International Journal of Mechanical Sciences 60 (2012) 54–58

Contents lists available at SciVerse ScienceDirect

International Journal of Mechanical Sciences

0020-74

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

The frequency, amplitude and magnetic field dependent torsional stiffnessof a magneto-sensitive rubber bushing

Peter Blom, Leif Kari n

KTH Royal Institute of Technology, The Marcus Wallenberg Laboratory for Sound and Vibration Research, 100 44 Stockholm, Sweden

a r t i c l e i n f o

Article history:

Received 10 July 2006

Received in revised form

16 April 2012

Accepted 27 April 2012Available online 6 May 2012

Keywords:

Audio frequency

Magneto-sensitive

Non-linear

Rubber isolator

03/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.ijmecsci.2012.04.006

esponding author. Tel.: þ46 8 7907974; fax:

ail address: leifkari@kth.se (L. Kari).

a b s t r a c t

A dynamic torsional stiffness model of a magneto-sensitive circular annular rubber bushing is

presented where influences of frequency, amplitude and magnetic field dependence are included. This

is achieved by employing a newly developed non-linear magneto-sensitive audio frequency constitu-

tive equation in an engineering formula for the torsional stiffness of a rubber bushing. The engineering

stiffness formula predicts the frequency and amplitude dependent stiffness in a simple way, based on

geometric dimensions and the shear modulus. The shear modulus is provided by the rubber model. The

results from these calculations predict and clearly display the possibility of controlling the magneto-

sensitive rubber bushing stiffness over a large frequency range, through the application of a

magnetic field.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Rubber bushings are traditional and effective isolation compo-nents in myriad engineering applications subjected to shocks andvibrations. Their utility stems from the fact that they act partly asmirrors reflecting energy backwards and partly as damperstransforming vibration energy into heat; thus hindering thepropagation of vibrations. However, their performances are verydependent on conditions; they are both frequency and amplitudedependent [1]. Typically, a vehicle for instance, is subjected tovibrations of largely varying frequencies whereas isolators arenormally optimized with respect only to low frequency require-ments. Hence, the high frequency properties are liable to bepoor over large regions. The ability to reversibly, rapidly andeffectively control the properties of a component would providean effective means to meet high frequency requirements too, thusreducing noise and vibrations which is a sought-after quality inindustrial applications. This is in fact the envisaged benefit ofmagneto-sensitive rubber; the material has proven to possess thecharacteristics suitable to the purpose; while strongly magneto-sensitive—reacting strongly, rapidly and reversibly to an exter-nally applied field—it retains the classical rubber properties forwhich rubber was originally chosen. The application of a magneticfield gives rise to a magnetic dipole–dipole interaction betweenthe iron particles causing the apparent increase in stiffness.

While some existing rubber models include amplitude depen-dence [2–4] none of them includes magneto-sensitivity which is

ll rights reserved.

þ46 8 7906122.

the case also for general bushing models [5–8]. Conversely, eventhough recent years have witnessed a steep rise in the numberof research reports on magneto-sensitive fluids and solidsalike [9–35], a large number focuses merely on the quasi-staticproperties [10–23] often excluding the amplitude dependence.Nevertheless, in recent publications [24–35] the amplitudedependence is revealed to be an important feature for magneto-sensitive rubber even at small strains. Moreover, the magneticsensitivity is shown to be strongly amplitude dependent, which iswhy the inclusion of such effects in a model is essential toaccurately reflect the physical phenomena present. In this work,an existing, however within an application’s context, yet unusedconstitutive model [36], treating all the mentioned effects is usedin combination with an engineering formula [37] to predict thedynamic stiffness of a magneto-sensitive rubber torsional isolator.Indeed, the results predict, and clearly display, the possibility ofcontrolling the magneto-sensitive rubber bushing stiffness over alarge range through the application of a magnetic field.

2. Model

In order to arrive at an engineering formula for the dynamicstiffness in the frequency domain of a magneto-sensitive circularannular rubber bushing with inner radius a, outer radius b andlength L, subjected to a torsion angle jðtÞ, an equivalent isolatorshear strain is first calculated [37]. The classical linear theory ofelasticity gives the shear strain

gryðtÞ ¼2b2a2

b2�a2

1

r2jðtÞ, ð1Þ

P. Blom, L. Kari / International Journal of Mechanical Sciences 60 (2012) 54–58 55

where t signifies time and the subscripts r and y denote cylind-rical coordinate directions. From this an average strain can becalculated through a volume average

gryðtÞ ¼4b2a2jðtÞðb2�a2Þ

2loge

b

a

� �: ð2Þ

This formula has been shown [37] to be satisfactorily accurate forwide and thin bushings yielding a very small error. The magneto-sensitive rubber bushing can be viewed in Fig. 1.

Before proceeding to the constitutive relations, certainassumptions need first to be made concerning the material. Themagneto-sensitive rubber is homogeneous at the length scaleconsidered, isotropic at no applied magnetic field and non-ageingwhile confined to isothermal conditions at room temperature.Furthermore, the isolator is subjected only to small strainsexcluding finite deformations, as the small strain audio frequencyapplications at no static prestrain are of interest. Upon applicationof a magnetic field perpendicular to the shear direction—theoptimal direction for as large a change as possible of the shearmodulus in the shear direction—the material becomes anisotro-pic as the shear modulus is dependent on both strength anddirection of the magnetic field.

The subsequent step consists in introducing a constitutiverelation between stress t and strain g [36]. The total stressresponse is assumed to be additively decomposable into threeparts depending on time t,

t¼ teþtveþtf : ð3Þ

The elastic stress teðtÞ is linearly related to the instantaneousengineering strain gðtÞ through the following relation:

te ¼ Geg, ð4Þ

where Ge is the elastic shear modulus.The viscoelastic stress tveðtÞ is linearly related to the history of

the strain rate [38],

tve ¼m

Gð1�aÞd

dt

Z t

�1

gðsÞðt�sÞa

ds, 0oar1, m40, ð5Þ

where G is the gamma function and a and m are materialconstants.

Cylindrical outer iron shell subject to a torsion angle

Iron rod

Magneto-sensitive rubber

Magnetic flow

Fig. 1. Picture of the axially symmetric magneto-sensitive rubber bushing.

The friction stress tf ðtÞ is nonlinearly related to the strain[2,39]

tf ¼ tfsþg�gs

g1=2 1�signð _gÞ tfs

tfmax

� �þsignð _gÞ½g�gs�

½tfmax�signð _gÞtfs�,

ð6Þ

where the maximum friction stress developed tfmax and g1=2 aremodel constants with signð _gÞ yielding the direction of the dis-placement. The parameters tfs and gs are updated each time thereis a change in shear direction at _gs ¼ 0 as tfs’tf 9 _g ¼ 0 andgs’g9 _g ¼ 0.

Magnetic sensitivity is introduced in connection with g1=2,tfmax and Ge according to the following relations:

Ge ¼ 1þM

Ms

� �2

d1

" #Ge0, ð7Þ

g1=2 ¼g1=20

1þM

Ms

� �2

d2

, ð8Þ

tfmax ¼ 1þM

Ms

� �2

d3

" #tfmax0, ð9Þ

where M represents magnetization perpendicular to the sheardirection, Ms saturation magnetization and d1, d2 and d3, real andpositive material constants. The parameters Ge0, g1=20 and tfmax0

represent respectively the zero state values. The reader is referredto Blom and Kari [36] for details regarding the magnetic fielddependence of the constitutive model. Isolator dimensions will bechosen in such a way that an approximately homogenous mag-netic field can be assumed across the isolator.

Introducing the equivalent shear strain in the constitutiveequations yields an equivalent stress response. Taking the Fouriertransform of both stress and strain, transformed quantities aredenoted by ~ð�Þ, and linearizing, that is dropping overtones in thestress response, an expression for the strain averaged frequencyand amplitude dependent shear modulus is obtained,

G ¼~tðoÞ~gðoÞ

: ð10Þ

From this relation the torsional dynamic stiffness of themagneto-sensitive rubber isolator can be calculated as

kdyn ¼4pLa2b2GðoÞ

b2�a2

: ð11Þ

Since the shear modulus derives from a strain average thatyields negligible errors for thin (9b�a9 small) and wide (9aþb9=2large) bushings, the formula becomes more accurate for suchdimensions. Furthermore, the same condition holds for theassumption of a homogenous magnetic field across the isolator.

3. Results

The constitutive equation parameters are assigned values thatare displayed in Table 1. These have previously been successfullyoptimized to fit experimental results for magneto-sensitive rub-ber consisting of natural rubber containing 33% iron [36]. Isolatordimensions are ascribed values that can be viewed in Table 2. It isverified that b�aol=6, where l is the wavelength for the highestfrequency 1000 Hz. This is in order to avoid wave effects.Accordingly, the loss factor is equal to Imag kdyn=Real kdyn.Furthermore, the values of a and b are chosen to permit ahomogenization of the magnetic field across the bushing (herea/b¼0.8). Moreover, this ratio grants a very small error in the

Table 1Parameter values.

Parameters Values Units

Ge,0 1.35 MN m�2

m 0.12 MN m�2 sa

a 0.36

g1=20 0.00103

tfmax0 0.9 kN m�2

d1 2.22

d2 6.94

d2 1.53

m0Ms 0.6 T

Table 2Isolator dimensions.

Dimensions Values Units

L 0.025 m

a 0.02 m

b 0.025 m

102 1030

500

1000

1500

2000

2500

3000

3500Torsional angle amplitude 0.000019 rad

Frequency (Hz)

Mag

nitu

de (N

m/ra

d)

0.8T0.55T0.3T0T

Fig. 2. Torsion stiffness magnitude versus frequency at induced magnetic field of

0, 0.3, 0.55 and 0.8 T and amplitude of 1:9� 10�5 rad.

1031020

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Torsional angle amplitude 0.000019 rad

Loss

fact

or

Frequency (Hz)

0.8T0.55T0.3T0T

Fig. 3. Torsion stiffness loss factor versus frequency at induced magnetic field of 0,

0.3, 0.55 and 0.8 T and amplitude of 1:9� 10�5 rad.

102 1030

500

1000

1500

2000

2500

3000

3500Torsional angle amplitude 0.000048 rad

Frequency (Hz)

Mag

nitu

de (N

m/ra

d)

0.8T0.55T0.3T0T

Fig. 4. Torsion stiffness magnitude versus frequency at induced magnetic field of

0, 0.3, 0.55 and 0.8 T and amplitude of 4:8� 10�5 rad.

102 1030

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Torsional angle amplitude 0.000048 rad

Frequency (Hz)

Loss

fact

or

0.8T0.55T0.3T0T

Fig. 5. Torsion stiffness loss factor versus frequency at induced magnetic field of 0,

0.3, 0.55 and 0.8 T and amplitude of 4:8� 10�5 rad.

P. Blom, L. Kari / International Journal of Mechanical Sciences 60 (2012) 54–5856

average strain measure since it becomes negligible for wide andthin bushings. Introducing these values into the equations yieldsthe dynamic stiffness that can be viewed in Figs. 2–9. Thedifferent aspects that need to be captured to aptly predict themagneto-sensitive rubber isolator behaviour in this frequencyrange are all treated; this can be realized by studying the graphs.The material behaves viscoelastically for all amplitudes as theisolator becomes stiffer with frequency and the loss factor growsaccordingly. The dependence upon the magnetic field is reflectedin the results through the fact that the stiffness can be seen toincrease with the magnetic field. The amplitude dependence ontorsional angle amplitude, yielding a higher stiffness the smallerthe amplitude and vice versa, can be viewed by comparing curvesat different amplitudes and no applied magnetic field, shown inthe zero field curves in Figs. 2 and 8. Although the amplitudesappear to be very small, they nevertheless may give rise to highsound pressure levels. Yet another phenomenon present in theresults is the magnetic sensitivity dependence on torsional angleamplitude. This can be observed in the graphs by noting that thesmaller the amplitude the larger the stiffness change withmagnetic field. The calculations behind the results are straightfor-ward and well established, and should therefore yield physically

sound results in a computationally rapid manner. Furthermore,the results highlight the possibility of controlling the magneto-sensitive rubber bushing stiffness over a large frequency range,

102 1030

500

1000

1500

2000

2500

3000Torsional angle amplitude 0.00015 rad

Frequency (Hz)

Mag

nitu

de (N

m/ra

d)

0.8T0.55T0.3T0T

Fig. 6. Torsion stiffness magnitude versus frequency at induced magnetic field of

0, 0.3, 0.55 and 0.8 T and amplitude of 1:5� 10�4 rad.

102 1030.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Torsional angle amplitude 0.00015 rad

Frequency (Hz)

Loss

fact

or

0.8T0.55T0.3T0T

Fig. 7. Torsion stiffness loss factor versus frequency at induced magnetic field of 0,

0.3, 0.55 and 0.8 T and amplitude of 1:5� 10�4 rad.

102 1030

500

1000

1500

2000

2500

3000Torsional angle amplitude 0.00048 rad

Frequency (Hz)

Mag

nitu

de (N

m/ra

d)

0.8T0.55T0.3T0T

Fig. 8. Torsion stiffness magnitude versus frequency at induced magnetic field of

0, 0.3, 0.55 and 0.8 T and amplitude of 4:8� 10�4 rad.

102 1030.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Torsional angle amplitude 0.00048 rad

Loss

fact

or

Frequency (Hz)

0.8T0.55T0.3T0T

Fig. 9. Torsion stiffness loss factor versus frequency at induced magnetic field of 0,

0.3, 0.55 and 0.8 T and amplitude of 4:8� 10�4 rad.

P. Blom, L. Kari / International Journal of Mechanical Sciences 60 (2012) 54–58 57

through the application of a magnetic field. In fact, for the smalleramplitudes, an almost 100% increase in stiffness is obtainable; forexample making it possible to rapidly and adaptively increase the

resonance frequency of a simple mass–spring system with morethan 40%. This opens up exciting possibilities for developing anew generation of bushings consisting of magneto-sensitiverubber, capable of combatting unwanted noise and vibrationsvery effectively, granting among other things smoother rides andless harshness on vehicles.

4. Conclusions

A dynamic torsional stiffness model of a magneto-sensitiverubber bushing is presented. It includes influences of frequency,amplitude and magnetic field dependence. This is achieved byemploying a newly developed non-linear magneto-sensitive audiofrequency rubber model in an engineering formula for the torsionalstiffness of a rubber bushing. The engineering formula yields thefrequency and amplitude dependent stiffness based on geometricdimensions and the shear modulus. The latter is provided from thenon-linear audio frequency constitutive equation for magneto-sensitive rubber. The results from these calculations predict, andclearly display, the possibility of controlling the magneto-sensitiverubber bushing stiffness over a large range through the applicationof a magnetic field.

Acknowledgements

The financial support provided from the Swedish ResearchCouncil is gratefully acknowledged (Contract no: 621-2002-5643).

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