july 27, 2021 silent engineering

July 27, 2021

Silent Engineering(Lecture 7)

Setting Damping Materials toReduce Sound Radiation Power

- Noise reduction with constraintand non-constraint dampers -

Tokyo Institute of TechnologyDept. of Mechanical EngineeringSchool of Engineering

Prof. Nobuyuki Iwatsuki

１. Sound Power Reduction with Non-constraint Damper

Let’s assume a dual layer plate covered with viscoelastic materialas a pure single plate with synthesized property calculated with properties of steel plate and viscoelastic material layer based onforce balance.

1.1 Analysis of two layers plate with viscoelastic damping material

Two layers plate with viscoelastic material

hd

hpThickness

Density ρp，Young’s modulus Ep ，Poison’s ratio νp

Viscoelasticmaterial

Steel plateNeutral plane

Density ρd，Young’s modulus Ed ，Poison’s ratio νd

Synthesized density:pd

ppdd

hhhh

+

+=

ρρρ (1)

])1()1[(21

ddpp hht ξξ −++=Position of neutral plane: (2)

])11(31[

])1(31[22

22

tdd

tpp

RD

RDD

+++

++=

ξ

ξ

Synthesized bending stiffness:

(3)

Synthesized Poison’s ratio:

pp

pd

d

d

ppp

pdd

d

d

hE

hE

hE

hE

22

22

11

11

νν

νν

νν

ν

−+

−

−+

−= (4)

Modal total loss factor:i

iddi Π

Πηη ,= (5)

Properties of damped pure plate can be calculated.

where

::

:

,)1(12

,)1(12

,

,21

,21

,

,1

,1

1

,

2

3

2

3

22

i

id

d

d

ddd

p

ppp

p

dt

d

d

dd

p

p

pp

d

p

dp

hEDhE

D

hhR

RhE

RhE

Π

Πη

νν

νσ

νσ

σσ

ξ

ξξ

ξξ

ξ

−=

−=

=

−=

−=

=

+=

+=

Loss factor of viscoelastic material

Modal strain energy in viscoelastic material

Modal total strain energy

(6)

Experiment to measure properties of viscoelastic material with thecantilever method

Electromagnetic exciter

Noncontact displacement sensor

Specimen

Constant temperature bath

Amplifier Generator

PCCharge Amp.

1.2 Properties of viscoelastic damping materials

Nomograph to calculate loss factor and Young’s modulus

1kHz

40

Experimental setup for rectangular cantilever plate

Experimental modal analysis and measurement of sound radiation power

Sound Intensity Microphone

1.3 Validation of sound power reduction

Estimation of sound power radiating from rectangularcantilever plate with viscoelastic damping material

Estimation of sound power radiatingfrom pure rectangular cantilever plate

Peak values of sound radiation power can be estimated with an adequate accuracy.

×：Measured

(2,2)

(0,3)

(2,1)(1,2)(2,0)

(0,2)

(1,0)

(0,1)(0,0)

(1,1)

The relation between thickness of damper layer and natural frequency

1.4 Sound power reduction characteristics

×：Measured

(0,3)(2,1)(2,2)

(0,1)

(0,0)

(0,2)(1,1)

(2,0)(1,2)(1,0)

The relation between thickness of damper layer and total loss factor

Damping of complex modesincreases asdamper thickness

×：Measured

(0,1)(1,1)(0,2)(1,0)(0,0)

(0,3)

(1,2)

(2,2)

(2,0)

The relation between thickness of damper layer and radiation loss factor

×：Measured

(0,1)

(0,0)

(0,2)(1,1)

(1,0)(2,0)

(2,1)(2,2)(1,2)

(1,3)

Damping increases asdamper thickness.

The relation between thickness of damper layer and estimation parameter

Let’s have a 10 minutes break here!

If possible, would you please answer to 2021 2QCourse Survey of Study Effectiveness for this course‘Silent Engineering’ now?

The web-site to answer the survey is as follows:

https://www.ks-fdcenter.net/fmane_titech/Ans?ms=t&id=titech&cd=Bab3CZus

The deadline to answer is August 7, Saturday.

2. Sound Power Reduction with Constraint Damper

2.1 Analysis of three layers beam with the constraint layer theory

The constraint layer theory:“Analysis of three layers beam”

Constraint layer (Elastic layer)

Base layer (Elastic layer)

Damping layer(Viscoelastic layer)

L

h3h2h1

x

y

Damping by share deformation

γ

3uxw

∂∂1u

Assumptions:(1)No slippage between layers(2)Same lateral displacement in each layer(3)Young’s modulus of damping layer is negligible.(4)Energy loss is generated by only share deformation

22

)(1

)22

(

32

1

312

32

1312

hhhh

xwhuu

h

xwhhhuuh

++=

∂∂

+−=∴

∂∂

+++−=

γ

γShare strain:

where

(7)

Strain energy per unit width:

Base layer:

12,

21

241

21

21

211

311

1111

0

2

2

2

1

21

1

0 0

2

2

2311

21

11

0 0

2

2

2

11

21

111

hEDhEK

dxxwD

xuK

dxxwhEdx

xuhE

dxxwIEdx

xuAE

b

L

L L

L L

==

∂∂

+

∂∂

=

∂∂

+

∂∂

=

∂∂

+

∂∂

=

∫

∫ ∫

∫ ∫

Π

where

(8)

Constraint layer:

12,

21

333

3333

0

2

2

2

3

23

33

hEDhEK

dxxwD

xuK

L

==

∂∂

+

∂∂

= ∫

Π

where

(9)

Viscoelastic layer:

　　 ∫

∫ ∫

∂∂

+−=

=−

L

L h

h

dxxwhuu

hG

dxdG

0

2

312

2

02

2

222

21

21 2

2ξγΠ

(10)

　　　 ∫ ∫

∂∂

+−+

∂∂

++

∂∂

+

∂∂

=

++=∴

L Ldx

xwuu

hG

xwDD

xuK

xuK

0 0

231

2

2

2

2

2

31

23

3

21

1

321

)()(21

ΠΠΠΠ

(11)

Kinetic energy per unit width:

332211

0

2

0

2

332211

21

)(21

hhh

dxtw

dxtwhhhT

L

L

ρρρµ

µ

ρρρ

++=

∂∂

=

∂∂

++=

∫

∫

where

Equations of motion with respect to u1,u3,w

(12)

Lagrange’s equation of motion:

wT

wT

dtd

uT

uT

dtd

uT

uT

dtd

∂−∂

=

∂−∂

∂−∂

=

∂−∂

∂−∂

=

∂−∂

)()(

)()(

)()(

33

11

ΠΠ

ΠΠ

ΠΠ

(13)

tjtjtj weweuueuu ωωω === ,, 3311

Harmonic solution can be assumed as

Non-dimensional parameters are adopted as

Lxx

Lww

huu

huu ==== ,,, 3

31

1

(14)

DL

KhLGgh

DKY

KKK

KKk

KKK

KKk

DDDKK

KKK

422

2

222

1

3133

3

3111

3131

31

,,

,,

,,

µωΩ ===

+==

+==

+=+

=

(15)

g-parameter

0)(

0)(

0)(

3123

2

3

3121

2

1

22

231

4

4

=+−+

=+−−

=−+−−

xdwduug

xdudk

xdwduug

xdudk

wxdwd

xdud

xdudgY

xdwd Ω

Equation of motion

(16)

xxx WeweUueUu λλλ === ,, 3311

By assuming the mode shape as(17)

[ ]

0)(0)(

0)(

312

33

312

11

2231

4

=+−+

=+−−

=−+−−

λλ

λλ

Ωλλλλ λ

WUUgUkWUUgUk

eWWUUgYW x

We obtain

(18)

=

−−−

−−−

000

3

1

23

21

224

WUU

ggkggggk

gYgYgY

λλλλ

ΩλλλλThe matrix form can be obtained as

(19)

Therefore

[ ]0

)1()()(

det

22246231

2231

4231

63131

831

23

21

224

=+−+−=

+−−++−=

−−−

−−−

ΩλΩλλλ

λΩλΩλλ

λλλλ

Ωλλλλ

gYgkkgkkkkgYkkkkkk

ggkggggk

gYgYgY

(20)

(21)1)()(

)( 231

231

31

231

1

31

3

31

31

31 =++

=+

++

+

=+

KKKK

KKKK

KKK

KKK

kkkk

00)1(

2

22246

=

=+−+−

λ

ΩλΩλλ gYgTherefore we obtain the equations with respect to λ as

(22)

(23)

0)3,2,1(,

)3,2,1(

1

=

=±±±=±

=

λβαα

αλ

iji

ii

i

　　Three real rootsOne real root andtwo imaginary roots

By solving the above equations, we obtain

(24)

(25)

+++++

+++=

===

∑

∑

=

−−

=

−

3

211

3

1

3311

)coscos(

)()(

)(),(),(

11

ii

xii

xi

xx

i

xi

xi

DxCxeBxeAeBeA

DxCeBeAxf

xWfwxfUuxfUu

ii

ii

ββ αααα

αα

Therefore

: Eivenvectors which will be calculated with ΩWUU ,, 31

(26)

+++++

+++=

===

∑

∑

=

−−

=

−

3

211

3

1

3311

)coscos(

)()(

)(),(),(

11

ii

xii

xi

xx

i

xi

xi

DxCxeBxeAeBeA

DxCeBeAxf

xWfwxfUuxfUu

ii

ii

ββ αααα

αα

(27)

By taking account of boundary conditions, natural angular frequency, Ω,can be determined.

1,0=xAt both ends of the beam,

0,0,0,0

0,0,0,0

0,0,0,0

0,0,0,0

312

2

3

3

312

2

31

31

====

====

====

====

uxd

duxd

dwxd

dwxd

d

uxd

duwxd

dw

uxd

duwxd

dw

uuwxd

dw

　　　All layers are clamped:

Only base layeris clamped:

Only base layer is simply supported：

All layers are free:

(28)

(29)

(30)

(31)

+−=

+++=

+−=

+++=

+−=

+++=

∑

∑

∑

∑

∑

∑

=

−

=

−

=

−

=

−

=

−

=

−

3

133

3

133

3

111

3

111

3

1

3

1

)(

)(

)(

)(

)(

)(

i

xi

xii

i

xi

xi

i

xi

xii

i

xi

xi

i

xi

xii

i

xi

xi

CeBeAUuxd

d

DxCeBeAUu

CeBeAUuxd

d

DxCeBeAUu

CeBeAWwxd

d

DxCeBeAWw

ii

ii

ii

ii

ii

ii

αα

αα

αα

αα

αα

αα

α

α

α

Example: In case where a base layer is clamped at both ends andall λ’s are real numbers.

(32)

=

−−−

−−−

−−−

−−−

−−

−−−

−−−

−−−

00000000

0111011101110111

3

3

2

2

1

1

333332323131

111111

332211

333332323131

111111

332211

332211

332211

332211

332211

DCBABABA

UeUeUeUeUeUeUeUeUeUeUeUe

WeWeWeWeWeWeWeWeWeWeWeWeUUUUUU

UUUUUUWWWWWW

WWWWWW

αααααα

αααααα

αααααα

αααααα

αααααα

αααααα

αααααα

αααααα

Mode shapes can then be calculated.

(33)

[ ]

=

00000000

)(

3

3

2

2

1

1

DCBABABA

M Ω

(34)

By solving this equation, we can calculatenatural angular frequencies.

[ ] 0))(det( =∴ ΩM

Loss factor:

2.2 Calculation of loss factor of three layers beam

321

2

ΠΠΠΠ

ηη

++=

d

Loss factor can be calculates as a ratio of strain energyin viscoelastic layer to total strain energy as

(35)

Natural angular frequency and loss factor of three layers beam simply supported at both ends

2.3 Examples of calculation of three layers beam

2

22

KhLGg =

Loss factor takes peak value for g-parameter

Non

-dim

ensi

onal

Nat

ural

ang

ular

freq

uenc

yLo

ss fa

ctor

Loss factor takes peak value for g-parameter

Natural angular frequency and loss factor of three layers beam clamped at both ends

Non

-dim

ensi

onal

Nat

ural

ang

ular

freq

uenc

yLo

ss fa

ctor

Natural angular frequency and loss factor of cantilever three layers beam

Loss factor takes peak value for g-parameter

Non

-dim

ensi

onal

Nat

ural

ang

ular

freq

uenc

yLo

ss fa

ctor

2.4 Experimentalvalidation Measured

Natural angular frequency and loss factor of cantilever three layers beam

〇 ：Measured by Okazaki, Gifu University

Natural angular frequency and loss factor of three layers beam whichis clamped at both ends and is partially covered with constraint damper

The calculated data agreed verywell with the measured one.

Loss factor takes peak value when the beam is not fully covered

〇 ：Measured by Okazaki, Gifu University

Natural angular frequency and loss factor of three layers beam which base layeris simply supported at both ends and is partially covered with constraint damper

The calculated data agreed verywell with the measured one.

Loss factor takes peak value when the beam is not fully covered

2.5 Application to noise reduction of circular plate

〇 ：Measured by Okazaki, Gifu University

The calculated data agreed verywell with the measured one.

Loss factor takes peak value when the plate is not fully covered

Natural angular frequency and loss factor of three layers circular plate whichis clamped at outer circumference and is partially covered with constraint damper

Constraint damper adhesion pattern

Sound power reduction of circular plate with a solid shaft

The calculated natural angular frequency of a circular plate with a solid shaft with constraint damper

Modal total loss factor of a circular plate witha solid shaft with constraint damper

Experimental setup for sound power measurement

Estimation and measurement of sound power radiating from a circular plate with a solid shaft with constraint damper

Estimation of modal total lossfactors of a circular plate witha solid shaft with constraintdamper

Estimation of sound powerreduction of circular plate with a solid shaft with a constraint damper

Estimated

Measured

3. Concluding remarksSound power reduction by using constraint/non-constraint damper is introduced.(1)The properties of rectangular plate with non-

constraint damper are derived. The reduced sound radiation power can easily be estimated.

(2)Vibration analysis of a plate covered with a constraint damper can be derived with the constraint layer theory.

(3)The sound power reduction of thin strips and circular plates covered with a constraint dampercan be estimated.

Concluding Remarks of Lecture(1)Sound radiation power of thin plate can be estimated

based on modal analysis, forced vibration analysis and sound radiation analysis.

(2)Sound radiation power can be characterized with three estimation parameters: total loss factor, radiation loss factor and input power.

(3)Several method to reduce sound power with the structural optimization by adding ribs or hollows

(4)Several method to reduce sound power with constraint/non-constraint dampers

Thank you for your attention!

Concluding Remarks of Lecture

Subject of final report - Reminder

I am looking forward to your excellent report!

Calculate the frequency spectrum of sound power radiating froma thin/thick plate or shell under the following conditions.(1)You can arbitrarily choose vibrating plate or shell.(2)You are expected to execute modal analysis, forced vibration

analysis and estimation of sound radiation power howeveryou can assume any mode shapes or natural frequencies if it will be difficult to calculate exact or approximated vibrationmodes.

(3)You can select mechanical point excitation or sound excitationwith arbitrary frequency spectrum of excitation force.

(4) You also assume modal total loss factors even if they areconstant.

(5)Please illustrate mode shapes with natural frequencies andfrequency spectra of loss factors, input power and soundradiation power.

The report will be summarized in A4 size PDF with less than 10pages and sent to Prof. Iwatsuki via OCW-i by August 14, 2021.