Structural‐Acoustics Tutorial

Part 1 ‐Fund

amentals

Dr. Step

hen A. H

ambric

ASM

E IM

ECE 2009

Orlando

, Florida

Overview

•Instructor

•Structural Vibratio

ns–Mod

es in

 beams and plates

–Mob

ility

•Soun

d radiated

 by structural waves

–Ra

diation efficiency

•Structural waves gen

erated

 by im

pinging soun

d–Transm

ission

 loss

•Brief introdu

ction to num

erical m

etho

ds–FE, B

E, SEA

Instructor

•Dr. Step

hen A. H

ambric

–App

lied Re

search Lab, Pen

n State University

–Also Professor in Pen

n States’ G

radu

ate Program in

 Acoustics

–Associate Dire

ctor, Pen

n State Ce

nter fo

r Acoustics 

and Vibration (CAV

)

•Accom

panying materials:

–Structural Acoustics Tutorials, Parts 1 and

 2, from 

Acoustics Toda

y magazine

–Dow

nload at www.ham

bricacou

stics.com

http://www.hambricacoustics.com/

Motivation

•Structures can

 amplify

 (and

 atten

uate) sou

nd 

sources substantially

Soun

d Source

Transfer Fun

ction

Soun

d at Receiver

dBdB

dB

Structural Acoustic

 Transfer F

unctions

•Define transfer fu

nctio

n:

•How

 is it defined

?

2rad

P F

2 22

2ρ

ρ=

rad

oo

ao

rd

oPcAv

F

v Fc

PA

Soun

d po

wer 

radiation 

efficiency

Soun

d po

wer

App

lied force2

Surface 

averaged

 mob

ility

Fluid 

impe

dance

Surface Averaged

 Mob

ility

•Averaged

 surface vibratio

n am

plitu

des caused

 by know

n forces (m

obilitie

s)

2 2

v F

Peaks in th

e mob

ility 

functio

n are caused

 by 

mod

es of reson

ance

Mod

es of R

eson

ance

•Supe

rposition

 of forward and backward traveling waves

•Exam

ple for fle

xure of sim

ply supp

orted be

am:

•For o

ther bou

ndary cond

ition

s, waven

umbe

rs change, 

e.g., for free

 bou

ndaries:

22

22π

ωρ

ρ=

=m

mEI

mEI

kA

aA

22

2

(21)

4π

ωρ

−≅

mm

EIa

A

Mod

es of R

eson

ance

•Can also be compu

ted with

 Finite

 Element (FE) 

mod

els or m

easured

–Measured and simulated

 resonance freq

uencies seldom

 match exactly

Glass plates 

with

 free

 bo

undary 

cond

ition

s

Mob

ility as a Summation of M

odes

•A structure’s m

obility fu

nctio

n is a sim

ple 

series sum

mation of m

odal re

spon

ses to a 

driving force

–Exam

ple:  sim

ply supp

orted rectangular p

late

()

02

11

02

,sin

sin

4

1sin

sin

,(

)ω

ωπ

πω

ρπ

π∞

∞

==

⎛⎞

⎡⎤

⎡⎤

=×

×⎜

⎟⎢

⎥⎢

⎥⎛

⎞⎣

⎦⎣

⎦ ⎠⎝

⎜ ⎝−

⎟ ⎠

∑∑

mo

mo

nn

mx

ny

xy

ab

hab

xy

mx

ni

yv

aF

b

Mod

al m

ass 

(fraction of 

static m

ass)

Differen

ce 

betw

een drive 

and resonance 

freq

uencies

Mod

e shape at 

respon

selocatio

n

Mod

e shape at 

drivelocatio

n

Mod

al Respo

nse Peaks

•Why isn’t m

odal re

spon

se infin

ite at 

resonance freq

uencies?

•Be

cause 

ωmnis com

plex, due

 to dam

ping

–For lightly dam

ped system

s:

whe

re η

is th

e loss fa

ctor

()

22

1, ,

()

ωω

∝−

oo

mn

vxy

Fxy

12η

ωω

⎛⎞

≅+

⎜⎟

⎝⎠

mn

mn

i

Dam

ping

 Mechanism

s

•Thin she

ets of rub

ber adhe

red 

to th

e surface, or sand

wiche

d be

tween structures

•Adjoining

 structures

–Co

upling losses

–Joint losses (friction)

•Soun

d radiation

Effects of Dam

ping

 on

 Mod

al Respo

nse Peaks

•Peak (and

 ‘anti‐p

eak’) amplitu

des de

crease with

 increasing

 dam

ping

–Mob

ility app

roache

s the mean levels of an infin

ite plate

How

 can

 mob

ility 

of an infin

ite plate 

be com

puted?

Infin

ite Structure M

obilitie

s•

A structure is effectively infin

ite whe

n waves re

flected

 from

 bou

ndaries are 

very weak with

 respect to the original 

waves emanating from

 a sou

rce

•For plates:

•Useful for:

–scaling mob

ilitie

s

–pe

rforming en

gine

ering estim

ates of 

prop

osed

 material changes

–Ch

ecking

 mob

ility m

easuremen

t accuracy

()

inf

inf

1/

8ρ

==

YvF

Dh

Infin

ite Structure M

obility 

Scaling Exam

ple

•Tw

o pane

ls with

 identical geo

metries and

 different m

aterials

–Lexan (plastic) and

 Aluminum

Infin

ite Structure M

obility 

Scaling Exam

ple

•Measured mob

ilitie

s of both pane

ls

Infin

ite Structure M

obility 

Scaling Exam

ple

•Aluminum

 panel m

obility scaled to th

at of Lexan / /ρ ρ

==

AlAl

AlAl

Lexan

Lexan

Lexan

Lexan

Ef

cf

cE

Radiation Efficiency

2ρ

rad

ooP cAv

ff c

1.0 σ (f)

Critical Frequ

ency, w

here 

bend

ing and acou

stic 

wavespe

edsmatch

Structure radiates 

poorly

Peak ra

diation

σ = 1

Coincide

nce

Bend

ing waves are 

dispersive, slower th

an 

acou

stic waves at low

 freq

uencies (sub

sonic), 

and faster at h

igh 

freq

uencies (sup

ersonic)

Radiating Flat Baffle

d Pane

lBe

low Coinciden

ce

Pressures

Odd

/Odd

 >Odd

/Even >

Even

/Even

Radiating Flat Baffle

d Pane

l –Increasing

 Frequ

ency

Intensities

(propo

rtional 

to squ

are of 

pressure)

Radiation Efficiency –Effects of 

Coincide

nce Freq

uency Shifts

ff c

1.0 σ (f)

f c

Stiffer, lighter 

structure with

 faster waves

Highe

r σ

f c

Radiation Efficiency –Effects of 

Coincide

nce Freq

uency Shifts

ff c

1.0 σ (f)

Flim

sier, heavier 

structure with

 slow

er waves

Lower σ

Soun

d Po

wer Transfer Functio

ns

•Co

mbining

 surface averaged mob

ility with

 radiation efficiency tells us ho

w well a 

structure radiates sou

nd whe

n driven

 by a 

know

n force

(1,1) m

ode ‐> 

‘loud

speaker’ m

ode

Acoustic

 Waves Im

pinging on

 Structures

Soun

d Transm

ission

 Loss 

of an Infin

ite Panel [

](

)(

)

()(

)

20

0

4

4

2

0

0

2

00

si

2sin

s

2s

n

ni

i

n

ρφ

τω

ρ

ωρ

η

ω

φ

φ

φ=

⎡⎤

++

⎣⎦

⎡⎤

−⎣

⎦

Dk

h

c

Dk

c

Dam

ping

 im

portant a

t coincide

nce

Mass im

portant 

below 

coincide

nce 

(mass law)

Stiffne

ss 

impo

rtant a

bove 

coincide

nce

Dips occur whe

n mass an

d stiffne

ss 

term

s cancel

φ= 0

φ= 90

Soun

d Transm

ission

 near C

oinciden

ce

Finite Element (FE) A

nalysis

•Used gene

rally to

 mod

el 

structures

–Plates, beams, and

 solids

•Also sometim

es used to 

mod

el acoustic

 region

s, 

usually inside

 an ob

ject

•Many commercial 

software packages 

available

Boun

dary Element (BE

) Analysis

•Used to m

odel th

e bo

unda

riesof acoustic

 region

s–Inside

 or o

utside

 a vibratin

g surface

•Once surface pressures and 

velocitie

s are know

n, th

e acou

stic field anyw

here m

ay 

be com

puted

•Some commercial softw

are 

packages are available (not 

as m

any as FE packages)

Coup

led FE/BE Analyses

•Im

pedance matrices of structures (from FE) and

 acou

stic re

gion

s (from BE) m

ay be coup

led

–Enforce continuity of n

ormal fluctuating velocity along

 bo

undary

Statistical Ene

rgy Analysis (SEA

)

•SEA m

odels the en

ergy exchange be

tween large 

grou

ps of reson

ances in intercon

nected

 structures and

 acoustic

 region

s–Usually used at high freq

uencies, whe

re m

ode coun

ts 

are very high

–Exchange of e

nergy is m

odeled

 statistically

•Get calculatio

ns averaged over large region

s, and

 over w

ide 

freq

uency (usually one

‐third octave) bands

–Fast com

putatio

ns, ind

epen

dent of increasing 

freq

uency

–Va

‐One

 softw

are

Structural-Acoustics Tutorial�Part 1 - FundamentalsOverviewInstructorMotivationStructural Acoustic Transfer FunctionsSurface Averaged MobilityModes of ResonanceModes of ResonanceMobility as a Summation of ModesModal Response PeaksDamping MechanismsEffects of Damping �on Modal Response PeaksInfinite Structure MobilitiesInfinite Structure Mobility �Scaling ExampleInfinite Structure Mobility �Scaling ExampleInfinite Structure Mobility �Scaling ExampleRadiation EfficiencyCoincidenceRadiating Flat Baffled Panel�Below CoincidenceRadiating Flat Baffled Panel – Increasing FrequencyRadiation Efficiency – Effects of Coincidence Frequency ShiftsRadiation Efficiency – Effects of Coincidence Frequency ShiftsSound Power Transfer FunctionsSlide Number 24Acoustic Waves Impinging on StructuresSound Transmission Loss �of an Infinite PanelSound Transmission near CoincidenceFinite Element (FE) AnalysisBoundary Element (BE) AnalysisCoupled FE/BE AnalysesStatistical Energy Analysis (SEA)

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