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Dynamic stiffness of ageing rubber vibration isolators
Leif Kari
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Structure-borne sound
Source Receiver
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Physical principle
”Hard”
”Hard”
”Soft”
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fwithout FwithFwithFwithout
Force transmissibility TF =
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
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
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
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fwithout Fwith
Nonrigid foundation
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
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
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Acoustic radiation
Wall
Wave fronts
1 W/m2 ⇔ 120 dB !
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Fideal
Fideal
ideal isolator
k
m
non-ideal isolator
Fin
Fout
m
uin uout
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
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]
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
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]
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Physical ageing
[Cangialosi et al Soft Matter 2013]
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Physical ageing cont
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
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]
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]
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]
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling chemical ageing
Scission of polymer chains
�� � �, �� � = 1 − \H]^ ��$<H]^ _D�`ab_ \H]^ = 1 − \H]^<H]^ = <̂H]^e cdefg#̀ ab
�
\H]^ = 1 − (_ − �� �<H]^_
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Modelling chemical ageing contPlus reformation of new polymer links
<C�h iD�`abi \C�h = 1 − \C�h<C�h = <̂H]^e cjbkg#̀ ab
�
\C�h = 1 − (i − �� �<C�hi
�� � �, �� � = 1 − \H]^ + \C�h\C�hl ��$
[Kari 2016a,b]
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]
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
Vibration isolator
[Kari et al. 2001]
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
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]
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
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
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
DMTA measurements and modelling
[Kari et al. 2001]
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]
The Marcus Wallenberg Laboratory
for Sound and Vibration Research
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)
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