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Novel Empirical Model of Rubber Bushing in Automotive
Suspension System
Zhang Lijun, Yu Zengliang, Yu Zhuoping College of Automotive
Engineering, Tongji University Cao’an Road, 4800, 201804 Shanghai,
China
email: [email protected]
Abstract It is very important to establish appropriate dynamical
model of rubber bushing for the suspension vibration reduction,
vehicle noise refinement and vehicle maneuverability enhancement.
This paper aims to present a novel empirical modeling method of
rubber bushing used in double wishbone suspension system. Firstly,
the mathematical equations of both Berg and Dzierzek models are
derived, and the theoretical methods employed to identify
parameters are discussed. Secondly, static and dynamical
characteristics of the rubber bushing were tested and the
prediction precisions of the both models are analyzed with the
parameters identified from the experimental data. Based on the
detailed analysis of dynamical stiffness and damping, a novel
empirical model is established, taking advantage of both Berg and
Dzierzek models. The novel model can gain reasonable compromise
between prediction accuracy, identification difficulty and
computational effort, and it can be suitable tool for automotive
chassis dynamics simulation and analysis.
1 Introduction
Various rubber bushings have been used in modern automobile
suspension systems, whose elasticity property and inherent damping
enhance vehicle driving performance significantly. They can improve
suspension kinematics and elastic kinematics by acting as elastic
joints, and effectively reduce vehicle body vibration induced by
road roughness. So the design and selection of rubber bushing for
automotive chassis has attracted attention from scientists and
engineers all over the world. It is very important to take the
characteristics of rubber bushing into account, from the viewpoint
of system engineering, when men design automotive suspension
system, predict suspension dynamic behavior or optimize system and
each component. So it has become one of the most important works to
understand the mechanical properties and to establish or choose
appropriate rubber bushing model for component behavior prediction
and suspension dynamics analysis [1].
However, it is not an easy task. As we know, the viscoelasticity
of rubber bushings is usually nonlinear and displays obvious
preload, frequency, dynamic amplitude and temperature dependence.
To develop a simple but effective model to represent rubber bushing
dynamical properties and to use it for fast vehicle dynamics
simulation is very desirable. Till now, many rubber bushing models
have been built to represent
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the linear viscoelastic properties, such as Maxwell model,
Kelvin Voigt model, multi-parameters Maxwell model, standard linear
solid model. A better description of rubber characteristics can be
obtained by using a summation of Maxwell models [2, 3]; however it
correspondingly increases the model parameters and the difficulty
of parameters identification. To simulate frequency dependence of
rubber bushing, Kari and Sjöberg [4-7] establish a fractional
derivatives model, which can predict frequency dependent
characteristics with two parameters. Amplitude dependence is
significant for rubber components filled with carbon black. One way
to simulate the amplitude effect is to adopt a friction component
in addition to the elastic and viscous forces. As the commonly used
linear Coulomb friction model plays jagged behavior, Berg [8]
presented a more sophisticated description of friction giving more
smooth friction behavior. Dzierzek [9] carried out a further trial
to build radial-normal coupled model by extending linear elastic
force to nonlinear regime, combined with nonlinear frequency and
amplitude dependence components. All these works significantly
improve the model accuracy of cylindrical, carbon black filled
rubber bushings.
This paper is to extend and improve previous works, particularly
taking advantage over the works done in references [8] and [9],
attempting to develop a novel, nonlinear dynamic model of a
cylindrical, carbon black filled rubber bushing. At the same time,
the model accuracy, parameter identification method will also be
studied and discussed.
2 Model Formulations and Parameter Identification
2.1 Model Formulation and Characteristics
2.1.1 Berg Model
The model proposed by Berg [8] (depicted in Figure 1a) is
one-dimensional, contains five parameters and is based on a
superposition of elastic, friction and viscous forces. The elastic
part reflects the amplitude dependence. The included friction force
means that an increased stiffness at small displacements as well as
rate-independent hysteresis can be considered. And the viscous
force can describe increasing stiffness with increasing frequency
as well as rate-dependent hysteresis. This model represents a
reasonable compromise between accuracy and computational effort and
is used for railway vehicle dynamics analysis.
As shown in Figure 1(a), the Berg model describes the
relationship between force and motion through the superposition of
three forces
vfe FFFF ++= (1)
Where , and are the elastic, friction and viscous forces
respectively. eF fF vF
For steady harmonic excitation txx ωsin0= , the amplitude of
each force can be expressed as:
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0 1 0
max 2 20 2 0 2 0 2
22
20Re 2 02
2
0Im 022
( 62
( / )1 ( / )
11 ( / )
e
ff
v
v
F K xF
0 )F x x x x x xx
C KF K xC K
F CxC K
ωω
ωω
=⎧⎪⎪ = + + −⎪⎪⎨
=⎪ +⎪⎪
=⎪ +⎩
−
(2)
Where ,0x ω and t are displacement amplitude, circular
excitation frequency and time. is the
maximum friction force, whereas the displacement is the
displacement required to gain the friction
force as large as . denotes the linear elastic stiffness
constant corresponding to linear
elastic behavior. and represent the spring stiffness constant
and the damping coefficient of
viscous branch. and are the real and imaginary force component
of complex viscous force.
maxfF
2x
2/maxffF F= 1K
2K C
Re0vF Im0vF
The energy losses per cycle of elastic, viscous and friction
branch are then given by
2 2 0 0max 0 2 0
2 0
202
2
0(1 ) 22 (2 (1 ) ln
(1 )
1 ( / )
e
f f
v
E
)x a xE F x x ax a
CE xC Kπωω
⎧⎪ =⎪⎪ + +
= − +⎨ +⎪⎪
=⎪+⎩ (3)
The total force amplitude and energy per cycle 0F E can be
written as
2 20 0 0 0Re 0Im( ) (e f v v
v f
F F F F F
E E E
⎧ = + + +⎪⎨
= +⎪⎩
)
(4)
Hence the dynamic stiffness K and loss angle δ of the rubber
bushing can be gained as
0 0
0 0arctan( )K F x
E F xδ π=⎧
⎨ =⎩ (5)
Figure 1b shows the typical results of rubber bushing force
versus harmonic displacement based on Berg model. It can be seen
that there is stiffness increase after the displacement changes in
direction. The hysteresis loops also clearly indicate that dynamic
stiffness decreases as the displacement amplitude increases, which
behavior is known as the Payne effect.
The characteristics of dynamic stiffness and loss angle versus
amplitude and frequency are plotted in Figure 1c and 1d. It is
concluded that:
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Dynamic stiffness is significantly affected by excitation
amplitude and frequency. Dynamic stiffness increases with the
increase of frequency, but decreases as amplitude increases. This
results from the breaking of the filler structure. The structure is
composed of carbon black aggregates held together by van der Waals
bonds [10].
Loss angle is also greatly affected by excitation amplitude and
frequency. In frequency domain, rubber materials energy dissipation
normally experiences rubber, transition and glassy region in turn.
The loss angle increases with frequency in the rubber region, and
reaches a maximum in the transition region, and then decreases in
the glassy region. Berg model displays this theoretical property of
loss angle throughout broad frequency range. For various
amplitudes, the loss angle decreases with amplitude increasing
except for the very low excitation amplitude region, where the loss
angle increases with increasing amplitude, which coincides with the
measurement results.
maxfF2x
1K2K
F x
C
(a) (b)
(c) (d)
Figure 1: Berg model structure and mechanical
characteristics
2.1.2 Dzierzek Model
The Berg model neglects the nonlinear elastic force
characterization. It is well known that the nonlinear elastic force
contributes to rubber bushing mechanical performance especially
under large amplitude conditions.
Stawomir Dzierzek [9] presented a semi-empirical model based on
measurement observations as depicted in Figure 2a. The Dzierzek
model is also one-dimensional model composed of three major parts,
i.e.
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non-linear elastic force, viscoelastic force and friction force
branch. According to [9], the analytical description of the built
model is written as
4
3 5 2
2 21 1 2 2
Re 1 22 21 1 2 2
1 2Im 2 2
1 1 2 2
2 tan2
2 tan2
( / ) ( / )( )1 ( / ) 1 ( / )
( )1 ( / ) 1 ( / )
te t
t
C
tf t t
t
v
v v
d xF kd
d x xF c k k x cd x xx
c k c kF kc k c k
c cF x cc k c k
ππ
ππ
ω ωω ω
ωω ω
⎧ =⎪⎪⎪ ⎡ ⎤⎛ ⎞⎪ ⎢ ⎥= − +⎜ ⎟⎪ ⎢ ⎥⎪ ⎝ ⎠ −⎣ ⎦⎨⎪
= +⎪+ +⎪
⎪= +⎪
+ +⎪⎩
&
& &
k x
+
&
(6)
Where is the non-linear elastic force, is the friction force,
and are the real and
imaginary force component of complex viscoelastic force. is the
stiffness coefficient, i.e. the elastic
curve inclination at x=0, is the characteristic thickness
representing the location of elastic curve
asymptote. , , are the viscous damping coefficients and , are
the stiffness coefficients.
, are the dimensionless friction coefficients and is the static
frictional force at x=0.
eF fF RevF ImvF
tk
td
1c 2c vc 1k 2k
3c 4c 5c
Energy loss per cycle for viscous branch is calculated by
2 1 22 2
1 1 2 2
( )1 ( / ) 1 ( / )v v
c cE x cc k c k
πωω ω
= ++ +
+ (7)
The energy dissipation of friction force can be solved using
integral method
0( )
T
f fE F dx t= ∫ (8) Figure 2b shows the typical results of rubber
bushing force versus harmonic displacement based on Dzierzek model.
It can be seen that there is stiffness nonlinearity. The hysteresis
loops also clearly indicate that dynamic stiffness and loss angle
are dependent to amplitude and frequency.
The characteristics of dynamic stiffness and loss angle versus
amplitude and frequency are plotted in Figure 2c and 2d. It is
concluded that:
(1) Dynamic stiffness dependence with frequency and amplitude is
similar to those of Berg model, except that the sharp decline of
stiffness at low amplitude region and the tiny stiffness increase
at large amplitude range. This results from the non-linear
elasticity.
(2)Loss angle generally increases with frequency, whereas the
increasing grad decreases gradually. Meanwhile, loss angle reduces
with amplitude increase, which is contradicted with physical test
results especially for the lowest amplitude.
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(a) (b)
(c) (d)
Figure 2: Dzierzek model structure and mechanical
characteristics
2.2 Identification of Model Parameters
2.2.1 Berg Model
The Berg model is mainly composed of a friction element and a
three-parameter Maxwell element, the model parameters
identification can be divided into two steps:
(1) To retrieve friction parameters based on larger amplitude
and triangular waveform
excitation static test.
maxfF
(2)To identify viscoelastic element parameters by high frequency
and low amplitude dynamic sinusoidal test results.
Detailed procedures are listed as follows:
(1) To evaluate the friction parameter and . From Figure 3a, the
two dashed lines
representing the stiffness are parallel and the vertical
distance between them equals approximately
twice of the . So can be determined and the friction
displacement is
maxfF 2x
eK
maxfF maxfF 2x
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max2
max e
FxK K
=− (9)
(a) Berg model (b) Dzierzek model
Figure 3: Frictional element parameters evaluation method
(2) To determine the contribution of friction force to model
dynamic stiffness and energy loss. According to Equation 2 and 5
the contribution to dynamic stiffness yields
00 xFK ff = (10)
(3) To determine the corrected dynamic stiffness and energy loss
and . corrK corrE
corr dyn f
corr f
K K KE E E
= −⎧⎨ = −⎩ (11)
(4) To set the relationship between dynamic property and
component parameters for viscoelastic element by summarizing
Equation 4 and 5
22 22
2 12 22 2
22
2
( / )( ) (1 ( / ) 1 ( / )
1 ( / )
corr
corr
C k CK k kC k C k
CE xC K
ω ωω ω
πωω
⎧= + +⎪
+ +⎪⎨⎪ =⎪ +⎩
)
(12)
(5) To establish error function to optimize the best
parameters.
, 2 2
1 ,exp exp
min (( 1) ( 1) )i ikdyn theor theori i
i dyn
KK
δφδ=
= − +∑ − (13)
Where denotes the number of test cases used for fitting
procedure, is the analytical
calculation value of dynamic stiffness, is the experimental
value of dynamic stiffness, similarly
and represent analytical and measured values of loss angle,
respectively.
k i theordynK ,
iexpdynK ,
itheorδ
iexpδ
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(6) To implement the fitting procedure by minimizing the error
function with Matlab fmincon
algorithm , in order to determine the viscous element parameters
, and . As there are three model
parameters to be identified in this step, at least two dynamic
test cases are needed.
1k 2k 1c
2.2.2 Dzierzek Model
Dzierzek model consists of 12 parameters, which increases the
difficulty of parameter identification and the number of physical
test required.
(a) (b)
Figure 4: Influence of and , on parameter identification tk 3c
4c
The detailed procedure includes:
(1) To determine the nonlinear elastic force parameters and
through static tests. As shown in
Figure 3b, man can evaluate the two parameters value through
least-square method based on the static force-displacement
loop.
tk td
(2) To determine the sub-parameters p , , q r of stiffness
coefficient . According to the hyperbolic
function between and these sub-parameters, three sub-parameters
can be gained by using data fitting. A
sample hyperbolic curve is depicted in Figure 4a. Total 9 static
tests with different amplitude are needed to look for the
appropriate sub-parameters.
tk
tk
(3) To gain the friction force parameter . From the definition
of , it is determined from the static
force-displacement loop at zero displacement position shown in
Figure 3b.
5c 5c
(4) To determine the contribution of friction force to dynamic
stiffness and energy loss. From the description of the friction
element, it is found that the friction force is proportional with
deformation rate, so the contribution is focused on the energy
dissipation and the influence of friction force to dynamic
stiffness can be neglected.
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Additionally, the first part in round brackets of ⎟⎟⎠
⎞⎜⎜⎝
⎛− xk
dxdkc t
t
tt
2tan23
ππ
mainly appears in highly
nonlinear elastic range of hysteresis loop. To ‘isolate’ the
exponential parameters and , lower
amplitude excitation tests are chosen for identification. Hence
the friction force can be simplified as
3c 4c
5 2f
xF cx xx
=−
&
& &&
k
(14)
And then the energy dissipation due to friction force can be
calculated by using integral method. The corrected dynamic
parameters are written as
corr dyn t
corr f
k kE E E
= −⎧⎨ = −⎩ (15)
(5) To formulate the contributions of of viscous element to
dynamic stiffness and energy loss as
2 22 21 1 2 2 1 2
1 22 2 21 1 2 2 1 1 2 2
2 1 22 2
1 1 2 2
( / ) ( / )( ) (1 ( / ) 1 ( / ) 1 ( / ) 1 ( / )
( )1 ( / ) 1 ( / )
corr
corr v
c k c k c ck k kc k c k c k c k
c cE x cc k c k
ω ω ω ωω ω ω ω
πωω ω
⎧= + + +⎪
+ + + +⎪⎨⎪ = + +⎪ + +⎩
2 )
(16)
(6) To set the error function as same as Equation 13 and to
evaluate viscous element
parameters , , , and through Matlab fmincon algorithm. Since
five parameters will be
determined in this step, three dynamic tests are needed for
equation solving.
1k 2k 1c 2c vc
(7) To get the exponential parameters and . From Figure 4b it
can be seen that the ‘half-done
model’ curve without and differs from that of ‘complete model’.
The difference results from
4c
3c 4c
4
, 3 2
2 tan2
C
tdiff theor t t
t
d x xF c k k xd x xx
ππ
⎛ ⎞= −⎜ ⎟
⎝ ⎠ −
&
& && (17)
If man establishes error function based on Equation 17, which is
read in Equation 18, the two parameters can be obtained through
Matlab fmincon algorithm.
,exp ,
1min ( ( ) ( ))
k
diff i diff theor ii
F x F xφ=
= −∑ (18)
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3 3 Berg and Dzierzek Model Validations
3.1 Experimental
In order to identify the parameters of two models proposed
above, a series of experiments are conducted. A cylindrical, solid
rubber bushing is tested, which is made of carbon black-filled
rubber and is used in vehicle suspension system. The rubber bushing
is tested on a servo-hydraulic tester, which can excite the tested
object with displacement input. The tests are carried out with
sinusoidal waves, whose frequency ranges from 1 to 40Hz, and the
displacement amplitude ranges from 0.1mm to 3mm. At the same time,
triangular wave is used for quasi-static tests. The amplitude rate
is only 0.05mm per second and the amplitude is from 0.2mm to
4mm.
The dynamic characteristics of the tested rubber bushing are
shown in Figure 5. It can be seen that the dynamic stiffness and
loss angle are both amplitude and frequency dependent. Detailed
analysis is as follows
The dynamic stiffness and loss angle both decreases with
increasing amplitude, but the loss angle shows different states
under low amplitude excitation with 25Hz and 30Hz frequency
The dynamic stiffness and loss angle both increases with
increasing frequency, but the loss angle shows very complex cases
in high frequency range.
3.2 Model Validation
(a) (b)
Figure 5: Amplitude and frequency dependency of tested rubber
bushing
The parameters of the proposed model are identified from
experimental data according to the described procedures. And the
two models are validated through comparisons between measurement
results and prediction results of dynamic stiffness and loss
angle.
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3.2.1 Berg Model
Figure 6 illustrates the prediction accuracy of Berg model for
dynamic stiffness and loss angle. It can be found that: (1) The
average deviation of dynamic stiffness is about 8%, the highest
value is around 16%. And the smallest error appears when the
amplitude is 1mm. The reason is that the 1mm amplitude tests are
used for parameters identification. No matter the amplitude
increases or decreases, the deviations increase always. These mean
that the Berg model can basically capture the dynamic stiffness
behavior of the tested rubber bushing but lacking of robustness to
identification tests’ variation. (2)The energy loss behavior under
high frequency is predicted more accurate than that of low
frequency. For the low frequency and high amplitude condition, the
deviation is as large as 42%.
(a) Dynamic Stiffness Deviation (b) Loss Angle Deviation
Figure 6: Berg model prediction deviation
(a) Dynamic Stiffness Deviation (b) Loss Angle Deviation
Figure 7: Dzierzek model prediction deviation
3.2.2 Dzierzek Model
From Figure 7a and 7b, it can be seen that: (1) The Dzierzek
model can predict both dynamic stiffness and loss angle with higher
accuracy than Berg model except some special case; (2) For the
special test case, typically for low amplitude and high frequency
excitation situation, the error of loss angle is fast 100%, which
means that this model can hardly predict the real damping
behavior.
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25Hz 30Hz
Figure 8: Dzierzek model amplitude dependency validation for
specific frequencies
25Hz 30Hz
Figure 9: Berg model amplitude dependency validation for
specific frequencies
4 4 Novel Model Establishment
4.1 Motivation
Based on the above analysis, we can summarize the advantage and
disadvantage of the two models as that the Dzierzek model shows
high accuracy for most of test conditions, however it can hardly
capture the energy dissipation amplitude dependence under low
amplitudes. While Berg model describes the component behaviors
fairly well with respect to the mean values. It has advantages over
modeling energy dissipation amplitude dependency in low amplitudes
.But its main drawback is the unreasonable dynamic stiffness
prediction ability due to the influence of identification test
selection.
More detailed analysis to the dynamics stiffness and loss angle
prediction accuracy is carried out. And the results for are higher
specific frequencies plotted in Figure 8 and Figure 9 as 2D plots
of dynamic stiffness and loss angle versus amplitude. It can be
seen that the predicted loss angle by Dzierzek model shows opposite
tendency in low amplitude range, which is contrasted to the
measured results. But, the Berg model gets much better accuracy in
these areas. At the same time, it can be found that the Dzierzek
model has better accuracy for dynamical stiffness prediction than
Berg model.
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Taking into the identification tests requirement, Dzierzek model
has 12 parameters and requires 9 static tests and 3 dynamic tests
to identify all the parameters. However, Berg model contains only 5
parameters, which can be determined from 1 static test and 2
dynamic tests.
Based on the above discussion on the advantages and limitations
of the two models, the authors tend to propose a novel model, which
can take advantage of the two models.
4.2 Novel Model Formulation
maxfF2K
F x
2x
Kt1K
1C 2Cdt
(a) (b)
(c) (d)
Figure 10: Novel model structure and characteristics
The newly proposed model is shown in Figure 10. The novel model
totally includes 10 parameters, which is a combination of nonlinear
elastic part derived from Dzierzek model, friction part adopted
from Berg model and two parallel Maxwell elements represent viscous
part. From Figure 10 it is found that:
(1) The novel model contains the frictional element
characteristics and appears obvious nonlinear elastic property at
the higher amplitude area, as displayed in Figure 10b.
(2) The dynamic stiffness declines with increasing amplitude and
increases with frequency increasing. The loss angle increases with
increasing amplitude under lower amplitudes, and then decreases
with amplitude for larger amplitudes. The loss angle dependence to
frequency appears firstly increasing and then decreasing with
frequency.
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According to the previous analysis, the mathematical equations
of forces and energies for the novel model are
max 2 2
2 0 2 22
2 21 1 2 2
0Re 1 22 21 1 2 2
1 20Im 2 2
1 1 2 2
2 tan2
( 6 )2
( / ) ( / )( )1 ( / ) 1 ( / )
( )1 ( / ) 1 ( / )
te t
t
ff
v
v
d xF kd
FF x x x x x x
x
c k c kF k kc k c kc c
x
F xc k c k
ππ
ω ωω ω
ωω ω
⎧ =⎪⎪⎪
= + + − −⎪⎪⎨⎪ = +⎪ + +⎪⎪ = +⎪ + +⎩
(19)
2 1 2
2 21 1 2 2
2 2 0 0max 0 2 0
2 0
( )1 ( / ) 1 ( / )
(1 ) 22 (2 (1 ) ln(1 )
v
f f
c cE xc k c k
)x a xE F x x ax a
πωω ω
⎧ = +⎪ + +⎪⎨ + +⎪ = − +⎪ +⎩
(20)
4.3 Parameters Identification
Based on the parameter identification procedures described in
former sections, the parameters fitting process of the novel model
is as follow:
(1) To determine friction element parameters and according to
the same approach
mentioned above.
maxfF 2x
(2) To evaluate the nonlinear elastic branch parameters p , ,q r
and . td
(3) To determine the viscous parameters , , , . When setting
corrected dynamic stiffness and
loss angle, the friction and nonlinear elastic elements should
both be considered. The corrected function is written as:
1k 2k 1c 2c
corr dyn t fcorr f
k k k kE E E= − −⎧
⎨ = −⎩ (21)
The correlations between viscous element parameters with
corrected dynamic stiffness and energy loss are
2 22 21 1 2 2 1 2
1 22 2 21 1 2 2 1 1 2 2
2 1 22 2
1 1 2 2
( / ) ( / )( ) (1 ( / ) 1 ( / ) 1 ( / ) 1 ( / )
( )1 ( / ) 1 ( / )
corr
corr
c k c k c ck k kc k c k c k c k
c cE xc k c k
ω ω ω ωω ω ω ω
πωω ω
⎧= + + +⎪
+ + + +⎪⎨⎪ = +⎪ + +⎩
2 ) (22)
(4) To search the best fitting values of the four parameters
through Matlab subroutines.
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4.4 Model Validation
The prediction accuracy of the novel model has been validated
and the results are shown in Figure 11. From Figure 11 it can be
concluded that: (1) The novel model successfully described the
dynamic behavior of rubber bushing with satisfying accuracy. (2)
For dynamic stiffness prediction, the novel model kept lower
deviation for all test range. The average deviation of dynamic
stiffness is 5.2%, and the highest value is 12%. Meanwhile, it
eliminates the influence of identification test selection. (3) For
energy dissipation characteristic simulation, the novel model
inherits the good performance of Dzierzek model together with the
Berg model’s advantage of predicting amplitude dependence of loss
angle.
(a) Dynamic Stiffness Deviation (b) Loss Angle Deviation
Figure 11: The Novel model prediction deviation
stiffness average deviation
Stiffness max. deviation
Loss angle average deviation
Loss angle max. deviation
Required test
Berg Model 7.9% 16.7% 17% 42.5% Static: 1 Dynamic: 2
Dzierzek Model 4.6% 11% 13.2% 98.9% Static: 9 Dynamic: 3
Novel Model 5.2% 12% 9.3% 17% Static: 9 Dynamic: 2
Table 1: Accuracy and identification comparison
The novel model is a compromise between Berg model and Dzierzek
model. It contains 10 parameters which simplifies model structure
and reduces the difficulty of parameter identification. The
comparison results of accuracy and identification among the three
models are listed in Table. 1. Obviously, the novel model has the
advantages over the other two models.
5 Conclusions and Further Development
It can be concluded that the proposed novel model, based on Berg
model and Dzierzek model, has good agreement with measurement
results and represents higher accuracy for both dynamical stiffness
and loss angle prediction than Berg model and Dzierzek model. The
identification tests required are less than
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Dzierzek model. Such a model can gain reasonable compromise
between prediction accuracy, identification difficulty and
computational effort, and it can be suitable tool for automotive
chassis dynamics simulation and analysis.
Future possible extensions of the novel model are to account for
preload and temperature effect, to apply it to random displacement
excitation and to analyze multiplex characteristics under
multi-directional displacement excitation.
References
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viscoelastic model for one dimensional response of elastomeric
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analyse the non-linear dynamic behaviour of carbon-black filled
rubber components using standard FE codes. Proceedings of the
Second Conference on Constitutive Models for Rubbers, 2001:
231–235
[3] A. K. Olsson, P. E. Austrell. Finite element analysis of a
rubber bushing considering rate and amplitude dependent effects.
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Matias Sjoeberg. On Dynamic Properties of Rubber Isolators. PhD
Dissertation of Royal Institute of Technology, 2002
[4] M. Sjöberg, L. Kari. Nonlinear behavior of a rubber isolator
system using fractional derivatives. Vehicle System Dynamics,
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[5] Mattias Sjöberg. Nonlinear isolator dynamics at finite
deformations: An effective hyperelastic, fractional derivative,
generalized friction model. Nonlinear Dynamics, 2003(33):
323-336
[6] Matias Sjöberg. Rubber isolators – Measurements and
modelling using fractional derivatives and friction. SAE
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