Engineering rubber bushing stiffness

formulas including dynamic amplitude

dependence

Maria-Jose 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

Akademisk avhandling som med tillstand av Kungliga Tekniska Hogskolan i Stock-

holm framlaggs till offentlig granskning for avlaggande av teknologie licentiatexa-

men torsdagen den 15:e juni 2006, kl 10.30 i sal MWL 74, Teknikringen 8, KTH,

Stockholm.

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

c©Maria-Jose Garcia, June 2006

Abstract Engineering design models for the torsion and axial dynamic stiff-ness of carbon black filled rubber bushings in the frequency domain includingamplitude dependence are presented. They are founded on a developed ma-terial model which is the result of applying a separable elastic, viscoelasticand friction rubber component model to the material level. Moreover, therubber model is applied to equivalent strains of the strain states inside thetorsion or axial deformed bushing previously obtained by the classical lineartheory of elasticity, thus yielding equivalent shear moduli which are insertedinto analytical formulas for the stiffness. Therefore, unlike other simplifiedapproaches, this procedure includes the Fletcher-Gent effect inside the bush-ing due to non-homogeneous strain states. The models are implemented inMatlabr. In addition, an experimental verification is carried out on acommercially available bushing thus confirming the accuracy of these mod-els which become a fast engineering tool to design the most suitable rubberbushing to fulfil user requirements. Finally, they can be easily employed inmulti-body and finite element simulations.

Licentiate Thesis

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

Paper AM.J. Garcıa Tarrago, L. Kari, J. Vinolas and N. Gil-Negrete, 2006:“Torsion stiffness of a rubber bushing: a simple engineering design formulaincluding amplitude dependence”. Submitted to Journal of Strain Analysisfor Engineering Design.

Paper BM.J. Garcıa Tarrago, L. Kari and J. Vinolas, 2006: “Axial stiffnessof carbon black filled rubber bushings including frequency and amplitudedependence”. Submitted to Kautschuk Gummi Kunststoffe.

Contents

1 Introduction 1

2 Objectives 3

3 Methodology 4

3.1 Theoretical formulation . . . . . . . . . . . . . . . . . . 4

3.1.1 Rubber material model . . . . . . . . . . . . . . 5

3.1.2 Equivalent strain . . . . . . . . . . . . . . . . . 8

3.1.3 Engineering formulas for the stiffness . . . . . . 10

3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.1 Test object . . . . . . . . . . . . . . . . . . . . 12

3.2.2 Setup . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.3 Type of measurements . . . . . . . . . . . . . . 13

4 Results 14

4.1 Implementation in Matlabr . . . . . . . . . . . . . . 14

4.2 Experimental verification . . . . . . . . . . . . . . . . . 18

5 Conclusions 23

Acknowledgments 24

References

Paper A

Paper B

1 Introduction

Rubber isolators play an important role in the noise and vibration con-trol. They isolate the car body from the vibration source: wheel and tireunbalance, road input, powertrain, etc. There is a wide variety of designs de-pending on the geometry of the component and the disposition of the metallayers, chosen to fulfil user requirements. In particular, rubber bushings arethe object of this study. They consist of a rubber tube bonded on theirouter and inner surfaces to rigid metal layers. Their high axial and torsionflexibility, large radial and conical rigidity and their inherent damping makethem interesting for being used in primary suspensions, pivot arms and sev-eral types of mechanical linkage. Therefore, due to the increasing interestin predictions performed by multi-body simulations of complete vehicles orsubsystems, it is important to develop simple and effective models to repre-sent the dynamic stiffness of these rubber bushings. Simplicity is necessaryas the rubber bushing represents only a small part connecting componentsof more complex structures.

The main characteristic of rubber materials is their elasticity and capac-ity for dissipating vibration energy which is represented by the hysteresisloop enclosed by the loading and unloading curves in a stress–strain dia-gram. Furthermore, rubber materials are strongly dependent on frequencyand amplitude, as reviewed by Medalia (1). The latter dependence, knownas Fletcher–Gent effect (2), involves a non-linearity in the dynamic stress-strain behavior and it becomes more pronounced with the presence of rubberfillers, generally one of the many kinds of carbon black, as shown by Payneand Whittaker (3). But also this presence produces an increase in the shearmodulus and damping. For that reason, it is important to include the am-plitude dependence in the representation of the dynamic behavior of filledrubber bushings.

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

1

Fourier and Bessel functions (5) or through principal mode approaches (6).Lately, Horton et al. have developed formulas for radial, conical and torsionstatic stiffness (7; 8; 9) based on three-dimensional static elastic theory.Moreover, the static stiffness in many directions is provided in standardtextbooks (10; 11; 12; 13; 14). In addition, the structure-borne propertiesof a long rubber bush mounting in all directions are considered in Kari(15). However, none of these methods takes into account the amplitudedependence.

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

Furthermore, several authors have employed these material models inorder to calculate the dynamic stiffness of rubber bushings. Some of theminsert the models into finite element systems, like Austrell et al. (25; 26; 27)who represent frequency and amplitude dependencies adding integer deriva-tives to stick-slip friction components, similar to Gil–Negrete (28) exceptfor the use of fractional derivatives. However, finite element procedures re-quire a long-time overlay process to calculate the stiffness. In contrast, otherauthors work directly at the component level, like Berg (29) who presentsa five-parameters model which gives a good resemblance to the smoother-friction rubber behavior, also used by Sjoberg and Kari (30) together with arate-dependent part using fractional derivatives, and Misaji et al. (31) whotake into account amplitude dependence as the parameters of an ordinaryKelvin–Voigt model are updated continuously for every oscillation cycle.However, the latter methods neglect the amplitude dependence inside therubber bushing due to non-homogeneous strain states.

2

This work overcomes previous limitations by the development of sim-ple engineering formulas to represent the dynamic stiffness of rubber bush-ings in the frequency domain including amplitude dependence. They areachieved by applying a separable elastic, viscoelastic and friction materialmodel to equivalent strains of the non-homogeneous strain states inside arubber bushing, when it is subjected to torsion, and axial deformations, asshown in Paper A and B respectively, thus giving equivalent shear moduluswhich is inserted into analytical formulas for the stiffness. The novel materialmodel is presented in Paper A and results from applying a sound compo-nent level to the material level. Consequently, unlike other simplified modelsthese formulas include the amplitude dependence due to non-homogeneousstrain states and therefore they become a fast engineering design tool todetermine the most suitable rubber bushing. Moreover, they can be eas-ily employed in multi-body and finite element simulations. Experimentalverification has been carried out in Paper B for the axial stiffness of a com-mercial bushing, showing a good agreement between measured and modelstiffness over a frequency range at several amplitudes. Furthermore, for thetorsion case it is shown in Paper A—by dividing the bushing into severalslices and consequently each equivalent shear modulus is closer to the truevalue—that the approach of working with only one equivalent modulus forthe whole bushing is accurate enough.

2 Objectives

The main purpose of the presented research is to contribute to the work donein the field of modelling rubber isolators by developing simple engineeringdesign models for the dynamic stiffness of rubber bushings including am-plitude and frequency dependence when they are subjected to harmonicdisplacement amplitudes in the 1-100 Hz frequency range.

This objective involves the following tasks:

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

3

• Apply the classical theory of elasticity in order to obtain the strainstates 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 givingequivalent shear moduli that are inserted into engineering formulasfor the stiffness.

• Models implementation in Matlabr.

• Experimental verification on commercially available bushings.

3 Methodology

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

3.1 Theoretical formulation

Static engineering formulas are widely used to predict the behavior of rub-ber bushings. Nevertheless they neglect the dynamic properties of rubberleading to errors in the stiffness prediction. The solution presented in thiswork is based on the idea that the dynamic properties can be introducedinto engineering stiffness formulas by means of a dynamic equivalent mod-ulus which is the result of applying a rubber model to a harmonic strain.However, this modulus depends on the strain amplitude, due to the non-linear relation between stress and strain for rubber. Therefore, in order torepresent the behavior of the whole bushing, a global value for the shearmodulus is required.

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

4

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

3.1.1 Rubber material model

The simplest representation of rubber materials is the Kelvin–Voight modelwhere an elastic frequency independent spring is coupled in parallel with aviscous dashpot, as discussed by Knothe and Grassie (32). However, thismodel strongly overestimates both stiffness and damping for higher frequen-cies. Furthermore, the three-parameter Maxwell model, which is the resultof including a spring in series with the dashpot, represents better the stiff-ness at high frequencies. However, a better representation of the frequencydependence can be obtained by using a summation of Maxwell models, asdiscussed by Lodhia and Esat (33) and Betz (34). Nevertheless, this rep-resentation increases the complexity and number of parameters needed toaccurately describe material properties. An alternative to obtain a good de-scription of the frequency dependence while reducing the required numberof parameters consists of using time derivatives of non-integer order, knownas fractional derivatives, a method used by Sjoberg and Kari (30). However,the fractional derivative models neglect the amplitude dependence, an effectwhich appears when including fillers, such as carbon black, in a rubber mix-ture, thus introducing additional bonds within the material. These bondsexhibit a friction-like behavior as discussed in Ref. (35) and modelling itdemands 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 rub-ber friction behavior, a representation also used by Sjoberg and Kari (30).The material model employed in this work is developed in Paper A byapplying to the material level the component model presented by Sjobergand Kari (30), where the amplitude dependence is represented by a smooth

5

friction component while the frequency dependence is modelled by a frac-tional Kelvin–Voigt model, thus giving an effective model while using onlyfive parameters.

)(t o t a l ts )( te

a,mm2 2/1m a x , es f

Figure 1: Rubber material model.

Elastic stress The first branch in Fig. 1 represents the elasticity by alinear relation between the elastic stress σelast and strain ε

σelast(t) = 2µε(t), (1)

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

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

σfract(t) = mDαε(t), 0 < α ≤ 1, (2)

where σfract is the viscoelastic stress and Dα denotes the fractional timederivative of order α, defined through an analytical continuation of a frac-tional Riemann–Liouville integration (36). Numerically, the viscoelastic

6

component can be evaluated as

σfract(t) ≈ m∆t−α

Γ(−α)

n−1∑

j=0

Γ(j − α)Γ(j + 1)

εn−j , (3)

where εn−j = ε((n − j)∆t), εn is the final strain at time tn = n∆t, ∆t isa constant time step applied in the estimation process and Γ denotes theGamma function (37) defined as

Γ(β) =∫ ∞

1xβ−1 e−xdx +

∞∑

j=0

(−1)j

j!(β + j), β 6= 0,−1,−2, ... (4)

In addition, the temporal Fourier transformation of the fractional timederivative branch leads to

σfract(ω) = m(iω)αε(ω), (5)

which is an expression numerically more easy to deal with while omittingthe Γ function, where (·) represents the temporal Fourier transform, ω theangular frequency and i the imaginary unit.

Frictional stress The third branch in the rubber model represents theamplitude dependence by a smooth friction component which enables a verygood fit to measured curves using only two parameters σfmax and ε1/2. Thefrictional stress σfrict develops gradually following the equation

σfrict(t) = σfs +

[ε(t)− εs

][σfmax − sign(ε) σfs

]

ε1/2

[1− sign(ε) σfs

σfmax

]+ sign(ε)

[ε(t)− εs

] , (6)

where the material parameters σfmax and ε1/2 are the maximum frictionstress developed and the strain needed to develop half of that stress, withsign(ε) denoting the sign of the strain rate. The values of σfs and εs areupdated each time the strain changes direction at ε = 0 as σfs ← σfrict |ε=0

and εs ← ε |ε=0.

7

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

σtotal(t) = σelast(t) + σfract(t) + σfrict(t). (7)

This Equation, with Equations (1), (2) and (6), represents a nonlinear rela-tion between the total stress and the strain and a value of the shear moduluscannot 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 totwo cylindrical metal layers at inner and outer radii a and b and subjected toa harmonic torsion deformation is studied in Paper A while the axial caseis analysed in Paper B. The classical linear theory of elasticity provides

Ø 2 a

Ø 2 b

jzur

qur

rur

L

q

r

z

xy

z

Figure 2: Rubber cylindrical bushing subjected to torsion deformation.

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

8

constitutive equations between the stress and strain have only elastic andfrequency dependence components, as the amplitude dependence is takeninto account later on when the rubber model is applied to the equivalentstrains. Furthermore, the strain states with torsion and axial deformationshave only one shear component and therefore the constitutive equation inboth cases reads

σt/a(t) = 2µεt/a(t) + mDαεt/a(t), (8)

where σt/a is the torsion/axial shear stress and εt/a is the torsion/axial shearstrain. In addition, working in the frequency domain as result of applying thetemporal Fourier transform makes calculations easier because the derivationdisappears and the total stress is proportional to the strain

σt/a(ω) =[2µ + m(iω)α

]εt/a(ω) = 2µ(ω)εt/a(ω). (9)

Secondly, the rubber is assumed homogeneous and isotropic and displace-ment gradients are considered small enough to apply the theory of elastic-ity. Furthermore, over the frequency range of interest wave effects withinthe rubber bushing are negligible. Consequently, when the bushing is sub-jected to a harmonic torsion angle at the outer surface relative to the innerϕ(t) = ϕ sin(ωst) with ωs the excitation frequency and ϕ the amplitude, thestrain state reads

εt(r, ω) =b2a2

b2 − a2

1r2

ϕ(ω), (10)

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

εa(r, ω) =1

2r loge(ba)

d(ω). (11)

Subsequently, in order to calculate the equivalent strains one idea would beto perform volume averages of the strain states. However, in the torsion andaxial deformation modes the strain states have only one shear component,but in other deformation modes it might be quite complex as it happens inthe radial case where there are both shear and traction/compression strains

9

inside the bushing, consequently performing a volume average of the strainstate is not a general methodology that could be extended to other defor-mation modes. Therefore, the solution here presented consists in workingwith equivalent strains obtained from energy density balances between thetorsion and axial deformation in a bushing and a simple shear specimen. Forinstance, as shown in Paper A the deformation energy in the torsion caseis equal to the distortion energy because the volume remains constant. Fur-thermore, the temporal Fourier transform of the volume averaged distortionenergy reads

Ud bushing ∝ 1V

∫σ∗t εtdV =

1V

∫2µ|εt|2dV, (12)

where V = π(b2 − a2)L is the volume of the bushing and µ is the frequencydependent shear modulus presented in Equation (9). In addition, the tem-poral Fourier transform of the distortion energy density in case of a simpleshear specimen made of the same material as the rubber bushing is

Ud specimen ∝ 2µε∗shεsh = 2µ|εsh|2, (13)

where εsh is the temporal Fourier transform of the homogeneous shear straininside the specimen. Consequently, the balance between both energy densi-ties gives the equivalent strain in the frequency domain which is transformedinto the time domain by applying the inverse Fourier transform

εequiv t(t) =b a

b2 − a2ϕ(t). (14)

Moreover, if the same methodology is conducted for the axial case the equiv-alent strain becomes

εequiv a(t) =d(t)√

2(b2 − a2) loge(ba)

ϕ(t). (15)

3.1.3 Engineering formulas for the stiffness

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

10

made up of three components: elastic, frequency dependent and frictionalstresses. Furthermore, the last component introduces the amplitude depen-dence and makes the rubber model nonlinear.

In order to switch to the frequency domain the excitation frequencyof the input signals ϕ(t) and d(t) is varied as ωs = s∆ω with ∆ω a con-stant frequency step and s ∈ N, and therefore s different equivalent strainsεsequiv t/a(t) and their corresponding total stresses σs

total t/a(t) are obtained.Furthermore, the temporal Fourier transform is applied to those signals,thus giving the frequency functions εs

equiv t/a(ω) and σstotal t/a(ω) which are

evaluated at ωs leading to the equivalent complex moduli in the frequencydomain

µequiv t/a(ωs) =σs

total t/a(ωs)

2 εsequiv t/a(ωs)

. (16)

This linearization process takes into account the non-linear relation betweenstress and strain at frequency ωs considering the first order response whileomitting the less important overtones (stress response at 3ωs, 5ωs,...). Fi-nally, the moduli are inserted into the stiffness formulas calculated previouslywith the classical linear theory of elasticity, thus giving a formula for thetorsion dynamic stiffness

Kdyn torsion(ωs) =4πa2b2L

b2 − a2µequiv t(ωs), (17)

while the axial stiffness reads

Kdyn axial(ωs) =2πL

loge(ba)

µequiv a(ωs). (18)

Consequently, two engineering formulas to represent in the frequency domainthe dynamic behavior of rubber bushings in the torsion and axial directionsincluding the amplitude dependence have been achieved.

3.2 Experiments

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

11

commercially available bushing in Paper B. Firstly, the material parame-ters involved in the formula of the stiffness, see Equation (18), are achievedfrom two measurements: elastic and amplitude dependence parameters areobtained from a large deformation quasi-static test while the viscoelasticparameters are determined by a steady state harmonic dynamic test per-formed 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 areconducted in order to demonstrate that the formula for the axial stiffnesspredicts well the bushing behavior including the amplitude and frequencydependence.

3.2.1 Test object

The test object is a commercial bushing manufactured by Trelleborg AVSunder 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 carbonblack filler is a combination of N660 and N774.

a = 2 0 m mb = 3 5 m m

L=90

mm

Figure 3: Test object.

12

3.2.2 Setup

The experiments are performed in a servo-hydraulic test machine consistingof a lower body with a hydraulic piston inside, two columns to the sidesand an upper rigid crosshead, as shown in Fig. 4. Measurements are per-formed with the inner surface of the bushing fixed and joined to the uppercrosshead through a shaft and one load sensor, while the outer surface of thebushing is subjected to axial displacements produced by the lower hydraulicpiston. The motion of the bushing is measured by two displacement sensorssymmetrically positioned on the outer surface of the bushing. The electricalsignals are conditioned in a 6-channels amplifier producing electrical inputsto the frequency analyzer which are processed subsequently in a personalcomputer. In order to achieve constant amplitude, the electric signal thatcomes from the displacement sensor of the test machine and goes throughthe amplifier is automatically adjusted in a control loop in the analyzer tosupply the exciter.

3.2.3 Type of measurements

The test component is mechanically conditioned before measurements toeliminate influence of the Mullins’ effect (38) which is shown when previ-ously unstrained rubber is subjected to strain cycles at constant peak valuereducing peak stress values at the first few oscillations.

Quasi-static test Measurements at 0.1 Hz and 1 mm displacement ampli-tude are used to fit the elastic and amplitude dependence parameters. Thelarge deformation assures that all friction has been developed in the loopand 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 amaximum 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 imple-mented in Matlabr, thus giving a fast engineering design tool to predictthe dynamic behavior of rubber components. Furthermore, the calculationof the torsion stiffness for a typical commercial bushing is carried out inPaper A, thus showing the expected behavior for a rubber bushing, whilePaper B presents an experimental verification for the axial stiffness per-formed on a commercially available bushing.

4.1 Implementation in Matlabr

The implementation in Matlabr is carried out for each deformation modeas follows: firstly, the equivalent strains in the time domain are calculated

14

as function of the geometry of the bushing and the boundary conditions, asshown in Equations (14) and (15). Secondly, the rubber model is applied tothose strains and the steady state of the resultant stresses are converted intofrequency functions through the temporal Fourier transform. Furthermore,the value of the frequency dependent shear moduli result from dividing thestress frequency functions by the strain frequency signals when both func-tions are evaluated at the excitation frequency, see Equation (16). In addi-tion, the amplitude of the displacement excitations is varied, as well as, thefrequency range that goes from 1 to 100 Hz, to check how well the modelrepresents both dependencies. Finally, the frequency and amplitude depen-dent 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 bush-ing.

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 Nsα/m2

b = 11.0× 10−3 m α = 0.40σfmax = 4.0× 103 N/m2

ε1/2 = 1.0× 10−3

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

15

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

0

10

20

30

40

50

Mag

nitu

de (

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 1000

0.1

0.2

Frequency (Hz)

Loss

Fac

tor

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 atthree 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 com-ponents, a procedure commonly used in engineering design, may induceerrors. Therefore, the developed engineering formulas become a fast andeffective tool to predict the behavior of rubber bushings including frequencyand amplitude dependence.

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

16

0

10

20

30

40

50M

agni

tude

(N

m/r

ad)

20Hz50Hz80Hz

10−4

10−2

0

0.1

0.2

Amplitude (radians)

Loss

Fac

tor

20Hz50Hz80Hz

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

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

The dynamic stiffness is the result of an iterative process which startswith an estimation of the torsion angles at the inner and outer radius of eachslice to, subsequently calculate the complex moment at each radius followingthe methodology explained previously. Furthermore, in order to fulfil the

17

continuity of the moment along the radius, the values of the torsion anglesare 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 commerciallyavailable bushing in Paper B. Calculations and graphical representationsare carried out in Matlabr. Elastic and amplitude dependence parametersare varied until the model hysteresis loop enclosed by the resultant forcein the bushing when is subjected to axial displacement at 0.1 Hz and 1mm 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 achievedwith the values: µ = 2.42 × 106 N/m2, σfmax = 1.92 × 104 N/m2 andε1/2 = 4.0× 10−3.

Moreover, viscoelastic parameters are obtained from the error minimiza-tion iteration process between the complex stiffness obtained with one dy-namic test performed from 1 to 100 Hz at 0.1 mm constant amplitude andthe stiffness calculated by Equation (18) at the same conditions. In par-ticular, when the viscoelastic parameters are assigned the values: m =6.0 × 105 Nsα/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 ofthe model when using the five material parameters calculated previously,displaying a good agreement over the whole frequency range, as presentedin Figs. 9 and 10, which verifies the accuracy of Equation (18) representingthe axial dynamic stiffness of a rubber bushing including amplitude andfrequency dependence.

18

−1.5 −1 −0.5 0 0.5 1

x 10−3

−3000

−2000

−1000

0

1000

2000

3000

Displacement (m)

For

ce (

N)

ModelMeasurement

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

19

0

1

2

3

4

5

x 106

Mag

nitu

de (

N/m

)

ModelMeasurement

0 20 40 60 80 1000

0.05

0.1

0.15

Frequency (Hz)

Los

s Fa

ctor

ModelMeasurement

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

20

0

2

4

6

Mag

nitu

de (

kN/m

m)

Model 0.04mmMeas. 0.04mmModel 0.1mmMeas. 0.1mmModel 0.35mmMeas. 0.35mm

0 20 40 60 80 1000

0.05

0.1

0.15

Frequency (Hz)

Los

s Fa

ctor

Model 0.04mmMeas. 0.04mmModel 0.1mmMeas. 0.1mmModel 0.35mmMeas. 0.35mm

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

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3

3.5

4

4.5

5

5.5

6

Mag

nitu

de (

kN/m

m)

Model 10HzMeas. 10HzModel 20HzMeas. 20HzModel 40HzMeas. 40Hz

0.05 0.1 0.15 0.2 0.25 0.3 0.350.08

0.1

0.12

0.14

0.16

0.18

Amplitude (mm)

Los

s Fa

ctor

Model 10HzMeas. 10HzModel 20HzMeas. 20HzModel 40HzMeas. 40Hz

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

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5 Conclusions

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

Paper A presents the model for the torsion stiffness and illustrates theresults in magnitude and loss factor at several amplitudes over a frequencyrange from 1 to 100 Hz when the model is applied to typical geometricand material data of rubber components, showing the expected behaviorof a rubber bushing. Furthermore, in order to check the accuracy of theapproach the same bushing has been divided into several slices thus usingan equivalent shear modulus closer to the true value for each slice. Thesmall differences in magnitude and loss factor between the dynamic stiffnessobtained when the bushing is divided into a single, two and three slices, showthat the amplitude dependence is enough well modelled while using a singleslice. In addition, an experimental verification of the axial stiffness modelis carried out on a commercially available bushing in Paper B. Firstly, themodel parameters are obtained from two tests: a quasi-static and dynamic.Subsequently, five new dynamic experiments are performed on the bushingto compare the results with those obtained with the engineering formulaat the same amplitude and frequency conditions, showing good agreementswhich verifie the accuracy of the model.

The simplicity of this model is very convenient for the dynamic analysisof complex structures in which rubber bushings are connecting components.Several advantages can be outlined. Firstly, a novel model has been pre-sented to represent the dynamic behavior of the rubber at the material levelincluding frequency and amplitude dependence with only five parameters.Secondly, the time invested in performing the calculations: Matlabr hasbeen 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 othersimplified models these formulas depend on the geometry of the bushingand the material properties and therefore they are useful and fast predic-tion tools to determine the most suitable rubber bushing to connect to otherstructures 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 mysupervisor Leif Kari for excellent and always encouraged guidance throughthe course of this project. My thanks also to Kent Lindgren and DaniloPrelevic for professional assistance in my experiments and to my colleaguesat MWL. Finally I thank my family and friends for being always there.

References

[1] Medalia, A. I. Effects of carbon black on dynamic properties of rub-ber. Rubber Chem. Tech., 1978, 51, 437–523.

[2] Fletcher, W. P. and Gent, A. N. Non-Linearity in the dynamicproperties of vulcanised rubber compounds. I.R.I. Transactions , 1953,29, 266–280.

[3] Payne, A. R. and Whittaker, R. E. Low strain dynamic propertiesof filled rubbers. Rubber Chem. Tech., 1971, 44, 440–478.

[4] Morman, K. N. and Pan, T. Y. Application of finite-element analy-sis in the design of automotive elastomeric components. Rubber Chem.Tech., 1988, 61, 503–533.

24

[5] Hill, J. M. Radical deflections of rubber bush mountings of finitelengths. Int. J. Eng. Sci., 1975, 13, 407–422.

[6] Adkins, J. E. and Gent, A. N. Load-deflexion relations of rubberbush mountings. Brit. J. Appl. Phys., 1954, 5, 354–358.

[7] Horton, J. M., Gover, M. J. C. and Tupholme, G. E. Stiffnessof rubber bush mountings subjected to radial loading. Rubber Chem.Tech., 2000, 73, 253–264.

[8] Horton, J. M., Gover, M. J. C. and Tupholme, G. E. Stiffnessof rubber bush mountings subjected to tilting deflection. Rubber Chem.Tech., 2000, 73, 619–633.

[9] Horton, J. M., Gover, M. J. C. and Tupholme, G. E. Stiffnessof spherical rubber bush mountings. Int. J. Solids Struct., 2005, 42,3289–3297.

[10] Lindley, P. B. Engineering design with natural rubber , 1992 (TheMalaysian Rubber Producers’ Research Association, Brickendonbury).

[11] Gent, A. N. Engineering with rubber , 1992 (Carl Hansen Verlag, Mu-nich).

[12] Gobel, E. F. Rubber springs design, 1974 (Newnes-Butterworths, Lon-don).

[13] Payne, A. R. and Scott, J. R. Engineering design with rubber , 1960(Interscience Publishers, New York).

[14] Freakley, P. K. and Payne, A. R. Theory and practice of engineeringwith rubber , 1978 (Applied Science Publishers, London).

[15] Kari, L. Dynamic stiffness matrix of a long rubber bush mounting.Rubber Chem. Tech., 2002, 75(4), 747–770.

[16] Kraus, G. Mechanical losses in carbon black filled rubbers. J. Appl.Polym. Sci., 1984, 39, 75–92.

25

[17] Ulmer, J.D. Strain dependence of dynamic mechanical properties ofcarbon black-filled rubber compounds. Rubber Chem. Tech., 1998, 69,637–667.

[18] Lion, A. Phenomenological modelling of strain-induced structuralchanges in filler-reinforced elastomers. Kautsch. Gummi Kunstst., 2005,58(4), 157–162.

[19] Gregory, M.J. Dynamic properties of rubber in automotive engineer-ing. Elastomerics, 1985, 117(11), 19–24.

[20] Coveney, V. A., Johnsson, D. E. and Turner, D. M. A triboelasticmodel for the cyclic behavior of filled vulcanizates. Rubber Chem. Tech.,1995, 68, 660–670.

[21] Kaliske, M. and Rothert, H. Constitutive approach to rate-independent properties of filled elastomers. Int. J. Solid. Struct., 1998,35(17), 2057–2071.

[22] Bruni, S. and Collina, A. Modelling of viscoelastic behaviour of elas-tomeric components : An application to the simulation of train-trackInteraction. Vehicle Syst. Dyn., 2000, 34, 283–301.

[23] Olsson, A. K. and Austrell, P. E. 2001, A fitting procedure for aviscoelastic-elastoplastic material model. Second European Conferenceon Constitutive Models for Rubber.

[24] Brackbill, C.R., Lesieutre, G.A., Smith, E.C. and Ruhl, L.E.Characterization and modelling of the low strain amplitude and fre-quency dependent behavior of elastomeric damper materials. J. Am.Helicopter Soc. , 2000, 45(1), 34–42.

[25] Austrell, P. E., Olsson, A. K. and Jonsson, M. 2001, A Methodto analyse the non-Linear dynamic behaviour of rubber componentsusing standard FE codes, Paper no 464. Conference on Fluid and SolidMechanics.

26

[26] Olsson, A. K. and Austrell, P. E. 2003, Finite element analysis ofa rubber bushing considering rate and amplitude dependence effects.Third European Conference on Constitutive Models for Rubber.

[27] Olsson, A. K.,Austrell, P. E. and Sandberg, G. 2005, Approx-imate viscoelastic FE procedures in frequency and time domain toaccount for the Fletcher-Gent effect. Fourth European Conference onConstitutive Models for Rubber.

[28] Gil-Negrete, N. 2004, On the modelling and dynamic stiffness pre-diction of rubber isolators. University of Navarra. Spain.

[29] Berg, M. A Non linear rubber spring model for rail vehicle dynamicsanalysis. Vehicle Syst. Dyn., 1998, 30, 197–212.

[30] Sjoberg, M. and Kari, L. Non-linear behavior of a rubber isolatorsystem using fractional derivatives, Vehicle Syst. Dyn., 2002, 37, 217–236.

[31] Misaji, K., Hirose, S. and Shibata, K. vibration analysis of rubbervibration isolators of vehicle using the restoring force model of powerfunction type, Jsme Int. J. C-Dyn. Con., 1995, 38(4), 679–685.

[32] Knothe, K.L. and Grassie, S.L. Modelling of railway track and vehi-cle/track interaction at high frequencies, Vehicle Syst. Dyn., 1993, 22,209–262.

[33] Lodhia, B.B. and Esat, I.I. Vibration simulation of systems incorpo-rating linear viscoelastic mounts using Prony series formulation, ASME,Engineering system design and analysis conference, 1996, 81(9), 171–176.

[34] Betz, E. and Spanier, J. Spring and dashpot models and their appli-cations in the study of the dynamic properties of rubber, Mechanicaland Chemical Engineering Transactions, 1996.

27

[35] Sjoberg, M.Rubber isolators - measurements and modelling usingfractional derivatives and friction, SAE PAPER No. 2000-01-3518 ,2000.

[36] Oldham, K. B. and Spanier, J. The Fractional Calculus, 1974 (NewYork and London: Academic press).

[37] Abramowitz, M. and Stegun., I. A. Handbook of MathematicalFunctionss, 1972 (New York: Dover Publication).

[38] Mullins, L. Softening of rubber by deformation. Rubber Chem. Tech.,1969, 42, 339–362.

[39] Spencer, A. J. M. Continuum Mechanics, 1980 (New York: LongmanPublishers).

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