

Z=(1+R)/(1-R)

• Transmission Loss:

Normal Incident PressureTransmission Coeff. &

Reflection Coeff.: T, R

TL=20 log 10 (1/|T|)

• Sound Absorption:

=1-|R|2

• Sound Dissipation:“a quick and convenient method

d = -

-or e erm n ng e un amen aacoustical properties”



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15/17

Absorption Mechanism –

Imaginary part of Impedance Absorption Coefficient

Membrane with finite-depth backing space

Man resonancesbecause of

membrane dynamics

Frequency [Hz] Frequency [Hz]

219.0 mkg

s PaT 75 0063.0 ml 1192.0

Backing n Membranenn Z Z Z ,,

- Absorption peaks when Im{ Zn } = 0

- Significant sound absorption in narrow frequency regions produced by

dissi ation in the membrane



16/17

Absorption Mechanism –

Membranes with various Tensions

Limp Membrane with finitedepth backing space(Conventional Picture) Absor tion Coefficient

0.7

0.8

0.9

1

To=25

To=75To=200

0

5

10

15

20

m (

Z )

PaT 0

0.3

0.4

0.5

0.6

200 400 600 800 1000 1200 1400 1600-20

-15

-10

-5

Frequency[Hz]

Duct only

Membrane Only

Sum

Im(Z)=0

I

200 400 600 800 1000 1200 1400 16000

0.1

0.2

Frequency[Hz]

0.5

0.6

0.7

0.8

0.9

1

Theory

C o e f f i c i e n t No loss mechanism

from the membrane

Only dissipationcomes from wall losses

219.0 mkg

s 0063.0 ml 1192.0200 400 600 800 1000 1200 1400 16000

0.1

0.2

0.3

0.4

Frequency[Hz]

A b s o r p t i o n eren n s o soun a sorp on

characteristics with various tension values



17/17

Conclusions & Future WorkTheoretical models for the various membrane systems were

,

membrane systems A sound absorption mechanism for a tensioned membrane was

impedance and sound absorption

The effects of various parameters in the sound absorptionperformance were presented, which can provide guidelines for

designing a membrane system to enhance its sound dissipation

Effect of membrane ermeabilit stiffenin b curvature and use

of light weight dissipative material in backing space will beconsidered in future


Modeling of Membrane Sound Absorbers

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