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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.
