silent engineering

July 6, 2021

Silent Engineering(Lecture 4)

Fundamental Equations to Estimate Sound Power Radiating from

Vibrating Plate and an Example of Estimation

- Energy balance in vibrating plate and parameters to estimate sound radiation power -

Tokyo Institute of TechnologyDept. of Mechanical EngineeringSchool of Engineering

Prof. Nobuyuki Iwatsuki

１. Energy Balance in Vibrating Plate and Loss Factors

Noise radiationVibration propagation to the air

Exciting forceMechanical excitationAcoustic excitation

Plate/shell structure

Vibration propagationto other structure

↓Multi-DOFvibration system

“Flexural vibrationdue to natural mode ofvibration”

Vibration energy and radiating sound power

Input power Win＝PV

Sound (noise) radiation power Wrad

Energy of steady vibration E

Power to other structure Wext（negligible if stiff connection）

Internal dissipated power Wint（Dissipated as heat）

Velocity at exciting point

Calculated with acceleration of vibrating plane

Viscous damping（depends on velocity

“Accurately calculatedwith vibration response”

Wrad＜Wint

Power balance: inradintdis WWWW =+= (1)

Total loss power is equal to input power and is represented as thesummation on internal dissipated power and sound radiation power.

Loss factors: Definition:

Loss factor is the ratio of the loss power to the vibration energyper 1 radian of vibration.

Total loss factor:

( ) EW

TEW disdis

dis ωπη ==

2//(2)

:::

ωTEwhere Vibration energy [J]

Angular velocity [rad/s]

Vibration period [s]

EW disdis ωη=∴　 (3)Total loss power:

:disη Total loss factor

Internal dissipated power:

EW intint ωη= (4) :ntiη Internal loss factor

Sound radiation power:

EW radrad ωη= (5) :radη Radiation loss factorAccording to the power balance, we obtain

in

dis

ntirad

ntirad

ntiraddis

WE

EEE

WWW

==

+=+=

+=

　　

　　

　　

　　

ωηωηηωηωη

)(

(6)

Therefore

ntiraddis ηηη += (7)

disrad ηη <<In general

(9)

For the angular frequency spectra, we obtain

)()()()( ω

ωηωηω in

dis

radrad WW = (10)

Fundamental equation to estimate sound radiation power“ We can calculate frequency spectrum of sound radiation power.”

(8)indis

radin

ntirad

radrad WWW

ηη

ηηη

=+

=

2. Frequency Spectra of Parameters to Estimate Sound Radiation Power

2.1 Total loss factor ηdis

Since vibration displacement can be given as the linear combinationof modes of vibration, the total loss power can be given as the summation of loss power of each mode as

∑∑

=

=

ii

iidis

disdis

E

WE

W

ω

ωω

ωωη

)(

)()(

,

　　　 (11)

Dissipated power of each mode

Vibration energy of each mode

Let’s consider a vibration system with 1 DOF.

kc

m

xPcosωt

Equation of motion:tPkxxcxm ωcos=++ (12)

wheremkc

mk

tmPxxx

n

nn

2,

cos2 2

==

=++

ζω

ωωζω

　　　　

(13)

(14)

where

Displacement:

221

2222

2tan

)cos()2()(

)cos(

ωωωζωφ

φωωζωωω

φω

−=

−=+−

−=

−

n

n

nn

tAm

tPx

　　　　

(15)

kc

m

xPcosωt

Total dissipated power:

[ ]

22

22

0

22

0

22

0

222

0

2

0

2

4)(2sin

2

2)(2cos1

)(sin

)sin(

ωωζ

ω

ωφωω

φωω

φωω

φωω

n

T

T

T

T

T

dis

mA

cAT

tTcA

T

dttcA

T

dttcA

T

dttAc

dtT

xxcW

=

=

−

−=

−−

=

−=

−−=

⋅=

∫

∫

∫

∫

　　

　　

　　

　　

　　

　　

(16)

kc

m

xPcosωt

Vibration energy:

　　

　　2

21

22

2max

nmA

xmE

ω=

=

(17)

Total loss factor:

　　

　　

2

22

22

n

n

disdis

mAmA

EW

ωω

ωωζω

η

=

=

(18)

Let’s assume that the response of multi-DOF system can be representedas a linear combination of displacement of 1 DOF system as

　　

)cos(),,( ii

i tFtyxw φω −= ∑ (19)

where

221

2222

2tan

),(),(

)2()(

ωωωωζφ

ωωζωω

−=

=

+−=

−

∫

i

iii

Si

iii

ii

dSyxpyxWQ

QF

　

(20)

(21)

(22)

when the i-th mode has a natural angular frequency ωi , damping ratio ζ i and modal mass mi

Total dissipated power of the i–th mode:22

, ωωζ iiiiidis mFW = (23)Vibration energy of the i–th mode:

　　

2

22iii

iFmE ω

= (24)

(25)

Total loss factor of multi-DOF vibration system can be given as

　　　　

　　　

ωω

ωη

ωω

ωωζ

ωωη

∑∑

∑

∑

∑∑

=

=

=

iiii

iiiiii

i

iii

iiii

i

ii

iidis

dis

Fm

Fm

Fm

mF

E

W

22

2

22

22

,

2

)(

where 　ii ζη 2=denotes modal total loss factor

(26)

Total loss factor can be represented with modal mass, natural angular frequency and driving angular frequency.

2.2 Radiation loss factor ηrad

Let mean square velocity in the vibrating plate be

[ ][ ]

　　

∫∫=

S

S

dS

dSyxww

22

),()(

ω (27)

　　

),( yxwdS

Vibration energy of the plate Ep can begiven as

[ ]2)(ωρ whSEp = (28)

Mass per unit area

Area

Mean square velocity

Note that < > means mean square value.

Let’s consider the sound radiation power in case where the plate oscillatesas a piston at the square average velocity, , in the plate.

The radiating sound power, Wpiston, in this case can be given as

[ ]　　

2)(ωw

[ ]　　

2)(ωρ wSvW airairpiston = (29)

where

　　

::

air

air

vρ Density of air [kg/m3]

Speed of sound in the air [m/s]

Next, let’s consider the radiation efficiency.Definition of radiation efficiency:

The ratio the sound power radiating from the real vibrating plate, Wrad,to the sound power radiating from a plate vibrating with a piston motion,Wpiston.

[ ]　　

2)()(

)()()(

ωρω

ωω

ωσwSv

WWW

airair

rad

piston

radrad

== (30)

Sound power in vibration of air column

Sound power due to real bending vibration

Sound power due to rigid vibration

By substituting Eqs.(28) and (29) into Eq.(4), the radiation loss factorcan be calculated as

p

radrad E

Wω

ωωη )()( =

Therefore we would like to preobtain sound power, Wrad(ω), which means the sound radiation power due to the unit excitation force, and mean square velocity, <[w(ω)]2>. ・

(31)

[ ]

[ ][ ]

)(

)(

)()(

)(

)()(

2

2

2

ωσωρ

ρ

ωωρ

ωρωσ

ωωρ

ωωσ

radairair

airairrad

pistonrad

hv

whS

wSv

whS

W

=

=

=

　　　　

　　　　

　　　　　

Mean square velocity:

[ ][ ]

∫∫=

S

S

dS

dSyxww

22

),()(

ω

Average value of mean square velocity in the area

　　

)sin(

)cos(

ii

i

ii

i

tFw

tFw

φωω

φω

−−=

−=

∑

∑

(33)

(34)

T: vibration period

(32)

　　

　　　　　

　　　　　

∫ ∫

∫

=

=

S

T

S rms

dtdSwTS

dSwS

0

2

2

11

1

222 )sin(

−= ∑ i

ii tFw φωω

Therefore

(35)

　　

　　　　　　

−

+

=

−=

−=

∑∑

∑∑

∑∑

∑

iiii

ii

iiii

ii

iiii

ii

iii

i

FFtt

FtFt

FtFt

ttF

φφωω

φωφωω

φωφωω

φωφωω

sincoscossin2

sincoscossin

sincoscossin

)sincoscos(sin

22

222

22

22

−

+

=

−

+

=

∫∑∑

∫∑∫∑

∑∑

∫ ∑∑

∫

tdttFF

tdtFtdtFT

dtFFtt

FtFtT

dtwT

T

iiii

ii

T

iii

T

ii

i

iiii

ii

T

iiii

ii

T

ωωφφ

ωφωφω

φφωω

φωφωω

cossinsincos2

cossinsincos

sincoscossin2

sincoscossin

1

0

0

22

0

222

0

22

22

2

0

2

　　　　　

　

　　　　　　

　

(36)

−

+

+

−

=

∫∑∑

∫∑∫∑

dttFF

dttFdttFT

T

iiii

ii

T

iii

T

ii

i

0

0

2

0

22

2sinsincos

22cos1sin

22cos1cos

ωφφ

ωφωφω

　　　　　

　　

=T/2 =T/2

=0

+

+

+

−

=

∑∑

∑∑T

iiii

ii

iii

T

ii

i

tFF

tFtFT

0

2

0

22

2cossincos

42sin

2sin

42sin

2cos

ωωφφ

ωωφ

ωωφω

　　　　

(37)

+

= ∑∑

222

sincos2 i

iiii

i FF φφω

(38)

[ ]

dSQ

QS

dSFFS

w

ii iii

i

S ii iii

i

Si

iiii

i

+−+

+−=

+

=

∴

∑

∫ ∑

∫ ∑∑

2

2222

2

2222

2

222

2

sin)2()(

cos)2()(2

1

sincos2

1

)(

φωωζωω

φωωζωω

ω

φφω

ω

　　　　　

　

Sound radiation power due to the unit excitation force:z

xy

O

θ

ϕ

R

dS(x,y)

d

Observation point P:)cos,sinsin,cossin( 　θϕθϕθ RRR

Vibrating plate

Sound field generated by vibration of plate“Let’s consider a set of vibrating point sources”

Acceleration w(x,y)‥

Sound pressure p

Sound pressure at the observation point due to the vibration of an infinitesimal element dS can be given as

　　

dSdwdp air

πρ2

= (39)

Sound pressure at the observation point due to the vibration of wholeplate can then be given as

　　

　　　　　　∫

∫=

=

Sair dS

dtyxw

dptRp

πρ

θϕ

2)',,(

),,,(

(40)

(41)

where

( ) ( )

　　

　

airvdttyxRRyx

RRyRxd

/')sincos(sin2

cossinsincossin222

2222

−=

+−++=

+−+−=

ϕϕθ

θϕθϕθ

(42)Phase delay due to propagation distance is taken into account.

Sound radiation power:

HdRIWHrad ∫= ),,()( θϕω

Sound power can be calculated by integrating sound intensity on whole hemisphere as

Sound intensity

Sound pressure

Integration on the hemisphereϕθθ

ρπ π

ddRv

p

airair

rms sin22

0

2/

0

2

∫ ∫=　　　　

ϕθθρ

θϕπ π

ddRv

dttRpT

airair

T

sin),,,(1

22

0

2/

0

0

2

∫ ∫∫

=　　　　

(43)

ϕθθρ

πρ

π πddR

v

dtdsd

yxwT

airair

T

Sair

sin2),(1

22

0

2/

0

0

2

∫ ∫∫ ∫

=

　　　　　

　　

)cos(2i

ii tFw φωω −−= ∑ (44)

Since

dSv

dFv

dF

dtpT

p

airi

ii

airi

iiS

air

T

rms

++

+

=

=

∑∑∫

∫22

2

0

2

2

sincos22

1

1

ωφωφωπ

ρ

　

We obtain

(45)

We can calculate the sound radiation power Wrad(ω).

2.3 Input power Win

Input power can also be given as summation of input power to each mode as

　　

∑=i

iinin WW , (46)

( ) ( )

( ) ( )　　

　　 i

iii

i

iii

iiT

iiin

Q

dttQtQT

W

φωωζωω

ω

ωωζωω

φωωω

sin22

1

2

)sin(cos1

2222

2

22220,

+−=

+−

−−⋅= ∫

where

(47)

Especially the excitation is point excitation at driving point (xd,yd) withamplitude P0, we obtain

( ) ( )　　

i

iii

ddiiin

yxWPW φωωζωω

ω sin2

),(21

2222

220

,+−

=

Real part of Driving point mobility

(49)

　　

　　　 20

20

)Re(21

)Re(21

PY

PYWi

iin

=

=∴ ∑

(48)

( ) ( )

　　

　　

　　

20

20

2222

2

)Re(21

2

sin),(21

PY

PyxW

i

iii

iddi

=

+−=

ωωζωω

φω

３. An Example of Sound Power Estimation- Sound Power Radiating from Peripherally Clamped

Rectangular Thin Plate Subjected to Point Excitation -

Peripherally clamped thin rectangular plate(Object to be estimated)

x

y

z

z

3.1 Peripherally clamped rectangular thin plate

3.2 Modal analysis with the Rayleigh-Ritz method and experimental validation

−−−⋅

−−−=

⋅=

=

∑

∑

by

by

by

by

ax

ax

ax

ax

C

yWxWCyxWtyxWtyxw

jyjyjy

jyjy

ixixix

ixix

jiij

jyji

ixij

,,,

,,

,,,

,,

,

,,

,

sinsinhcoscosh

sinsinhcoscosh

)()(),(cos),(),,(

λλα

λλ

λλα

λλ

ω

　　　　　　　　

　　　　

Eigenfunction:

Boundary conditions:

0,0;,0

0,0;,0

=∂∂

==

=∂

∂==

yWWby

xWWax

　

　Remember Lecture 2

Setup for experimental modal analysis

Experimental modal analysis:

w(ω)・

Vibration velocity

Exciting forceP(ω)

Mobility at various response points anddriving point are measured .

Mode Calculated[Hz] Measured[Hz] Error [%] Damping ratio[%](0,0) 57.1 49.0 +16.4 1.110(1,0) 78.3 68.7 +14.1 0.747(2,0) 115.6 102.8 +12.5 1.211(0,1) 145.9 126.1 +15.7 2.704(1,1) 166.3 147.2 +13.0 1.242(3,0) 168.3 150.2 +12.0 0.710(2,1) 201.4 178.9 +12.5 0.551(4,0) 235.7 209.0 +12.8 0.652(3,1) 251.5 225.7 +11.5 0.493(0,2) 279.8 * * *(1,2) 300.1 260.9 +15.0 1.057(4,1) 316.7 284.4 +11.3 0.544

(5,0) 317.4 282.4 +12.4 1.270(2,2) 334.4 295.9 +13.0 0.742(3,2) 383.3 342.1 +12.0 0.577(5,1) 396.6 358.1 +10.7 0.601(6,0) 413.1 356.5 +15.9 2.453(4,2) 446.7 400.8 +11.5 0.530

Results of the calculated and measured modes of vibration

The calculated natural frequencies are higherthan the measured values.

Stiffness at plate edgesis low.

Peripherally clamped conditions are notsatisfied.

It is difficult to realize completely clamping the plate edges with bolt fixing.

Number of

nodal lines

Therefore in order to obtain the correct natural frequencies which agreewith the measured values, suitable dimensions of the rectangular plateat which the plate is clamped are searched by evaluating error ofnatural frequencies.

The plate may be fixed between fixing bolts and edge of fixing blocks.Ideal boundary(Edge of fixing block)

Actual boundary

Fixing bolts

Since the eigenvalues of the coefficient matrix of approximated natural angularfrequency equation are function of the ratio, µ=b/a, of side lengths of the plate, the suitable side length a which makes the calculated natural frequency agree with the measured natural frequency at each mode is calculated for various µ as

2/12/1

/1

=

Dha i

ii ρ

λµω

(1)

Mean value of aiStandard deviation of ai

Minimum deviation b=749.5mm

a=1326.8mm

We can find the suitable dimensions of plate.

Mode Modified[Hz] Measured[Hz] Error [%] Damping ratio[%](0,0) 50.0 49.0 +2.07 1.110(1,0) 69.4 68.7 +1.01 0.747(2,0) 103.2 102.8 +0.32 1.211(0,1) 127.6 126.1 +0.71 2.704(1,1) 146.1 147.2 -0.75 1.242(3,0) 150.9 150.2 +0.47 0.710(2,1) 178.1 178.9 -0.56 0.551(4,0) 211.8 209.0 +1.34 0.652(3,1) 223.9 225.7 -1.12 0.493(0,2) 244.3 * * *(1,2) 262.7 260.9 +0.80 1.057(4,1) 283.1 284.4 -0.46 0.544

(5,0) 285.6 282.4 +1.13 1.270(2,2) 293.9 295.9 -0.68 0.742(3,2) 338.6 342.1 -1.02 0.577(5,1) 355.8 358.1 -0.64 0.601(6,0) 372.0 356.5 +4.35 2.453(4.2) 396.3 400.8 -1.12 0.530

Results of modes of vibration which are calculated with the modified dimensions

Natural frequency errors are minimized.

For a=1326.8mm, b=749.5mm, h=4.5mm, E=205GPa, ν=0.3, ρ=7860kg/m3

(0,0) mode

Calculated mode shapes and natural frequenciesof peripherally clamped thin rectangular plate (1)

(1,0) mode

(2,0) mode

(0,1) mode

(1,1) mode

(3,0) mode

Number ofnodal lines


(2,1) mode


(4,0) mode

(3,1) mode

(0,2) mode

(1,2) mode

(4,1) mode


(5,0) mode


(2,2) mode

(3,2) mode

(5,1) mode

(6,0) mode

(4,2) mode

3.3 Parameters to estimate sound radiation power

Fundamental equation to estimate sound radiation power:

[ ] )(21)(Re

)()(

)()()()(

2 ωωωηωη

ωωηωηω

PY

WW

dis

rad

indis

radrad

⋅⋅=

=

　　　　

For point excitation

(50)

Total loss factor:

　ωω

ωηωη

∑∑

=

iiii

iiiiii

dis Fm

Fm

22

2

)( (51)

10-1

10-2

10-3

10-4

TOTAL LOSS FACTOR η d

is(ω)

Almost of total loss factors at natural frequencytake modal total loss facotor.

Driving point：(-500,-225)

Radiation efficiency:

[ ]　

2)()()(ωρ

ωωσwSv

W

airair

radrad

= (52)

100

10-1

10-4

RADIATION EFFICIENCY

σ rad(ω)

10-2

10-3

10-5


Radiation loss factor: (53))()( ωσ

ωρρωη rad

airairrad h

v=

10-2

10-5

RADIATION LOSS FACTOR η r

ad(ω)

10-3

10-4

10-7

10-6

Radiation loss factor is smaller than total loss factor.


Driving point mobility:

(54)( ) ( )　　

∑+−

=i

iii

iddi yxWY2222

2

2

sin),()](Re[ωωζωω

φωω

10-2

10-5

DRIVING POINT MOBILITY Y(

ω) m/Ns 10-3

10-4

10-7

10-6

Real part

Imaginary part

10-2

10-5

10-3

10-4

10-7

10-6

Driving point mobility takes peak values at natural frequency.


3.4 Estimation and measurement of sound radiation power

Sound Intensity microphone

Setup to measure sound power with the sound intensity method

EXC

ITN

GFO

RC

EP

rms

N

0.2

0.1

0.0D

RIV

ING

PO

INT

MO

BILI

TY

Re(

Y)

m/N

s 10-2

10-4

10-6

10-8

INPU

T PO

WER

W

indB

re

f 1pW

80

60

40

20

LOSS

FAC

TOR

Sη d

is, η

rad

10-2

10-3

10-4

10-5

10-6

10-7

SOU

ND

RAD

IATI

ON

POW

ER

Wra

ddB

ref

1pW

60

40

20

50

30

ηrad

ηdis

Estimated

Measured

Radiation loss factor is smaller than total loss factor.

Driving pointmobility strongly affects input power.

Driving point mobility also strongly affects sound radiationpower.


3.5 Discussion of sound radiation power due to driving point

Various driving points

Driving point 2

Driving point 1

Driving point 3

Total loss factor:

TOTAL LOSS FACTOR

η dis(ω)

10-1

10-2

10-3

10-4

Driving point:(-225,-500)(0,0)(-225,0)

Driving point hardly affects total loss factor.

Radiation loss factor: RADIATION LOSS FACTOR η r

ad(ω)

10-2

10-3

10-4

10-5

Driving point:(-225,-500)(0,0)(-225,0)

10-6

10-7

Driving point hardly affects radiation loss factor.

Real part of driving point mobility:

Driving point strongly affects mobility.

Driving point:(-225,-500)(0,0)(-225,0)

DRIVING POINT MOBILITY Re[Y(

ω) ] m/Ns

10-2

10-5

10-3

10-4

10-7

10-6

EXC

ITN

GFO

RC

EP

rms

N

0.2

0.1

0.0

DR

IVIN

G P

OIN

TM

OBI

LITY

R

e(Y

)m

/Ns 10-1

10-3

10-5

10-7

INPU

T PO

WER

W

indB

re

f 1pW

90

70

50

30

LOSS

FAC

TOR

RAT

IOη r

ad/η

dis

110-1

10-2

10-3

10-4

10-5

SOU

ND

RAD

IATI

ON

POW

ER

Wra

ddB

ref

1pW

80

40

20

60

Driving point:(-225,-500)(0,0)(-225,0)

White noise

Peak noise can be reduced by arranging driving pointjust on nodal lines.

4. Concluding remarks

Fundamental equations to estimate sound radiation power were explained and one example of estimation was shown. (1)Frequency spectrum of sound radiation power

can be calculated with frequency spectra of total loss factor, radiation loss factor and input power.

(2)Frequency spectra of total loss factor, radiationloss factor and input power can be derived withresults of forced vibration analysis.

(3)Frequency spectrum of sound power radiating from peripherally clamped thin rectangular platecan be estimated and is experimentally validated.

(4)Frequency spectrum of driving point mobility which depends on the position of driving pointstrongly affects sound radiation power. Therefore there exists a possibility to reduce noise by arranging driving point on nodal line of vibrating plate.

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