10 vac statistical energy analysis -...

David HerrinUniversity of Kentucky

An Introduction to Statistical Energy Analysis

Vibro-Acoustics Consortium

August 17, 2020


August 17, 2020

2

• Basic Theory• Fundamentals• Measuring SEA Parameters• Examples

Overview


August 17, 2020

Room Acoustics Theory

Direct FieldReverberant Field

Direct Field Reverberant Field

𝐿 𝐿 10 logΓ

4𝜋𝑟4𝑅

𝑝 20 10 Pa𝑊 1.0 10 W

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August 17, 2020

In the Reverberant Field

𝐿 𝐿 10 log4𝑅

𝐿 10 log𝑝𝑝

10 log𝑊𝑊 10 log

𝑅4

𝑝𝑝

Input Power

𝑊𝑊

𝑅4

𝑝𝑝

Acoustic Energy“Loss Factor”

𝑊 𝜂 𝜔𝐸

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Theory Coupled Simple Oscillator

Burroughs et al., 1997

𝑀 𝑀𝐾

𝑥 𝑥

𝑅

𝐾𝐾

𝑅𝐵

Gyroscope

Power transferred from subsystem 1 to 2 (for a frequency band)

Constant which is a function of the oscillator properties

Energy in subsystems 1 and 2 (for a frequency band)

𝐹 𝑡

Assumption: Forces are assumed to be broadband and incoherent.

𝐹 𝑡𝐸 𝑀 𝑣 𝐸 𝑀 𝑣

𝑊 𝛽 𝐸 𝐸

𝑊

𝛽

𝐸 ,𝐸


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Theory Coupled Simple Oscillator


𝑀 𝑀𝐾

𝑥 𝑥

𝑅

𝐾𝐾

𝑅𝐵

Gyroscope

• Power flow is proportional to the differences in the (modal) energies.• The constant relating the power flow to energy difference is a function of the

coupling parameters and oscillator properties.• Power flows from the oscillator with higher (modal) energy to the oscillator

with lower (modal) energy.

𝐹 𝑡 𝐹 𝑡𝐸 𝑀 𝑣 𝐸 𝑀 𝑣

𝑊 𝛽 𝐸 𝐸


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Theory Coupled Multi-Resonant Case


𝑁 modes or oscillators 𝑁 modes or oscillators

𝐸𝐸𝑁

𝑊 𝛽 𝑁 𝑁 𝐸 𝐸

𝐸𝐸𝑁


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Theory Coupled Multi-Resonant Case

Assumption More modes implies greater energy storage potential.

Coupling Loss Factors


𝑊 𝛽 𝑁 𝑁 𝐸 𝐸

𝑊 𝜔 𝜂 𝐸 𝜂 𝐸

𝜂𝛽 𝑁

𝜔 𝜂𝛽 𝑁

𝜔

𝜂𝑁 𝜂𝑁


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Overview


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What is SEA?

• SEA is a lumped parameter approach for vibro-acoustic analysis that accounts for the flow of energy throughout the system based on statistical coupling of the system modes.

• Lumped parameter: each component or subsystem is a single entity having its own energy, either acoustic or vibrational.

• Energy flow: steady-state averages are determined from energy balances for each component or subsystem.

• Statistical averages: sound pressure and vibration velocity are values averaged over a band of frequencies and over space.


August 17, 2020

Frequency Averaging

0 100 200 300 400

Response

Frequency (Hz)

FE or BE

SEA

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August 17, 2020

What is SEA? A Simple Example

Flexible wall, velocity 𝑣2

Cavity volume 𝑉, sound pressure 𝑝

Absorption coefficient

Physical World

Oscillating force

Sound source

SEA World

Power from force Sound Power

Power absorbedPower damped

𝐸vib 𝑚 𝑣 𝐸acoust𝑝 𝑉𝜌𝑐

Flexible Wall Acoustic Cavity

Rigid walls

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August 17, 2020

Water Tank Analogy

(Usually, 𝐸 and 𝐸 are our unknowns)

𝑊 , 𝑊 ,

Known input powers

𝑁𝑁𝐸vib

𝑁𝑚 𝑣𝑁 𝐸acoust

𝑁1𝑁

𝑝 𝑉𝜌 𝑐

𝑊out,damp 𝜂𝜔 𝐸vib 𝑊out,absorb𝑆𝑐𝛼4𝑉 𝐸acoust

Wva

𝑊in,force 𝑊out,damp 𝑊va 𝑊in,sound 𝑊out,absorb 𝑊va

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Assumptions Weak Coupling

Fahy, 1994

• Energy is uniform everywhere within a subsystem.• More energy is dissipated in a subsystem than is transmitted to other

subsystems.

Room 1 Room 2


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YES No

Plate Beam

Assumptions Weak Coupling


August 17, 2020

Understanding Loss Factors

Fahy, 1994

Damping Loss Factor – Energy lost by structural damping and acoustic radiation damping. Coupling Loss Factor – Energy lost by transmission across a junction.

Room 1 Room 2

Energy absorbed at junction is included in the damping loss factors (not the coupling loss factor). Some damping at junctions is beneficial to SEA since it promotes weak coupling.

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Overview


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Measuring Input Power

𝑊 Re 𝐹 𝑣∗ 𝐹 Re 𝑌

𝑌 is the mobility at the input location. Excitation Panel

AccelerometerLoad Cell Electro

Magnetic Shaker


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Measuring Damping Loss Factors

𝜂𝛾

27.3𝑓

Accelerometer

Hammer

Decay Test

where 𝛾 is the initial slope of the transient response.

-40

-30

-20

-10

0

10

20

30

40

50

60

1.127 1.147 1.167 1.187 1.207 1.227 1.247Time (sec)

Tran

sien

t Res

pons

e (d

B)

Lyon and Dejong (1995)


August 17, 2020

Measuring Energy

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Structural Subsystems

𝐸 𝑀 𝑉

Acoustical Subsystems𝐸

𝑉𝜌𝑐 𝑝

Electro Magnetic Shaker

Load Cell

Accelerometer

indicates spatial averaging.


August 17, 2020

Subsystem 1

Measuring Coupling Loss Factors

Subsystem 2(Heavily Damped)

𝐸 – Energy of Response Panel 𝑗 with respect to Excitation Panel 𝑖

𝑊 – Input power for subsystem 𝑖

𝜂1𝜔

𝐸𝐸

𝑊𝐸

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August 17, 2020

Subsystem 1

Step 1 Excite Subsystem 1


Measure accelerations (spatially average over several points) to find 𝐸 and 𝐸 .

𝜂1𝜔

𝐸𝐸

𝑊𝐸

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August 17, 2020

Subsystem 1

Step 2 Excite Subsystem 2


Measure 𝑊 using impedance head and 𝐸 accelerometer.

𝜂1𝜔

𝐸𝐸

𝑊𝐸

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Overview


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4

1

3

2

56

5 mm Plastic

3 mm Steel

0.5 m x 0.5 m

0.7 m x 0.5 m

0.8 m x 0.1 m

0.5 m x 0.1 m

1

3

2

4

5

6

Test Article

Subsystems used for experimental SEA

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VA-One Model

Measured Coupling Loss Factors

Measured Input Power

Semi Infinite Fluid


August 17, 2020

1.E-15

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

Ener

gy (N

-m)

Frequency (Hz)

Measurement

SimulationExcitation

Response

Single Input Flexural Energy

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August 17, 2020

1.E-15

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

Ener

gy (N

-m)

Frequency (Hz)

Measurement

Simulation Excitation

Response


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August 17, 2020


Excitation

Response

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

Ener

gy (N

-m)

Frequency (Hz)

MeasurementSimulation

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August 17, 2020

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

Ener

gy (N

-m)

Frequency (Hz)

Measurement

Simulation

Response

Two Inputs Flexural Energy

Excitation

Excitation

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August 17, 2020

Two Inputs Flexural Energy

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

Ener

gy (N

-m)

Frequency (Hz)

Measurement

Simulation

Response

Excitation

Excitation

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August 17, 2020

Experiment Setup

• Electromagnetic shaker is attached on steel panel as input source• Average sound pressure level inside the acoustic cavity is measured• Radiated sound power from one side of the enclosure is measured

Electro Magnetic Shaker

62 4

3

57

Radiated Power1

Impedance Head

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August 17, 2020

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Cavity Sound Pressure Level Comparison

30

40

50

60

70

80

Soun

d Pr

essu

re L

evel

(dB)

Frequency (Hz)

Measurement

Simulation


August 17, 2020

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Radiated Sound Power Level Comparison

0

10

20

30

40

50

60

Soun

d Po

wer

Lev

el (d

B)

Frequency (Hz)

MeasurementSimulation


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Treatment 1 Structural Damping

• Damping applied on panels 1, 2, and 6.

62 4

3

57

Radiated Power1


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0

10

20

30

40

50

60

Soun

d Po

wer

Lev

el (d

B)

Frequency (Hz)

BaselinePanel 1 Extra DampingPanel 2 Extra DampingPanel 6 Extra Damping


Simulation


August 17, 2020

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0

10

20

30

40

50

60

Soun

d Po

wer

Lev

el (d

B)

Frequency (Hz)

BaselinePanel 1 Extra DampingPanel 2 Extra DampingPanel 6 Extra Damping

Measurement



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Sound Absorbing Material

1 inchFiber

attached inside cavity

Treatment 2 Sound Absorption

Leak Path


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Disconnect SIF from cavityCavity gap is blocked by lagging material

Treatment 3 Barrier in Leak


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Effect of Acoustical Treatments

0

10

20

30

40

50

60

Soun

d Po

wer

Lev

el (d

B)

Frequency (Hz)

BaselineAbsorptive LiningBarrier Material

Simulation


August 17, 2020

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0

10

20

30

40

50

60

Soun

d Po

wer

Lev

el (d

B)

Frequency (Hz)

Baseline

Absorptive Lining

Barrier Material

Effect of Acoustical Treatments

Measurement


August 17, 2020

Experiment Sample Boxes

19.6 x 19.6 in2

(1/16 in steel)

36 x 36 x 36 in3 wood box(1/2 inch thick)

26.4 x 19.6 x 19.6 in3 steel box

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August 17, 2020

Experiment Approach

• Upper box rests on lower box.• Source is placed in the lower box.• Average sound pressure level

(SPL) of two boxes measured.

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August 17, 2020

Experiment Approach

• Loudspeaker source placed in larger box.• Two microphones roved inside of boxes to identify spatially averaged SPL.

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August 17, 2020

Noise Reduction Comparison

0

10

20

30

40

50

160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000

Noi

se R

educ

tion

(dB)

Frequency (Hz)

Experiment

Simulation SEA

Simulation FEM

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August 17, 2020

SEA Model (No Flanking Path)

Diffuse Acoustic

Field

Simulation SEA VA One

SEA Model (With Flanking Path)

Diffuse Acoustic

Field

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August 17, 2020

Noise Reduction Comparison

0

10

20

30

40

50

160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000

Noi

se R

educ

tion

(dB)

Frequency (Hz)

ExperimentSEA (Flanking Path Included)SEA (No Flanking Path)

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Overview


August 17, 2020

References

• C. B. Burroughs, R. W. Fischer, and F. R. Kern, “An Introduction to Statistical Energy Analysis,” J. Acoust. Soc. Amer., Vol. 101, No. 4, pp. 1779-1789 (1997).

• F. J. Fahy, “Statistical Energy Analysis: A Critical Review,” Phil. Trans. R. Soc. Lond. A, Vol. 346, pp. 431-447 (1994).

• R. H. Lyon and R. G. DeJong, Theory and Application of Statistical Energy Analysis, RH Lyon Corp. (1998).

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10 vac statistical energy analysis -...

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