Design of a MR Hydraulic Power Actuation Systemterpconnect.umd.edu/~jhyoo/my_paper/SPIE01.pdfDesign of a MR Hydraulic Power Actuation System Jin-Hyeong Yoo, Jayant Sirohi, and Norman

Design of a MR Hydraulic Power Actuation System

Jin-Hyeong Yoo, Jayant Sirohi, and Norman M. Wereley

Alfred Gessow Rotorcraft Center, Department of Aerospace Engineering,

University of Maryland, College Park, MD 20742

ABSTRACT A magnetorheological (MR) fluid-based hydraulic power system is analyzed and experimentally validated by testing a prototype. A set of MR valves is proposed to implement within a Wheatstone bridge hydraulic power circuit to drive a hydraulic actuator using a pump. The MR valves are used in place of conventional mechanical servo valves. The proposed use of MR valves in hydraulic actuator systems has many advantages. First, MR valves have no moving parts, enhancing reliability. Second, the MR valves operate at the same speed as the actuation bandwidth (typically below twenty Hz in our applications). Third, the actuator relies on flow rates for a given pump speed, and avoids, to a large degree fluid compliance. Fourth, if a change in stroke direction is required, the flow through each of the MR valves can be controlled smoothly via changing the applied magnetic field. The performance of the Wheatstone bridge with MR valves is theoretically derived using three different models of the MR fluid behaviors: an idealized model, a Bingham-plastic model and a biviscous model. The analytical system efficiency in each case is compared, and departures from ideal behavior are recognized. The driving force and efficiency will be evaluated in the MR hydraulic power actuator system for both Bingham plastic and biviscous flows. An MR valve is designed using a magnetic finite element analysis. The magnetic flux density developed in the MR valve are verified by analytical and experimental methods. The yield stresses achieved in the MR valve due to the applied current are also measured to validate the design methodology. The overall performance of the MR fluid based hydraulic power system is described using the experimental MR valve performance data. Keywords: Hydraulic actuator, Magnetorheological valve, Bingham-plastic model, Biviscous model

1. INTRODUCTION In this paper, magnetorheological (MR) valves and bridges[1] will be analyzed and designed to develop a compact hydraulic power actuation system with a nominal force of 300N and a stroke of approximately 4mm. This system is envisaged to meet the actuation requirements of a trailing edge flap system[2] on a smart rotor model. Driving force, stroke, cut-off frequency and efficiency are the evaluation parameters in a general hydraulic actuator[3]. Durability and miniaturization are stumbling blocks to expand the application area for conventional mechanical valves. These problems can be overcome by replacing the mechanical valves by MR valves. There are many advantages of using MR valves in hydraulic actuation systems. First, the valves have no moving parts, eliminating a lot of the complexity and durability issues of conventional mechanical valves. Second, the actuator relies on flow rates for a given impeller speed, and avoids, to a large degree, fluid compliance. Third, if a change in direction is required, the flow through each of the valves can be controlled smoothly via changing the applied magnetic field. This is much better than the abrupt opening and closing of mechanical valves that can lead to high frequency disturbances in the controlled output. Above all, most important advantages of MR valve will be the miniaturization and compactness compared to a mechanical valve. This miniaturization can expand the application area to the aerospace industry, making it a feasible means of actuating trailing-edge flaps in helicopter blades[2]. Two disadvantages are the block force and the cut-off frequency of this actuator. The block force is dependent on the yield stress of the MR fluid and cut-off frequency relies on the response time of MR fluid. The steady and transient pressure responses of MR fluid controlled by MR valve are of significant importance in the estimation and the controllability of the induced force. There are several study to determine the bandwidth of ER valve which is similar to MR valve, Zheng, et al.[1] consider the mass effect of ER fluid in the valve. Nakano and Yonekawa[4] investigate transient pressure response assuming the compressibility of ER fluid. Whittle, et al.[5] try to formulate transient velocity profile inside ER valve. In the case of ER/MR actuator, Zheng et al.[1] analyzed and evaluated ER valves and bridges experimentally in its ability to control the flow and pressure condition. Wendt and Büsing[6] built an actuator which is controlled by an ER valve-block and Nakano et al.[7] developed a miniature bellows actuator driven by a pair of PWM (Pulse Width Modulator) controlled ER valves. The response characteristics might mainly depend upon the MR effects of MR fluid flowing through an MR valve and the magnetic circuit efficiency in the MR valve. Therefore in this paper

magnetorheological valves and bridges will be analyzed and evaluated experimentally in its ability to control the flow and pressure condition. Also the magnetic circuit analysis will be conducted to enhance the electro-magnetic performance. Typically, as is done in this paper, Bingham plastic analysis is used to quantify MR fluid performance. However, we also examine the effects of biviscous flow[9] on overall actuator performance. The driving force and efficiency will be evaluated in the MR hydraulic power actuator system for both Bingham plastic and biviscous flows. An MR valve was designed using a magnetic finite element analysis. The magnetic flux density developed in the MR valve are verified by analytical and experimental methods. The yield stresses achieved in the MR valve due to the applied current are also measured to validate the design methodology. The overall performance of the MR fluid based hydraulic power system is described using the experimental MR valve performance data.

2. MR VALVES MR valves used in this study consist of a core and a flux return, and an annulus through which the MR fluid flows, as shown in Figure 1. The core is wound with insulated wire. A current applied through the wire coil around the bobbin creates a magnetic field in the gap between the core and the flux return. The magnetic field increases the yield stress of the MR fluid between the core and flux return. This increase in yield stress alters the velocity profile of the fluid in the gap and raises the pressure difference required for a given flow rate. We now consider the approximate parallel plate analysis of the flow mode valve system containing MR fluid. The 1D axisymmetric analysis is given in Kamath, Hurt and Wereley[8]. For Newtonian flow, the volume flux through the annulus, Q is

PL

bdQa

∆−=µ12

3 (1)

For Bingham-plastic flow, the velocity profile in each region of flow is determined by the boundary condition. The typical velocity profile is illustrated in Figure 2. The total volume flux is

PL

bdQa

∆+−−= )2/1()1(12

23

δδµ

(2)

where the non-dimensionalized plug thickness d/δδ = has been introduced. The biviscous model solution for the total volume flux through the annulus from Wereley and Lindler[9] is

−++−−= δδ

µµ

δδµ

)1(23

)2/1()1(12

223

pr

po

apoLbdQ (3)

Figure 1. Schematic of the valve. Figure 2. Typical velocity profile for the Bingham plastic model. To verify the performance of the valve, an MR bypass damper was designed and fabricated. A schematic of a flow mode damper is shown in Figure 3. The damper consists of four main parts: a double rod hydraulic cylinder, industrial tube fittings, an accumulator and an MR bypass valve. The volume flux displaced by the hydraulic cylinder head is proportional to the cylinder head velocity, 0v , or

0vAQ p= (4)

where pA is the area of the cylinder head minus the area of the shaft. Solving for the force acting at the shaft, the equation for the Bingham-plastic model could be expressed as

023

2

)2/1()1(12

vbd

ALF pa

δδµ

+−= (5)

If we define Bingham number, Bi , as

dvBi y

/0µτ

= (6)

then we can rewrite Equation (5) as a polynomial in δ as

011623

21 3 =+

+− δδ

BiAA

d

p (7)

where, dA is the cross sectional area of annulus. Given the input velocity, 0v , the Bingham number, Bi , is calculated, and

the non-dimensional plug thickness, δ can be determined either using a root finding algorithm or analytically. Once we get F and 0v experimentally, we can get δ from

012

123

21 0

3

23 =

−+−Fv

bdAL paµ

δδ (8)

From Newtonian flow equation (1), it is easily shown that 3

212bd

AL paµα = is the slope of F - 0v relation.

Figure 3. Test set-up for the valve. Figure 4. Conceptual diagram of hydraulic actuator system with MR valves.

3. A DOUBLE-ROD HYDRAULIC ACTUATOR Figure 4 shows the schematic diagram of the hydraulic actuator system where the load attached to the cylinder generates a force F. To evaluate the performance of the hydraulic actuator with MR valves, the MR fluid behavior is considered as 1) Idealized MR valve, 2) Bingham-plastic model, and 3) Biviscous model. From the above, system efficiency can be derived.

Comparison of analytical models

1) Simplified system with idealized MR valves.

Assuming an ideally operated MR valve ( 0=ai and ∞=bi ) then gives the following equations:

FAPPuAPQQQQuAPPQQ

pba

pbNpasN

=−

=====−=

)(

,0 ,)( 4321 (9)

where, a

N LbdQ µ12

3= , d is the gap of MR valve and b is the width of the valve. It therefore follows that the steady-

state pressures are given by:

p

spa A

FPAP

20+

= , p

spb A

FPAP

20−

= , sba PPP =+ 00 (10)

−=

−=

MaxMax

spp

sNF

FuPA

FAPQu 11

2 (11)

−=

Max

MaxF

Fux 1ω

where, ω is frequency in rad/sec.

2) Actuator system with MR valves based on Bingham-Plastic model. Assuming that the MR valve follows the Bingham-Plastic model behavior, then gives the following equations:

bNbsN

aNasN

PQQPPQQ

PQQPPQQ

=−+−=

+−=−=

42

3

221

),)(2/1()1(

)2/1()1( ),(

δδ

δδ (12)

FAPPuAQQQQ

pba

p

=−

=−=−

)(3421

(13)

It therefore follows that the steady-state pressures of equation (10) and the velocity of the actuator will be:

{ }FPAAQu sp

p

N )1()1(2 112 ∆+−∆−= (14)

where, )2/1()1( 21 δδ +−=∆ .

If we define the Bingham number, Bi as

uBd

Bi y

µτ )(

=

The polynomial equation for non-dimensional plug thickness will be

0216)(22

321 3 =+

−+− δ

τδ

BiAA

BLdP

d

p

ya

s (15)

3) Actuator system with MR valves based on biviscous model. If the MR valve acts as biviscous model then the following equations hold:

bNbsN

aNasN

PQQPPQQ

PQQPPQQ

=−

−++−=

−++−=−=

422

3

2221

,)()3/1(23)2/1()1(

)3/1(23)2/1()1( ,)(

δδµδδ

δδµδδ (16)

where, apo

N LNbdQ µ12

3= , and

pr

poµ

µµ = .

It therefore follows that the steady-state pressures of equation (10) and the velocity of actuator will be:

{ }FPAAQu sp

p

N )1()1(2 222 ∆+−∆−= (17)

where, δδµδδ )3/1(23)2/1()1( 22

2 −++−=∆ .

If we define the Bingham number, Bi as

uBd

Bipo

y

µτ )(

=

The polynomial equation for non-dimensional plug thickness will be

0216)(2

)1(23)1(

21 3 =+

−+−−− δ

τµδµ

BiAA

ELdP

d

p

ya

s (18)

The system efficiency. The system efficiency, defined as the power transferred to the load divided by the MR valve supply power, is given by:

)(

)()( valvesupply toPower

load todeliveredPower 31

3421

QQPQQPQQP

s

ba

+−−−==η (19)

From the above, system efficiency of each model can be derived as follow:

• Idealized MR valve

spPAF=η (20)

• MR valve based on Bingham Plastic model

)1()1(/)1()1(

11

12

1

∆−−∆+∆+−∆−=

FPAPAFF

sp

sPη (21)

where, )2/1()1( 21 δδ +−=∆ .

• MR valve based on biviscous model

)1()1(/)1()1(

22

22

2

∆−−∆+∆+−∆−=

FPAPAFF

sp

sPη (22)

where, δδµδδ )3/1(23)2/1()1( 22

2 −++−=∆ .

In the case of idealized MR valve, it may then be deduced that maximum power is transferred to the load when:

2spPA

F = (23)

And the resulting power is 8

2s

MaxQPP = .

4. VALVE DESIGN To validate this non-dimensional analysis, a compact valve was constructed. The valve consists of four main parts: a flux return, dumbbell shaped bobbin core, a pair of spoke disk to maintain the gap between the bobbin core and flux return, coil winding, as pictured in Figure 5. The brass end adapters of the MR valve use standard SAE threaded ports, so that they can be fitted to the hydraulic cylinder with standard tube fittings. Cone shaped fixture at the core helps to reduce the flow losses. To briefly determine the number of turns in magnetic circuit, magnetic circuit analysis concept was adopted. The magnetic field, H

�

, can be thought of as a magnetic stress on space, and the magnetic flux density, B�

, a magnetic strain. In other words, the magnetic field, H

�

, tends to magnetize space, and the magnetic flux density, B�

, is the total magnetic effect that results. Magnetic flux density is an important quantity in magnetic circuit because magnetic flux, of which B

�

gives the spatial distribution, plays an important role in electromechanical energy conversion. Magnetic flux lines encircle the currents that generate them. Thus, magnetic flux lines exist only in closed loops. This means that magnetic flux is conserved in any closed spatial region. The magnetic flux is a scalar quantity with units of Webers (Wb) and the average flux density would be the flux divided by the area. Average magnetic field is the average flux density divided by the permeability of the medium.

(Wb)totalR

IN ×=Φ , (Tesla)i

i AB Φ= , (Amp./m)

i

ii

BHµ

= (24)

Using the permeability of the MRF-132LD fluid (Lord Corporation, 1999), the relation between magnetic flux density of the gap and the number of turns for various applied current are shown in Figure 6. Using this magnetic circuit analysis, the saturation phenomenon effect is ignored. Figure 5. Photograph of the valve parts. Figure 6. Magnetic flux density at the gap. If we fix the bobbin length as 30mm, we can make about 160 turns with 2-layers of #26 wire. The core length is set at 2 mm per each side. Increasing this length result decreasing the magnetic flux density at the gap. Decreasing the length further, may however, makes it susceptible to damage from the high-pressure fluid. To design the small valve, the outer diameter of the flux return is assumed to be limited to 20mm. The wall thickness of the flux return was set to 4.5 mm to allow threads for adapter. The core outer diameter should be 10 mm considering the 0.5 mm air gap at the annulus. So the maximum diameter of the bobbin shaft will be 8 mm to leave adequate space for the coil.

5. MAGNETIC FIELD ANALYSIS MR fluid has changing characteristics of its yield stress according to the applied magnetic field. Therefore, the applied magnetic field to the MR fluid must be exactly correlated by prediction or measurement. In this paper, the magnetic field analysis for the valve system is conducted using ANSYS/Emag 2D, which is a FEM software for magnetic field. The purpose of the analysis is to identify the effect of the saturation phenomenon in the magnetic circuit of the valve. To verify the theoretical result, the measurement of magnetic field is also performed using a Hall sensor. For the analysis, the FE model was modeled using Plane 53, 2D 8-node magnetic solid element with axisymmetric option. As shown in Figure 7 of an analysis result, the bottleneck of the magnetic circuit is bobbin diameter. Increasing the bobbin shaft

diameter, the magnetic flux density at the annulus gap will be increased. The average magnetic flux density at the annular gap resulting from the applied current changing from 0.1-ampere to 3 ampere is shown in Figure 8 and compared with the experimental result. The magnetization curve of the Permalloy[10] steel material using in this analysis is shown in Figure 9. If we consider the sensing error of the hall sensor, the results in Figure 8 are in good agreement with each other. Especially the saturation trend according to the applied current is identical. As shown in Figure 8, the maximum magnetic flux density of this valve system is about 0.41 Tesla. Actually, the permeability at the annulus gap is assumed as air permeability in this experiment and analysis. Practically, the magnetic field measurement in the MR fluid at the annulus gap is relatively difficult and the results will not be so accurate. So, in this analysis the magnetic flux density for the annulus gap filled with MR fluid will be calculated by ANSYS/Emag 2D according to the former results. At that case the maximum magnetic flux density will go up to 0.56 Tesla with the applied current 1.6 ampere. Figure 7. A magnetic field analysis result. Figure 8. Test and simulation result for the magnetic flux density at the gap.

6. ANALYTICAL RESULTS Due to the nonlinearity of the MR-fluid, determination of the performance of the valve and hydraulic actuator requires numerical analysis. Our object is to design hydraulic actuators meeting a specified performance level. In the following analysis we use a commercially available MR fluid and numerically examine the valve performances and hydraulic actuator performance with the valve. For the valve performance, the plug thickness and velocity-force relation will be examined based on bypass damper test. In the case of hydraulic actuator performance, force-amplitude relation of the hydraulic actuator will be examined according to the applied magnetic flux, which is the result from the ANSYS/Emag 2D analysis. For the Bingham plastic model, the required material properties are dynamic yield stress, yτ , and plastic viscosity, µ , as a function of magnetic field. We will examine a commercially available MR fluid, namely MRF-132LD (Lord Corporation, 1999). The typical data supplied by a MR-fluid manufacturer is the shear stress versus magnetic field diagram, as shown in Figure 10. The dynamic yield stress for this fluid can be approximated by a cubic equation of the magnetic field, so that

012

23

3 aBaBaBay +×+×+×=τ The polynomial coefficients were determined by least-squares fit of the dynamic yield stress data as a function of magnetic field from Lord Corporation (1999), and are 0a =-0.877 kPa, 1a =17.42 kPa/Te, 2a =122.56 kPa/Te2 and 3a =-86.51 kPa/Te3. To simplify the analysis, the MR fluid is assumed to have a nominal plastic viscosity of 0.6 Pa s. The nominal design of the actuator is as follows: the bore size is 25.4 mm (1 inch) and the shaft diameter is 15.875 mm (5/8 inch). The active length of the valve is 4 mm. The inner core of the valve has outer radius of 5 mm. The gap, d, is 0.5 mm, unless the gap is explicitly varied. As shown in former analysis of magnetic flux, the valve is not so efficient as a magnetic circuit because it has a bottleneck at the bobbin shaft as shown in Figure 7. The maximum magnetic flux will be about 0.56 Tesla.

1) Performance of the valves In the actuator, the performance is strongly dependent on the plug thickness. Figure 11 illustrates the dependence of the plug thickness on applied velocity of the bypass damper shaft and magnetic field. As the velocity increases for a given magnetic field, the plug thins, and the pressure difference is reduced. For a given velocity, the plug can be thickened and the pressure

difference is increase by applying higher magnetic fields. The force versus velocity diagram from the flow mode damper analysis is shown in Figures 12 and 13, which illustrates the typical Bingham plastic and biviscous model characteristics. As increasing the applied currents, the force differences decrease, because the magnetic core tends to become saturated. Figure 9. Normal magnetization curves for Permalloy. Figure 10. Shear stress versus magnetic induction diagram for a typical MR fluid, MRF-132LD. (Lord Co., 1999) Figure 11. Plug thickness as a function of the applied Figure 12. Force versus velocity diagram for the damper current and applied velocity of the shaft. model based on Bingham plastic model.

2) Performance of the actuators Figure 14 shows the maximum performance of the actuator, as a function of driving frequency. The applied pressure is 250 PSI. As the frequency is increased, the maximum amplitude will be decreased but the maximum force remains constant. Considering the efficiency in this case, the maximum efficiency will occur at the maximum force, i.e. N 532≈= ps APF . Up to that point, the efficiency will linearly increase according to the applied force at the actuator shaft. Assuming that the MR valve follows the Bingham plastic model behavior, the results will be shown in Figure 15 at 10 Hz operations. Figure 15 says that the actuator can achieve 2 mm stroke and 100 N force at 10 Hz operation with the valve driving current of 0.8 ampere. Figure 16 shows the efficiency results based on equation (21). The force-amplitude relation for the biviscous model is shown in Figure 17. The driving current is held constant as 1.6 ampere. As the slope ratio increases, the performance of the hydraulic actuator and the efficiency will decrease, as shown in Figure 18.

7. EXPERIMENTAL RESULTS To validate this non-dimensional analysis using the nonlinear Bingham-plastic shear flow, a high stroke ( cm20≈ ) MR damper was constructed. The damper consists of four main parts: a hydraulic cylinder, industrial tube fittings, an accumulator and an MR bypass valve as pictured in Figure 19. The steel hydraulic cylinder is a modified hydraulic actuator with a 2.54cm (1 inch) bore, 5/8-inch shaft diameter and a maximum stroke of 8 inches. The main element of the MR valve is made from dumbbell shaped core, on which the coil wounded and steel cylinder shaped flux return. The gap between core and flux return was chosen to be 0.5 mm and the active length of the core was designed to be total 4mm. The damper is charged with

MR fluid, MRF-132LD (Lord Corporation). The accumulator connected to the cylinder increase the overall pressure inside the damper for the purpose of reducing the amount of air bubbles inside and protecting cavitations. For the experimental validation of the flow mode valve equations, force measurement from triangular displacement cycles were recorded on a MTS machine. The MTS machine shown in Figure 20 with the MR bypass valve damper mounted in the clevises, triangularly oscillates the shaft of the damper, measures the shaft displacement using an LVDT and measures the applied load using a 250 lb load cell. Figure 13. Force versus velocity diagram for the damper model Figure 14. Maximum force-amplitude characteristic based on biviscous model of 05.0=µ . of the hydraulic actuator as frequency varies. Figure 15. Force-amplitude characteristic of a hydraulic actuator Figure 16. Efficiency characteristics of the MR valve based on Bingham plastic model. (10 Hz) controlled hydraulic actuator. (10 Hz, Bingham-plastic model) Figure 17. Force-amplitude characteristic of a hydraulic actuator Figure 18. Efficiency characteristics of the MR valve based on biviscous model. (10 Hz) controlled hydraulic actuator. (10 Hz, Biviscous model)

Figure 19. Photograph of experimental setup for MR damper. Figure 20. Photograph of the construction for experimental setup.

Figure 21. A case of the experimental force and displacement response.

Figure 22. The experimental force versus velocity curves for Figure 23. The experimental force versus velocity curves for the MR damper is compared to theory based on the MR damper is compared to theory based on Bingham plastic model. biviscous model of 05.0=µ .

In Figure 21, a case of the experimental force and displacement response are plotted and the final processed results of force-velocity diagram for the valve are plotted in Figures 22 and 23 compared with analytical results. Figure 22 shows the results of Bingham-plastic model and Figure 23 shows the results of biviscous models assuming that 05.0=µ . The magnetic flux density of the valve according to the applied current is calculated by ANSYS/Emag 2D module. The yield stress data of MR fluid is applied from LORD corporation data sheet. To simplify the analysis, the MR fluid is assumed to have a nominal plastic viscosity, µ of 0.6 Pa sec.

8. CONCLUSION The driving force and efficiency are evaluated in the MR hydraulic power actuator system for both Bingham plastic and biviscous flows model. An MR valve is designed using a magnetic finite element analysis. The magnetic flux density developed in the MR valve are verified by analytical and experimental methods. As a result, the maximum magnetic flux density will go up to 0.56 Tesla. As you can see in Figure 7, the bottleneck in the magnetic flux will be the core shaft diameter. To increase the magnetic flux density at the gap, the shaft diameter of the bobbin should be increased. The yield stresses achieved in the MR valve due to the applied current are also measured to validate the design methodology. The overall performance of the MR fluid based hydraulic power system is described using the experimental MR valve performance data. If you refer the Figure 15, 2-mm stroke and 10 N will be achieved at 10 Hz with 0.8-ampere current input based on Bingham plastic model. In spite of there is no adjusting the material properties of field-dependent dynamic yield stress, Figure 22 shows that the simulation results are well agree with experimental results.

ACKNOWLEDGEMENT This work was supported under a NSF grant. The authors thank Monique Gabriel and Dr. Mark Jolly of Lord Corporation (Cary, NC) for providing the MR-fluid (MRF-132LD, 1999) used in this study.

REFERENCES

1. Zheng Lou, Robert D. Ervin and Frank E. Filisko, “Behaviors of Electrorheological Valves and Bridges,” Proceedings of the International Conference on ER fluid, 15-16 October 1991, pp.398-423.

2. Milgram, J. H. and Chopra, I., “Helicopter Vibration Reduction with Trailing-edge Flaps,” Proceedings of the 36th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference (New Orleans, LA), April 995, pp.601-612.

3. John Watton, Fluid Power System, Prentice Hall, 1989. 4. Nakano M. and Yonekawa T., “Pressure responses of ER fluid in a Piston Cylinder – ER valve system,” Proceedings of

4th International Conference on ER fluid, 20-23 July 1993, pp.477-489. 5. Whittle M. Atkin R. J. and Bullogh W. A., “Dynamics of an Electrorheological Valve,” Proceedings of the 5th

International Conference on ER fluid, 10-14 July 1995, pp.100-117. 6. Eckhard Wendt and Klaus W. Büsing, “A New Type of Hydraulic Actuator Using Electrorheological Fluids,”

Proceedings of 6th International Conference on ER fluid, 22-25 July 1997, pp.780-786. 7. Masami Nakano, Shuji Minakawa and Katsuya Hagino, “PWM Flow Rate Control of ER valve and Its Application to

ER Actuator Control,” Proceedings of 6th International Conference on ER fluid, 22-25 July 1997, pp.772-779. 8. Kamath, G. M., Hurt, M. K., and Wereley, N. M., “Analysis and Testing of Bingham Plastic Behavior in Semi-Active

Electrorheological Fluid Dampers,” Smart Materials and Structures, Vol. 5, No. 5, pp.576-590. 9. Wereley, N. M. and Lindler, J., “Biviscous Damping Behavior in Electrorheological Dampers,” Adaptive Structures and

Materials Systems, AD-Vol. 59/MD-Vol.87, ASME symposium, Nashville, TN, 14-19 November 1999, pp.67-75. 10. Herbert C. Roters, Electromagnetic Devices, John Wiley & Sons, Inc., 1941.