Quasi-steadyBinghamBiplasticAnalysisofElectrorheological...

Quasi-steady Bingham Biplastic Analysis of Electrorheologicaland Magnetorheological Dampers

GLEN A. DIMOCK,* JIN-HYEONG YOO AND NORMAN M. WERELEY

Smart Structures Laboratory, Department of Aerospace Engineering, University of Maryland, College Park MD 20742 USA

ABSTRACT: Electrorheological (ER) and magnetorheological (MR) fluids are characterizedby an increase in dynamic yield stress upon application of a magnetic field. The Binghamplastic model has proven useful in modeling flow mode dampers utilizing ER and MR fluids.However, certain MR fluids can exhibit shear thinning behavior, wherein the fluid’s apparentplastic viscosity decreases at high shear rates. The Bingham plastic model does not account forsuch behavior, resulting in overprediction of equivalent viscous damping. We present aBingham biplastic model that can account for both shear thinning and shear thickeningbehaviors. This approach assumes a bilinear postyield viscosity, with a critical shear ratespecifying the region of high shear rate flow. Furthermore, the model introduces non-dimensional terms to account for the additional parameters associated with shear thinning andthickening. A comparison is made between Bingham plastic and Bingham biplastic forceresponses to constant velocity input, and equivalent viscous damping is examined with respectto nondimensional parameters.

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INTRODUCTION

MAGNETORHEOLOGICAL (MR) fluids and electro-rheological (ER) fluids have been proposed for a

wide variety of engineering devices requiring semiactivedamping (Carlson et al., 1999). These fluids, demon-strated to be qualitatively similar, are characterized by afield-dependent yield stress (Weiss et al., 1994). Becauseof their field-dependent properties, ER and MR fluidshave been utilized in a number of control studies toreduce transmissibility in civil structures (Spencer et al.,1998; Hiemenz and Wereley, 1999), automotive systems,and other dynamical systems (Sharp and Hassan, 1986;Jeon et al., 1999). This paper will focus on MR fluids,which respond to a magnetic field and have achievedgreater commercial success in practical devices than theirER counterparts (Stanway et al., 1996; Jolly et al., 1999).

Magnetorheological fluids are known to exhibit anumber of nonlinear phenomena. A wide variety ofnonlinear models have been used to characterize MRfluids and devices, including the Bingham plastic model(Phillips, 1969), biviscous models (Stanway et al., 1996),hysteretic biviscous models (Wereley et al., 1998), andmechanism-based models (Kamath and Wereley, 1997;Sims et al., 1999). One nonlinear phenomenon, shearthinning, refers to the reduction in apparent viscosity athigh shear rates (Jolly et al., 1999) and has traditionally

been modeled with power-law or exponential functions(Wolff-Jesse and Fees, 1998; Wang and Gordaninejad,1999). These models are complex, and a less complexalternative is to extend the widely-accepted Binghamplastic model.

The Bingham plastic model has been successfully usedin previous studies to model field dependent rheologicalfluids (Block and Kelly, 1988; Kamath et al., 1996). Thismodel considers a fluid with a dynamic yield stress,beyond which a linear plastic viscosity is observed.When applied to flow between two parallel plates, theBingham plastic model predicts a region of fluid thatdoes not shear, having not reached the dynamic yieldstress. Approaching the walls, the fluid shear rateincreases as the shear stress increases. For high rodvelocity and small gap size, this shear rate can reachvalues known to cause shear thinning (Jolly et al., 1999).The Bingham plastic model predicts a constant plasticviscosity for all shear rates, but this assumption can beinaccurate at high shear rates.

This paper proposes a modified Bingham plasticmodel to account for shear thinning. Rather than fit anexponential curve to the postyield viscosity, the Binghambiplastic approach considers biplastic postyield beha-vior. This approach, while clearly a simplification, maybe accurate enough to model MR fluid in mostsituations. Furthermore, the resulting model is simpleenough to readily compare to the Bingham plastic case.For the new model, two parameters are added to theBingham plastic model. Shear thinning is assumed to

*Author to whom correspondence should be addressed.

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 00—November 2002 1

1045-389X/02/00 0001–10 $10.00/0 DOI: 10.1106/104538902030906� 2002 Sage Publications

+ [6.11.2002–3:43pm] [1–10] [Page No. 1] FIRST PROOFS i:/Sage/Jim/JIM-30906.3d (JIM) Paper: JIM-30906 Keyword

occur at a fixed critical shear rate, after which the fluidexhibits a high shear rate viscosity, and both of theseparameters are assumed to be independent of field orshear rate. The new parameters are nondimensionalizedfor comparison to the Bingham plastic model. Finally,the Bingham biplastic relations are shown to reducedown to Bingham plastic equivalents when limitingcases suggest Bingham plastic flow.

DAMPER ANALYSIS

We will consider dampers, with an annular duct orvalve in the piston head, that operate in the flow modeof operation, that is Poiseuille flow.

Flow Mode

Figure 1 depicts a magnetorheological damper oper-ating in the flow mode. A force F is applied to thedamper shaft, resulting in a pressure differential �Pacross an annular valve in the piston head. As a result,fluid flows through the annular valve, resulting in shaftmotion of velocity V0. First, we assume that the annulargap, d, is very small relative to the inner radius of theannulus, R, so that the annular duct may be approxi-mated by a rectangular duct or two parallel plates. Thewidth of the rectangular duct is denoted by b, and isrelated to the circumference of the centerline of theannular duct as below:

b � 2� Rþd

2

� �ð1Þ

A further consequence of the small gap assumptionis that the velocity profile across the annular gapin response to a linear pressure gradient must besymmetric across the valve. Thus, we make the followingassumptions: (1) the gap is assumed to be small relativeto the annular radius (as above), (2) the fluid isincompressible; (3) the flow is fully developed along

the entire finite active length of the valve, that is,the length over which the field is applied, so thatwe assume a 1-D problem; (4) the flow is assumed to besteady or quasisteady, so that acceleration terms can beneglected; (5) a linear (shaft) axial pressure gradient isassumed, so that the pressure gradient is the pressuredrop, �P, over the length of the valve, L. Therefore, weuse a simplified form of the governing equation forPoiseuille flow in a rectangular duct as below (Wereleyand Pang, 1998)

d�

dy¼ �

�P

Lð2Þ

where �P ¼ Pin � Pout. Here, Pin and Pout are thepressures at the inlet and outlet of the MR valve,respectively.

Newtonian Flow

The Newtonian constitutive relationship is

� ¼ �du

dyð3Þ

Differentiating Equation (3) with respect to y andsubstituting the result into the governing Equation (2)yields

�d 2u

dy2¼

�P

Lð4Þ

Integrating Equation (4) twice with respect to y andapplying the velocity boundary conditions at the wallsof the valve,

u �d

2

� �¼ 0 ð5Þ

ud

2

� �¼ 0 ð6Þ

leads to the familiar Newtonian parabolic velocityprofile

uðyÞ ¼�P

8�Ld2 � 4y2� �

ð7Þ

Integrating the governing equation, Equation (2), andapplying the shear stress boundary condition, �ð0Þ ¼ 0,gives the shear stress profile between the two parallelplates of the valve as

�ðyÞ ¼ ��P

Ly ð8Þ

Figure 1. Typical magnetorheological damper.

2 GLEN A. DIMOCK, JIN-HYEONG YOO AND NORMAN M. WERELEY


Integrating the velocity profile across the gap yields thetotal volume flux through the annulus

Q ¼ b

Z d=2

�d=2

uðyÞ dy

¼bd 3�P

12�Lð9Þ

Noting thatQ ¼ Apv0, and that�P ¼ F=Ap, Equation (9)can be expressed as

F ¼ Cv0 ð10Þ

where the Newtonian viscous damping, C, is given by

C ¼12A2

p�L

bd 3ð11Þ

Bingham Plastic Flow

A Bingham plastic material is characterized by adynamic yield stress, �y. In the presence of applied field,the fluid particles align with the applied field (Figure 2).Figure 3 shows an optical photomicrograph of MRparticle chain formation. According to the idealizedBingham plastic constitutive relationship, if the shearstress is less than the dynamic yield stress, then the fluidis in the preyield condition. In this preyield condition thefluid is assumed to be a rigid material. Shearing does notoccur until the local shear stress, �, exceeds the dynamicyield stress, �y. Once the local shear stress exceeds thedynamic yield stress, the material flows with a plasticviscosity of �. Therefore, the postyield shear stress canbe expressed as

� ¼ �ysigndu

dy

� �þ �

du

dyð12Þ

Thus, a Newtonian fluid can be viewed as a Binghamplastic material with a dynamic yield stress of zero(Figure 4). For an MR fluid, the dynamic yield stresscan be approximated by a power law function of themagnetic field.

Two distinct flow regions (Figure 4) arise. The centralplug region, or Region 1, is characterized by the localshear stress being less than the fluid yield stress �y, sothat the shear rate or velocity gradient is zero. Thiswidth of the preyield region is denoted by the plugthickness, �, which is nondimensionalized with respect tothe gap between the two parallel plates of the valve as

�� ¼2L�yd�P

ð13Þ

The second region is the postyield region where the localshear stress is greater than the yield stress of the fluid.The velocity profile for Bingham plastic flow betweenparallel plates with uniform field is (Wereley andPang, 1998):

uðyÞ ¼

�P

2�L

d � �

2

� �2

if jyj � �=2

�P

2�L

d � �

2

� �2

� y��

2

� �2" #

if jyj > �=2

8>>>><>>>>:

ð14Þ

Figure 3. Optical photomicrograph of particle chain formation in amagnetorheological fluid under a passive magnetic field.

Figure 2. A schematic of particle chain formation in a magneto-rheological fluid and traditional shear stress diagram for Binghamplastic model.

Figure 4. Bingham plastic shear stress and velocity profiles acrossthe gap in a MR rectangular duct valve.

Quasi-steady Bingham Biplastic Analysis of ER and MR Dampers 3


The total Bingham plastic volume flux is

Q ¼bd 3�P

12�Lð1 � ��Þ2 1 þ

��

2

� �ð15Þ

Equating the volume flux displaced by the piston headto that through the annular duct, leads to

F ¼ Ceqv0 ð16Þ

where the equivalent viscous damping is given by

Ceq ¼1

ð1 � ��Þ2ð1 þ ��=2Þð17Þ

Bingham Biplastic Flow

Shear thinning refers to the decrease of apparentviscosity at high shear rates and is an importantcharacteristic of MR fluids, as shown in Figure 5(Jolly et al., 1999). To account for this behavior, we willconsider a Bingham biplastic material with two distinctlinear postyield viscosities. A critical shear rate, _��t,

marks the boundary between the low shear rateviscosity, �0, and the high shear rate viscosity, �1.Figure 6 shows the effects of shear thinning and shearthickening in this analysis.

By calculating force equilibrium across the gap, theshear stress can be obtained in terms of pressure drop,�P, as

� ¼ ��P

Ly ð18Þ

This expression is useful when calculating the velocityprofile. By inspection of Figure 6, the shear stress canalso be written in terms of the Bingham biplastic fluidparameters as

�¼

�ysigndu

dy

� �þ�0

dudy if 0<

du

dy

��< _��t

½�yþð�0��1Þ _��tsigndu

dy

� �þ�1

du

dy

� �if

du

dy

��> _��t

8>>><>>>:

ð19Þ

Note that the high shear rate equation specializes tothe Bingham plastic case for equal low and high shearrate viscosities, �0 and �1. Also, this analysis canaccount for shear thickening, where �1 > �0.

Bingham biplastic flow consists of three regions, eachsymmetric about the gap-center axis (Figure 7). Thecentral plug region or Region 1 or the preyield region, isidentical to the Bingham plastic preyield plug, where thelocal shear stress has not yet exceeded the dynamic yieldstress. Region 2, the postyield, low shear rate region,consists of fluid that has yielded but has not reached the

(a) (b)

Figure 6. Bingham biplastic shear stress vs. shear rate for: (a) shear thinning; (b) shear thickening.

Figure 5. Viscosity as a function of shear rate at 25� for MRX-126PD(�), MRX-242AS (f), MRX-140ND (œ), and MRX-336AG (g).Figure provided courtesy of Dr. Mark Jolly (Lord Corporation).

Figure 7. Bingham biplastic shear stress and velocity profilesacross the gap in a MR rectangular duct valve.



critical shear rate, _�t�t, or the onset of shear thinning orthickening. Region 3, the postyield, high shear rateregion, contains fluid exhibiting shear thinning orthickening behavior at high shear rates. The velocityprofile is obtained by summing forces and applying theappropriate boundary conditions in each region.

In calculating the velocity profile, y is chosen tooriginate at the gap center, and the flow is examined ononly one side of the gap. Because the velocity profile issymmetric about the gap-central axis in a parallel plateanalysis, the resulting volume flux equations aredoubled to obtain total volume flux. Note that thisapproach is a change from the original Bingham plasticanalysis, where the y axis was measured from one side ofthe gap (Wereley and Pang, 1998).

Force equilibrium yields distances yy and yt, whichdefine the flow regions as below

yy ¼�y2¼

�yL

�Pð20Þ

yt ¼�t2¼

_�t�t�0L

�Pþ�yL

�Pð21Þ

It can be noted that yt does not depend on the highshear rate viscosity, �1, and that this value approachesyy as critical shear rate _��t approaches zero.

Note that �y is equivalent to the Bingham plastic plugthickness, �, and �t represents the low shear rate region.These parameters are nondimensionalized with respectto the gap width as

��y ¼2�yL

d�Pð22Þ

��t ¼2 _��0L

d�Pþ

2�yL

d�Pð23Þ

REGION 3: POSTYIELD AND HIGH SHEAR RATEThe boundary condition for Region 3 dictates zero

flow at the walls

u3d

2

� �¼ 0 ð24Þ

In this region, Equation (19) gives the shear stress as

� ¼ �½�y þ ð�0 � �1Þ _�t�t þ �1du

dyð25Þ

The velocity profile in Region 3 is obtained byintegrating Equation 25 across the region and substitut-ing Equation (18) for �. Applying the aforementionedboundary condition gives

u3 yð Þ ¼2y� d

2�1

� ��P

4Lð2yþ d Þ þ �y þ _�t�tð�0 � �1Þ

�ð26Þ

REGION 2: POSTYIELD AND LOW SHEAR RATEIn Region 2, the boundary condition requires flow

continuity between Regions 2 and 3:

u2ð ytÞ ¼ u3ð ytÞ ð27Þ

Because Region 2 does not experience shear thinning,Equation (19) gives the shear stress as

� ¼ ��y þ �0du

dyð28Þ

Note that the shear stress in Region 2 corresponds tothat of a Bingham plastic material.

The Region 2 velocity profile is calculated byintegrating Equation (28) across the region and applyingthe flow-continuity boundary condition. Equations (20)and (21) define the locations of Regions 2 and 3, and weobtain

u2ð yÞ ¼1

�0�y �

�P

2Ly

� �y

þ1

2�1

�P

4d 2 � dL½�y þ _�t�tð�0 � �1Þ

�

þL2

�Pð�y þ _�t�t�0Þ

2ð1 � ��Þ

�ð29Þ

where the viscosity ratio, ��, is defined as

�� ¼�1

�0ð30Þ

REGION 1: PREYIELDIn Region 1, the fluid flows with a uniform velocity,

because the local shear stress throughout this region hasnot exceeded the yield stress. Thus, given the boundarycondition of flow continuity between Regions 1 and 2,the preyield plug velocity may be found by substitutingEquation (20) into Equation (29) to give

u1ðyÞ ¼�2yL

2�0�Pþ

1

2�1

��P

4d2 � L½�y þ _�t�tð�0 � �1Þd

þL2

�Pð�y þ _�t�t�0Þ

2ð1 � ��Þ

�ð31Þ

FLOW RATETotal flow rate will be examined in terms of the flow

properties, ��y and ��t. These properties, in turn, dependon fluid parameters and applied pressure differential.Therefore, with knowledge of the fluid properties anddamper geometry, flow rate can be predicted in terms ofapplied force.



The velocity profile is related to the flow properties bysubstituting Equations (20) and (21) into the velocityprofile relations to obtain

u1ð yÞ ¼�P

8L

ðd � �tÞ2

�1þð�t � �yÞð2d � �y � �tÞ

�0

�

u2ð yÞ ¼�P

8L

ðd � �tÞ2

�1�

4yð y� �yÞ þ 2dð�t � �yÞ þ �2t

�0

�

u3ð yÞ ¼�P

8L

ðd � 2yÞð2yþ d � 2�tÞ

�1þ

2ðd � 2yÞð�t � �yÞ

�0

�

ð32Þ

Integrating the velocity profile across each regiongives the flow rate in that region. Each region existstwice in the symmetrical gap, so the results are doubledto yield total flow.

Q1 ¼ 2b

Z �y

0

u1ð yÞ dy

Q2 ¼ 2b

Z �t

�y

u2ð yÞ dy

Q3 ¼ 2b

Z d=2

�t

u3ð yÞ dy

ð33Þ

Integrating after substitution of Equation (32) yields

Q1 ¼bd 3�P

8L

ð1� ��tÞ2

�1þð ��t � ��yÞð2� ��t � ��yÞ

�0

��y

Q2 ¼bd 3�P

8L

ð1� ��tÞ2

�1þ

2ð ��t � ��yÞð3� 2 ��t � ��yÞ

3�0

�ð ��t � ��yÞ

Q3 ¼bd 3�P

8L

2ð1� ��tÞ

3�1þð ��t � ��yÞ

�0

�ð1� ��tÞ

2

ð34Þ

The total volume flux is then

Q¼bd 3�P

12�0Lð1� ��yÞ

2 1þ��y2

� ��ð1� ��tÞ

2 1þ��t2

� �1�

1

��

� � �ð35Þ

The velocity profile in the case of shear thinning isplotted as a function of field and viscosity ratio, ��, inFigure 8. Note the effects of these parameters on thesize of the preyield region, �y, and the location of theextent of shear thinning, the region outside �t.Increasing field widens the preyield region, �y, and ahigher critical shear rate increases the size of low shearrate region, �t.

CONSTANT VELOCITY INPUT

For flow mode damper analysis, it is useful toconsider force response for a given velocity. ForBingham plastic flow, this response depends on plasticviscosity, �, yield stress, �y, and the damper geometry.However, in the event of shear thinning, several newparameters must be considered. The high shear ratebehavior depends both on critical shear rate, _��t, and onthe high shear rate viscosity, �1.

By noting that Q ¼ Apv0 and that �P ¼ F=Ap,Equation (35) can be manipulated to give a polynomial

(b)

Figure 8. The velocity profile of a Bingham biplastic material with:(a) shear thinning ð �� ¼ 0:5Þ; (b) shear thickening ð �� ¼ 5Þ; comparedto that of a Bingham plastic material.

(a)



expression for applied force F with respect to rodvelocity v0

F 3 �bd 3

12�1A2pL

!

þ F 2 v0 þbd2

4�1Ap½�y þ _��tð�0 ��1Þ

� �

�bApL

2

3�0

� �½�3y þ _��tð�0 ��1Þð3�

2y þ 3 _��t�y � _��2

t �20Þ ¼ 0

ð36Þ

The roots of this polynomial are solved numerically,and the single valid root is determined. For a Binghamplastic material, the fluid yield stress causes a yield forcein the damper, and the postyield plastic viscosity causesnearly linear postyield damping. Shear thinning modi-fies this profile, due to a bilinear postyield viscosity,characterized by a critical shear rate.

Figure 9 depicts the theoretical effects of shearthinning and thickening on a typical MR damper filledwith Lord MRF-336AG MR fluid. For this simulation,we chose nominal damper dimensions of d ¼ 0:8mm(gap), L ¼ 10:2mm (active length) and b ¼ 28:4mm(piston head diameter). The Lord MR fluid has nominalproperties of �y ¼ 58 kPa (yield stress at 1T) and� ¼ 1 Pa s (viscosity at a 550/s shear rate). The steady-state force response is plotted with respect to velocity,and the results are compared to the Bingham plastic casein various shear thinning/thickening scenarios. Bothplots show the effects of shear thinning and thickening( �� ¼ 0:5 and �� ¼ 5, respectively) occuring at a criticalshear rate of _��t ¼ 7000 /s.

NONDIMENSIONAL ANALYSIS

For damper sizing purposes, we examine equivalentviscous damping, Ceq, in nondimensional terms,with and without shear thinning or thickening. The

equivalent viscous damping is nondimensionalizedwith respect to Newtonian damping, Cn, and specialconsiderations are made for shear thinning. Thenondimensional damping term is compared to variousnondimensionalized flow properties as follows.

Bingham Plastic Flow

Noting thatQ ¼ Apv0, and that �P ¼ =FAp, Equation(15) can be expressed as

F ¼ Ceqv0 ð37Þ

where Ceq represents the equivalent viscous dampingfor a Bingham plastic flow. The ratio of Ceq to theNewtonian damping constant, C, is

Ceq

C¼

1

ð1 � ��Þ2ð1 þ ��=2Þð38Þ

The damping coefficient Ceq=C indicates the dynamicrange of the damper because it is the ratio of the fielddependent damping to the zero field damping. Note thatas the shear rate becomes very high or the appliedmagnetic field becomes very low, the nondimensionalplug thickness approaches zero, and the damper exhibitsa dynamic range of 1. In this case, the damper behavesas a passive viscous damper. Similarly, at low shearrates or high applied magnetic fields, the plug thicknessincreases, giving the damper a high dynamic range.

Bingham Biplastic Flow with Newtonian Normalization

To determine the Bingham biplastic damping coeffi-cient, an appropriate value of C must be chosen for thenormalization. In the Bingham plastic case, Ceq is simplynormalized with respect to Newtonian damping, C,which also corresponds to the zero-field Bingham plasticcase. If we normalize Bingham biplastic damping, Ceq, t,

(a) (b)

Figure 9. Force vs. velocity for varying yield force, critical shear rate, and viscosity ratio for: (a) zero field; (b) 1 T.



with respect to Newtonian damping as illustrated inFigure 10, we obtain the relation

Ceq;t

C¼

1

ð1� ��yÞ2ð1þ ��y=2Þ � ð1� ��tÞ

2ð1þ ��t=2Þð1� 1= ��Þ

ð39Þ

where Newtonian damping, C, is given by Equation (11),and we have set the Newtonian viscosity equal to the lowshear rate viscosity, �0. Note that this normalization isineffective at high shear rates, where it indicates anunrealistic dynamic range for the damper. As the shearrate gets very large ( ��y�y ! 0), the damper should exhibitpure viscous damping, with a damping coefficient of 1.However, because this normalization does not allow forshear thinning in the Newtonian damping value, Ceq, t

does not approach C at high shear rates.

Bingham Biplastic Flow with Zero-field Damper

Normalization

Bingham biplastic damping, Ceq, t, is more appro-priately normalized with respect to the zero-fieldbiplastic case, Ct, which is defined as

Ct ¼ Ceq;t as ��y ! 0 ð40Þ

This normalization accurately predicts the damper’sdynamic range at all shear rates, as illustrated inFigure 11.

When Ceq;t is compared to the equivalent zero-fieldcase with shear thinning, Ct, the damping coefficientbecomes

Ceq;t

Ct¼

1 � ð1 � ��tÞ2ð1 þ ��t=2Þð1 � 1= ��Þ

ð1 � ��yÞ2ð1 þ ��y=2Þ � ð1 � ��tÞ

2ð1 þ ��t=2Þð1 � 1= ��Þ

ð41Þ

Variation of Biplastic Parameters

Bingham biplastic flowdepends on two additional non-dimensional parameters. First, the ratio of low and highshear rate viscosity is defined by the viscosity ratio, ��.Secondly, critical shear rate, _��, will be nondimensiona-lized with a new parameter, ��.

From Equation (22), it can be seen that the size of thelow shear rate region depends on critical shear rate butnot on the high shear rate viscosity. Therefore, anondimensional term including �t will provide informa-tion about _��t independently of ��. We define a ratio, ��, as

�� ¼1 � ��t

1 � ��yð42Þ

(a) (b)

Figure 11. Normalization of damping force and damping coefficient with respect to zero-field damping with shear thinning Ct: (a) normalizationwith Ceq;t and Ct; (b) damping coefficient using zero-field Ct.

(a) (b)

Figure 10. Normalization of damping force and damping coefficient with respect to Newtonian damping C: (a) normalization with Ceq;t and C;(b) damping coefficient using Newtonian C.



where �� represents the extent of shear thinning orthickening in the gap, which is the ratio of fluidexhibiting shear thinning to the total amount ofpostyield fluid. When �� approaches 1, all postyieldfluid is undergoing shear thinning. Similarly, as ��approaches 0, no fluid exhibits shear thinning. It isimportant to use �� instead of ��t when nondimensionaliz-ing the damper properties, because ��t can never besmaller than ��y ; shear thinning does not apply withinthe preyield region. This would restrict the scope of thenondimensional analysis. Therefore, ��t is incorporatedinto a parameter that can vary independently of ��y.

By substituting Equation (42) into Equations (39) and(41), the damping coefficients can be expressed in termsof �� as

Ceq;t

C¼

1

ð1 � ��yÞ2f1 þ ��y=2 � ��2½3=2 � ��=2ð1 � ��yÞg

�1

ð1 � 1= ��Þð43Þ

Ceq;t

Ct¼

1 � ��2ð1 � ��yÞ2½3=2 � ��=2ð1 � ��yÞ

ð1 � ��yÞ2f1 þ ��y=2 � ��2½3=2 � ��=2ð1 � ��yÞg

�ð1 � 1= ��Þ

ð1 � 1= ��Þ�Þð44Þ

Given these relations, shear thinning behavior may becompared to Bingham plastic behavior, which isrepresented on all plots by a dashed line. Figure 12shows variation in viscosity ratio, �� for low and highvalues of the shear thinning ratio, ��. This plot uses theðCeq=CÞt normalization and demonstrates the effect ofshear thinning extent on damping coefficient. As wouldbe expected, shear thinning and thickening are morenoticeable at high values of ��, where they substantiallyimpact the dynamic range of the damper. Note thatshear thinning lowers the damping coefficient, whileshear thickening raises the damping coefficient.

LIMITING CASES

The shear thinning relations can be shown to collapsedown to equivalent Bingham plastic equations invarious limiting cases.

Extreme Shear Rates

When the shear rate, du=dy, has not yet reached thecritical shear rate, _��t, the fluid exhibits only two velocityprofile regions. Here, ��t ¼ 1, and all the Region 3 termsin the shear thinning equations approach zero.Therefore, these equations collapse down to Binghamplastic equivalents. Indeed, it can be seen from Figure 6that the fluid behaves as a Bingham plastic material atlow shear rates.

When the shear rate is small, it may be seen fromEquation (22) that �t approaches �y. In this case, thefluid is dominated by shear thinning, as Region 2disappears. Similar to the low shear rate case, high shearrates result in a Bingham plastic material, where ��t ¼ ��y.Again, the shear thinning equations collapse down toBingham plastic equivalents, where the fluid exhibits aplastic viscosity �1.

Colinear Postyield Viscosities

When the low shear rate viscosity, �0, approaches thehigh shear rate viscosity, �1, Equation (30) demon-strates that the viscosity ratio, ��, approaches 1.Furthermore, it can be shown that all of the Binghambiplastic relations collapse down to equivalent Binghamplastic equations with viscosity �0 ¼ �1.

CONCLUSIONS

Magnetorheological fluids have been successfullymodeled with the Bingham plastic model, but thesefluids are known to exhibit shear thinning. In this

(a) (b)

Figure 12. ðCeq=CÞt vs. ��y�y for varying �� and ��: (a) high critical shear rate ð �� ¼ 1:0Þ; (b) low critical shear rate ð �� ¼ 0:5Þ.



analysis, the Bingham plastic model is extended toaccount for shear thinning or thickening, which is therespective decrease or increase of viscosity at high shearrates. Shear thinning or thickening over a given shearrate range is approximated by low and high shear rateregions, each of which is characterized by a uniquelinear plastic viscosity. A critical shear rate divides thetwo regions.

Nondimensional parameters are introduced to accountfor the ratio of low and high shear rate viscosity, as wellas critical shear rate. When coupled with Bingham plasticnondimensional parameters, these relationships show theexpected effects of shear thinning or thickening on MRdamper performance. The damping coefficient indicatesthe dynamic range of the damper and is compared to theBingham plastic case. With shear thinning, we observethat the damper exhibits a lower dynamic range and mustbe operated at higher magnetic fields or lower speeds toachieve a given damping coefficient. The opposite is trueof shear thickening.

ACKNOWLEDGMENTS

Research support was provided under a grant bythe U.S. Army Research Office Young InvestigatorProgram, contract no. 38856-EG-YIP. Laboratoryequipment was provided under a grant by the FY96Defense University Research Instrumentation Program(DURIP), contract no. DAAH-0496-10301. For bothgrants, Dr. Gary Anderson served as technical monitor.Glen A. Dimock was partially supported by a scholarshipfrom the Vertical Flight Foundation.

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