designofahigh-efficiencymagnetorheologicalvalvejhyoo/my_paper/jim-31988.pdf · fluid mechanics. the...

Design of a High-efficiency Magnetorheological Valve

JIN-HYEONG YOO AND NORMAN M. WERELEY*

Alfred Gessow Rotorcraft Center, Department of Aerospace Engineering, University of Maryland,College Park, Maryland 20742 USA

ABSTRACT: A high efficiency design was explored for meso-scale magnetorheological (MR)valves (< 25 mm OD with an annular gap < 1 mm). The objective of this paper is tominiaturize the MR valve while maintaining the maximum performance of the MR effect inthe valve. The main design issues in the MR valve involve the magnetic circuit and nonlinearfluid mechanics. The performance of the MR valve is limited by saturation phenomenon in themagnetic circuit and by the finite yield stress of the MR fluid. When field is applied to themagnetic circuit in the MR valve, a semisolid plug (as a result of particle chain formation)forms perpendicular to the flow direction through the valve, and a finite yield stress isdeveloped as a function of field. The resulting plug thickness is used to control flow ratethrough, and pressure drop across, the MR valve. The nondimensional plug thickness isevaluated as a basis for evaluating valve efficiency. Design parameters of the MR valve arestudied and an optimal performance was designed using steel (Permalloy) material in themagnetic circuit. A maximum magnetic flux density at the gap was achieved in the optimizedvalve design based on a constraint on the outer diameter limitation. Valve performance wasverified with simulation. A flow mode bypass damper system was fabricated and was used toexperimentally validate valve performance.

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INTRODUCTION

THE substantial field-induced yield stresses exhibitedby magnetorheological (MR) fluids make possible

numerous industrial applications (Carlson et al., 1996).Magnetorheological fluids can be implemented in avariety of semiactive smart actuation systems (Stanwayet al., 1996) including optical polishing (Kordonski andGolini, 2000), fluid clutches (Lee et al., 2000), aerospace(Kamath et al., 1999), automotive (Lindler and Wereley,1999; Gordaninejad and Kelso, 2000), and civil dampingapplications (Dyke et al., 1998; Gavin et al., 2001a).Furthermore, several studies have focused on thedevelopment of active devices utilizing electrorheologi-cal (ER) (Lou et al., 1991; Choi et al., 1997, 2002) or MRfluids (Yoo et al., 2001) in hydraulic actuation systems.Almost all of these applications use relatively largevalves.

Using MR valves in hydraulic actuation systemsaccrues many advantages, including: (1) valves have nomoving parts, and (2) electronic flow control via anelectromagnet. The most important advantages of anMR valve will be weight savings and reduction incomplexity and moving parts as compared to amechanical valve. A Wheatstone bridge-based hydraulic

actuator is being developed at the University ofMaryland for compact actuation in such applicationsas unmanned air vehicles and helicopters. The MR valveis a key component of the actuation system. However, asweight is a key issue in aerospace systems, smallerdiameter valves are the focus of this study. Suchreductions in size and volume may make actuationsystems based on MR valves a feasible means ofactuating such devices as trailing-edge flaps in helicopterblades (Milgram and Chopra, 1998). Two limitations ofsuch an actuation scheme are the block force and thecut-off frequency of the actuator. The block force is afunction of the yield stress of the MR fluid, and the cut-off frequency is a function of the response time of theMR fluid.

The objective of this paper is to design and test ameso-scale MR valve while exploiting the maximumfield dependent yield stress of the MR fluid. This entailsdesigning an effective magnetic circuit in the valve: twokinds of steel material are examined – a low permeabilitysteel, Permalloy and a high permeability Hiperco 50-A.The MR valves will be analyzed and evaluatedexperimentally to assess controllability of axial flowrate and pressure drop in the valve. Also, magnetic fieldanalysis will be utilized to optimize electromagneticperformance with given material properties. The pres-sure drop achieved across the MR valve is alsomeasured as a function of applied current to validate*Author to whom correspondence should be addressed.

E-mail: [email protected]

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

1045-389X/02/00 0001–7 $10.00/0 DOI: 10.1106/104538902031988� 2002 Sage Publications

+ [18.12.2002–9:25am] [1–8] [Page No. 1] FIRST PROOFS i:/Sage/Jim/JIM-31988.3d (JIM) Paper: JIM-31988 Keyword

the design methodology. The performance of the MRvalve is then evaluated using the nondimensional plugthickness (Wereley and Pang, 1998; Lindler andWereley, 1999) as a metric of the valve efficiency.

MR VALVES

The MR valve in this study consists of a core and aflux return forming an annulus through which the MRfluid flows, as shown in Figure 1(a). The bobbin shaft iswound with insulated wire. A current applied throughthe wire coil around the bobbin creates a magnetic fieldin the gap between the bobbin flanges and the fluxreturn. The magnetic field increases the yield stress ofthe MR fluid between the bobbin flanges and the fluxreturn. This increase in yield stress alters the velocityprofile of the fluid in the gap by creating plug flow,which decreases the volume flux, Q, and raises thepressure drop, �P, required for a given flow rate. Wenow consider the approximate rectangular duct analysisof the flow mode valve system containing MR fluid,which is assumed to behave as a Bingham-plasticmaterial. Gavin (2001b) has carefully examined theapproximation errors when estimating annular ductbehavior using the rectangular duct approximation.However, the rectangular duct approximation yieldsinteresting and important insights in many situations, aslong as the valve radius, R, is much greater than theannular gap, d. The typical velocity profile is illustrated

in Figure 1(b). The total volume flux (Wereley and Li,1998) is

Q ¼ �bd 3

12�Lað1 � ��Þ2ð1 þ ��=2Þ�P ð1Þ

where the nondimensional plug thickness is �� ¼ �=d and�� ¼ 0 for Newtonian flow. Also, La is the active lengthin the MR valve, which is the sum of the three bobbinflange thicknesses. Here, b is the mid annulus circum-ference of the MR valve and � is the differential post-yield viscosity. To verify the performance of the valve,an MR bypass damper was designed and fabricated. Aschematic of the flow mode bypass damper is shown inFigure 2. The volume flux displaced by the hydrauliccylinder head is proportional to the cylinder headvelocity, vp, or Q¼Apvp, where Ap is the area of thecylinder head minus the area of the shaft. Solving for theforce acting at the shaft, the pressure drop becomes

�P ¼F

Ap¼

12�LaAp

bd 3ð1 � ��Þ2ð1 þ ��=2Þvp ð2Þ

Measuring shaft force, F and velocity, vp, we obtain ��from (Wereley and Pang, 1998)

1

2�� 3 �

3

2��þ 1 �

12�LaA2p

bd 3

vpF

� �¼ 0 ð3Þ

In evaluating the valve performance, the nondimen-sional plug thickness in Equation (1) plays a substantialrole. In the case of �� ¼ 1, there is no flow through thevalve as in the case of an ideal valve (fully closed withinfinite blocking pressure). Both flow rate, Q, andpressure drop, �P, are a function of nondimensionalplug thickness, ��. Thus, the nondimensional plug

42

25.4

Details in (b)3 15

R8.

0

R8.

5

CW CCW

42

25.4

Details in (b)3 15

R8.

0

R8.

5

CW CCW

(a) Valve cross section. All units in mm

(b) Velocity Profile

φ

Figure 1. Schematic of the valve: (a) valve cross section; (b) velocityprofile. Figure 2. Test configuration for the MR valve.

2 JIN-HYEONG YOO AND NORMAN M. WERELEY


thickness, which is a measure of the valve constrictionand has a value range from �� ¼ 0 (open) to �� ¼ 1(closed), is an appropriate measure of valve efficiency.

MAGNETIC CIRCUIT

The primary components of the MR valve design arepictured in Figure 3. The yield stress of the MR fluid canbe varied as a function of the applied magnetic field.Therefore, the magnetic field applied to the MR fluidmust be correlated with prediction or measurement. Themain design parameters of the magnetic circuit are gapdistance, bobbin shaft diameter, bobbin flanges length,thickness of the flux return and number of windings inthe coil, which is related to the length of the bobbinshaft. To achieve an efficient magnetic circuit, the areaof the path of magnetic flux should be maintainedconstant. Theoretically, a smaller gap distance is betterbecause the permeability of the MR fluid in the gap ismuch less than those of the iron-based bobbin and fluxreturn. Practical gaps typically range from 0.25 to 2 mmfor ease of manufacture and assembly. A constant gapbetween the core and flux return was maintained foruniformity of the magnetic flux in the gap. Furthermore,the gap must be considered in the flow analysis. The gapdetermines the flow rate for a given pressure differenceas shown in Equation (1). In this study, the gap will beset to 0.5 mm. The flux return (or hydraulic cylinder)has two functions: one is as an element in themagnetic circuit and the other is as a connector tothe hydraulic cylinder cap with a seal as shown inFigure 3. In our small valve design, the minimum wallthickness of the flux return is set by the height of thethreads around the hydraulic cylinder used to connect tothe caps.

A magnetic field finite element analysis for the valvesystem was conducted using ANSYS/Emag 2D. Thepurpose of the analysis was to identify saturationphenomenon in the magnetic circuit and to evaluatethe effect of the design parameters on the magnetic

behavior of the valve. The magnetization data forPermalloy steel (Roters, 1941) and Hiperco 50-A(Harner, 1999) material were used in this analysis. TheMR fluid also has a saturation phenomenon in the yieldstress as a function of applied magnetic field.Considering the shear stress versus magnetic inductanceof MRF-132LD (Lord Corp., 1999), 0.8 T was the fieldat which magnetic saturate occurred in the MR fluidand the yield stress of the material was maximized.Throughout this analysis, an optimized MR valve wasinvestigated to achieve smaller outer diameter andlonger active length with maintaining magnetic fluxdensity of 0.8 T at the gap.

Bobbin Diameter

The bobbin shaft radius is the most sensitive designparameter limiting themagnetic performance. In Figure 4,the averaged magnetic flux density along the bobbinflanges is plotted versus the bobbin shaft radius for alow permeability Permalloy steel, and a high perme-ability Hiperco 50-A powder metallurgical alloy. It isdesired to have as high a magnetic flux as possible. Bothmaterials provide adequate magnetic flux for bobbinshaft radii of at least 4 mm. However, the lowpermeability Permalloy rolls off below 4 mm muchfaster than the high Permeability Hiperco 50-A, due tosaturation. Therefore, to reduce the bobbin shaft andhence, the valve diameter further, more costly higherpermeability magnetic materials must be used.

Bobbin Flange Length

Figure 5 shows the trend of magnetic flux density atthe gap as a function of bobbin flange length, L. In thecase of Permalloy, as the bobbin flange length decreases,the magnetic flux density at the gap tends to increasebecause the magnetic flux density at the bobbin shaftdecreases with decreasing the flange length. Decreasing

2 3 4 5Bobbin shaft radius (mm)

0.02

0.04

0.06

0.08

0.1

Mag

netic

flux

den

sity

(T

e)

Hiperco 50APermalloy

Figure 4. Magnetic flux density at the gap as a function of thebobbin shaft radius (gap¼0.5 mm, air).Figure 3. Photograph of the MR valve parts.

Design of a High-efficiency Magnetorheological Valve 3


the flange length resulted in higher magnetic flux densityat the gap by reducing saturation at the bobbin shaft. Toverify this effect, the magnetic flux density at the gapwas calculated for cases of 2, 4, and 10 mm flange lengthper each, as shown in Figure 6. Clearly, a length of 2 mmhas the maximum magnetic flux density of the three.Figure 7 shows the magnetic flux density along the airgap with various active lengths. In this narrow range ofthe flange lengths, as the length increases, the uniformityof the magnetic field across the active length improves,but the level of magnetic flux density also decreases.Based on the results of Figure 7, a bobbin flange length,L¼ 3 mm per each flange was chosen as our optimaldesign because we achieve an average magnetic field of0.8 T in the bobbin flange, while maximizing the corelength and thus the blocking pressure or yield force.

Magnetic Material

The magnetic material has a limited achievablemagnetic flux density at the gap due to magneticsaturation. This saturation phenomenon must beconsidered along with the MR fluid yield stress

saturation, itself to achieve optimized magnetic field inthe MR valve. Considering the shear stress versusmagnetic inductance of MRF-132LD (Lord Corp.,1999) and magnetization curve for Permalloy, 0.8 Tis the maximum magnetic flux density achievable withthese materials. The averaged maximum magnetic fluxdensity of our valve (dimensions are shown in Table 1) isabout 0.8 T with 1.6-A input current, as shown in Figure 8.When high permeability material, Hiperco 50-A, is usedas shown in Figures 4 and 5, the bobbin shaft diametercan be reduced, and the core length increased, whilestill obtaining the same performance as a larger valvemade of lower permeability Permalloy steel. From theseresults, the valve can be reduced in volume on scale,while maintaining the same performance, by using abobbin/flux return manufactured from a higher perme-ability material.

-20 0 20Relative position along the air gap (mm)

0

0.2

0.4

0.6

0.8

1

Mag

netic

Flu

x D

ensi

ty (

Te) (b) in detail

(a)

-4 -2 0 2 4

0.2

0.4

0.6

0.8M

agne

tic F

lux

Den

sity

(T

e)

L=1mm

L=2mm

L=3mm

L=4mm

(b) Zoom on circled region in Fig. (a)

Figure 7. Magnetic flux density along the gap with various bobbinflange lengths (Permalloy, I¼1 A).

2 4 6 8 10Bobbin flange length (mm)

0.12

0.14

0.16

0.18

0.2

Mag

netic

flux

den

sity

(T

e)

Hiperco 50APermalloy

Figure 5. Magnetic flux density at the gap as a function of thebobbin flange length (gap¼0.5 mm, air).

0 1 2 3Applied current (A)

0

0.2

0.4

0.6

Mag

netic

flux

den

sity

(T

e) L = 2 mm

L = 4 mm

L = 10 mm

Figure 6. The magnetic flux density at the gap with various bobbinflange lengths (Permalloy).

Table 1. Valve dimensions.

Outer diameter 25.4 mmBobbin diameter 14 mmFlange length/each 3 mmAir gap 0.5 mmNumber of windings 160 turnsMaximum tesla at the gap 0.80 T



MEASUREMENT OF THE MAGNETIC FLUX

DENSITY

A thin film (FH-301-060, F.W.BELL) Hall sensor wasused to measure the magnetic flux density at the gap anda hand-held Gauss meter (F.W.BELL, model 5080) wasused to calibrate the Hall sensor. Figure 8 compares theexperimental data with the analytical prediction fromANSYS/Emag 2-D for the valve with an air gap. Takingthe error of the Hall sensor into consideration, theresults in Figure 8 are in good agreement with eachother. With these results, we conclude that simulationusing ANSYS/Emag 2-D will be sufficiently accurate topredict the steady state performance of the magneticfield at the gap when filled with MR fluid. Thus, at 1.6 Aof applied current, we will have induced a magnetic fluxof 0.8 T, which is sufficient for our purposes.

MAGNETORHEOLOGICAL FLUID

CONSTITUTIVE MODEL

Due to the nonlinearity of the MR fluid, determina-tion of the performance of the valve requires numericalanalysis. We will evaluate the valve performance bycalculating the pressure difference and flow rate. Weused a commercially available MR fluid, namely MRF-132LD (Lord Corp., 1999) in our experiments. For theBingham-plastic model, the MR fluid properties ofdynamic yield stress, �y, and plastic viscosity, �, wererequired as a function of applied magnetic field. Thedynamic yield stress for this fluid was approximated bya cubic equation of the magnetic field, B, so that

�y ¼ a3B3 þ a2B

2 þ a1Bþ a0:

The polynomial coefficients were determined byleast-squares fit of the dynamic yield stress data as afunction of magnetic field from data supplied byLord Corporation (1999), and are: a0 ¼ �0:877 kPa,a1 ¼ 17:42 kPa=T, a2¼ 122:56 kPa=T2 and a3¼

�86:51 kPa=T3: To simplify the analysis, the MR fluidis assumed to have a nominal plastic viscosity of 0.3 Pa s.

EXPERIMENTAL RESULTS

To validate our nondimensional analysis using thenonlinear Bingham-plastic shear flow and the magneticcircuit design with ANSYS/Emag 2-D, a high stroke(� 20 cm) MR damper was constructed. The damperconsists of four main parts: a hydraulic cylinder, indus-trial tube fittings, an accumulator and an MR bypassvalve as pictured in Figure 9. The damper was chargedwith MR fluid, MRF-132LD (Lord Corporation). Theaccumulator connected to the cylinder was used to

MTS Hydraulic Actuator

Digital OscilloscopeMTS Controller

DC Power supply

MR

Val

ve

Accumulator

Load cell

Hyd

raul

ic

Cyl

inde

r

SignalConditioningAmplifier

Figure 9. Experimental setup for performance measurement of the MR valve.

0 2 4 6Applied current (A)

0

0.2

0.4

0.6

0.8

1

Mag

netic

flux

den

sity

(T

e)

Test result, air

Simulation, air

Simulation, MR

Figure 8. Performance of the magnetic flux density at the gap withMR fluid permeability (simulation) and with air permeability (test andsimulation, Permalloy).



pressurize the MR fluid inside the damper for purposesof compressing adsorbed air bubbles in the fluid, andpreventing cavitation.

For the experimental validation of the flow modebypass valve equations, force measurement from aconstant velocity amplitude square wave on an MTSservo-hydraulic testing machine were recorded on acomputer. The MR bypass damper was mounted in theclevises, and the shaft of the damper was oscillated. Theshaft displacement was measured using an LVDT andthe applied load was measured using a 250 lb load cell.

The shear stress of the MR fluid in the case of thevalve is shown in Figure 10. The maximum shear stressis about 47 kPa with 1.6-A input current. There are twosaturation effects accounted for simultaneously in thisdiagram, the magnetic flux density and the yield stress ofthe MR fluid. The pressure drop versus flow ratediagram is shown in Figure 11 with experimental results.The analysis, based on the quoted MR fluid data,tended to overpredict the experimental pressure differ-ence data. At low currents this overprediction wassubstantial, almost 30% error. We will be studying thereasons for such overpredictions in future studies. Withthis valve configuration, nominally 250 psi of pressuredrop can be achieved.

Figure 12 compares the test results for the non-dimensional plug thickness to analysis in the valve, for a

constant current input. This data demonstrates that aplug thickness of from 91% to just over 72% can beachieved over the pressure range of 1644–2085 kPa.From these data, we deduce that 1497.7 kPa of blockpressure with 1.6-A input current can be achieved.Figure 13 shows the time response of the valve withinput step current. With this result, we can expect thatthe maximum drive frequency will be about 100 Hz.

CONCLUSION

Considering the shear stress versus magnetic induc-tance of the MR fluid (MRF-132LD, Lord Corp., 1999)and magnetization curve for Permalloy, a maximummagnetic flux density at the gap was achieved with anoptimized design and was verified with simulation andexperiment. Based on these results, we conclude thefollowing:

1. Using low permeability Permalloy steel material,1497.7 kPa block pressure can be achieved with ourMR valve design with a 25.4 mm outer diameter.

2. Using high permeability material, the size of the valvecan be reduced and the active core length can be

0 0.01 0.02Time (sec)

0

200

400

600

Pre

ssur

e R

espo

nse

(kP

a)

Figure 13. The time response of normalized pressure difference ofthe valve (test results).

0 0.4 0.8 1.2 1.6Applied current (Amp.)

0

10

20

30

40

50

She

ar S

tres

s (k

Pa)

Figure 10. Shear stress of the MR fluid at the gap (Permalloy).

1500 2000Pressure Difference (kPa)

0.6

0.8

1

Plu

g T

hick

ness

,δ[

1]

Test

Simulation

Figure 12. The nondimensional plug thickness, �� (I¼1.6 A).

0 10 20 30 40 50 60Flow Rate (cc/sec)

0

500

1000

1500

2000

2500

Pre

ssur

e D

iffer

ence

(kP

a) 1.6 A1.2 A0.8 A0.6 A0.4 A

Figure 11. Flow characteristics (symbols: test and lines: simulation).



increased for a higher blocking pressure. The Hiperco50-A steel core has same magnetic flux density at thegap with only 60% of the shaft radius for thePermalloy steel case.

3. The nondimensional plug thickness is a usefulmeasure of valve efficiency, and the valve configura-tion of this paper achieves up to 90% efficiency.

4. The valve design based on the magnetic analysis inthis paper achieves 100 Hz of dynamic range includ-ing the response time of MR fluid.

ACKNOWLEDGMENT

The authors thank Monique Gabriel and Dr. MarkJolly of Lord Corporation (Cary, NC) for providing theMR fluid (MRF-132LD, 1999) used in this study.

REFERENCES

Carlson, J. D., Catanzarite, D. M. and Clair, K. A. S. 1996.‘‘Commercial Magnetorheological Fluid Devices Technology,’’International Journal of Modern Physics Part B, 10(22–23):2857.

Choi, S.-B., Cheong, C.-C., Jung, J.-M. and Choi, Y.-T. 1997.‘‘Position Control of an ER Valve-Cylinder System via NeuralNetwork Controller,’’ Mechatronics, 7(1):37–52.

Choi, S.-B., Sung, K. -G. and Lee, J. -W. 2002. ‘‘The Neural NetworkPosition Control of a Moving Platform Using ElectrorheologicalValves,’’ ASME Journal of Dynamic Systems, Measurement andControl, 124(3):435–442.

Dyke, S. J., Spencer, B. F., Sain, M. K. and Carlson, J. D. 1998.‘‘Experimental Study of MR Dampers for Seismic Protection,’’Smart Materials and Structures, 7(5):693–703.

Gavin, H., Hoagg, J. and Dobossy, M. 2001a. ‘‘Optimal Design of MRDampers,’’ In: Proceedings U.S.-Japan Workshop on SmartStructures for Improved Seismic Performance in Urban Regions,14 August 2001, Seattle WA, pp. 225–236.

Gavin, H. P. 2001b. ‘‘Annular Poiseuille Flow of Electrorheologicaland Magnetorheological Materials,’’ Journal of Rheology,45(4):983–994.

Gordaninejad, F. and Kelso, S. P. 2000. ‘‘Fail-safe Magneto-Rheological Fluid Dampers for Off-Highway, High-PayloadVehicles,’’ Journal of Intelligent Material Systems and Structures,11(5):395–406.

Harner, L. L. 1999. A Simplified Method of Selecting Soft MagneticAlloys, Carpenter Technology Corporations Technical Articles.(http://carpenter.idesinc.com/TechArtcles/TA00005.htm)

Kamath, G. M., Wereley, N. M. and Jolly, M. R. 1999. ‘‘Character-ization of Magnetorheological Helicopter Lag Damper,’’ Journal ofthe American Helicopter Society, 44(3):234–248.

Kordonski, W. I. and Golini, D. 2000. ‘‘Fundamentals ofMagnetorheological Fluid Utilization in High PrecisionFinishing,’’ Journal of Intelligent Material Systems and Structures,10(9):83–689.

Lee, U., Kim, D., Hur, N. and Jeon, D. 2000. ‘‘Design Analysis andExperimental Evaluation of an MR Fluid Clutch,’’ Journal ofIntelligent Material Systems and Structures, 10(9):701–707.

Lindler, J. and Wereley, N. M. 1999. ‘‘Analysis and Testing of Electro-rheological Bypass Dampers,’’ Journal of Intelligent MaterialSystems and Structures, 10(5):363–376.

Lou, Z., Ervin, R. D. and Filisko, F. E. 1992. ‘‘Behaviors of Electro-rheological Valves and Bridges,’’ In: Proceedings of the InternationalConference on Electrorheological Fluids: Mechanics, Properties,Structure, Technology and Applications, 15–16 October, 1991,World Scientific Publishing Co., Carbondale, Illinois, pp. 398–423.

Milgram, J. H. and Chopra, I., 1998. ‘‘Parametric Design Study forActively Controlled Trailing Edge Flaps,’’ Journal of the AmericanHelicopter Society, 43(2):110–119.

Roters, H. C. 1941. Electromagnetic Devices, John Wiley & Sons, Inc.

Stanway, R., Sproston, J. L. and El-Wahed, A. K. 1996. ‘‘Applicationsof Electro-Rheological Fluids in Vibration Control: A Survey,’’Smart Materials and Structures, 5(4):464–482.

Yoo, J.-H., Sirohi, J. and Wereley, N. M. 2001. ‘‘Design of an MRHydraulic Power Actuation System,’’ In: Proceedings of SPIE –The International Society for Optical Engineering, Vol. 4327, 2001,Smart Structures and Materials 2001 – Smart Structures andIntegrated Systems – 5–8 March, 2001, Newport Beach, CA,pp. 148–158.

Wereley, N. M. and Pang, L. 1998. ‘‘Nondimensional Analysis ofSemi-active Electrorheological and Magnetorheological Dampersusing Approximate Parallel Plate Models,’’ Smart Materials andStructures, 7(5):732–743.



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