Engineering rubber bushing stiffness formulas including dynamic amplitude dependence

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Engineering rubber bushing stiffness formulas including dynamic amplitude

dependence

Maria-Jos´e Garcia

Stockholm 2006

Licentiate Thesis Royal Institute of Technology School of Engineering Sciences

Department of Aeronautical and Vehicle Engineering

The Marcus Wallenberg Laboratory for Sound and Vibration Research

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Akademisk avhandling som med tillst˚and av Kungliga Tekniska Högskolan i Stock- holm framläggs till offentlig granskning för avläggande av teknologie licentiatexa- men torsdagen den 15:e juni 2006, kl 10.30 i sal MWL 74, Teknikringen 8, KTH, Stockholm.

TRITA-AVE -2006:36 ISSN -1651-7660

c

°Maria-Jos´e Garcia, June 2006

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AbstractEngineering design models for the torsion and axial dynamic stiffness of carbon black filled rubber bushings in the frequency domain including amplitude dependence are presented. They are founded on a developed material model which is the result of applying a separable elastic, viscoelastic and friction rubber component model to the material level. Moreover, the rubber model is applied to equivalent strains of the strain states inside the torsion or axial deformed bushing previously obtained by the classical linear theory of elasticity, thus yielding equivalent shear moduli which are inserted into analytical formulas for the stiffness. Therefore, unlike other simplified approaches, this procedure includes the Fletcher-Gent effect inside the bushing due to non-homogeneous strain states. The models are implemented in Matlab^r. In addition, an experimental verification is carried out on a commercially available bushing thus confirming the accuracy of these models which become a fast engineering tool to design the most suitable rubber bushing to fulfil user requirements. Finally, they can be easily employed in multi-body and finite element simulations.

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Licentiate Thesis

This licentiate thesis consists of this summary and two appended papers listed below and referred to as Paper A and Paper B.

Paper A

M.J. Garc´ıa T´arrago, L. Kari, J. Vi˜nolas and N. Gil-Negrete, 2006:

“Torsion stiffness of a rubber bushing: a simple engineering design formula including amplitude dependence”. Submitted to Journal of Strain Analysis for Engineering Design.

Paper B

M.J. Garc´ıa T´arrago, L. Kari and J. Vi˜nolas, 2006: “Axial stiffness of carbon black filled rubber bushings including frequency and amplitude dependence”. Submitted to Kautschuk Gummi Kunststoffe.

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1 Introduction

Rubber isolators play an important role in the noise and vibration control. They isolate the car body from the vibration source: wheel and tire unbalance, road input, powertrain, etc. There is a wide variety of designs de- pending on the geometry of the component and the disposition of the metal layers, chosen to fulfil user requirements. In particular, rubber bushings are the object of this study. 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, pivot arms and several types of mechanical linkage. Therefore, 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. Simplicity is necessary as the rubber bushing represents only a small part connecting components of more complex structures.

The main characteristic of rubber materials is their elasticity and capac- ity for dissipating vibration energy which is represented by the hysteresis loop enclosed by the loading and unloading curves in a stress–strain dia- gram. Furthermore, rubber materials are strongly dependent on frequency and amplitude, as reviewed by Medalia (1). The latter dependence, known as Fletcher–Gent effect (2), 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, as shown by Payne and Whittaker (3). But also this presence produces an increase in the shear modulus and damping. For that reason, it is important to include the amplitude dependence in the representation of the dynamic behavior of filled rubber bushings.

Firstly, the behavior of these components has been defined by their static stiffness as calculated by finite element analysis (4), by using truncated

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Fourier and Bessel functions (5) or through principal mode approaches (6).

Lately, Horton et al. have developed formulas for radial, conical and torsion static stiffness (7; 8; 9) based on three-dimensional static elastic theory.

Moreover, the static stiffness in many directions is provided in standard textbooks (10; 11; 12; 13; 14). In addition, the structure-borne properties of a long rubber bush mounting in all directions are considered in Kari (15). However, none of these methods takes into account the amplitude dependence.

To date, several models have been devised to represent this property, like Kraus (16) who explains the amplitude dependence as due to the continuous breaking and reforming of van der Waals bonds between carbon-black ag- gregates, while modifications and experimental application are carried out by Ulmer (17) and Lion (18) presenting a time domain formulation of the Kraus model. Moreover, the Fletcher–Gent effect is also characterized us- ing friction models, like Gregory (19), Coveney et al. (20) and Kaliske and Rothert (21) proposing the Prandtl element, a Coulomb damper in series with a elastic spring; a model expanded by Bruni and Collina (22), Olsson and Austrell (23) and Brackbill et al. (24).

Furthermore, several authors have employed these material models in order to calculate the dynamic stiffness of rubber bushings. Some of them insert the models into finite element systems, like Austrell et al. (25; 26; 27) who represent frequency and amplitude dependencies adding integer derivatives to stick-slip friction components, similar to Gil–Negrete (28) except for the use of fractional derivatives. However, finite element procedures re- quire a long-time overlay process to calculate the stiffness. In contrast, other authors work directly at the component level, like Berg (29) who presents a five-parameters model which gives a good resemblance to the smoother- friction rubber behavior, also used by Sj¨oberg and Kari (30) together with a rate-dependent part using fractional derivatives, and Misaji et al. (31) who take into account amplitude dependence as the parameters of an ordinary Kelvin–Voigt model are updated continuously for every oscillation cycle.

However, the latter methods neglect the amplitude dependence inside the rubber bushing due to non-homogeneous strain states.

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This work overcomes previous limitations by the development of simple engineering formulas to represent the dynamic stiffness of rubber bushings in the frequency domain including amplitude dependence. They are achieved by applying a separable elastic, viscoelastic and friction material model to equivalent strains of the non-homogeneous strain states inside a rubber bushing, when it is subjected to torsion, and axial deformations, as shown in Paper A and B respectively, thus giving equivalent shear modulus which is inserted into analytical formulas for the stiffness. The novel material model is presented in Paper A and results from applying a sound component level to the material level. Consequently, unlike other simplified models these formulas include the amplitude dependence due to non-homogeneous strain states and therefore they become a fast engineering design tool to determine the most suitable rubber bushing. Moreover, they can be easily employed in multi-body and finite element simulations. Experimental verification has been carried out in Paper B for the axial stiffness of a commercial bushing, showing a good agreement between measured and model stiffness over a frequency range at several amplitudes. Furthermore, for the torsion case it is shown in Paper A—by dividing the bushing into several slices and consequently each equivalent shear modulus is closer to the true value—that the approach of working with only one equivalent modulus for the whole bushing is accurate enough.

2 Objectives

The main purpose of the presented research is to contribute to the work done in the field of modelling rubber isolators by developing simple engineering design models for the dynamic stiffness of rubber bushings including amplitude and frequency dependence when they are subjected to harmonic displacement amplitudes in the 1-100 Hz frequency range.

This objective involves the following tasks:

• Create a rubber material model by extending a sound component model to the material level.

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• Apply the classical theory of elasticity in order to obtain the strain states inside a bushing subjected to torsion and axial deformations, to subsequently calculate an equivalent strain for each state. Fur- thermore, the rubber model is applied to these strains, thus giving equivalent shear moduli that are inserted into engineering formulas for the stiffness.

• Models implementation in Matlab^r.

• Experimental verification on commercially available bushings.

3 Methodology

The methodology to achieve the objects presented in previous section is subsequently described. Two main procedures are carried out: firstly, a theoretical formulation of the model and secondly, an experimental verification.

3.1 Theoretical formulation

Static engineering formulas are widely used to predict the behavior of rubber bushings. Nevertheless they neglect the dynamic properties of rubber leading to errors in the stiffness prediction. The solution presented in this work is based on the idea that the dynamic properties can be introduced into engineering stiffness formulas by means of a dynamic equivalent modulus which is the result of applying a rubber model to a harmonic strain.

However, this modulus depends on the strain amplitude, due to the nonlinear relation between stress and strain for rubber. Therefore, in order to represent the behavior of the whole bushing, a global value for the shear modulus is required.

The model is developed as follows: Firstly, a rubber material model is created by extending a sound component model to the material level. Sec- ondly, the classical theory of elasticity is applied to obtain the strain states inside the bushing when it is subjected to torsion and axial deformation, while considering that the relation between stress and strain contains only elastic and fractional derivative components. Subsequently, the calculation

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of energy density balances between the bushing subjected to each deformation and a simple shear specimen made of the same material gives the equivalent strains. Finally, the rubber model is applied to those strains, thus giving the equivalent complex shear moduli which are inserted into the formula for the torsion and axial stiffness calculated previously with the classical theory of elasticity.

3.1.1 Rubber material model

The simplest representation of rubber materials is the Kelvin–Voight model where an elastic frequency independent spring is coupled in parallel with a viscous dashpot, as discussed by Knothe and Grassie (32). However, this model strongly overestimates both stiffness and damping for higher frequencies. Furthermore, the three-parameter Maxwell model, which is the result of including a spring in series with the dashpot, represents better the stiffness at high frequencies. However, a better representation of the frequency dependence can be obtained by using a summation of Maxwell models, as discussed by Lodhia and Esat (33) and Betz (34). Nevertheless, this representation increases the complexity and number of parameters needed to accurately describe material properties. An alternative to obtain a good de- scription of the frequency dependence while reducing the required number of parameters consists of using time derivatives of non-integer order, known as fractional derivatives, a method used by Sj¨oberg and Kari (30). However, the fractional derivative models neglect the amplitude dependence, an effect which appears when including fillers, such as carbon black, in a rubber mixture, thus introducing additional bonds within the material. These bonds exhibit a friction-like behavior as discussed in Ref. (35) and modelling it demands for friction models. Austrell et al. (25) employ stick-slip com- ponents, an approach also used by Bruni and Collina (22). Furthermore, Berg (29) presents a model giving better resemblance to the smoother rubber friction behavior, a representation also used by Sj¨oberg and Kari (30).

The material model employed in this work is developed in Paper A by applying to the material level the component model presented by Sj¨oberg and Kari (30), where the amplitude dependence is represented by a smooth

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friction component while the frequency dependence is modelled by a fractional Kelvin–Voigt model, thus giving an effective model while using only five parameters.

)(

t o t a l

t

s )(t e

a, m

m2

2/1m a x

, e s

f

Figure 1: Rubber material model.

Elastic stress The first branch in Fig. 1 represents the elasticity by a linear relation between the elastic stress σ_elast and strain ε

σ_elast(t) = 2µε(t), (1)

where µ is the elastic shear modulus and t is time.

Frequency dependent stress The second component in Fig. 1 indicates the frequency dependence and it is modeled by using fractional derivatives which increases the ability to adjust to measured characteristics while keep- ing the number of parameters to only two: a proportionality constant m and the time derivative order α

σ_fract(t) = mD^αε(t), 0 < α ≤ 1, (2)

where σ_fract is the viscoelastic stress and D^α denotes the fractional time derivative of order α, defined through an analytical continuation of a frac- tional Riemann–Liouville integration (36). Numerically, the viscoelastic

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component can be evaluated as

σ_fract(t) ≈ m∆t^−α Γ(−α)

n−1X

j=0

Γ(j − α)

Γ(j + 1)ε_n−j, (3)

where ε_n−j = ε((n − j)∆t), ε_n is the final strain at time t_n = n∆t, ∆t is a constant time step applied in the estimation process and Γ denotes the Gamma function (37) defined as

Γ(β) = Z _∞

1

x^β−1 e^−xdx + X∞ j=0

(−1)^j

j!(β + j), β 6= 0, −1, −2, ... (4) In addition, the temporal Fourier transformation of the fractional time derivative branch leads to

˜

σ_fract(ω) = m(iω)^αε(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.

Frictional stress 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 σ_{f max} and ε_1/2. The frictional stress σ_frict develops gradually following the equation

σ_frict(t) = σ_{f s}+

h

ε(t) − ε_s ih

σ_{f max}− sign( ˙ε) σ_{f s} i

ε_1/2 h

1 − sign( ˙ε) _σ^σ^{f s}

f max

i

+ sign( ˙ε) h

ε(t) − ε_si , (6)

where the material parameters σ_{f max} and ε_1/2 are the maximum friction stress developed and the strain needed to develop half of that stress, with sign( ˙ε) denoting the sign of the strain rate. The values of σ_{f s} and ε_s are updated each time the strain changes direction at ˙ε = 0 as σ_{f s} ← σ_frict|_˙ε=0 and ε_s← ε |_˙ε=0.

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Total stress Finally, the rubber behavior is represented by a total stress σ_total which is the sum of the three stresses

σ_total(t) = σ_elast(t) + σ_fract(t) + σ_frict(t). (7) This Equation, with Equations (1), (2) and (6), represents a nonlinear relation between the total stress and the strain and a value of the shear modulus cannot be obtained directly as would be the case with the elastic component.

3.1.2 Equivalent strain

The strain state inside the rubber bushing of length L in Fig. 2 bonded to two cylindrical metal layers at inner and outer radii a and b and subjected to a harmonic torsion deformation is studied in Paper A while the axial case is analysed in Paper B. The classical linear theory of elasticity provides

Ø 2 a

Ø 2 b

jz

ur

q

ur

r

ur

L

qr

z x y

z

Figure 2: Rubber cylindrical bushing subjected to torsion deformation.

the strain states inside the rubber bushing while considering firstly, that the

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constitutive equations between the stress and strain have only elastic and frequency dependence components, as the amplitude dependence is taken into account later on when the rubber model is applied to the equivalent strains. Furthermore, the strain states with torsion and axial deformations have only one shear component and therefore the constitutive equation in both cases reads

σ_t/a(t) = 2µε_t/a(t) + mD^αε_t/a(t), (8) where σ_t/ais the torsion/axial shear stress and ε_t/ais the torsion/axial shear strain. In addition, working in the frequency domain as result of applying the temporal Fourier transform makes calculations easier because the derivation disappears and the total stress is proportional to the strain

˜

σ_t/a(ω) = h

2µ + m(iω)^α i

˜

ε_t/a(ω) = 2ˆµ(ω)˜ε_t/a(ω). (9) Secondly, the rubber is assumed homogeneous and isotropic and displacement gradients are considered small enough to apply the theory of elasticity. Furthermore, over the frequency range of interest wave effects within the rubber bushing are negligible. Consequently, when the bushing is subjected to a harmonic torsion angle at the outer surface relative to the inner ϕ(t) = ϕ sin(ω_st) with ω_sthe excitation frequency and ϕ the amplitude, the strain state reads

˜

ε_t(r, ω) = b²a² b²− a²

1

r² ϕ(ω),˜ (10)

where r is the radius. Whereas if a harmonic axial displacement d(t) = d sin(ω_st) is applied at the outer surface the strain state becomes

˜

ε_a(r, ω) = 1

2r log_e(^b_a) d(ω).˜ (11) Subsequently, in order to calculate the equivalent strains one idea would be to perform volume averages of the strain states. However, in the torsion and axial deformation modes the strain states have only one shear component, but in other deformation modes it might be quite complex as it happens in the radial case where there are both shear and traction/compression strains

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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 here presented consists in working with equivalent strains obtained from energy density balances between the torsion and axial deformation in a bushing and a simple shear specimen. For instance, as shown in Paper A the deformation energy in the torsion case is equal to the distortion energy because the volume remains constant. Fur- thermore, the temporal Fourier transform of the volume averaged distortion energy reads

U˜_{d bushing}∝ 1 V

Z

˜

σ_t^∗ε˜_tdV = 1 V

Z

2ˆµ|˜ε_t|²dV, (12) where V = π(b²− a²)L is the volume of the bushing and ˆµ is the frequency dependent shear modulus presented in Equation (9). In addition, the temporal Fourier transform of the distortion energy density in case of a simple shear specimen made of the same material as the rubber bushing is

U˜_{d specimen}∝ 2ˆµ˜ε^∗_shε˜_sh= 2ˆµ|˜ε_sh|², (13) where ˜ε_shis the temporal Fourier transform of the homogeneous shear strain inside the specimen. Consequently, the balance between both energy densi- ties gives the equivalent strain in the frequency domain which is transformed into the time domain by applying the inverse Fourier transform

ε_{equiv t}(t) = b a

b²− a² ϕ(t). (14)

Moreover, if the same methodology is conducted for the axial case the equivalent strain becomes

ε_{equiv a}(t) = d(t) q

2(b²− a²) log_e(^b_a)

ϕ(t). (15)

3.1.3 Engineering formulas for the stiffness

In order to obtain the equivalent shear modulus, the rubber model in Fig. 1 is applied to the equivalent strains thus giving the total stresses σ_{total t/a}(t)

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made up of three components: elastic, frequency dependent and frictional stresses. Furthermore, the last component introduces the amplitude dependence and makes the rubber model nonlinear.

In order to switch to the frequency domain the excitation frequency of the input signals ϕ(t) and d(t) is varied as ω_s = s∆ω with ∆ω a con- stant frequency step and s ∈ N, and therefore s different equivalent strains ε^s_{equiv t/a}(t) and their corresponding total stresses σ^s_{total t/a}(t) are obtained.

Furthermore, the temporal Fourier transform is applied to those signals, thus giving the frequency functions ˜ε^s_{equiv t/a}(ω) and ˜σ^s_{total t/a}(ω) which are evaluated at ω_s leading to the equivalent complex moduli in the frequency domain

ˆ

µ_{equiv t/a}(ω_s) = σ˜^s_{total t/a}(ω_s)

2 ˜ε^s_{equiv t/a}(ω_s). (16) This linearization process takes into account the non-linear relation between stress and strain at frequency ω_s considering the first order response while omitting the less important overtones (stress response at 3ω_s, 5ω_s,...). Fi- nally, the moduli are inserted into the stiffness formulas calculated previously with the classical linear theory of elasticity, thus giving a formula for the torsion dynamic stiffness

Kdyn torsion(ω_s) = 4πa²b²L

b²− a² µˆ_{equiv t}(ω_s), (17) while the axial stiffness reads

K_{dyn axial}(ω_s) = 2πL

log_e(_a^b)µˆ_{equiv a}(ω_s). (18) Consequently, two engineering formulas to represent in the frequency domain the dynamic behavior of rubber bushings in the torsion and axial directions including the amplitude dependence have been achieved.

3.2 Experiments

The accuracy of the developed models representing the dynamic stiffness of rubber components while including amplitude dependence is experimental verified. Specifically, the model for the axial stiffness is validated by a

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commercially available bushing in Paper B. Firstly, the material parameters involved in the formula of the stiffness, see Equation (18), are achieved from two measurements: elastic and amplitude dependence parameters are obtained from a large deformation quasi-static test while the viscoelastic parameters are determined by a steady state harmonic dynamic test performed over a frequency range from 1 to 100 Hz at an amplitude of 0.1 mm.

Secondly, five new dynamic tests at 0.013, 0.04, 0.17, 0.25 and 0.35 mm are conducted in order to demonstrate that the formula for the axial stiffness predicts well the bushing behavior including the amplitude and frequency dependence.

3.2.1 Test object

The test object is a commercial bushing manufactured by Trelleborg AVS under the trade name of VP 40/70120 with a rubber hardness of 60 IRH.

The rubber type is a mixture between SMR 10 and SMR GP and the carbon black filler is a combination of N660 and N774.

a = 2 0 m m b = 3 5 m m

L=90 mm

Figure 3: Test object.

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3.2.2 Setup

The experiments are performed in a servo-hydraulic test machine consisting of a lower body with a hydraulic piston inside, two columns to the sides and an upper rigid crosshead, as shown in Fig. 4. Measurements are performed with the inner surface of the bushing fixed and joined to the upper crosshead through a shaft and one load sensor, while the outer surface of the bushing is subjected to axial displacements produced by the lower hydraulic piston. The motion of the bushing is measured by two displacement sensors symmetrically positioned on the outer surface of the bushing. The electrical signals are conditioned in a 6-channels amplifier producing electrical inputs to the frequency analyzer which are processed subsequently in a personal computer. In order to achieve constant amplitude, the electric signal that comes from the displacement sensor of the test machine and goes through the amplifier is automatically adjusted in a control loop in the analyzer to supply the exciter.

3.2.3 Type of measurements

The test component is mechanically conditioned before measurements to eliminate influence of the Mullins’ effect (38) which is shown when previously unstrained rubber is subjected to strain cycles at constant peak value reducing peak stress values at the first few oscillations.

Quasi-static test Measurements at 0.1 Hz and 1 mm displacement amplitude are used to fit the elastic and amplitude dependence parameters. The large deformation assures that all friction has been developed in the loop and the low frequency guarantees that all viscous effects are eliminated.

Dynamic test The bushing is excited by stepped sine displacements start- ing at 1 Hz and increasing with a constant frequency step of 1 Hz to a maximum frequency of 100 Hz, with amplitude held constant at 0.1 mm.

Viscoelastic parameters are the results of an estimation iterative minimiza- tion process between the measured and the model complex stiffness.

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D I S P L A C E M E N T S E N S O R

P I S T O N

F I X E D S E R V O - H Y D R A U L I C

T E S T M A C H I N E

D E S K C O N T R O L L E R

P E R S O N A L C O M P U T E R F R E Q U E N C Y A N A L Y Z E R

I N P U T O U T P U T

L O A D S E N S O R

6 - C H A N N E L S A M P L I F I E R

R U B B E R B U S H I N G

Figure 4: Measurement setup.

4 Results

The methodology to obtain the torsion and axial dynamic stiffness is implemented in Matlab^r, thus giving a fast engineering design tool to predict the dynamic behavior of rubber components. Furthermore, the calculation of the torsion stiffness for a typical commercial bushing is carried out in Paper A, thus showing the expected behavior for a rubber bushing, while Paper B presents an experimental verification for the axial stiffness performed on a commercially available bushing.

4.1 Implementation in Matlab^r

The implementation in Matlab^ris carried out for each deformation mode as follows: firstly, the equivalent strains in the time domain are calculated

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as function of the geometry of the bushing and the boundary conditions, as shown in Equations (14) and (15). Secondly, the rubber model is applied to those strains and the steady state of the resultant stresses are converted into frequency functions through the temporal Fourier transform. Furthermore, the value of the frequency dependent shear moduli result from dividing the stress frequency functions by the strain frequency signals when both functions are evaluated at the excitation frequency, see Equation (16). In addition, the amplitude of the displacement excitations is varied, as well as, the frequency range that goes from 1 to 100 Hz, to check how well the model represents both dependencies. Finally, the frequency and amplitude dependent shear moduli are inserted into the formulas of the stiffness, Equations (17) and (18).

Table 1: Typical geometric and material data for a commercial rubber bushing.

Geometric data Material data L = 22.0 × 10⁻³ m µ = 2.0 × 10⁶ N/m² a = 5.0 × 10⁻³ m m = 2.0 × 10⁵ Ns^α/m² b = 11.0 × 10⁻³ m α = 0.40

σ_{f max} = 4.0 × 10³ N/m² ε_1/2= 1.0 × 10⁻³

Paper A presents the torsion stiffness for a typical commercial available bushing whose data is given in Table 1. The torsion angle is varied harmonically with amplitudes going from 1.74 × 10⁻⁶ to 1.74 × 10⁻¹ rad over a frequency range from 10 to 100 Hz. Fig. 5 displays the variation in magnitude and loss factor of the torsion stiffness versus frequency at three angle amplitudes while Fig. 6 shows the variation in magnitude and loss factor of the stiffness versus angle amplitude at three frequencies. Both fig- ures demonstrate that the stiffness calculated with the presented formula represents the expected behavior for a rubber bushing: the magnitude and the loss factor increase with frequency, while versus the angle amplitude the

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magnitude of the stiffness decreases and the loss factor initially increases and then decreases.

0 10 20 30 40 50

Magnitude (Nm/rad)

6.3 x 10⁻⁶ rad 2.9 x 10⁻⁴ rad 1.3 x 10⁻² rad

10 20 30 40 50 60 70 80 90 100

0 0.1 0.2

Frequency (Hz)

Loss Factor

6.3 x 10⁻⁶ rad 2.9 x 10⁻⁴ rad 1.3 x 10⁻² rad

Figure 5: Torsion stiffness magnitude and loss factor versus frequency at three amplitudes.

Consequently, the displayed frequency and amplitude stiffness depen- dency highlights the need of models which take into account both effects.

Moreover, the use of static models to represent the behavior of rubber components, a procedure commonly used in engineering design, may induce errors. Therefore, the developed engineering formulas become a fast and effective tool to predict the behavior of rubber bushings including frequency and amplitude dependence.

In addition, in Paper A the sample bushing is divided into one, two

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0 10 20 30 40 50

Magnitude (Nm/rad)

20Hz 50Hz 80Hz

10⁻⁴ 10⁻²

0 0.1 0.2

Amplitude (radians)

Loss Factor 20Hz

50Hz 80Hz

Figure 6: Torsion stiffness magnitude and loss factor versus amplitude at three frequencies.

and three slices to show that even if there are several slices and therefore the equivalent shear moduli are closer to the true value than dealing with only one equivalent modulus for the whole bushing, there are only small differences in the magnitude and loss factor of the dynamic stiffness between all cases, and therefore working with only one equivalent shear modulus for the whole bushing is accurate enough.

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, subsequently calculate the complex moment at each radius following the methodology explained previously. Furthermore, in order to fulfil the

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continuity of the moment along the radius, the values of the torsion angles are modified until the moment differences at each radius are negligible.

4.2 Experimental verification

The model for the axial stiffness is experimental validated for a commercially available bushing in Paper B. Calculations and graphical representations are carried out in Matlab^r. Elastic and amplitude dependence parameters are varied until the model hysteresis loop enclosed by the resultant force in the bushing when is subjected to axial displacement at 0.1 Hz and 1 mm amplitude fits well with the hysteresis loop obtained from one quasi- static test performed on the bushing at the same displacement conditions.

The good agreement between the two loops displayed in Fig. 7, is achieved with the values: µ = 2.42 × 10⁶ N/m², σ_{f max} = 1.92 × 10⁴ N/m² and ε_1/2 = 4.0 × 10⁻³.

Moreover, viscoelastic parameters are obtained from the error minimiza- tion iteration process between the complex stiffness obtained with one dynamic test performed from 1 to 100 Hz at 0.1 mm constant amplitude and the stiffness calculated by Equation (18) at the same conditions. In par- ticular, when the viscoelastic parameters are assigned the values: m = 6.0 × 10⁵ Ns^α/m² and α = 0.20, the measured stiffness fits well in mag- nitude and loss factor with the model stiffness, as shown in Fig. 8.

In addition, five new dynamic tests are performed at amplitudes of 0.013, 0.04, 0.17, 0.25 and 0.35 mm and the results are compared to those of the model when using the five material parameters calculated previously, displaying a good agreement over the whole frequency range, as presented in Figs. 9 and 10, which verifies the accuracy of Equation (18) representing the axial dynamic stiffness of a rubber bushing including amplitude and frequency dependence.

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−1.5 −1 −0.5 0 0.5 1 x 10⁻³

−3000

−2000

−1000 0 1000 2000 3000

Displacement (m)

Force (N)

Model Measurement

Figure 7: Comparison between measured and model hysteresis loop at 1 mm amplitude and 0.1 Hz.

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0 1 2 3 4 5

x 10⁶

Magnitude (N/m) Model

Measurement

0 20 40 60 80 100

0 0.05 0.1 0.15

Frequency (Hz)

Loss Factor Model

Measurement

Figure 8: Comparison between measured and model dynamic stiffness in magnitude and loss factor over a frequency range from 1 to 100 Hz when amplitude is held constant at 0.1 mm

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0 2 4 6

Magnitude (kN/mm)

Model 0.04mm Meas. 0.04mm Model 0.1mm Meas. 0.1mm Model 0.35mm Meas. 0.35mm

0 20 40 60 80 100

0 0.05 0.1 0.15

Frequency (Hz)

Loss Factor

Model 0.04mm Meas. 0.04mm Model 0.1mm Meas. 0.1mm Model 0.35mm Meas. 0.35mm

Figure 9: Measured and model dynamic stiffness in magnitude and loss factor versus frequency at three amplitudes.

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3 3.5 4 4.5 5 5.5 6

Magnitude (kN/mm)

Model 10Hz Meas. 10Hz Model 20Hz Meas. 20Hz Model 40Hz Meas. 40Hz

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.08

0.1 0.12 0.14 0.16 0.18

Amplitude (mm)

Loss Factor

Model 10Hz Meas. 10Hz Model 20Hz Meas. 20Hz Model 40Hz Meas. 40Hz

Figure 10: Measured and model dynamic stiffness in magnitude and loss factor versus amplitude at three frequencies.

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5 Conclusions

Simple engineering models for the torsion and axial dynamic stiffness of filled rubber bushings including amplitude and frequency dependence have been presented. They have been developed by applying a rubber material model to the equivalent strains of the strain states inside the deformed bushing, thus providing equivalent shear moduli which are inserted into the analytical formulas for the stiffness.

Paper A presents the model for the torsion stiffness and illustrates the results in magnitude and loss factor at several amplitudes over a frequency range from 1 to 100 Hz when the model is applied to typical geometric and material data of rubber components, showing the expected behavior of a rubber bushing. Furthermore, in order to check the accuracy of the approach 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, an experimental verification of the axial stiffness model is carried out on a commercially available bushing in Paper B. Firstly, the model parameters are obtained from two tests: a quasi-static and dynamic.

Subsequently, five new dynamic experiments are performed on the bushing to compare the results with those obtained with the engineering formula at the same amplitude and frequency conditions, showing good agreements which verifie the accuracy of the model.

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 behavior of the rubber at the material level including frequency and amplitude dependence with only five parameters.

Secondly, the time invested in performing the calculations: Matlab^r has been used to carry out the process without requiring large amount of mem- ory due to discretization process, just by applying the novel rubber model

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to one equivalent strain for the whole bushing. Additionally, unlike other simplified models these formulas depend on the geometry of the bushing and the material properties and therefore they are useful and fast prediction tools to determine the most suitable rubber bushing to connect to other structures to fulfil user requirements.

Acknowledgments

The European Community is gratefully acknowledged for the financial sup- port through contract No: MEST-CT-2004-503675, for a research train- ing project European Doctorate in Sound and Vibration Studies (EDSVS) within the framework of the European Community’s Scheme Improving Hu- man Research Potential. I would like to express my special thanks to my supervisor Leif Kari for excellent and always encouraged guidance through the course of this project. My thanks also to Kent Lindgren and Danilo Prelevic for professional assistance in my experiments and to my colleagues at MWL. Finally I thank my family and friends for being always there.

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