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Dynamic stiffness of ageing rubber vibration isolators
Leif Kari
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Structure-borne sound
Source Receiver
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Physical principle
”Hard”
”Hard”
”Soft”
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fwithout FwithFwithFwithout
Force transmissibility TF =
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fwith
FwithFwithout
TF =Fwith
Fe
− ω2mue = Fe - Fwithue ≠ 0
Ideal isolator
k
k
m
Fwith = k ue
1
1−= ω2
ω02
ω02 = k /m
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
100
101
102
103
104
10−8
10−6
10−4
10−2
100
102
Fo
rce
tran
smis
sib
ilit
y
Frequency [Hz]
No isolatorIdeal isolator
Rigid foundation – ideal isolator
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
100
101
102
103
104
10−8
10−6
10−4
10−2
100
102
Fo
rce
tran
smis
sib
ilit
y
Frequency [Hz]
No isolatorIdeal isolator
η
Rigid foundation – ideal isolator
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fwithout Fwith
Nonrigid foundation
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
uf
Foundation stiffness
FfFf
uf
kf = Ff / uf → ∞)1(12
8ωi 2f
ff2ff ν
ρ
−=
Ehk
hf
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
100
101
102
103
104
10−8
10−6
10−4
10−2
100
102
Fo
rce
tran
smis
sib
ilit
y
Frequency [Hz]
No isolatorNonrigid foundationRigid foundation
Nonrigid foundation – Ideal isolator
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Acoustic radiation
Wall
Wave fronts
1 W/m2 ⇔ 120 dB !
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fideal
Fideal
ideal isolator
k
m
non-ideal isolator
Fin
Fout
m
uin uout
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fideal
Fideal
ideal isolator
k
non-ideal isolator
Fin
Fout
uin uoutue
Fideal = k ueFin = kinin uin + kinout uoutFout = koutin uin+ koutout uout
with kinout = koutin
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Spherical part
Constitutive preliminaries
tr � = 3���(, � �, �� �)div �
dev � = 2��(, � �, �� �) dev �� + � ��(, � �, �� �; � − �) �dev ��(�)�� d�� �
Deviatoric part
limt→∞ µ1 = 0
limt→∞ µ = µ∞
[Kari 2016a,b]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Equilibrium elastic modulus
Density
�� , � �, �� � = "#$"$ �� � �, �� � ,
ρT
(equlibrium)≈ (1 − α ∆T)ρ0
α = −1ρ
�ρ�
∆T = T − T0
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Specific relaxation function
�� , � �, �� �; � = ∆() −∆ ��*+, �+#) ℎ(�)
() . =/ .0Γ(1 + β3)�
04$
+# = 10 67∆#689∆#
Non-dimensional relaxation intensity ∆ » 1
0 < β ≤ 1
[Kari 2016a,b]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Physical ageing
[Cangialosi et al Soft Matter 2013]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Physical ageing cont
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling physical ageing
[Greiner & Schwarzl 1984, Kovacs 1963, Doolittle 1953, Cohen & Turnbull 1959]
:# = ;# − ;$;$:#� = limt→∞:# = ;#� − ;$;$
<# d:#d� = :#� − :#<# = <̂ exp @:#
:#� = :#A� + BC�D∆BC�D = BCEFF�CG − B D�HHG
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling physical ageing modified
<# d:#d� = :#� − :# <# �I )D�) :# = :#� − :#
D�) :# = 1Γ(1 − K)� 1(� − �))�$
d(:#(�))d� d�
<# �I = <̂ exp @:# = <̂10MNOP
@Q = @log�$ e = @0.434294…
[Kari 2016a,b]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
WLF shift function
+# = 10 67∆#689∆#
:#� = :#A� + BC�D∆
<#� �I = <̂10 MNOPY
<#A� �I = <̂10MNOPAY
+# = <#� �I<#A� �I
Z� = @Q:#A� Z[ = :#A�BC�D
[Greiner & Schwarzl 1984, Kovacs 1963, Doolittle 1953, Cohen & Turnbull 1959]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Cont
+, � = <# �I<#� �I = 10
67 OPYOP �
<#� �I = �*+#∆�)
+, �+# = <# �I<#� �I
<#� �I<#A� �I =<# �I<#A� �I
�� , � �, �� �; � = ∆() −∆ ��*+, �+#) ℎ(�)
[Kari 2016a,b]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling chemical ageing
Scission of polymer chains
�� � �, �� � = 1 − \H]^ ��$<H]^ _D�`ab_ \H]^ = 1 − \H]^<H]^ = <̂H]^e cdefg#̀ ab
�
\H]^ = 1 − (_ − �� �<H]^_
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling chemical ageing contPlus reformation of new polymer links
<C�h iD�`abi \C�h = 1 − \C�h<C�h = <̂H]^e cjbkg#̀ ab
�
\C�h = 1 − (i − �� �<C�hi
�� � �, �� � = 1 − \H]^ + \C�h\C�hl ��$
[Kari 2016a,b]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling chemical ageing cont
Scission and reformation of new polymer links
�� � �, �� � = (_ − �� �<H]^_ + 1 − (i − �� �<C�h
i \C�hl ��$
[Kari 2016a,b]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Vibration isolator
[Kari et al. 2001]
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The Marcus Wallenberg Laboratory
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Fideal
Fideal
ideal isolator
k
non-ideal isolator
Fin
Fout
uin uoutue
Fideal = k ueFin = kinin uin + kinout uoutFout = koutin uin+ koutout uout
with kinout = koutin
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling approaches- Wave-guides
Traction free surface
Infinite beam
Wave equations Bessel
Trig.
Exp. harm.
Satisfy traction free B.C:s � Dispersion relation
[Kari 2001a,b, Östberg et al. 2011]
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The Marcus Wallenberg Laboratory
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100
101
102
103
104
10−8
10−6
10−4
10−2
100
102
Forc
e tr
ansm
issi
bil
ity
Frequency [Hz]
No isolatorReal isolatorIdeal isolatorIdeal isolator − Rigid foundation
Nonrigid foundation – Real isolator
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
DMTA measurements and modelling
[Kari et al. 2001]
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The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Cont
101
102
103
104
103
104
105
106
107
a)
Tra
nsfe
r S
tiffn
ess [
N/m
] -60ºC
-25ºC
0ºC
+25ºC
+60ºC
101
102
103
104
102
104
106
b)
Drivin
g P
oin
t S
tiffn
ess [N
/m]
Frequency [Hz]
-60ºC
-25ºC 0ºC
+25ºC
+60ºC
[Kari et al. 2001]
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The Marcus Wallenberg Laboratory
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References• Cangialosi, D., Boucher, V.M., Alegria, A., Colmenero, J.: Physical aging in polymers and polymer nanocomposites:
recent results and open questions. Soft Matter 9, 8619–8630 (2013)
• Cohen, M.H., Turnbull, D.: Molecular transport in liquids and glasses. J. Chem. Phys. 31, 1164–1169 (1959)
• Doolittle, A.K.: Studies in newtonian flow. II. The dependence of the viscosity of liquids on free-space. J. Appl. Phys. 22, 1471–1475 (1951)
• Greiner, R., Schwarzl, F.R.: Thermal contraction and volume relaxation of amorphous polymers. Rheol. Acta 23, 378–395 (1984)
• Kari, L.: On the waveguide modelling of dynamic stiffness of cylindrical vibraitnoso iltaors. Part I: The model, solution and experimental comparison. J. Sound. Vib. 244, 211–233 (2001a)
• Kari, L.: On the waveguide modelling of dynamic stiffness of cylindrical vibration isolators. Part I: The dispersion relation solution, convergence analysis and comparison with simple models. J. Sound. Vib. 244, 235–257 (2001b)
• Kari, L.: Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range. Part 1: Constitutive equations. Continuum Mech. Thermodyn. Submitted (2016a)
• Kari, L.: Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range. Part 2: Waveguide solution. Continuum Mech. Thermodyn. Submitted (2016b)
• Kari, L., Eriksson, P., Stenberg, B.: Dynamic stiffness of natural rubber cylinders in the audible frequency range using wave guides. Kaut. Gummi Kunstst. 54, 106–111 (2001)
• Kovacs, A.J., Aklonis, J.J., Hutchinson, J.M., Ramos, A.R.: Isobaric volume and enthalpy recovery of glasses. II. A transparent multiparameter theory. J. Polym. Sci., Part B: Polym Phys 17, 1097–1162 (1979)
• Odegard, G.M., Bandyopadhyay, A.: Physical aging of epoxy polymers and their composites. J. Polym. Sci., Part B: Polym Phys 49, 1695–1716 (2011)
• Östberg, M., Kari, L.: Transverse, tilting and cross-coupling stiffness of cylindrical rubber isolators in
• the audible frequency range—the wave-guide solution. J. Sound. Vib. 330, 3222–3244 (2011)