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Torsion stiffness of a rubber bushing: A simple engineering design formula
including the amplitude dependence
Article in The Journal of Strain Analysis for Engineering Design · January 2007
DOI: 10.1243/03093247JSA246
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Vehicle System Dynamics
Vol. 00, No. 00, April 2006, 1–11
Torsion stiffness of a rubber bushing: a simple effective engineering formula including
amplitude dependence
MARÍA J. GARCÍA TÁRRAGO∗ †‡, LEIF KARI†, JORDI VIÑOLAS‡ and NERE GIL-NEGRETE‡
†The Marcus Wallenberg Laboratory for Sound and Vibration Research, Department of Aeronautical and Vehicle
Engineering, KTH, 100 44 Stockholm, Sweden.
‡Ceit and Tecnun (University of Navarra), Manuel de Lardizábal 15, 20018 San Sebastián, Spain.
(Received 00 Month 200x; In final form 00 Month 200x )
An effective engineering formula for the torsion stiffness of a filled rubber bushing in the frequency domain including amplitude
dependence is presented. It is developed by applying a novel separable elastic, viscoelastic and friction material model to an equivalent
strain of the strain state inside the bushing, thus leading to an equivalent shear modulus which is inserted into an analytical formula for
the torsion stiffness. The novel rubber model is the result of extending a sound component model to the material level, therefore, unlike
simplified previous methods, this procedure takes into account the variation of the properties inside the bushing due to non-homogeneous
strain states. Moreover, as this formula depends on the bushing geometry in addition to the material properties, it is an effective and
fast tool to determine the most suitable rubber bushing to fulfil user requirements. Furthermore, it is shown—by dividing the considered
bushing into several slices, consequently each equivalent shear modulus is closer to the true value—that the approach of working with
only one equivalent shear modulus for the whole bushing is accurate enough.
Keywords: Filled rubber bushing, torsion stiffness formula, amplitude dependence, equivalent modulus.
1
Introduction
Rubber bushings have been used in vehicle suspension systems and transmission axles for many years.
Usually they consist of a rubber tube bonded on their outer and inner surfaces to rigid metal layers. Their
high axial and torsion flexibility, large radial and conical rigidity and their inherent damping make them
interesting for being used in primary suspensions of railway vehicles where they have a critical influence
in the behaviour of the vehicle and its dynamic stability. Also on most cars, bushings are mounted inside
the suspension control arm where torsion and axial stiffness play an important role. Hence, due to the
increasing interest in predictions performed by multi-body simulations of complete vehicles or subsystems,
it is important to develop simple and effective models to represent the dynamic stiffness of these rubber
bushings including also their amplitude dependence. Simplicity is necessary as the rubber component
represents only a small part when the full model is built and effectiveness refers to the precision of the
model when it is compared to the real vehicle.
Firstly, the behaviour of these components has been defined by their static stiffness as calculated by
finite element analysis [1], by using truncated Fourier and Bessel functions [2] or through principal mode
approaches [3]. Lately, Horton et al. have developed formulas for radial, conical and torsion static stiffness
[4–6] based on three-dimensional static elastic theory. Moreover, the static stiffness in many directions
is provided in standard textbooks [7–11]. In addition, the structure-borne properties of a long rubber
bush mounting in all directions are considered in Kari [12], however, it does not take into account their
amplitude dependence.
The dynamic properties of rubber are dependent on frequency and amplitude. The latter dependence,
known as the Fletcher–Gent effect [13], involves a non-linearity in the dynamic stress–strain behavior and
it becomes more pronounced with the presence of rubber fillers, generally one of the many kinds of carbon
black. But also this presence produces an increase in the shear modulus and damping. Filled rubbers are
∗ Corresponding
author. Email: mjgarcia@ceit.es
Vehicle System Dynamics
c 2006 Taylor & Francis
ISSN 0143-1161 print / ISSN 1366-5901 online °
http://www.tandf.co.uk/journals
DOI: 10.1080/00423110xxxxxxxxxxxxx
2
M.J. GARCIA TARRAGO, L. KARI, J. VIÑOLAS and N. GIL-NEGRETE
widely used in bushings within vehicle suspension systems reducing the energy transmission that goes
inside the vehicle coming from the road, which means an increase in comfort.
To date different models have been developed in order to represent the dynamic properties of rubber,
either at the material level, such as Coveney et al. [14] who represent this amplitude dependence focusing
on quasistatic behavior and Kraus [15] who develops micromechanically motivated models; or at the
component level. Within the latter group some authors obtain the dynamic stiffness of rubber components
after a long-time process by inserting material properties into tedious finite elements models, like Austrell
et al. [16] who add frequency dependencies using integer derivatives to stick-slip friction components, an
approach also used by Bruni and Collina [17] and Gil–Negrete [18, 19] except for the use of fractional
derivatives; while other authors model the component without considering the material level, such as
Berg [20] who presents a five-parameter model giving a better resemblance to the smoother rubber friction
behavior, a model also used by Sjöberg and Kari [21] together with a rate-dependent part using fractional
derivatives; whereas Misaji et al. [22] take into account amplitude dependence as the parameters of an
ordinary Kelvin–Voigt model are updated continuously for every oscillation cycle.
This paper presents a simple and effective engineering formula to represent in the frequency domain
including amplitude dependence, the dynamic torsion stiffness of a cylindrical bushing of highly filled
rubber as function of the dimensions of the bushing and the material properties of the rubber. It is
developed by applying a novel separable elastic, viscoelastic and friction material model to an equivalent
strain of the strain state inside the bushing, thus yielding an equivalent complex shear modulus that is
inserted into a torsion-stiffness analytical formula. The novel material model is obtained by extending a
sound component model to the material level. Consequently, it takes into account the variation of the
properties inside the bushing due to non-homogeneous strain states. Compared to previous models this
formula is an effective and fast tool to choose the most suitable rubber bushing to fulfil user requirements
and it can be used easily in multi-body and finite element simulations.
2
Formula foundations
Due to the growing interest in modeling rubber bushings several formulas have been developed to represent
the behaviour of these components. Such is the case of the simple analytical formula developed by Adkins
and Gent [3] for the torsion stiffness of a cylindrical rubber bushing of length L with inner and outer radii
a and b and elastic shear modulus µ
Kt =
M
4πa2 b2 L
= 2
µ,
ϕ
b − a2
(1)
where M is the moment and ϕ the torsion angle applied to the outer surface relative to the inner
one. This formula uses the static shear modulus µ and consequently leaves out the material dynamic
characteristics which are dependent on frequency and amplitude. In this and following section, a new
formula is developed to overcome the previous limitations.
s
2 m
to ta l
(t)
m , a
s
e (t)
f m a x
,e
Figure 1. Rubber material model.
1 2
Torsion stiffness of a rubber bushing
3
Different material models are interesting to represent the dynamic properties of rubber. For instance, the
relation in the time domain between stress and strain inside the rubber can be determined by extending
the component model presented in Sjöberg and Kari [21] to the material level; as a result three parallel
components are employed to represent the dynamic behaviour of the rubber including frequency and
amplitude dependencies by using only five parameters. This novel material model is subsequently applied
in this paper. Firstly, in Figure 1 the elasticity is modelled by a linear relation between the elastic stress
σelast and strain ε where µ is the elastic shear modulus and t is the time
σelast (t) = 2µε(t).
(2)
The second branch in Figure 1 indicates the frequency dependence and it is modelled by using fractional
derivatives which increases the ability to adjust to measured characteristics while keeping the number of
parameters to only two: a proportionality constant m and the time derivative order α
σfract (t) = mDα ε(t)
0 < α ≤ 1,
(3)
where Dα denotes the fractional time derivative of order α, defined through an analytical continuation of
a fractional Rieman–Liouville integration [23]. Numerically, the viscoelastic component can be evaluated
as
n−1
σfract (t) ≈ m
∆t−α X Γ(j − α)
εn−j ,
Γ(−α)
Γ(j + 1)
(4)
j=0
where εn−j = ε((n − j)∆t), εn is the final strain at time tn = n∆t, ∆t is a constantRtime step applied
∞
in the estimation process and Γ denotes the Gamma function [24] defined as Γ(β) = 0 sβ−1 e−s ds for
β > 0. In addition, the temporal Fourier transformation of the fractional time derivative branch leads to
σ̃fract (ω) = m(iω)α ε̃(ω),
(5)
which is an expression numerically more easy to deal with while omitting the Γ function, where (˜·) represents the temporal Fourier transform, ω the angular frequency and i the imaginary unit. The third branch
in the rubber model represents the amplitude dependence by a smooth friction component which enables
a very good fit to measured curves using only two parameters σf max and ε1/2 . The friction stress develops
gradually following the equation
σfrict (t) = σf s +
ε1/2
h
1 − sign(ε̇)
h
i
ε(t) − εs
i
h
i σf max − sign(ε̇) σf s ,
σf s
σf max + sign(ε̇) ε(t) − εs
(6)
where the material parameters σf max and ε1/2 are the maximum friction stress developed and the strain
needed to develop half of that stress, with sign(ε̇) denoting the sign of the strain rate. The values of σf s
and εs are updated each time the strain changes direction at ε̇ = 0 as σf s ← σfrict |ε̇=0 and εs ← ε |ε̇=0 .
Finally, the rubber behaviour is represented by a total stress which is the sum of the three stresses
σtotal (t) = σelast (t) + σfract (t) + σfrict (t).
(7)
Since Equation (7) does not represent a linear relation between the total stress and strain, a value of the
shear modulus cannot be obtained directly as would be the case with the elastic component, Equation
(2). Moreover, the stress–strain relation that can be derived from Equation (7) is valid at a specific strain,
but in order to represent the behaviour of the whole bushing a ”global” value for the shear modulus is
required. The simplified and effective solution developed in this paper consists in calculating an equivalent
4
M.J. GARCIA TARRAGO, L. KARI, J. VIÑOLAS and N. GIL-NEGRETE
shear modulus for the whole bushing using the classical theory of elasticity and the novel rubber model
presented in this section. This equivalent modulus is then used for the calculation of the torsion stiffness,
thus providing a modified version of Equation (1).
3
Methodology to obtain the dynamic stiffness
The methodology is developed as follows: Firstly, the classical theory of elasticity is applied to obtain the
strain state inside the bushing while considering that the relation between stress and strain contains only
elastic and fractional derivative components. Secondly, the calculation of a energy density balance between
the torsion deformation of the bushing and a simple shear specimen made of the same material gives an
equivalent strain which leads to an equivalent complex shear modulus by applying the novel rubber model
presented in Figure 1 and Section 2. Finally, the equivalent modulus is inserted into the formula of the
torsion stiffness obtained by the classical theory of elasticity.
3.1
Classical linear theory of elasticity
The rubber bushing of length L in Figure 2 bonded to two cylindrical metal layers at inner and outer radii
a and b is studied. When the bushing is subjected to a torsion angle ϕ(t) applied to the outer surface
Ø
2 a
z
u
L
r
j
z
r
z
u
u
Ø
q
r
y
q
x
r
r
2 b
Figure 2. Rubber cylindrical bushing.
relative to the inner one, displacement components in cylindrical coordinates and in time domain, at a
general point a ≤ r ≤ b and 0 ≤ z ≤ L are
ur = 0,
uθ = f (r, t),
uz = 0,
(8)
where f is an arbitrary function satisfying the displacement boundary conditions f (a, t) = 0 and f (b, t) =
bϕ(t). The rubber is assumed homogeneous and isotropic. The displacement gradient is small enough to
apply the classical linear theory of elasticity giving a non-zero shear strain
εrθ (r, t) =
1 ³ ∂f (r, t) f (r, t) ´
−
.
2
∂r
r
(9)
Torsion stiffness of a rubber bushing
5
The constitutive equation in the time domain which relates stress and strain is obtained from the rubber model presented in Section 2. In what follows only the elastic and the fractional derivative parts of
Equation (7) will be considered, as the amplitude dependence will be taken into account later on when
the rubber model is applied to the equivalent strain. Hence, the stress is the sum of the elastic part which
is proportional to the strain and the viscoelastic part which is proportional to the α time order derivative
of the strain, see Equations (2) and (3)
σrθ (r, t) = 2µεrθ (r, t) + mDα εrθ (r, t).
(10)
Working in the frequency domain as a result of applying the temporal Fourier transform to the constitutive
equation makes calculations easier because the derivation disappears and the total stress is proportional
to the strain
h
i
σ̃rθ (r, ω) = 2µ + m(iω)α ε̃rθ (r, ω) = 2µ̂(ω)ε̃rθ (r, ω)
(11)
with
ε̃rθ (r, ω) =
1 ³ ∂ f˜(r, ω) f˜(r, ω) ´
−
.
2
∂r
r
(12)
The equilibrium equation in the tangential direction [25] for a frequency range of interest where wave
effects within the rubber bushing are negligible is
∂ σ̃rθ
2
+ σ̃rθ = 0.
∂r
r
(13)
Substitutions of Equations (11) and (12) into (13) leads to a second order differential equation
∂ 2 f˜(r, ω) 1 ∂ f˜(r, ω) f˜(r, ω)
−
+
= 0,
∂r2
r
∂r
r2
(14)
k2
f˜(r, ω) = k1 r + ,
r
(15)
with a general solution
where the arbitrary constants k1 and k2 are calculated by satisfying the displacement boundary conditions
ũθ (r = a) = 0,
ũθ (r = b) = bϕ̃.
(16)
Finally, the shear strain state is given by
ε̃rθ (r, ω) =
b2 a2 1
ϕ̃(ω).
b2 − a2 r2
(17)
Furthermore, considering a cylindrical surface of radius r and length L with axis along the z-axis, the
moment is
Z LZ
2π
M̃ (ω) =
0
=
0
σ̃rθ (r, ω) r dθdz =
0
Z LZ
0
2
2π
Z LZ
0
0
2π
2µ̂(ω) ε̃rθ (r, ω) r2 dθdz =
i
h b2 a2 1
4πLa2 b2 µ̂(ω)
2
ϕ̃(ω)
r
dθdz
=
ϕ̃(ω),
2µ̂(ω) 2
b − a2 r2
b2 − a2
(18)
6
M.J. GARCIA TARRAGO, L. KARI, J. VIÑOLAS and N. GIL-NEGRETE
where the shear modulus goes out of the integral because, as expressed in Equation (11), it is constant for
the whole of the bushing volume, since the amplitude dependence has not yet been taken into account.
Hence, the dynamic stiffness of the rubber bushing in Figure 2 when subjected to a torsion angle ϕ at the
outer surface becomes
4πa2 b2 L
M̃ (ω)
= 2
µ̂(ω),
ϕ̃(ω)
b − a2
(19)
which is similar to the static formula developed by Adkins and Gent, see Equation (1), except for the
shear modulus because Equation (19) includes a dynamic frequency dependent shear modulus. Sections
3.2 and 3.3 explain the steps to be followed to obtain a dynamic stiffness which also includes the amplitude
dependence.
3.2
Equivalent strain
This section proposes the criteria for the calculation of an equivalent strain which leads to the equivalent
shear modulus. One idea would be to perform a volume average of the strain state. However, in the torsion
deformation mode the strain state inside the bushing has only one shear component, see Equation (17),
but in other deformation modes it might be more complex as in the radial case where there are both shear
and traction/compression strains inside the bushing. Consequently, performing a volume average of the
strain state is not a general methodology that could be extended to other deformation modes. Therefore,
the solution presented in this paper consists of working with an equivalent strain obtained from an energy
density balance between the torsion deformation in a bushing and a simple shear specimen case.
The deformation energy is equal to the sum of the energy needed to change the volume and the distortion
energy. In the present case of a bushing subjected to a torsion deformation, the volume is constant and the
total energy density is equal to the distortion energy density [25], which after being applied the temporal
Fourier transform reads
ŨT ∝ Ũd ∝
3 X
3
X
∗
σ̃ij
ε̃ij ,
(20)
i=1 j=1
where ŨT and Ũd are respectively the total and the distortion energy densities; and σ̃ij and ε̃ij are the stress
and strain components, while ∗ denotes complex conjugate. Furthermore, the volume averaged distortion
energy density within the bushing becomes
Ũd
bushing
1
∝
V
Z
∗
σ̃rθ
ε̃rθ dV
1
=
V
Z
2µ̂|ε̃rθ |2 dV,
(21)
where ε̃rθ is the strain state provided by Equation (17), V = π(b2 − a2 )L is the volume of the bushing and
µ̂ is the frequency dependent shear modulus displayed in Equation (11). In case of a simple shear specimen
made of the same material as the rubber bushing, the distortion energy density is
Ũd
specimen
∝ 2µ̂ε̃∗sh ε̃sh = 2µ̂|ε̃sh |2 ,
(22)
where ε̃sh is the temporal Fourier transform of the homogeneous shear strain inside the specimen. Consequently, an energy density balance gives the equivalent shear strain in the frequency domain
Z
1
ε̃∗rθ (r, ω)ε̃rθ (r, ω) dV =
V
Z b³ 2 2
³ b a ´2
1
b a
1 ´2 ∗
=
ϕ̃
(ω)
ϕ̃(ω)2πrL
dr
=
ϕ̃∗ (ω)ϕ̃(ω),
π(b2 − a2 )L a b2 − a2 r2
b2 − a2
|ε̃equiv (ω)|2 = |ε̃sh (ω)|2 =
(23)
Torsion stiffness of a rubber bushing
7
which depends, at this level of approximation, only on the geometric characteristics of the bushing and the
angle applied to it, but not on material properties and where ω is the angular frequency. The time-domain
equivalent strain is straight-forward obtained by applying the inverse Fourier transform
εequiv (t) =
3.3
b2
ba
ϕ(t).
− a2
(24)
Applying the rubber model to the equivalent strain
Now it is time to include the amplitude dependence of the rubber material properties inside the bushing.
For that reason the separable elastic, viscoelastic and friction rubber model presented in Section 2, is
applied to the time-domain equivalent strain obtained in Equation (24). The total stress is the sum of the
three stresses of Equations (2), (3) and (6) where ε(t) is identified with the equivalent strain εequiv (t). The
frictional component makes the rubber model nonlinear. In order to switch to the frequency domain the
temporal Fourier transform is applied to the total stress and the equivalent strain. The equivalent complex
modulus is obtained as
µ̂equiv (ωn ) =
σ̃total (ωn )
,
ε̃equiv (ωn )
(25)
where the two variables are evaluated at frequency ωn . Finally, this modulus is inserted into the formula
of the torsion stiffness calculated above by the linear classical theory of elasticity
kdyn (ωn ) =
4πa2 b2 L
µ̂equiv (ωn ),
b2 − a2
(26)
which represents the dynamic behaviour in the frequency domain of a bushing subjected to a torsion
deformation including amplitude dependence. This linearization process takes into account the non-linear
relation between stress and strain at frequency ωn considering the first order response while omitting the
less important overtones (stress response at 3ωn , 5ωn ,...).
4
Results
The variation with frequency and amplitude of the torsion dynamic stiffness given by Equation (26) as
function of the geometry of the bushing and the dynamic material properties of the rubber is studied in this
section with typical data of commercial bushings. The torsion angle varies harmonically with amplitudes
going from 1.74 × 10−6 to 1.74 × 10−1 rad over a frequency range from 10 to 100 Hz. The geometric and
material data of the carbon-black filled rubber bushing analysed in this work are presented in Table 1.
Table 1. Typical geometric and material data for a
commercial rubber bushing.
Geometric data
10−3
L = 22.0 ×
m
a = 5.0 × 10−3 m
b = 11.0 × 10−3 m
Material data
µ = 4.0 × 106 N/m2
m = 3.0 × 105 Nsα /m
α = 0.40
σf max = 8.0 × 102 N/m2
ε1/2 = 5.0 × 10−4
8
M.J. GARCIA TARRAGO, L. KARI, J. VIÑOLAS and N. GIL-NEGRETE
Magnitude (Nm/rad)
120
100
80
60
40
20
1.74 x 10
−6
rad
8.10 x 10
−5
rad
3.76 x 10
−3
rad
0
0.25
Loss Factor
0.2
0.15
1.74 x 10−6 rad
0.1
8.10 x 10−5 rad
0.05
0
10
3.76 x 10−3 rad
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 3. Torsion stiffness in magnitude and loss factor versus frequency for 3 different amplitudes.
Magnitude (Nm/rad)
120
100
80
60
20Hz
50Hz
80Hz
40
20
0
0.25
Loss Factor
0.2
0.15
0.1
20Hz
50Hz
80Hz
0.05
0
10
−5
−4
10
−3
10
Amplitude (radians)
−2
10
−1
10
Figure 4. Torsion stiffness in magnitude and loss factor versus torsion angle amplitude for 3 different frequencies.
Torsion stiffness of a rubber bushing
9
Calculations are carried out using Matlabr following the steps described in Section 3. Firstly, L, a and
b—the dimensions of the bushing—are inserted into Equation (24) together with the torsion angle in order
to obtain the equivalent strain in the time domain. Then, the rubber model using the material parameters
in Table 1 with the elastic, viscoelastic and friction components (see Equations (2), (3) and (6)) is applied
to that equivalent strain and the total stress of Equation (7) is achieved. Finally, the equivalent complex
shear modulus in the frequency domain is the result of dividing the temporal Fourier transform of the
total stress by the temporal Fourier transform of the equivalent strain, see Equation (25), a modulus that
is inserted into the torsion stiffness formula presented in Equation (26).
Figure 3 displays the variation in magnitude and loss factor of the stiffness versus frequency for three
different angle amplitudes and Figure 4 shows the variation in magnitude and loss factor of the stiffness
versus angle amplitude for three different frequencies. Both figures demonstrate that the stiffness calculated
with the presented formula represents the expected behaviour for a rubber bushing: the magnitude and
the loss factor increase with frequency, while versus the angle amplitude the magnitude of the stiffness
decreases and the loss factor increases until a value where the magnitude has decreased around seventy
percentage, then the loss factor starts decreasing.
5
Convergence of the stiffness
In this section the bushing is divided into several slices and therefore the equivalent shear modulus obtained
for each slice is closer to the true value than dealing with only one equivalent modulus for the whole bushing.
The dynamic stiffness is the result of an iterative process which starts with an estimation of the torsion
angles at the inner and outer radius of each slice to calculate the complex moment at each radius following
the methodology explained in Section 3. Furthermore, in order to fulfil the continuity of the moment along
the radius, the values of the torsion angles are modified until the moment differences at each radius are
negligible.
q
N
r
q
i- 1
i
r
q
1
r
r
N + 1
1
2
Figure 5. Rubber bushing divided into N slices.
This iteration process is carried out with the same bushing used in the Section 4. The analysis is
implemented in an amplitude range from 1.74 × 10−6 to 1.74 × 10−1 rad at a frequency of 50 Hz. Figure
6 shows that the differences of the dynamic stiffness in magnitude and loss factor when the bushing is
divided into one, two and three slices are small enough to consider that working with only one equivalent
shear modulus for the whole bushing is accurate enough.
10
M.J. GARCIA TARRAGO, L. KARI, J. VIÑOLAS and N. GIL-NEGRETE
Magnitude (Nm/rad)
120
100
80
60
3 slices
2 slices
1 slice
40
20
0
Loss Factor
0.2
0.15
0.1
3 slices
2 slices
1 slice
0.05
0
−5
10
−4
10
−3
10
Amplitude (radians)
−2
10
−1
10
Figure 6. Torsion stiffness in magnitude and loss factor versus angle amplitude for a 50Hz signal when the rubber bushing is divided
into one, two and three slices.
6
Conclusions
This paper concentrates on the torsion deformation of a bushing and the process is developed by applying a
novel rubber material model to an equivalent strain of the strain state inside the bushing, thus providing an
equivalent shear modulus that is inserted into an analytical formula. As a result, a simple, fast and effective
model to obtain the dynamic torsion stiffness of a filled rubber bushing including amplitude dependence
as function of the dimensions of the bushing and the material properties of the rubber is obtained.
Section 4 illustrates the results obtained with the model: the dynamic stiffness for a bushing sample
under several amplitudes and over a wide frequency range, showing the expected behaviour of a rubber
bushing. In order to check the accuracy of the approach, in Section 5 the same bushing has been divided
into several slices thus using an equivalent shear modulus closer to the true value for each slice. The
small differences in magnitude and loss factor between the dynamic stiffness obtained when the bushing
is divided into a single, two and three slices, show that the amplitude dependence is enough well modelled
while using a single slice. In addition, as future work, some experiments and correlation with theoretical
results will be carried out on commercial bushings at different conditions to find the limits of this approach.
The simplicity of this model is very convenient for the dynamic analysis of complex structures in which
rubber bushings are connecting components. Several advantages can be outlined. Firstly, a novel model
has been presented to represent the dynamic behaviour of the rubber at the material level including
frequency and amplitude dependence with only five parameters. Secondly, the time invested in performing
the calculations: Matlabr has been used to carried out the process without requiring large amount of
memory due to discretization process, just by applying the novel rubber model to one equivalent strain
for the whole bushing. Additionally, unlike previous models this formula depends on the geometry of the
bushing and the material properties and therefore it is a useful and fast prediction tool to determine the
most suitable rubber bushing to connect to other structures to fulfil user requirements.
The torsion stiffness is the first step to be able to predict the behaviour of a rubber bushing in all
directions. A possible extension of this work is to develop similar formulas for axial and radial deformations
in order to create a simple model representing the dynamic stiffness in all directions. Furthermore, while
Torsion stiffness of a rubber bushing
11
analysing these simplified formulas it might be possible to find relations between axial, radial and torsional
directions aiming at an additional simplification of the all-directions dynamic model of the bushing.
View publication stats
7
Acknowledgements
The European Community is gratefully acknowledged for the financial support through contract No:
MEST-CT-2004-503675, for a research training project European Doctorate in Sound and Vibration Studies (EDSVS) within the framework of the European Community’s Scheme Improving Human Research
Potential.
References
[1] Morman, K.N., and Pan, T.Y., 1988, Application of finite-element analysis in the design of automotive elastomeric components.
Rubber Chemistry and Technology, 61(3), 503-533.
[2] Hill, J.M., 1975, Radical deflections of rubber bush mountings of finite lengths. International Journal of Engineering Science, 13,
407-422.
[3] Adkins, J.E., and Gent, A.N., 1954, Load-deflexion relations of rubber bush mountings. British Journal of Applied Physics, 5,
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