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M ICHELL NYLUND S IZAR SHAMOON
4B1015
Examensarbete inom teknisk akustik
Örebro 2011
Marcus Wallenberg Laboratoriet för Ljud- och Vibrationsforskning
Master thesis on factors affecting the dynamic
properties of rubber isolators
Michell Nylund and Sizar Shamoon
Godkännes
Datum:
Ansvarig handledare:
ISP 09-06
4B1015 Examensarbete inom teknisk akustik
i
Abstract Rubber is a highly complicated material. Its properties are strongly dependent of
temperature, preload, frequency, amplitude and filling materials. In order to have
better knowledge about how rubber reacts for these different parameters, a number of
different papers that treats these parameters have been reviewed. Main focus is on
dynamic properties of carbon-black filled rubber. Mechanical models of rubber
isolator with different kinds of possible solution have been suggested.
A basic mathematical equation is derived to estimate the static and dynamic stiffness
of a rubber isolator depending on the preload for a geometry that is commonly used
by Atlas Copco Rock Drills AB. Tests of a rubber isolator with the same geometry is
also conducted to validate the model. It is shown that the model corresponds very well
to measurement data.
ii
Table of content Introduction .............................................................................................................................. vii
Part I: Literary Study .................................................................................................................. 1
1 Rubber properties ............................................................................................................. 2
1.1 Chemical properties of rubber ..................................................................................... 2
1.2 Mechanical properties of rubber ................................................................................. 3
1.2.1 Frequency .......................................................................................................... 3
1.2.2 Temperature ...................................................................................................... 4
1.2.3 Amplitude .......................................................................................................... 4
1.2.4 Preload ............................................................................................................... 5
1.3 Natural frequency of rubber isolator ........................................................................... 5
1.4 Geometry dependence ................................................................................................. 7
2 Rubber Modeling ............................................................................................................... 9
2.1 Modeling components ................................................................................................. 9
2.1.1 Elastic part ....................................................................................................... 10
2.1.2 Viscous part ..................................................................................................... 10
2.1.3 Frictional part .................................................................................................. 12
2.2 Existing models ........................................................................................................... 12
2.2.1 Generalized Maxwell model ............................................................................ 12
2.2.1.1 Implementation of the generalized Maxwell model ............................... 15 2.2.2 Zener model..................................................................................................... 16
2.2.3 Three way Maxwell model .............................................................................. 17
2.2.3.1 Implementation of the three way Maxwell model ................................. 18 2.2.4 Elastic force model .......................................................................................... 20
2.2.4.1 Implementation of the elastic force model ............................................. 20 3 Summary literary study ................................................................................................... 21
Part II: Experimental Part ........................................................................................................ 22
1 Introduction ..................................................................................................................... 23
2 Isolators and geometry.................................................................................................... 24
3 Method ............................................................................................................................ 25
3.1 Experimental .............................................................................................................. 25
3.1.1 Theory .............................................................................................................. 25
3.1.2 Dynamic measurement ................................................................................... 27
3.1.2.1 Uncertainty estimation of dynamic measurement ................................. 28 3.1.3 Static measurement ........................................................................................ 29
3.1.3.1 Uncertainty estimation of static measurement ...................................... 30
iii
3.2 Mathematical ............................................................................................................. 31
3.2.1 Isolator Stiffness with no preload ................................................................... 32
3.2.2 Isolator Stiffness with preload ......................................................................... 33
4 Result and discussion ...................................................................................................... 36
4.1 Evaluation of mathematical model ............................................................................ 36
5 Conclusion ....................................................................................................................... 39
Appendix A .................................................................................................. A-A Appendix B .................................................................................................... B-A Appendix C .................................................................................................... C-A Appendix D .................................................................................................. D-A
iv
Notation Introduction: DIL
F [dB] Insertion loss
Fdist [N] Disturbing force
LFafter
[dB] Vibration level after isolation
LFbefore
[dB] Vibration level before isolation
c [Ns/m] Damping
k [N/m] Spring stiffness
m [kg] Mass
u [m] Displacement
u [m/s] Velocity
u [m/s2] Acceleration
Part I: A [m
2] Cross sectional area
AB [m2] Bulge area
AL [m2] Load area
Cc [Ns/m] Critical damping level
E [N/m2] Elasticity modulus
E0 [N/m2] E-modulus, Young’s modulus
E∞ [N/m2] Bulk modulus
Ec [N/m2] Effective compression modulus
Et [N/m2] Spring rate in tension
F [N] Force
Fc [N] Force applied in compression direction
Fs [N] Force applied in shear direction
G [N/m2] Shear modulus
G∞ [N/m2] Long term shear modulus
K [N/m] Spring rate
Kc [N/m] Spring rate in compression
Ks [N/m] Spring rate in shear
Ib [kg∙m2] Moment of inertia of the cross sectional area
M [kg] Mass of supporting load
Mf [kg] Total mass of vibration isolator
S [-] Shape factor
T [m] Thickness
TABS [-] Transmissibility
Y [N/m2] Maximal frictional stress
c [Ns/m] Damping
dc [m] Compression displacement
ds [m] Shear displacement
fn [Hz] Natural frequency for undamped system
fnd [Hz] Natural frequency for damped system
l [m] Length
r [-] Ratio of exciting vibration- to the system natural-frequency
rg [m] Radius gyration of the cross-sectional area about neutral axis of
bending
t [s] Time
v
tr [s] Relaxation time
u [m] Displacement
u [m/s] Velocity
β [-] Loss factor in vibration isolator
ε [-] Strain
δ [-] Damping factor
ε [-] Loss factor
κ [-] Shear strain
κy [-] Yield shear strain
μ´ [-] Intensity of resonance caused by standing waves
ζ [N/m2] Stress
ηy [-] Yield shear strain
η [N/m2] Shear stress
θ [-] Elastomers compression coefficient
ω0 [Hz] Fundamental frequency
Part II: A [m
2] Loaded area
F [N] Force
Fscrew [N] Force from the screw joint
Fload [N] Force from the applied load
G [N/m2] Shear moduli
H [m] Hight of undeformed isolator
K [N/m] Stiffness
Kdyn [N/m] Dynamic stiffness of isolator, theory
Kdynmeas[N/m] Dynamic stiffness of isolator, measurement data
Kst [N/m] Static stiffness of isolator, theory
Kstmeas [N/m] Static stiffness of isolator, measurement data
Lkav [dB] Level of frequency-band-averaged dynamic transfer stiffness
Ry [m] Outer radius of isolator, loaded area
Ri [m] Inner radius of isolator, average inner radius
S [-] Shape factor
h [m] Hight of deformed isolator
Δl [m] Length difference
u [m] Deformation
u0 [m] A given deformation
uscrew [m] Deformation from screw joint
uover [m] Deformation on the isolator mounted over the support plate
uload [m] Deformation on the isolator mounted under the support plate
δins [dB] Uncertainty caused by instrumentation
δrep [dB] Uncertainty caused by installation repeatability of test element
δrig [dB] Uncertainty caused by the test rig
δlin [dB] Uncertainty caused by approximate linear behavior
ζins [%] Estimate of standard deviation, instrumentation
ζmeth [%] Estimate of standard deviation, measuring method
ζtot [%] Estimate of standard deviation, totally
χ [-] Dynamic factor
vi
Appendix: Cij [] Represent linear dependence
Ii [] Strain invariant
W [-] Energy strain function
αi [N/m2] Material constant
ni [-] Principal eigen vector
tr(B) [-] Trace of (B)
λi [-] Principal eigen value
μi [N/m2] Shear modulus
vii
Introduction
When mounting a vibrating device on a structure it is desired to isolate the vibrations
from spreading on to the structure for different reasons. One of these reasons is that if
vibrations are allowed to spread the structure will emit sound to the environment.
Examples of this are shown for several application used in industries over the world
where engines (vibrating devices) are mounted on chassis of different kind, see
FigA 1.
FigA 1 Engine mounted on a chassis with vibration isolators
The most common components used to isolate the vibrations are called rubber or
vibration isolators. They should not only be able to carry the load of the component
which is to be isolated but also reflect and dissipate as much of the vibrations as
possible. A suitable measure of vibration isolation is the insertion loss
afterbefore
IL FF
FLLD , [dB] EqA 1
where LFbefore
and LFafter
is the force level of a vibrating device acting on a foundation
before and after vibration isolation is added, see FigA 2.
viii
FigA 2 Vibrating device mounted on foundation before and after isolation
If the mechanical system is idealized as presented in FigA 4, the equation of motion
from solid mechanics is
distFkuucum
, EqA 2
where m is the mass, c a viscous constant, u the displacement and k is the stiffness.
They are shown in FigA 3.
FigA 4 A simplified mechanical system of the dynamic system in FigA 2
Knowledge about the spring stiffness and damping will provide information needed to
obtain good vibration isolation. But rubber is an advanced material where stiffness
and damping depend on frequency, amplitude, temperature and preload.
The basis for this thesis is to explore the dynamic properties of rubber isolators. It is
basically divided into two parts. The first part is a literary study about how rubber
isolator’s dynamic properties depend on different factors to give a better insight on
the material. Later some basic engineering formulas and models of rubber are
presented. The second part is based on experimental measurements which are used to
evaluate a mathematical rubber isolator model.
1
Part I: Literary Study
2
1 Rubber properties
Rubber is a special material with various features.
1.1 Chemical properties of rubber
Rubber is a material which is widely used in engineering applications because of its
unique features, and is the most common material for vibration and shock isolation
between stiff structures and mounting joints.
Rubber is a denotation used for a big group of materials with different
chemical compositions but similar molecular structure and mechanical properties.
Due to the high elasticity features, rubber is often called elastomers. Elastomers are
polymers with very long molecular chains. They are made of a wide variety of
organic substances for example from latex, which is the sap of a tropical rubber tree,
especially Hevea brasiliensis [1]. It coagulates in thin sheets and bundles into bales.
The important process of vulcanization, which was discovered by Charles Goodier
(Charles Goodyear) in 1839, converts the plastic raw elastomeric material into a solid
and elastic consistency. Vulcanization is a chemical process where the long molecular
chains are linked together and thereby from a stable and more solid molecular
structure. There are many kinds of rubber mixtures. The cross linking is enabled by a
small amount of sulfur that is mixed with raw material. When the mixture is heated to
about 170 C the vulcanization process starts and cross links are formed, connecting
the molecular chains, see Fig 1-1.
Fig 1-1 Microstructure of a Carbon-black filled rubber vulcanized Gray circles: Carbon particles.
Solid lines: Polymer changes. Dash and zigzag-lines: Cross links [2]
3
Natural rubber used in some specific engineering applications needs to be reinforced
by addition of carbon-black filling (very small particles of carbon 20μm-50 μm).
These particles are added to natural rubber before vulcanization. The carbon-black
filling increases the stiffness of the material and its wear resistance.
The rubber phase forms a constituent network, and the filler material forms
agglomerates inside the rubber network. The material is thus a two-phase material
made from constituents with completely different mechanical properties [2].
1.2 Mechanical properties of rubber
The two basic and most discussed features of rubber are stiffness (modulus) and the
ability to dissipate energy (loss factor). The loss factor is a quantity that measures the
total mechanical energy losses in the rubber isolator. These losses are coupled to
frictional and viscous effects that are generated in the rubber isolator when it is
subjected to vibrating loads. Modulus and loss factor are dependent on frequency,
amplitude, preload and temperature.
1.2.1 Frequency
At low frequencies rubber is soft and gets stiffer as the frequency increases. The loss
factor has its peak value where the change of the stiffness is at its maximum, see Fig
1-2. A jump in the modulus can be witnessed when going from static to quasi-static
frequencies. For filled rubber the phenomenon is more prominent [3].
Fig 1-2 Stiffness and loss factor dependence on frequency [4]
4
1.2.2 Temperature
At low temperature rubber is stiff and as the temperature rises it goes through a
transition region before it reaches the area where the material acts like rubber. As with
the frequency dependence the loss factor is at its highest when the absolute change of
the stiffness is maximum, see Fig 1-3.
Fig 1-3 Stiffness and loss factor dependent on temperature [4]
1.2.3 Amplitude
The Payne effect is an amplitude dependence which is more pronounced for filled
rubber isolators. When the excitation amplitude increases the dynamic modulus
decreases. Both loss angle and amplitude are presented in Fig 1-4 [3, 4, 5].
The Mullin’s effect is also a phenomenon coupled to the amplitude
dependence, it is an effect that occurs when rubber is subjected to an initial load.
There is a reduction in the peak stress values after the first oscillations. This is
believed to be the result of rearrangements in the filler molecules. Mullin’s effect is
more pronounced in rubber with carbon-black filling [6, 4, 7, 8].
Fig 1-4 Stiffness and loss angle dependent on amplitude, Payne effect [4]
5
1.2.4 Preload
For most rubber components the dynamic stiffness within the quasi-static frequency
range gets higher as the preload increases. An example of a rubber isolator subjected
to three different load cases is shown in Fig 1-5. At frequencies above 300Hz one can
observe a change in the curve shape with applied preload.
Fig 1-5 Measured results (b) transfer stiffness magnitude (c) and phase angle as a function of
frequency and different preloads. 0N (solid thin), 2100N (dashed line) and 3350N (dotted line) [9]
1.3 Natural frequency of rubber isolator
At high frequencies there are resonances when the dimensions of the rubber isolator
overlap with multiples of a half-wavelength. There are some simple assumptions
made to roughly estimate these resonance frequencies.
Ref. [10] assumes a rubber isolator as a mass less spring and plots the transmissibility
vs. frequency from a cylindrical rubber isolator. It is found that the fundamental
natural frequency is fn=30Hz and the first high-frequency peak is observed at
fn’=500Hz. The first natural frequency corresponds well to calculations but the
assumption does not hold from the first high-frequency resonance.
Another assumption is that the isolator can be designed as a cylindrical rod
with uniformly distributed mass. The high-frequency wave resonance for the isolator
occurs at
....3,2,1,0i ii , Eq 1-1
where ω0 is the fundamental natural frequency and γ is
AlMMM // f , Eq 1-2
where M is the mass of the object supported by the isolator and Mf is the total mass of
the isolation element, ρ is its density, A cross sectional area and l is its length [10].
It is also possible to calculate the high resonance frequency intensity with
tan
2.0
tan
2.0' f
M
M, Eq 1-3
6
where β is the loss factor of the vibration isolator. It is obvious from previous
formulas that both the positioning and the intensity of high resonance peaks is
depending on the ratio of the mass for the isolator and mass supported by the isolator.
The conclusion that the high-frequency resonance peaks will shift for different loads
can be made. The natural frequency of an undamped spring-mass system is
M
Kf
2
1n , Eq 1-4
where K is the spring rate of the carrying component.
One can also derive a corresponding equation for the damped system. The
derivation of this equation is based on the equation of motion for the undamped
system. The solution is based on that the transmissibility. The transmissibility
depends on both system characteristics and the vibrations see
)1(
12ABS
rT
, Eq 1-5
where r is the ratio between the excitation frequency and the system natural frequency
and
nf
fr . Eq 1-6
The natural frequency for almost all real systems varies slightly with amount of
damping in the system. This is the reason for introducing a variable ζ which is the
damping factor. The damping factor is approximately half of loss factor
2/ . Eq 1-7
The natural frequency for a damped system is
2
nnd 1 ff . Eq 1-8
It is obvious from the equation that the natural frequency for the damped system is
related to the undamped system. The damping factor is
cC
C , Eq 1-9
where the critical damping level Cc for the damped vibratory system is
KMC c , Eq 1-10
and the equation for absolute transmissibility of a damped system is
7
2
22
2
ABS
1)2(
)2(1
rr
rT
. Eq 1-11
Fig 1-6 Transmissibility for a damped and undamped system [11]
The Fig 1-6 above shows the transmissibility for a damped and undamped system. It
can be recognized from the figure that the damping reduces the transmissibility in
regions around r=1. It can also be noticed that the damping reduces the amount of
protection in region where r> 2 [11].
In real life, the isolator systems do not follow the above named model perfectly
due to some non-linearity, especially for elastomers which are very sensitive to
vibration level, frequency, temperature and more.
1.4 Geometry dependence
As mentioned before the mass/spring system is the most basic model when finding the
fundamental frequencies in rubber isolators. The key design parameter for this model
is K, which define the spring stiffness (sometimes called spring rate). K is
d
FK , Eq 1-12
where F is the applied force and d is the deflection. K varies with load deflection
(shear, compression, tension)
T
AGK s , Shear stiffness Eq 1-13
T
AEK c
c Compression stiffness Eq 1-14
8
and
T
AEK t
t , Tension stiffness Eq 1-15
where AL is effective load area, T is the thickness of the undeformed elastomer. The
parameters G, Ec and Et stand for the shear, compression and tension modules of the
elastomer.
For the shear load Eq 1-13 could be used for simple shear calculations of
sandwich models (rubber bonded with metal). The equation is only valid when
deformation due to bending is not taken into account. When the ratio of length to
thickness of the element exceeds 0.25, the argument about bending shear is not valid
anymore and the stiffness could be calculated according to
2
g
2
s
ss
36/1
1
rTT
AG
d
FK , Eq 1-16
where
2/1
bg )/( AIr , Eq 1-17
is the radius of gyration of the cross-sectional area about the neutral axis of bending,
and Ib is the moment of inertia of the cross-sectional area A about the neutral axis of
bending [7].
For the compression load case named above, Kc is calculated according to Eq 1-14.
To get a reliable result for this calculation, Ec (effective compression modulus) must
be known. Ec is a function of both material properties and component geometry.
To calculate the spring rate in compression Kc, there are different analytical
techniques used. For instance, Ec for a simple sandwich block could be calculated as
)21( 2
0c SEE For a bidirectional strain Eq 1-18
or
)1(33.1 2
0c SEE , For one-dimensional strain Eq 1-19
(long, thin compression strip)
where E0 is young’s modulus θ is elastomers compression coefficient (see TableC 1
in Appendix C) and S is the shape factor
B
L
A
AS , Eq 1-20
where AL is the load area and AB is the bulge area.
9
The formulas named above can be used to estimate the spring rate roughly. For more
accurate result, some more advanced analytical techniques or finite element analysis
are required.
2 Rubber Modeling
Many different rubber models have been developed with the approach of determining
attributes of interest e.g. dynamic stiffness, damping and loss factor. The models are
built up of almost the same elements, viscous dampers, elastic and friction elements
which are coupled together in different ways. An example is the common three way
Maxwell model seen in Fig 2-1.
Fig 2-1 Rubber model existing of elastic, viscous and frictional elements
Not all models can give a good physical representation. Some models are better at
representing amplitude dependence and some models are better at predicting the
frequency dependence. There are other variables that affect the properties of rubber
namely temperature, filling type and preload. It is difficult to include all these
parameter in one single model. Consequently the researchers often try to test one
single parameter at a time.
2.1 Modeling components
The different elements that build up some of the basic models used in scientific
literature will now be presented in a short introduction. Physical representation and
derivation of the fundamental equations that describes the model components will be
presented as well.
10
2.1.1 Elastic part
The single linear spring is one of the basic elements that a model could consist of, see
Fig 2-2.
Fig 2-2 Basic elements for rheological models [5]
It represents the linear elastic part of the model. The linear elastic part is not capable
to describe a complete model. There are some models which are more complicated
and include some non-linearity by using non-linear components, see Fig 2-3. And the
force depends non-linear on the displacement.
Fig 2-3 Non-linear elastic element [4]
2.1.2 Viscous part
The viscous part can also be modeled by various elements. One possible model is a
single linear dashpot, see Fig 2-4.
Fig 2-4 Basic dash-pot element for rheological models [5]
The force over the dashpot can be written as
ucFv Eq 2-1
and it is proportional to the velocity.
The dashpot element causes a time delay when cyclic load is applied i.e. the
force and displacement will be out of phase. If a plot is made with force versus
displacement a hysteresis loop is displayed, see Fig 2-5. This is obviously valid for all
dynamic systems [8].
11
Fig 2-5 Force and displacement [4]
The dynamic behavior of a viscoelastic material is analogical with a corresponding
spring, dashpot combination. The rheological models represent the elastic behavior of
spring and viscous behavior of the dashpot. Another possible model of viscous
element called spring-pot is presented in Fig 2-6.
Fig 2-6 The spring-pot element [4]
The spring-pot model treats the constitutive equation with fractional time derivatives,
while the dash-pot model uses the standard integer order derivative.
12
2.1.3 Frictional part
The frictional part is usually represented by two different elements. An element with
smooth characteristic is presented as a frictional part shown in Fig 2-7.
Fig 2-7 Smooth frictional component [4]
This element takes account for the amplitude dependence (Payne effect) during the
simulation[4]. Another element is the frictional solid element. It consists of a spring
and frictional blocks see Fig 2-8.
Fig 2-8 Frictional solid element [5]
An advantage of modeling with smooth frictional component is that you need only
one element, which gives two parameters to determine to get a good representation.
Not like the frictional solid element shown in Fig 2-8 where two or three elements are
needed which gives four to six parameters for good model fitting to experimental
data.
2.2 Existing models
Rubber has been used in the industry for over a hundred years. Many models have
been developed during this time to describe the characteristic. From the beginning
only simple models which consisted of dashpots and springs built up different
structures that could simulate the behavior of rubber e.g. Kelvin-voigt model, which is
a dashpot in parallel with a spring but it does not give a realistic physical behavior [2].
2.2.1 Generalized Maxwell model
The Maxwell element is shown in Fig 2-9.
Fig 2-9 Maxwell element [5]
The generalized Maxwell model contains a spring which is parallel with Maxwell
elements, see Fig 2-10.
13
Fig 2-10 The generalized Maxwell model [5]
The accuracy of the model depends on the amount of Maxwell elements. The total
stress
n ....1 , Eq 2-2
where n denotes the stress over n:th element.
The Maxwell elements are considered as viscoelastic fluid model. The
relaxation behavior can be generalized as Er(t) which is a time dependent fundamental
function that describes the behavior of viscoelastic material. For a single Maxwell
element with a spring and dashpot serial coupled, the requirement is that the total
strain is
springdashpot . Eq 2-3
The time derivative are
springdashpot
, Eq 2-4
dashpot Eq 2-5
and
E
spring . Eq 2-6
14
This gives
EE
, Eq 2-7
which is the differential equation that describes the strain-stress relation for a
Maxwell model. The normalized relaxation behavior derives by solving this equation
for a step strain. When t>0 the time derivative for strain is 0, inserting this in Eq 2-7
gives
0
E
at t > 0. Eq 2-8
When the step strain is applied, the dashpot acts like a rigid part due to the infinity
strain at t=0, see Fig 2-11. The initial stress is given only by the elastic spring and the
initial condition will be ζ(0)=Eε0. Using this to solve Eq 2-7 yields
tE
eEt
0)( . Eq 2-9
The relaxation modulus for the model can now be defined as
r)(R
t
t
EetE
, Eq 2-10
where the relaxation time is tr=ε/E. The relaxation behavior of this element is
illustrated in Fig 2-11.
Fig 2-11 Maxwell model’s relaxation behavior [5]
15
2.2.1.1 Implementation of the generalized Maxwell model
The generalized Maxwell model gives a good representation of the frequency
dependence of rubber components. The representation gets better with a growing
amount of Maxwell elements implemented. However this results in a larger amount of
parameters to be determined.
In FE-codes the viscoelastic part are modeled with different strain energy
functions e.g. Neo-Hookean or Mooney-Rivlin. All the physical properties have to be
determined by experimental measurements. A step by step method of implementing
frequency dependence and the amplitude dependence in commercial FE-code is
presented in ref. [3]. It is done in two ways. The first way is that a single generalized
Maxwell model is used to model the whole rubber isolator. The second way is to let
several generalized Maxwell models build up the isolator representing different parts
of the component depending on different strain levels, this way of modeling is proven
to give a better representation but is harder to implement.
The step by step method is schematically showed in Fig 2-12. If amplitude
dependence is ignored (e.g. non-filled or soft rubber) the shear and bulk modules are
extracted from experimental measurements and is inserted in the FE-code, the
dynamic properties can be obtained from applying a dynamic step. If amplitude
dependence is included (e.g. filled or hard rubber) the calculations get more
complicated as an equivalent strain needs to be calculated from quasi-static
measurements at the specific amplitude range of interest.
Fig 2-12 Practical procedure of the simplified methodology [3]
16
2.2.2 Zener model
Zener model is very similar to the generalized Maxwell. It consists of only two
elements, one single spring and one Maxwell element which are parallel coupled, see
Fig 2-13.
Fig 2-13 Zener model [5]
The Zener model is a simple viscoelastic model which acts like a broad frequency
range. It has reasonably good relaxation behavior but limited creep behavior. If we
now look at the stress in the different elements in this model and call the stress over
spring to ζ∞ and the stress over the Maxwell element to ζM, the total stress shown in
Fig 2-13 will be
M . Eq 2-11
Rewriting this equation gives
M Eq 2-12
and the time derivative is
M . Eq 2-13
Inserting this relation to the differential equation derived for the Maxwell model
follows in
E , Eq 2-14
E
EE
Eq 2-15
and
EE
EEE
)( . Eq 2-16
The last equation is the differential equation for the Zener model. It defines the
relations between stress and strain. This equation could be solved for a strain step
history. For further details please check ref. [5]. The relaxation behavior of a Zener
model is illustrated in Fig 2-14.
17
Fig 2-14 Relaxation behavior of Zener model [5]
2.2.3 Three way Maxwell model
The three way Maxwell model is also called the generalized viscoelastic elastoplastic
model, see Fig 2-15.
Fig 2-15 Generalized viscoelastic elastoplastic model [5]
There are two quite similar options of this model. The first option can be observed in
Fig 2-15, where the total stress is
fve . Eq 2-17
The second option has an elastic part consisting of a non linear spring. The first term
in Eq 2-17 is different in the non-linear viscoelastic elastoplastic model. The total
stress in this model is
fve )( E . Eq2-18
The previous expression describes the stress for only three elements. The total stress
in the whole model can be expressed as
fv
e )( iiii E , Eq 2-19
18
where i=1, 2, 3 … and subscripts v and f denotes the viscoelastic elements and
frictional elements.
2.2.3.1 Implementation of the three way Maxwell model
Ref. [6] has a good suggestion for implementing such a model in ABAQUS. The
model is shown in Fig 2-16.
Fig 2-16 One dimensional generalized viscoelastic elastoplastic model [6]
As mentioned before in this case shear quantities are used
N
j
j
M
i
i
1
ep
1
veeepvee , Eq 2-20
where is the shear stress and the subscripts e, ve and ep stand for elastic, viscoelastic
and elastoplastic elements.
The total stress can be obtained of the summation of the distribution of single
elements in the model. The amplitude dependence and the rate dependence can be
considered to be independent. It gives rather good results for some kind of rubber
materials, especially rubber with filling.
For the three dimensional model the total stress is achieved for the total
distribution of single stresses. For the finite element model the summation is obtained
by an overlay of finite element meshes, see Fig 2-17.
19
Fig 2-17 Overlay method Finite element meshes [6]
The main idea of the finite element model is to achieve the total stress tensor from the
parallel contribution of finite element meshes. This method is called the Overlay
method and the total stress tensor is obtained from the separate finite elements which
is included in the model.
The modeling of the elastic and viscoelastic parts is obtained by implementing
a hyperelastic model (Yeoh or Neo-Hooke), which is already existing in ABAQUS.
The elastoplastic part however could not be modeled directly because of the lack of a
piece wise kinematic hardening model in ABAQUS/explicit, instead an elastoplastic
model based on a hypoelastic description was used. The stress summation is achieved
by assembling each layer of elements into one set of nodes. This approach yields a
model able to represent the combined rate and amplitude dependence.
The main drawback with this solution is the amount of parameters to decide. To avoid
this problem a general step by step method is suggested to extract the model
parameters using an error function and the steps are presented bellow.
• Firstly a rough guess of the material parameters is made. Both
yield strains and relaxation times are given a logarithmic
distribution over the measured amplitudes and frequencies
respectively.
• Secondly, the shear moduli of the elastic and the elastoplastic
elements are fitted to only the lowest frequency for which the
influence of the viscoelastic elements may be neglected. The
yield strains for the elastoplastic contributions are unchanged at
this stage.
• Thirdly is to fit all of the shear moduli to all test data. After this
step the model should be fairly accurate.
• Fourthly is to fit all material parameters to all test data,
resulting in a minor adjustment of the material model.
These four steps together with an analytical approach give material parameters that
correspond well with measurement results.
20
2.2.4 Elastic force model
Fig 2-18 The elastic force model [4]
The elastic force model consists of three components, much like the three way
Maxwell model with the difference that nonlinear effects are taken into account. It
gives a good representation of a rubber component with only five parameters to
determine. It is a model relating component compression with component force and
accounts for both frequency and amplitude dependence, while neglecting temperature
and preload.
It is described with three elements, from the left in Fig 2-18. The first element
is an elastic spring showing nonlinear behavior to simulate the geometrical and
material nonlinearities that are associated with a real rubber component. The second
element is used to describe the frequency dependence through a fractional derivative,
spring pot. The third element is associated with amplitude dependence through a
smooth frictional component.
2.2.4.1 Implementation of the elastic force model
The elastic stiffness at different compressions is
)(81
)(4
3 2
2
2
elastxLL
D
xL
LDK
, Eq 2-21
this gives the compression force in
)(16
)2(1
4
32
22
elastxLL
xLD
xL
xDF
, Eq 2-22
where x is the compression ( Lx 0 ), D is the diameter, L is the original height and
μ is the static shear modulus.
The second part is associated with the frequency dependence. It is modeled by
a part named spring-pot. This model relates the force to displacement through a non-
integer time derivative. The relation is shown in
xbDF freq , Eq 2-23
21
where α (0< α<1) and b>0 are model constants. The relation can be rewritten through
a Grünwald definition as
jn
n
j
nn xj
jtbxbDtF
1
0
freq)(
)(
)()(
, Eq 2-24
the Γ is the Gamma function. This model is dependent on the displacement history
this suits rubber isolators very well because of the memory that is displayed by
rubber.
The third part accounts for the amplitude dependence. The force of the friction
is defined in two different directions when x is increasing
)()()1(
fsfmax
2
fsfrict FFxxx
xxFF
s
s
, Eq 2-25
holds and when x is decreasing
)()()1(
fsfmax
s2
sfsfrict FF
xxx
xxFF
, Eq 2-26
is accurate, where Ffmax and the displacement parameter x2 (the rate of friction force
development) are defined by the user. The total force response is a super positioning
of the three contributions according to
))(())(())(()( frictelastfreqtot nnnn txFtxFtxFtF . Eq 2-27
Later a model that describes frequency, preload and temperature dependence is
presented, for more insight reading of ref. [4] is recommended.
3 Summary literary study
The aim of this part of the master thesis has been to give further knowledge about
rubber and to study different models of rubber isolators and decide the accuracy of
these models and eventually suggest some appropriate ways to calculate the dynamic
behavior of rubber isolators.
Rubber is an advanced material. The selection of rubber model depends on the
application of interest. There are mainly two models that consider both frequency and
amplitude dependence that are presented. The models are named the three way
Maxwell model and the elastic force model. Both give good representation with an
advantage for the elastic force model because of less parameters to determine. On the
other hand the three way Maxwell model presented in ref. [6] shows a step by step
method implemented with existing FE-code which may lead to faster establishment
for industry use.
22
Part II: Experimental Part
23
1 Introduction
Previous studies show that the static and dynamic properties given by the sub-
suppliers of rubber isolators are not always correct. Measurements of a rubber isolator
that is commonly used in the industry have been conducted. Later a model is derived
that gives the static stiffness with regards to preload and geometry. The dynamic
stiffness can be obtained by introducing a dynamic factor.
24
2 Isolators and geometry
The geometry of the investigated isolator, CB2205-2, is shown in FigII 2-1 and Table
2-1. It should be mentioned that the mount consists of two pieces. Illustration of the
mounted isolator can be seen in FigII 2-2.
FigII 2-1 Geometry of the investigated rubber isolator [12]
Table 2-1 The dimensions [mm] of the isolator that is investigated [12]
Series Number A B C D E F G
CB-2205 124 27 38,1 85,9 64,8 31,8 37,8
FigII 2-2 Mounted rubber isolator
25
3 Method
Now follows a description of the experimental theory and a derivation of the static-
and dynamic-stiffness with geometry and preload dependence.
3.1 Experimental
Rubber is an advanced material and many precautions have to be made before
conducting measurements. In the upcoming sections some information about the
theory behind the measurement and also how the experiment, both static and dynamic,
has been carried out will be presented.
3.1.1 Theory
One applies a force and measures the displacement when calculating the stiffness, see
FigII 3-1. The stiffness is
l
FK
, EqII 1-1
where F is the applied force and Δl is the change in thickness.
FigII 3-1 Illustration of basic stiffness
A cylindrical metal plate is mounted in between the two pieces of the vibration
isolator. A cylindrical metal tube with one closed and one opened end is mounted on
top of the isolator resulting in that the excitation force can be led through to the
support plate, see FigII 3-2.
26
FigII 3-2 A exploded view of the vibration isolator assemble
An excitation force is applied to the cylindrical metal tube when calculating the
dynamic stiffness. The movement of the support plate relative to the underlying
surface is measured, see FigII 3-3.
FigII 3-3 Force and displacement positions
The force and displacement are measured in the time domain and then Fourier
transformed to the frequency domain before the operation resulting in the dynamic
stiffness in
)(
)(dyn
l
FK
. EqII 1-2
During the experiment (both static and dynamic) the isolator is prestrained with listed
torque value of 500Nm that is used during assemble.
27
3.1.2 Dynamic measurement
The measuring chain for the dynamic measurement can be seen in FigII 3-4. A total
of six accelerometers where mounted, three above and three under the isolator. The
result from the three displacements was then averaged to get a final outcome. The
isolators are also preloaded with a load that is 80% of the maximum recommended
load listed in Table 3-5 before the dynamic tests are conducted.
A random burst signal with the frequency span of 0 to 5000Hz and executed
every two seconds with a pause of two seconds in between every burst. Isolators
consisting of blends of natural rubber, carbon black, sulfur and other chemical agents
where investigated. Thus the Mullin’s effect (see heading 1.2.3 in Part I for details)
should be more pronounced. To make sure that the Mullin’s hade dissipated some
excitations where conducted before the measurement commenced. The frequency
range of interest is 0-200Hz.
FigII 3-4 Measuring chain of the dynamic test setup
The acceleration is measured by six piezoelectric accelerometers where each pair is
symmetrically mounted. The used equipment when conducting the dynamic
measurement is shown in Table 3-1.
Table 3-1 The equipment used during measurements
Accelerometers S/N Charge Amplifiers S/N Data Collection S/N
Brüel & Kjaer 4369 615013 Brüel & Kjaer 2635 1117816 Agilent 8408A -
Brüel & Kjaer 4369 1015936 Brüel & Kjaer 2635 1799669 Agilent 8491B -
Brüel & Kjaer 4369 1015933 Brüel & Kjaer 2635 669742 Agilent E1432A -
Brüel & Kjaer 4369 963010 Brüel & Kjaer 2635 1212679 - -
Brüel & Kjaer 4369 1015932 Brüel & Kjaer 2635 638515 - -
Brüel & Kjaer 4367 597775 Brüel & Kjaer 2635 735433 - -
Force Transducer S/N Brüel & Kjaer 2635 669794 - -
Kistler 9091 76297 - - - -
28
3.1.2.1 Uncertainty estimation of dynamic measurement
The uncertainty estimation of the dynamic measurements is based on ref. [14]. The
test method described is similar to the one conducted in this thesis.
The frequency-band-averaged dynamic transfer stiffness is
2
0
2
dynlg10
av
k
KLk , EqII 1-3
where k02=1[N/m]. With the introduction of error quantities, becomes
linrigrepinsavav
kk LL , [dB] EqII 1-4
where δins is the error caused by instrumentation, δrep an error caused by installation
repeatability of test element, δrig the error caused by the test rig and δlin is the error
caused by approximate linear behavior.
In this thesis the uncertainties are shown in Table 3-2. The sensitivity
coefficients are estimates.
Table 3-2 Uncertainty budget for dynamic measurements [14]
Quantity Estimate
dB
Standard
uncertainty
ui dB
Probability
distribution
Sensitivity
coefficient
ci
Uncertainty
contribution
ciui dB
Transfer
stiffness
level
avkL
0.3
Normal
1
0.3
δins 0 0.5 Normal 0.2 0.1
δrep 0 3/p Rectangular 0 0
δrig 0 0.5 Rectangular 0.5 0.25
δlin 0 0.5 Rectangular 0.5 0.25
The error is defined for a coverage probability of 95% and assumed to be a normal
distribution, the value of the expanded uncertainty is given by
5
1
2)(2i
iiucU . EqII 1-5
And in this case the expanded uncertainty is:
U=1.11 [dB]
29
3.1.3 Static measurement
The measurement chain and the equipment used are shown in FigII 3-5 and Table 3-3.
FigII 3-5 Measuring chain of the static test setup
Table 3-3 Equipment for statical measurement
Data Collection S/N Position Sensor S/N
Agilent 8408A - HBM 1-WA150MM-L 063910050
Agilent 8491B - HBM 1-WA150MM-L 063910056
Agilent E1432A - Low Pass Filter S/N
Force Transducer S/N HBM ML55B -
Instron Model8032 - HBM AB22A -
At the start of the measurement a load of 110N is applied and the position sensors are
zeroed. The load is increased a minimum of 300N and left under pressure a time of at
least 30 minutes to let the viscous effects of the rubber dissipate. The force and
displacement signal is sampled for 60 seconds, an average of this is the force at a
specific displacement.
30
3.1.3.1 Uncertainty estimation of static measurement
The uncertainty estimation of the static measurement is based on a summation of the
uncertainty of the instrumentation (force transducer, positioning sensors etc),
measuring method and other factors (viscous effects, fatigue etc). These estimations
are shown in Table 3-4.
Table 3-4 Estimation of standard deviation in percent
Standard
deviation
Estimate
[%]
Basis for estimation
ζins 0.8 Calculated from uncertainty given by the suppliers of the
measuring equipment.
ζmeth 1 The test method gives a low uncertainty due to the long
relaxation and sampling time.
ζother 2 Other factors that could affect the uncertainty.
ζtot 3.8 Summation of all standard deviations
0 0.5 1 1.5 2
x 10-3
0
1000
2000
3000
4000
5000
6000Force Statical measurement of CB2205-2
Deflection [m]
Forc
e [
N]
WRONG measurement
RIGHT Measurement
Extrapolation
FigII 3-6 Load against deformation of 2205-2 isolator. Blue line: Measurement with WRONG method
Crosses: Measurement with RIGHT method Red line: Extrapolation of RIGHT measurement points
If the viscous effects during the static measurements are not taken into account the
results are not correct. An example of this is illustrated in FigII 3-6. In this case a load
was applied with a step increase of 0.005mm/s (blue line), instead of waiting for the
viscous effects to disappear (red line).The slope of the line is the static stiffness. In
this case the isolator has a larger stiffness for a smaller load.
31
3.2 Mathematical
This model of rubber isolator is purposed to fit some isolators tested in KTH on
behalf of Atlas Copco Rock Drills AB, Rocktech Division. The model is specific for a
cylindrical rubber isolator with an opening in the middle of the specimen FigII 3-7.
FigII 3-7 Rubber investigated rubber isolator [12]
Stiffness of the isolator showed above will be discussed for two different cases, with
and without preload. Simple formulas for estimating the static and dynamic stiffness
for this isolator geometry in different load cases are derived.
The investigated isolator is produced by Lord Corporation is tabulated according to
Table 3-5.
Table 3-5 Data of isolators [12] (*all elastomers are blends of Natural Rubber,
Carbon Black, Sulfur and other chemical agents)
T-Thick support Plate
(recommended)
Part
Number
Elastomer Elastomer
code
Elastomer
nominal
ShoreA
Thickness T
[mm]
Load/
Deflection
[N/mm]
CB-2204-3 NR A079P 50 28.6 3110 at 2.3
CB-2204-5 NR A111P 58 28.6 4450 at 2.3
CB-2205-2 NR A060P 43 31.8 5340 at 2.3
CB-2205-3 NR A079P 50 31.8 6670 at 2.3
CB-2205-4 NR A091P 53 31.8 8010 at 2.3
CB-2205-5 NR A111P 58 31.8 9340 at 2.3
32
3.2.1 Isolator Stiffness with no preload
To make this derivation possible and easy, one has to define a simple geometry which
approximately describes the real specimen, see FigII 3-8.
FigII 3-8 Geometry approximation for rubber isolator with metal cap on top
The static stiffness of the rubber isolator shown in FigII 3-8 without any preload can
be written mathematically with equation
2
st 213
SH
GAK , EqII 2-1
where G is the shear modulus, H the height of the isolator (see TableC 1 in
Appendix C) and A is the load area which is
2
i
2
y RRA , EqII 2-2
where Ry and Ri (see Table 2-1) is the outer and inner radius of the undeformed rubber
isolator.
S is the shape factor defined as the loaded area divided by the surface area
HR
RRS
y
2
i
2
y
2
. EqII 2-3
Because the load area is different on the support plate side and the support bracket
side an average of these two inner radiuses is calculated. The static stiffness could
now be used to calculate the dynamic stiffness
stdyn KK , EqII 2-4
where χ is a dynamic factor given by the rubber producer or measured in a laboratory.
It is recommended that this factor is measured for each special case because of its
wide variety (1.3 - 5).
33
3.2.2 Isolator Stiffness with preload
In this section, a similar case as in the previous section will be discussed but with
respect to preload. When a load is applied upon the rubber isolator it will bulge as
shown in FigII 3-9 (a) but mathematically it is easier to approximate the deformation
as is shown in FigII 3-9 (b). The approximation takes account for nonlinearity in
geometry, however the material is still treated linearly neglecting possible non-
linearity.
FigII 3-9 (a) Real deformation of a rubber isolator (b) Mathematical approximation of deformation
EqII 2-1 to EqII 2-3 is reused, the difference is that now there is deformation
dependence and the stiffness of the isolator will now be
)(21)(
)(3)( 2
st uSuh
uGAuK , EqII 2-5
where the load area and height is
2
i
2
y)( RruA EqII 2-6
and
uHuh )( . EqII 2-7
And the shape factor becomes
uHr
RR
hr
RRuS
y
2
i
2
y
y
2
i
2
y
22)(
. EqII 2-8
Assuming incompressibility yields
2
i
2
i
2
y2
y
2
i
2
y
2
i
2
y Rh
HRRrhRrHRR
. EqII 2-9
Eq 2-5 to Eq 2-9 gives
34
32
i
2
i
2
y
32
i
2
y
2
2
i
2
y
st))](()[(2
)(
)(
)(3)(
uHuHRHRR
RR
uH
RRGHuK . EqII 2-10
The dynamic stiffness is
stdyn )( KuK EqII 2-11
and the static stiffness is
)()(st udu
dFuK . EqII 2-12
At a given deformation 0u the total force is
0 0
0 0
st0 )()(
u u
duuKdudu
dFuF . EqII 2-13
FigII 3-10 Isolator with screw force and applied load
The force acting on the rubber specimen from the screw joint will be the same for the
isolator mounted over and under the support plate, see FigII 3-10. This specific force
will give a displacement on each isolator given by
2
2screw
DTHu
. EqII 2-14
When an external load is applied the total displacement of the isolators
loadscrewover uuu EqII 2-15
35
and
loadscrewunder uuu . EqII 2-16
The total force acting on the support plate can now be defined as
duuKduuKF
uu
)()(underover
0
st
0
stst . EqII 2-17
36
4 Result and discussion
The last part of this thesis consists of analyzing the measurement result that has been
obtained for an isolator of model 2205-2 produced by LORD. To verify that the
mathematical model in section 3.2 is valid, static- and dynamic test data from isolator
CB2205-2 is compared to the calculations.
The equipment used during measurements is listed in Table 3-1 and Table 3-3.
A personal computer provided with MATLAB (R2010a) is used to perform the
analysis.
4.1 Evaluation of mathematical model
The dimensions of the rubber isolators are defined in Table 2-1 and the shear-moduli
corresponding to shore value is presented in TableC 1 in Appendix C. Calculations of
three different shore-values are plotted together with measurement data of isolator
CB2205-2. The force is illustrated in FigII 4-1 and the stiffness is illustrated in FigII
4-2. The displacement in the plots are defined from Fload i.e. the screw joint is
mounted and the displacement is zero at this point.
0 0.5 1 1.5 2 2.5
x 10-3
0
1000
2000
3000
4000
5000
6000
7000
Displacement u [m]
Forc
e F
[N]
Measurement and mathematical FORCE of 2205-2
Math Model G=370000Shore=35
Math Model G=450000Shore=40
Math Model G=540000Shore=45
2205-2 extrapolation
Given by LORD cataloge
2200-5 measurement
FigII 4-1 Comparison of static measurement force and mathematical model
As the above figure shows the model corresponds well to the real isolator. The
measurement values are illustrated by the crosses (error bars based on the standard
deviation estimation are also shown) and the line is an extrapolation of the
measurement results. Values given by LORD and a value given from the wrongfully
37
conducted measurement is also marked, see section 3.1.3 and FigII 3-6. The
measurements seem to match with what has been seen in literature, namely that the
isolator gets stiffer with higher preload as is expected.
0 0.5 1 1.5 2 2.5
x 10-3
2.2
2.4
2.6
2.8
3
3.2
3.4
3.6
3.8x 10
6
Displacement u [m]
Sta
tic S
tiff
ness K
[N/m
]
Measurement and mathematical STIFFNESS of 2205-2
Math Model G=370000Shore=35
Math Model G=450000Shore=40
Math Model G=540000Shore=45
2205-2 measurement
Given by LORD catalog
WRONG measurement
FigII 4-2 Comparison of measurement stiffness, mathematical stiffness and stiffness calculated from
LORD catalogue
It should also be noted that the shore value of 2205-2 is 43 but this value has a
deviation of 10-15% because of the accuracy in the production batches. While the line
in FigII 4-2 which follows the measurement data the best has a listed shore value of
40. As mentioned before it can be due to the deviation in batches during production.
Table 4-1 Comparison of load and static stiffness from mathematical, measurements and LORD
catalogue given data
2205-2 Load[N] at 2.3[mm] Static Stiffness[N/mm^2] at
2.3[mm]
Math. Model 7023 3,130
Measurements 6949 3,033
LORD 5340 2,322
WRONG
measurement
6200 2,364
In Table 4-1 the load and static stiffness at 2.3mm deflection is given. When
comparing the LORD results to the measurement conducted there is a difference of
roughly 77% both between load and static stiffness. One interesting observation is
that the wrongfully conducted experiment results are closer to the LORD catalog
given values.
38
20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
12
14x 10
6
X: 10.17
Y: 3.739e+006
Dynamic stiffness of 2205-2
Frequency [Hz]
Dym
anic
stiff
ness [
N/m
]
FigII 4-3 Dynamic stiffness of isolator 2205-2
When looking at FigII 4-3 one can observe peaks at 110Hz and 175Hz. The
conclusion can be made that these peaks are associated with resonances in the rig and
not the rubber isolator. Because the peaks are at the same frequencies for all rubber
isolators independent of their stiffness, the conclusion could be made from earlier
conducted measurements in the same test rig.
The measurement data was not stable under 10Hz, for this reason it was hard
to see the dynamic stiffness value of interest. To solve this problem an extrapolation
was made from 10Hz to 100Hz and is a black line displayed in FigII 4-3.
In FigII 4-3 the dynamic stiffness of 2205-2 is plotted. The dynamic stiffness is
evaluated at 10Hz and 80% of the force 5340N (Table 3-5) and gives the dynamic
factor:
24.16024.3
6739.3
stmeas
dynmeas
e
e
K
K [-]
This leads to, from Eq2-4:
679.36059.324.1dyn eeK [N/m^2]
This calculated dynamic stiffness can be used when calculating the first natural
frequency of the 1-Dof system before selecting which isolator to use in an assembly.
39
5 Conclusion
It is shown that the model presented is valid for an isolator in the CB2205 series. The
conclusion can be drawn that the model probably also will be valid for all isolators in
both CB2204- and CB2205-series because of the strikingly good correspondence
between test data and model in this evaluation. A comparison of all isolators against
the mathematical model would have been favorable, but because tests of the other
isolators had been delayed this could not be done at the time of writing this thesis.
The MATLAB code with all calculations included and is found in Appendix D.
1
References 1. P.B. Lindley. Engineering design with natural rubber. The Malaysian
rubber producer’s research association, 1964.
2. P.E. Austrell. Modeling of elasticity and damping for filled elastomers.
Division of structural mechanics LTH, Report TVSM-1009, 1997.
3. N. Gil-Negrete, J. Vinolas, L. Kari. A simplified methodology to predict
the dynamic stiffness of carbon-black filled rubber isolators using a finite
element code. Journal of sound and vibration 296 (2006) 757-767, 2006.
4. M. Sjöberg. On dynamic properties of rubber isolators. Department of
vehicle engineering, KTH, ISSN: 1103-470X, 2002.
5. F. Karlsson and A. Persson. Modeling non-linear dynamics of rubber
bushings parameter identification and validation. Division of structural
mechanics, LTH, ISSN: 0281-6679, 2003.
6. A. K. Olsson. Finite element procedures in modeling the dynamic properties
of rubber. Division of structural mechanics, LTH. ISSN: 0281-6679, 2007.
7. A. N. Gent. Engineering with rubber, 2nd
edition. ISBN: 1-56990-299-2,
2001.
8. P. E. Austrell. Konstruktionsberäkningar för gummikomponenter. Division
of structural mechanics, LTH. ISSN: 0281-6679, 2001.
9. L. Kari. On the dynamic stiffness of preloaded vibration isolators in the
audible frequency range: Modeling and experiments. MWL/Department of
Vehicle Engineering, KTH, 2002.
10. E. I. Rivin. Passive vibration isolation. ISBN: 1-86058-400-4, 2003.
11. Lord Corporation, Lord aerospace product design catalogue PC6116, Erie,
PA, 1990.
12. Lord Corporation, Lord industrial catalogue, 2009.
13. Sveriges Plast- och gummitekniska institutet, Konstruera med gummi,
ISBN 91-524-0831-0, 1986.
14. British Standard, Acoustics and vibration – Laboratory measurement of
vibroacoustic transfer properties of resilient elements. Part 2, BS EN ISO
10846-2:2008, 2008.
A-A
Appendix A
A.1 Search methodic Prior to the writing of this thesis a literary search was conducted. The search logic,
search engine and results are presented in
TableA 1. To narrow the search down terms were added to the search logic until a
manageable amount of titles were found. Only titles with relevance to rubber
modeling and behavior where chosen for the basis of the literary study.
TableA 1 Search results and search logic
Search engine: Compendex/Inspec (2000-
2010):
Scopus
Search logic: [(((("vibration control" ) WN
CV) OR ((NVH or "vibration
isolation" or "vibration
isolator" or "rubber mount" or
"rubber mounting" or "rubber
mounts") WN All fields))
AND (("dynamic stiffness")
WN All fields)), English only,
2000-2010];
[“vibration isolation” AND
“rubber isolator” AND
“dynamic stiffness”]
[(“vibration isolation” OR
“vibration control” OR “rubber
mount”) AND “dynamic
stiffness”]
Number of hits: 127hits (77hits/28hits) of
which 22hits were doubles;
57hits;
Interesting hits: 18hits 21hits
B-A
Appendix B
B.1 FEM implementation In Part I a short introduction of different kind of elements that could build up a
potential model is presented. The computer based solutions are the most efficient and
FEM. There are many FE-software that enables modeling of isolator parts. The elastic
and viscous parts are more common to model with FE-software. The frictional part is
thought more difficult to model e.g. there is no existing model for this part so the
researcher has to develop individual code.
B.1.1 Energy strain functions
For hyperelastic materials the stress could be expressed with strain energy functions.
These functions are presented in terms of principal stretches and invariants [5]. W
denotes the energy strain function and is assumed to depend on all strain components
),,,,,()( 321321 nnnB WWW . Eq B 1
If the tested material is assumed to be isotropic there is no direction dependence i.e. W
will only depend on the principal stretch
),,( 321 W . Eq B 2
To determine the roots of this equation it is easier to express Eq B 2 in terms of strain
invariant instead of principal stretch this is done with the relations
2
3
2
2
2
13
2
3
2
2
2
3
2
1
2
2
2
1
22
2
2
3
2
2
2
11
)det(
))(tr)(tr(2
1
)(tr
B
BB
B
I
I
I
, Eq B 3
which gives
),,( 321 IIIW . Eq B 4
For the incompressible material, there is no I3 dependence because it represents the
volume change, for rubber the bulk modulus is many times larger than the E-modulus
that is why incompressibility could be assumed [1].
Thus, the energy strain function only depends on two invariants
),( 21 IIWW . Eq B 5
This equation seems to indicate the possibility for all values I1, I2. But in reality, these
values are limited by the condition of incompressibility, see Fig B 1.
B-B
Fig B 1 Limitation due to incompressibility [2]
To eliminate one of the three principal stretches consider principal directions and use
the condition I3=1. This condition will give
21
3
1
Eq B 6
and
2
1
2
2
2
2
2
12
2
2
2
1
2
2
2
11
11
1
I
I
. Eq B 7
This will make it possible to choose independent values of 21, II .
B.1.2 Energy strain functions format There are different formats of energy strain functions which have been investigated by
scientists e.g. Mooney Rivlin, Ogden and Yeoh.
All the existing models will not be presented in detail in this thesis, references
to articles will be shown.
The polynomial form is one of most implemented forms in FE-software. The energy
strain functions for this model are expressed in forms of a series expansion
ji
ji
ij IICW )3()3( 21
0,0
, Eq B 8
where Cij is constant [5].
For the undeformed case, W is supposed to be zero and that is why the
equation for W is expressed in terms of (I1-3) and (I2-3). In spite of that the sum above
is written to infinity. Only the first few terms are usually used
B-C
)3( 110 ICW , Neo Hooke first term Eq B 9
)3()3( 201110 ICICW Rivlin first two terms Eq B 10
and
3
130
2
120110 )3()3()3( ICICICW . Yeoh Eq B 11
Ogden’s form is usually better than the polynomial. It fits the measured data better
and is generally more stabile. The strain energy density function for Ogden is
expressed as
N
i i
i iiiW1
3212)3(
2
, Eq B 12
where μi and αi are material constants.
The constants for this model are exponents. This makes Ogden’s model more
complicated than the polynomial model.
C-A
Appendix C
C.1 Tables and plots
TableC 1 Different modulus against the hardness of different carbon black filled rubber [13]
Hardness
◦IRH
Young’s
modulus
E0 [N/mm2]
Shear
modulus
G [N/mm2]
Elastomers
compression
coefficient φ[-]
Compression
Modulus
E∞ [N/mm2]
30
35
40
45
50
55
60
65
70
75
0.92
0.18
1.50
1.80
2.20
3.25
4.45
5.85
7.35
9.40
0.30
0.37
0.45
0.54
0.64
0.81
1.06
1.37
1.73
2.22
0.93
0.89
0.85
0.80
0.73
0.64
0.57
0.54
0.53
0.52
1000
1000
1000
1000
1030
1090
1150
1210
1270
1330
TableC 2 Bolts used and listed torque for assemble
Isolator series Bolt Tightening torque [Nm]
2204 M20 304
2205 M24 500
D-A
Appendix D
D.1 Implementation of mathematical model % % %Leif Kari's model for static and dynamic stiffness % clear all % close all % clc
%Geometry defenition for 2205 series isolators % In cat global G Ry Ri H Ry =(124e-3)/2; %Outer diameter (undeformed) A [m] Ria =(38.1e-3)/2; %Inner diameter ABOVE C [m] Riu=(64.8e-3)/2; %Inner diamter UNDER E [m] Ri=(Ria+Riu)/2; %The meanvalue [m] H=31.8e-3; %Hight of rubber isolator F [m] D=85.9e-3; %Hight of metal cylinder D [m] T=31.8e-3; %Thickness of support plate T [m]
% %The deflection that the Screw joint gives: u_m=((2*H+T)-D)/2;
% Isolator Rubber compound Shore(A) % CB-2204-3 A079P 50 % CB-2204-5 A111P 58 % CB-2205-2 A060P 43 The validated isolators % CB-2205-3 A079P 50 % CB-2205-4 A091P 53 % CB-2205-5 A111P 58
%ShoreA and Shear values from TableC.1 % Shore1= [30 35 40 45 50 55 60 65 70 75]; Shore1= [35 40 45]; % G1= [0.30 0.37 0.45 0.54 0.64 0.81 1.06 1.37 1.73 2.22].*10^6 G1= [0.37 0.45 0.54].*10^6;
%Here u1=displacement on the carrying side and u2=displacement on the %NONcarrying side is defined U=300; u_f=linspace(0,2.5e-3,U); u1=u_m+ u_f; u2=u_m- u_f;
Force_tot=[]; k= []; s=[]; for i=1:length(G1); G=G1(i); Shore=Shore1(i); %Here k1=stiffnes on the carrying side and u2=stiffness on the %NONcarrying side is defined k1=0.3e1 * G * (Ry ^ 2 - Ri ^ 2) * H ./ (H - u1) .^ 2 * pi .*
(0.1e1 + (Ry ^ 2 - Ri ^ 2) .^ 2 ... ./ ((Ry ^ 2 - Ri ^ 2) .* H ./ (H - u1) + Ri ^ 2) ./ (H - u1)
.^ 2 ./ 0.2e1); k2=0.3e1 * G * (Ry ^ 2 - Ri ^ 2) * H ./ (H - u2) .^ 2 * pi .*
(0.1e1 + (Ry ^ 2 - Ri ^ 2) .^ 2 ...
D-B
./ ((Ry ^ 2 - Ri ^ 2) .* H ./ (H - u2) + Ri ^ 2) ./ (H - u2)
.^ 2 ./ 0.2e1); %The total stiffness is calculated k(:,i)=k1+k2;
%The stiffness is integrated for i3=1:length(u_f); u=u1; Force1=quad(@stiffness,0,u1(i3)); u=u2; Force2=quad(@stiffness,0,u2(i3)); Force_tot1(i3)=Force1-Force2; end Force_tot(i,:)=Force_tot1; s1=['Math Model G=', num2str(G) 'Shore=', num2str(Shore)]; s{i}=s1; end
%% %Plots of calculated force and the static stiffness in comparison
with the model (for comparison with measurement data you need to load %the measurement, Atlas_2011_03-29_meas2205_3.m, so that "mu",
%"mforce", "u10","f" and "k_2205_2" ) close all figure(1) plot(u_f,Force_tot,u10,f,'c',2.3e-3,5340,'o',mu,mforce,'+b'); grid legend([s, '2205-2 extrapolation','Given by LORD cataloge','2200-5
measurement']) xlim([0 2.5e-3]) ylim([0 7500]) title('Measurement and mathematical FORCE of 2205-2'); xlabel('Displacement u [m]') ylabel('Force F[N]') hold on sigmaf=mforce*0.038; errorbar(mu,mforce,sigmaf,'+b')
figure(2) plot(u_f(1:length(k)),k,u10(1:length(k_2205_2)),k_2205_2,'k',2.3e-
3,5340/2.3e-3,'+k') legend([s,'2205-2 measurement','Given by LORD cataloge']) xlim([0 2.5e-3]) title('Measurement and mathematical STIFFNESS of 2205-2'); xlabel('Displacement u [m]') ylabel('Static Stiffness K[N/m]') grid disp(u_m)
D-C
D.2 Post processing of the data collected at the dynamic
measurement % FRFBurst.m % Code for calculating FRF with burst random excitation. % Analysis parameters provided by user % fs - sampling frequency [Hz] % fexc - fundamental frequency of excitation signal [Hz] % Nper - Number of bursts in time window % Input data % x - nominator signal in FRF "=" x/y % y - denominator signal in FRF "=" x/y % Output data % H1xy - FRF estimated as Sxx/Syx % H2xy - FRF estimated as Sxy/Syy % COHxy - Coherence function between x and y % % Flat-top window (no window) is used. Largest possible number of
averages % is used. Parameter Nper can be used to change the frequency
resolution at % the cost of number of averages. % % 2010-12-17, Ulf Carlsson, MWL KTH % analysis parameters fs = 5000; % Sampling frequency [Hz] %fexc = 1; % Fundamental frequency excitation signal [Hz] %fexc = 1; % Fundamental frequency excitation signal [Hz] %Nper = 10; % Number of excitation cycles in time window Nsampl = 20000; % Number of samples in one period Tw = Nsampl/fs; % Time window length [s]
%wh = hann(fs); %Hanning window Nave = floor(length(x)/Nsampl); % Maximum nr of averages freq = (0:Nsampl/2-1)/Tw; Xf = []; Yf = []; for n = 1:Nave X = x(1+(n-1)*Nsampl:Nsampl+(n-1)*Nsampl); Y = y(1+(n-1)*Nsampl:Nsampl+(n-1)*Nsampl); Xf(:,n) = fft(X); Yf(:,n) = fft(Y); end % Calculate cross- and autospectra XXf = Xf.*conj(Xf); YYf = Yf.*conj(Yf); XYf = Xf.*conj(Yf); % Calculate FRF and coherence XXF = mean(XXf.')/freq(2)/2/length(freq)^2; YYF = mean(YYf.')/freq(2)/2/length(freq)^2; XYF = mean(XYf.')/freq(2)/2/length(freq)^2; YXF = conj(XYF); H1xy = XXF./YXF; H2xy = XYF./YYF; COH1xy = XYF.*conj(XYF)./XXF./YYF; % Adjust frequency range H1xy = H1xy(1:Nsampl/2); H2xy = H2xy(1:Nsampl/2); COH1xy = COH1xy(1:Nsampl/2);
