Volume-constrained optimization of ...

INSTITUTE OF PHYSICS PUBLISHING

SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 13 (2004) 1303–1313

PII: S0964-1726(04)86098-1

Volume-constrained optimization of magnetorheological and electrorheological valves and dampers Nicholas C Rosenfeld and Norman M Wereley Smart Structures Laboratory, Alfred Gessow Rotorcraft Center, Department of Aerospace Engineering, University of Maryland, College Park, MD 20742, USA

Received 14 July 2004, in final form 14 July 2004 Published 27 September 2004 Online at stacks.iop.org/SMS/13/1303 doi:10.1088/0964-1726/13/6/004

Abstract This paper presents a case study of magnetorheological (MR) and electrorheological (ER) valve design within a constrained cylindrical volume. The primary purpose of this study is to establish general design guidelines for volume-constrained MR valves. Additionally, this study compares the performance of volume-constrained MR valves against similarly constrained ER valves. Starting from basic design guidelines for an MR valve, a method for constructing candidate volume-constrained valve geometries is presented. A magnetic FEM program is then used to evaluate the magnetic properties of the candidate valves. An optimized MR valve is chosen by evaluating non-dimensional parameters describing the candidate valves’ damping performance. A derivation of the non-dimensional damping coefficient for valves with both active and passive volumes is presented to allow comparison of valves with differing proportions of active and passive volumes. The performance of the optimized MR valve is then compared to that of a geometrically similar ER valve using both analytical and numerical techniques. An analytical equation relating the damping performances of geometrically similar MR and ER valves in as a function of fluid yield stresses and relative active fluid volume, and numerical calculations are provided to calculate each valve’s damping performance and to validate the analytical calculations. (Some figures in this article are in colour only in the electronic version)

Nomenclature A1 A2 A3 Ad Ap Bi b C Ceq c d F Foff

Bobbin core cross-sectional area Flux return cross-sectional area Flange interior area Fluid annulus cross-sectional area (=bd) Valve face area (=π R 2 ) Bingham number Mean annular circumference Newtonian damping Equivalent damping Yield stress parameter Annular gap width Damping force Off-state (Newtonian) damping force

0964-1726/04/061303+11$30.00 © 2004 IOP Publishing Ltd

Fon hc Iap L LA LP Q Q1, Q2, Q3 R Rw ta tb V v0 vp Printed in the UK

On-state damping force Coil height Applied current Total valve length Active length Passive length Volume flux through valve Volume flux through volumes 1, 2, 3 Valve radius Coil resistance Bobbin core radius Flange height Total annular fluid volume Mean fluid velocity though annulus Piston velocity 1303

N C Rosenfeld and N M Wereley

wc γ˙ p p1 , p2 , p3 pµ pτ δ¯ µ τ τy (•)ER (•)MR (•)max

Coil width Shear rate Total pressure drop across valve Pressure drop across volumes 1, 2, 3 Pressure drop due to viscosity across valve Pressure drop due to yield stress across valve Non-dimensional plug thickness Viscosity Shear stress Yield shear stress Electrorheological parameter Magnetorheological parameter Maximum value

1. Introduction Over the past decade, the use of magnetorheological (MR) and electrorheological (ER) fluids as the fluid component in smart damping devices has been given much consideration. The effectiveness of smart MR and ER dampers has been demonstrated for a variety of applications including wheeled and tracked land vehicles [1, 2], helicopter rotor systems [3– 5], seismic hazard mitigation [6, 7], and vibration isolation systems [8]. MR and ER fluids are similar in that they develop a controllable yield stress in the presence of magnetic and electric fields, respectively. However, differences in fluid properties and actuation requirements of ER and MR fluids warrant comparison between the two. Physically, MR fluids can produce yield stresses on the order of ten times greater magnitude than ER fluids. A general comparison of MR and ER fluids by Carlson [9] demonstrated that this difference in yield stress will allow an MR device to provide the same pressure response as a significantly larger ER device. Recent comparative studies have dealt, wholly or in part, with the actuation properties of smart dampers, though this field is still largely unexplored. Gavin et al have demonstrated that the power requirement for a switching MR damper is lower than that for an equivalent switching ER damper [10]. This is particularly true when the ER fluid is subjected to high temperature or operating current, which will cause dielectric breakdown of the fluid. Choi and Wereley have shown that ER fluids have faster non-dimensional time response characteristics, though MR fluids’ response characteristics are favorably comparable [11]. Choi et al’s comparison of similar PID-controlled ER and MR clutches showed that the MR clutch produced a significantly higher controllable torque, while the ER clutch had more favorable response characteristics for torque regulation and tracking tasks [12]. However, the nature of the actuation methods utilizing ER and MR fluids necessitates other comparisons from a design standpoint. A controllable MR valve requires a controllable magnetic flux across the active fluid, which is often produced with coils of magnet wire. Controllable ER valves, meanwhile, need a controllable electric field across the active fluid, achieved by electrifying an electrode on one side of the valve and grounding the opposite electrode. In many cases, an MR valve’s magnetic coils are housed within the valve, adjacent to the annulus [13, 14]. If constraints are placed on the volume 1304

Figure 1. MR and ER valve comparison.

of the valve, the addition of the magnetic coils will cause an MR valve to have less active fluid volume than a similarly constrained ER valve. This paper will present a comparison of MR and ER valves placed under such a constraint. In order to present an unbiased comparison of MR and ER valves, two valves with the same geometric configuration were constructed. A volume-constrained MR valve optimization routine was used to set the geometry of the MR valve, while the corresponding ER valve was defined by replacing the MR valve’s wire coils and flux return with tubular metallic electrodes (figure 1). The MR valve was chosen as the initial design because MR valves’ applied field characteristics were sensitive to bobbin and flux return geometry and material selection [15], whereas ER valves’ applied field actuation characteristics were dependent only on the valve gap. The MR and ER valves were used to perform both analytical and numerical comparisons between the two systems.

2. MR valve geometric design MR and ER fluid valves operate on the same basic principle as a mechanical valve: a high-pressure fluid is forced through a small duct, resulting in a drop in pressure of the outgoing fluid. The pressure drop across a valve is caused by energy loss in the fluid due to plasticity and viscosity. When MR or ER fluid is activated within the duct, the development of the yield stress binds a volume of fluid into a solid plug. The remaining unbound fluid then experiences greater shear forces, which in turn results in a greater pressure drop across the valve. MR and ER fluids act not only as the fluid component of the system, but also as a semi-active control element. MR and ER valves are attractive because they allow controllable performance with no moving parts. In the context of this study, the MR and ER valves are used as dashpots to provide a usable, controllable damping force.

Volume-constrained optimization of magnetorheological and electrorheological valves and dampers

Figure 2. Magnetic flux through an MR valve.

The framework of the optimization procedure was based on physical design needs. A damper was considered with a fixed cylindrical volume allotted for a valve. It was assumed that the off-state damping requirement had been fixed (e.g., by safety requirements for minimum damping), and that the damper should achieve the greatest possible on-state damping. The optimization procedure was therefore designed to find a geometry which maximized the damping coefficient Ceq /C, the ratio of the magnitude of on-state damping to off-state damping. In all analyses, quasi-steady operation is assumed. Typically, ER and MR valves have only one annular gap through which fluid is forced (as drawn in figure 1), but this is not a strict constraint. Analyses of the damping properties of flow through single rectangular [16] and annular ducts [17] are well known for ER and MR fluids. Configurations with multiple, concentric annular gaps have been suggested for both ER [18] and MR [19] valves. A disk-type valve with active flow through a radial volume has been suggested for MR valves [13, 20], and such a configuration could be easily extended for use in ER valves. In general, for any particular ER valve configuration, an analogous MR valve configuration can be constructed, and vice versa. Although this paper focuses on a rectangular duct configuration (as an approximation to an annular duct), the conclusions reached from this simple example will be generalizable to any configuration. This will be strictly true for comparisons between similar configurations of ER and MR dampers. Comparisons between differing ER and MR configurations would need to take the differing total fluid volumes of the ER valve and MR valve into account. The MR valve was constrained within a cylindrical volume of fixed height and diameter. Within this volume, candidate geometries were constructed on the basis of the number of wire coils desired in the valve. These candidate geometries were imported into an ANSYS magnetic finite element analysis (FEA) routine and the magnetic flux densities through the MR valve were calculated. The flux values in the active regions were then compared with the specific MR properties of Lord MRF-132AD fluid, from which damping coefficient and nondimensional plug thickness were calculated for each valve. These performance indicators were used to evaluate which candidate design gave the best overall performance. The MR valve was shaped to guide the magnetic flux axially through the bobbin, across the bobbin flange and fluid gap at one end, through the flux return, and across the fluid gap and bobbin flange again at the opposite end (figure 2). The volume of fluid though which the magnetic field passes was defined as the active volume; it is only within this active

Figure 3. MR valve geometry nomenclature.

volume that MR effects occur. In order to make the valve most effective, it was desirable to have a high magnetic flux density passing through a large active volume. However, producing large magnetic fields required large numbers of magnetic coils. For a valve with a constrained volume, more volume devoted to magnetic coils translated directly to a smaller active volume. Moreover, more volume devoted to the coils left less volume for the magnetically permeable carrier materials, which in turn meant that the valve could saturate at high fields, sapping effectiveness [15]. An optimized circuit would maintain a balance between the field produced and power required by the magnetic coils, and a valve design that would make best use of the field to activate the MR fluid yield stress. For a simplified valve configuration, candidate geometries were defined as a function of the size of the magnetic coils. A viable candidate geometry was one in which the various critical areas though which the magnetic field passes were the same size. This was necessary to keep the magnetic flux density constant throughout the circuit, which ensured that one region of the magnetic circuit did not saturate prematurely and cause a bottleneck effect. There are three critical areas in the magnetic circuit: the circular cross-section of the bobbin core A1 , the annular cross-sectional area of the flux return A2 , and the cylindrical area at the interior of the bobbin flanges A3 . The geometry of the magnetic circuit was defined by the following dimensions: radius R, length L, bobbin core radius ta , bobbin flange height tb , gap width d, coil width wc , and coil height h c (figure 3). Thus, the critical areas were defined by


A1 = πta2

(1)

 A2 = π R 2 − (ta + wc + d)2

(2)

A3 = 2πta tb .

(3)

The volume constraint on the circuit was specified by prescribing maximum values for R and L. Small changes in the valve gap, d, would drastically alter the performance of different valves, so a fixed gap was also prescribed to ensure an unbiased evaluation. For these constraints, the optimized geometry of the valve could be calculated algebraically. Setting (1) equal to (2) and rearranging into the quadratic form for positive ta yielded    ta = 12 −(wc + d) + 2R 2 − (wc + d)2 . (4) Since R and d were prescribed, (4) allowed ta to be solved for a given coil width wc . Setting (1) equal to (3) gave tb = 12 ta .

(5) 1305

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Table 1. Geometries of candidate valves. Wraps

L A (mm)

Total coils

Critical area (mm2 )

Active volume (mm3 )

8 10 12 14 16 18 20 22

11.3 10.7 10.1 9.42 8.73 8.03 7.3 6.55

128 170 228 280 336 414 480 572

401 360 320 279 239 202 167 135

1131 1100 1066 1014 959 899 831 757

Since the magnetic field crossed the MR fluid along a length tb at either end of the valve, the sum of these lengths was defined as the active length, L A : L A = 2tb .

(6) Figure 4. Comparison of magnetic permeability of Hiperco and silicon steel.

Finally, the coil height was calculated from h c = L − 2tb .

(7)

The procedure for constructing a candidate valve geometry was as follows: for a valve with fixed values of R, L, and d, a coil width wc was chosen. From the preceding equations, ta and tb were calculated. Lastly, the coil height h c was calculated. It was noted that wc was the only variable necessary to characterize the geometry of the valve. As a rule, as wc increased, the size of the magnetic coils increased, the size of the critical areas decreased, and the size of the active volume of MR fluid decreased. That is, as the width of the coils increased, the size of the coils, and the magnitude of the magnetic flux that they created, increased. At the same time, the cross-sectional area of the bobbin decreased, causing the magnetic flux density of the circuit to increase. The volume of active MR fluid decreased, meaning that less fluid in the valve was subject to the MR effect. A valve was sought that would balance the magnitude of the flux density with the volume of active MR fluid in order to achieve optimal performance. In practice, the width and height of the wire coils could not be varied continuously; these dimensions were integer multiples of the diameters of the individual wire strands from which they are made. In this analysis, all wires were sized as 24-gauge (diameter = 0.516 mm). For convenience, the width of the coil was expressed as the number of circumferential wire layers, or wraps; for example, a ‘12-wrap’ coil had a coil width equal to 12 wire diameters. When calculating the height of the coils, the largest multiple of the wire diameter which did not violate the prescribed valve length constraint was chosen. The geometry optimization was conducted with volume constraints of L = 20 mm and R = 20 mm, and a prescribed gap width d = 1 mm. Valve geometries were calculated for 8–22 wraps. Table 1 gives a comparison of relevant geometries for each valve.

3. MR valve magnetic optimization The candidate geometries were imported into an FEM routine to evaluate their static magnetic performance. A twodimensional model of the magnetic valve was created for axisymmetric analysis in the ANSYS FEM program. The 1306

model itself consisted of three volume-constrained circuits stacked axially, with MR fluid filling the fluid gap, and free space (air) at either end of the circuit. The bobbin and flux return materials were both modeled as high-magneticpermeability Hiperco alloy and moderately permeable silicon steel. The permeability curves of both there materials were plotted in figure 4. The MR fluid was modeled using the fluid properties of Lord MRF-132AD fluid [21]. The wire coils were assumed to have free-space permeability. Following a passive actuation method, the model circuit was actuated using a specified current input. A per-wire current density was distributed over the corresponding coil area in the model; the middle coil was given a negative current density to simulate a reverse-wound coil. The coils were actuated with input amperages in the range of 0.5–2.5 A; the upper value was chosen as a reasonable maximum current that can be safely carried through 24-gauge copper wire. A circuit’s MR performance was calculated as follows. The FEM analysis of the circuit was used to calculate the distributed magnetic flux density over the active volume. The distributed flux density was averaged across the thickness and length of the annulus to calculate an average flux density, which is used in all subsequent analyses. Though recent findings have shown that the distribution of flux density across the gap thickness can have significant effects on damper performance [22], these non-uniform effects were neglected in order to simplify analysis. The average flux density in the active volume was compared with an empirical characterization of Lord MRF-132AD fluid to determine an average yield shear stress τy for the fluid. The yield stresses for the test cases ranged from 8.54 kPa for an 8-wrap silicon steel bobbin powered at 0.5 A to 41.7 kPa for a 14-wrap Hiperco bobbin powered at 2.5 A. The yield stress for each case was used to create a Bingham plastic model of the fluid, characterized by the equation τ = τy sgn(γ˙ ) + µγ˙

(8)

where τ is the shear stress of the fluid, γ˙ is shear rate, and µ is fluid viscosity. For simplicity, viscosity was chosen to be constant, µ = 0.33 Pa s. This assumption allowed the

Volume-constrained optimization of magnetorheological and electrorheological valves and dampers

the central passive volume was defined as volume 2 (figure 5). The lengths of volumes 1 and 3 were each half the active length L A , and the length of volume 2 was defined as the passive length L P . All volumes were assumed to have the same annular cross-section area Ad . The volume flux for region 2 was approximated from Newtonian analysis for a rectangular duct as Ad d 2 p2 . (12) Q2 = 12µL P

Figure 5. Volumes for damping coefficient calculation.

calculation of a characteristic Bingham number, Bi , for the valve [16]: τy (9) Bi = v0 µd where v0 =

Ap vp . Ad

(10)

Here Ap is the area of the piston face (i.e., π R ), Ad is the cross-sectional area of the fluid annulus, v0 is the mean fluid velocity though the fluid annulus, and vp is the velocity of the piston. Ap and Ad were set by the valves’ geometries and the piston velocity was chosen to be constant vp = 0.05 m s−1 . The Bingham number was then used to determine two nondimensional parameters, non-dimensional plug thickness and damping coefficient, to allow comparison of the valves. Nondimensional plug thickness expressed the percentage of the gap width which behaved as a solid plug due to the MR or ER effect. This parameter was related to the activation of the MR or ER fluid itself; larger values indicate stronger induced MR or ER ¯ was calculated by effects. Non-dimensional plug thickness, δ, solving for the root of the following equation [16]:   3 6 ¯ 1 ¯3 δ − + δ+1=0 0
δ¯
1. (11) 2 2 Bi 2

The volume fluxes through volumes 1 and 3 were approximated as that for a Bingham plastic flux through a rectangular duct [16]:   2 1 Ad d 2 p1   1 1 − δ¯ 1 + δ¯ (13) Q1 = 2 12µ 2 L A Q3 =

  2 1 Ad d 2 p3   1 1 − δ¯ 1 + δ¯ . 2 12µ 2 L A

In all cases p is the pressure drop along the length of each respective volume. As an aside, note that Ad = bd for the rectangular duct approximation, so Q i ∝ d 3 —this strong relationship between volume flux and gap was the impetus for prescribing equal gaps for all candidate geometries. Continuity dictated that the volume flux through each volume was the same. Furthermore, the volume flux was the product of the cross-sectional area of the piston and the velocity at which the piston displaced fluid. Therefore, Q 1 = Q 2 = Q 3 = Q = Ap vp .

(15)

Substituting (15) into (12)–(14) and solving for the individual pressure drops yielded p1 =

4. Damping coefficient derivation Damping coefficient expresses the gain in the equivalent damping of an active MR valve compared to the zero-field (Newtonian) condition. Most analyses of damping coefficient consider the gain in the active region of the valve only. This is valid for ER valves, where ER effects can be activated in the entire fluid volume. Often, this characterization is extended to MR valves under the assumption that the passive volume of the value has a much wider gap than the active volume, so that the passive volume’s contribution may be neglected for both ‘off’ and ‘on’ states. For the simplified geometries considered in this study, such an assumption does not hold. Moreover, the emphasis of this study on optimizing valves with respect to a constrained total volume suggested that the performance of the valves be non-dimensionalized with respect to their common total length as opposed to their varying active lengths. These issues were addressed by deriving an expression for damping coefficients which included the effects of the passive volume. This expression is characterized by the ratio of active length L A to total length L. For ER valves this ratio is equal to unity, while MR valves have ratios less than unity. In order to derive the damping coefficient, three volumes of the valve are identified: the active volumes at either end of the valve were defined as volumes 1 and 3, respectively, and

(14)

12µL A 1 Ap vp  2  2Ad d 2 ¯ 1 − δ 1 + 12 δ¯

(16)

12µL P Ap vp Ad d 2

(17)

1 12µL A Ap vp  . 2  2Ad d 2 1 − δ¯ 1 + 12 δ¯

(18)

p2 = p3 =

The total pressure drop for the valve was the sum of the individual pressure drops; furthermore, the total pressure drop was equal to the damping force F over the piston area: p =

3

i=1

pi =

F . Ap

(19)

Solving (19) for damping force gave   12µL A A2P 12µL P A2P 1 + F= vp  2  Ad d 2 Ad d 2 1 − δ¯ 1 + 12 δ¯ = Ceq vp . (20) Noting that the sum of L A and L P is the total valve length L, the equivalent damping Ceq was simplified as 

12µL A2p 1 LA Ceq = 1+ − 1 . (21)  2  Ad d 2 L 1 − δ¯ 1 + 1 δ¯ 2

1307

N C Rosenfeld and N M Wereley

(a)

(b)

Figure 6. Non-dimensional plug thickness versus power density as a function of the number of wraps in the MR valve. (a) Hiperco and (b) silicon steel.

The term outside of the brackets in (21) is the Newtonian (offstate) damping C for the valve. Dividing by this term gave the damping coefficient 

Ceq 1 LA (22) =1+ −1 .  2  C L 1 − δ¯ 1 + 1 δ¯ 2

Examining the limiting cases, as the active length approached zero the damping coefficient approached unity: Ceq =1 →0 C

lim

LA L

(23)

which was to say that the valve provided only Newtonian damping. Similarly, as active length approached total length the equivalent damping coefficient approached lim

LA L

→1

Ceq 1 = 2  C ¯ 1 − δ 1 + 21 δ¯

(24)

which was the result for Bingham plastic flux though a fully active rectangular duct [16]. These results were both physically expected, further validating the analysis.

5. Optimization results and analysis Non-dimensional plug thickness and damping coefficient were compared as a function of power density for each valve. Power density was defined as the electric power applied to the coils per unit active volume of fluid. The electric power applied to the coils was calculated as follows. The length of wire in the coils was approximated by assuming each radial and axial coil was a circular loop of wire and summing the circumferences of all the circular loops. The resistance of the wire was then calculated using the empirical resistance-per-unit-length value of 17.2 /100 m for 24-gauge copper wire. Power density was then calculated from Power density = 1308

2 Iap Rw

Ad L A

(25)

where Iap is applied current, and Rw is the calculated wire resistance. Note that as wrap number was increased, Rw increased and both Ad and L A decreased. Therefore, in cases where applied current was the same for different valves, valves with higher wrap numbers had much greater power densities. The results of the analysis were plotted in figures 6 and 7. The non-dimensional plug thickness for the Hiperco valves was shown in figure 6(a), where each curve represented the performance of an individual valve as applied power was increased. Taken individually, each valve showed an increase in plug thickness as more power was applied. Looking at the maximum achievable plug thickness for all valves, it was seen that the maximum plug thickness increased with wraps for 8 to 12 wraps, peaked at 14 to 16 wraps, and slowly decreased with wraps larger than 18. The low-wrap trend indicated poor performance due to a deficit in actuation power—fewer magnetic coils produced a smaller magnetic field with which to activate MR effects. The high-wrap trend, conversely, indicated poor performance due to too much actuation power; specifically, the larger coils and smaller critical areas led to a magnetic flux density that saturated the magnetic permeability of the carrier material, which in turn led to decreased performance. Figure 6(b) shows the same values for the silicon steel valves, which had a lower magnetic permeability compared to the Hiperco valves. The same trends as were seen for the Hiperco valve were also evident here, but the high-wrap decrease in maximum plug thickness was more pronounced. This difference provided more evidence that magnetic flux saturation was the limiting factor for large wrap numbers. The damping coefficient for the Hiperco valves was plotted in figure 7(a). The damping coefficient trends were qualitatively similar to the plug thickness trends: individual valves showed increased performance with increased applied power, and the maximum performance of the valves first increased, then decreased, with increasing number of wraps. However, valuable insight into the importance of valve geometry was gained by comparing the damping coefficients with their respective plug thicknesses. The high-wrap drop-off of the damping coefficient was more pronounced and began at a lower wrap number compared to the similar trend in plug

Volume-constrained optimization of magnetorheological and electrorheological valves and dampers

(a)

(b)

Figure 7. Damping coefficient versus power density as a function of the number of wraps in the MR valve. (a) Hiperco and (b) silicon steel.

Figure 8. Maximum damping coefficients and plug thicknesses versus candidate geometry.

thickness. This indicated that the decrease in the active length associated with high wrap numbers was quite significant to the overall performance of the valve. Specific examples provided more evidence of this effect. The 14-, 16-, and 18-wrap valves had essentially the same maximum plug thickness, but their damping coefficient decreased significantly as the number of wraps increased and the active length decreased. Furthermore, the 12-wrap valve, which had a smaller maximum plug thickness but a larger active length than the 14–18-wrap valves, had the largest damping coefficient of all test cases. Also, the 8-wrap valve, which had the lowest maximum plug thickness but the longest active length, outperformed the 18-, 20-, and 22-wrap valves in terms of maximum damping coefficient. The damping coefficients for the silicon steel valves were plotted in figure 7(b), and again, similar trends were seen with a more pronounced high-wrap drop-off compared to Hiperco. To provide clarity, the maximum damping coefficients and plug thicknesses for all candidate valves were plotted with respect to wrap number in figure 8. The optimized valve was chosen as the candidate geometry that gave the greatest damping coefficient. Coincidentally, for both the Hiperco and silicon steel valves, the 12-wrap valve is the optimized valve geometry. Plots of the non-dimensional plug thickness and

damping coefficient for the Hiperco and silicon steel optimized valves were plotted in figure 9. This MR design analysis showed that striking the correct balance between available actuation power and available actuation volume was key to optimizing the performance of a volume-constrained MR valve. In general, designing a volume-constrained valve to obtain only maximum active fluid volume (small wrap number valves) or only maximum actuation power (large wrap number valves) will not result in an optimal design. Furthermore, characterization parameters must be chosen which illustrate these optimization strategies. The damping coefficient contained a term which reflected the effect of the active length on the overall valve performance. Had this contribution been neglected by considering the damping ratio of the active regions only (i.e., normalizing the results over the active volume), the design procedure would have resulted in a non-optimal valve configuration.

6. ER/MR valve comparison Once the 12-wrap MR valve was chosen as the optimal valve geometry, it was constructive to compare the MR valve with a similar ER valve. ER and MR fluids, while qualitatively similar in their development of a field-controllable yield stress, differ in actuation methods and physical fluid properties. While MR fluids are actuated by guiding a magnetic field perpendicular to the fluid flow direction, ER fluids are actuated by applying an electric field perpendicular to the fluid flow direction. In practice, this is achieved by applying a voltage to the inner tubular electrode of the valve and grounding the outer tubular electrode of the valve. Unlike MR valves, which have a passive interior region, ER fluid is activated along the entire length of the valve. However, this larger active volume is offset by ER fluid having a lower maximum yield stress compared to MR fluid. In order to compare ER and MR fluids, separate analytical and numerical analyses were performed. The analytical analysis sought a simple expression for comparing general ER and MR damping coefficients for geometrically similar valves in terms of the yield stresses and active volumes of the fluids. The numerical analysis was used to produce results specific 1309

N C Rosenfeld and N M Wereley

(a)

(b)

Figure 9. The influence of magnetic materials on optimal MR design. (a) Non-dimensional plug thickness, (b) damping coefficient.

pτ /pη > ∼100) [9]. For an active length that is not equal to the total valve length, (27) can be rewritten as   L A cτy L pτ = . (28) L d

Table 2. Properties of generic ER/MR fluids.

τy,max (kPa) µ (Pa s) L A /L

ER

MR

5 0.33 1

50 0.33 0.505

to the optimal valve and also to validate the analytical model. Generic ER and MR fluids were used for these calculations; table 2 shows the relevant valve geometries and fluid properties for each case. The yield stresses chosen for the ER and MR fluids are the maximum and minimum values, respectively, from the analysis performed by Carlson et al [9] (i.e., this analysis presents the most conservative performance mismatch between ER and MR fluids; any other values chosen would increase the disparity between the valves). In order to simplify analysis, the fluid viscosities µ of the two fluids were assumed to have the same value as that used in the MR optimization. All geometric properties of the MR and ER valves were the same with the exception of the ratio of active length L A to valve length L. Previously, Carlson et al showed that ER dampers would require an active fluid volume two to three orders of magnitude larger than that of an MR damper to achieve a desired control ratio pτ /pµ at a given flow rate Q and controlled pressure drop pτ [9]. Starting from the same initial equations, a similar analysis was performed to compare the damping coefficients of MR and ER dampers with equal valve fluid volumes. From Bingham plastic analysis, the pressure drops across each valve were divided into two components, a fieldindependent viscous component pµ and a field-induced yield stress component pτ . Approximating the annular valve as a rectangular duct, 12µQ L pµ = (26) bd 3 cτy L pτ = d

(27)

where c is a parameter with a minimum value of 2 (for pτ /pη < ∼1) and a maximum value of 3 (for 1310

An expression for total fluid volume V of the valve was found by manipulation of equations (26) and (28). Using the rectangular duct approximation, V = Lbd   2   L pτ 12 µ Qpτ . = 2 c τy2 LA pµ

(29)

The damping force of the valve in the off-state (Newtonian) condition equals the viscous pressure drop distributed across the piston face area, which also equals damping times piston velocity: (30) Foff = pµ Ap = Cvp . Solving (30) for viscous pressure drop gave pµ = C

vp . Ap

(31)

Likewise, for the on-state condition, the damping force equals the viscous plus controlled pressure drops distributed across the piston face, which is also equal to equivalent damping times piston velocity:  Fon = pµ + pτ Ap = Ceq vp . (32) Substituting in (31) allows (32) to be solved for controlled pressure drop as  vp . pτ = Ceq − C Ap

(33)

Substituting (31), (33), and the definition of flow rate from (15) into (29) yields     2 L 2 Ceq 12 µ Cvp2 . (34) − 1 V = 2 c τy2 LA C

Volume-constrained optimization of magnetorheological and electrorheological valves and dampers

This is the total fluid volume necessary to achieve a desired damping coefficient Ceq /C for a set off-state damping C at piston velocity vp . For an MR device, the total fluid volume is VMR

12 = 2 c





µMR 2 τy,MR

L LA

2  MR

Ceq C

2

 −1

CMR vp2 .

MR

(35) For an ER device which activates fluid across the entire length of the valve, the total fluid volume is     2 Ceq 12 µER − 1 CER vp2 . (36) VER = 2 2 c C ER τy,ER Dividing (35) by (36) and setting the total volumes of fluid equal yields  1=

µMR 2 τy,MR





L LA

µER 2 τy,ER

2  MR

 

Ceq C

Ceq C



ER

2 − 1 CMR MR . 2 − 1 CER



Figure 10. MR/ER valve comparison at vp = 0.05 m s−1 .

(37)

ER and MR fluids can be assumed to have equal viscosities [9], and by setting the off-state damping of the devices equal, (37) can be rearranged as 2     −1 τy,MR 2 L A 2 MR = . 2   τy,ER L MR Ceq − 1 C



Ceq C



(38)

ER

Taking the square root of both sides while noting that, physically, damping coefficients must be equal to or greater than unity, (38) is rewritten as   Ceq   − 1 τ C LA y,MR  MR = . (39) Ceq τ L MR y,ER −1 C

ER

It can be shown that the term in the numerator and denominator of the left-hand side, (Ceq /C)−1, is exactly equal to the control ratio pτ /pµ as defined by Carlson. This value expresses the ratio of a valve’s controllable effects to its passive effects. Equation (39) shows that geometrically similar MR and ER dampers having the same passive damping performance will have a ratio of controllable performance that is directly proportional to the ratio of the yield stresses and the ratio of the MR valve’s active length to its total length. Note that for damping coefficients (Ceq /C)  1—i.e., for conditions where total damping is approximately equal to controllable damping—(39) approaches   Ceq    C τy,MR LA  MR = . (40) Ceq τy,ER L MR C

ER

Examining the relationships in (39) and (40), we see that the MR damping coefficient will be greater than the ER damping coefficient for the condition     τy,MR L > . (41) τy,ER L A MR

For the chosen generic fluids, MR fluid’s maximum yield stress is ten times greater than ER fluid’s maximum yield stress. By equation (41), we see that an MR valve with an active length larger than 1/10th the total valve length can achieve a greater damping coefficient than an ER valve of the same geometry operating at the same flow rate. The optimum active length in this study’s example is approximately half of the active length, so the optimized MR damper will outperform the geometrically similar ER damper in all cases. The analytical analysis demonstrated the effects of yield stress and active length on the relative controllable damping of MR and ER valves, but it did not allow calculation of the actual damping performance of the individual valves. In order to accomplish this calculation, a numerical analysis of each valve was performed to allow direct comparison of the valves’ exact performance. For numerical analysis, a piston velocity of vp = 0.05 m s−1 was chosen and the Bingham number for each fluid was calculated from (9) for a range of yield stresses up to τy,max . The maximum Bingham numbers for these conditions were Bi ER,max = 25 and Bi MR,max = 254. Recall that these values represent the least possible disparity between the general ER and MR fluids. Noting that non-dimensional plug thickness could be expressed as a function of Bingham number, the damping coefficient calculation (22) for each valve was simplified as Ceq LA =1+ [ f (Bi ) − 1] C L

(42)

where the damping coefficient for a fully active duct, as given by (24), was approximated by the rational function [23]   Ceq 1 = 2  C L A =1 1 − δ¯ 1 + 1 δ¯ L = f (Bi )

2

5.17Bi ∼ . (43) = 1.019 + 0.172Bi + 129 + Bi The damping coefficient was plotted in figure 10 as a function of Bingham number. The active length ratio dictated the slope of the plot for each valve. The ER valve, having a larger active length ratio, had a steeper slope; for a given Bingham 1311

N C Rosenfeld and N M Wereley

Figure 11. MR/ER valve comparison at τy,MR = τy,MR,max and τy,ER = τy,ER,max . Table 3. MR/ER damping coefficient comparison. Numerical Analytical Numerical vp (m s−1 ) (Ceq /C)ER (Ceq /C)MR (Ceq /C)MR MR/ER ratio 0.05 0.1 0.5

6.25 3.67 1.56

24.8 13.3 3.65

27.5 14.5 3.83

3.97 3.62 2.34

number, the ER valve provided a greater damping coefficient than the MR valve. However, the larger achievable yield stress of MR fluid resulted in a greater achievable Bingham number, which in turn allowed the MR valve to attain a larger maximum damping coefficient. Operating at maximum yield stress, the volume-constrained MR valve achieved a maximum damping coefficient (Ceq /C)MR,max = 24.8, nearly four times greater the ER valve’s (Ceq /C)ER,max = 6.25. A second numerical comparison was conducted by setting the yield stress to τy,max and varying piston velocity to illustrate the maximum achievable damping coefficients for piston velocities of vp = 0.05, 0.1, and 0.5 m s−1 . This analysis was plotted in figure 11; note that the range of Bingham numbers was the same as for the previous analysis. As piston velocity increased, the Bingham number decreased. For all cases, the MR valve produced a higher damping coefficient than the ER valve at the same operating condition. These results were then used to validate the analytical analysis. For each numerically calculated ER damping coefficient, (39) was used to calculate the corresponding MR damping coefficient. The analytically calculated MR damping coefficients match well with the numerically calculated MR coefficients. Table 3 shows a comparison of all calculated values. In the limiting case as piston velocity approached infinity, the MR and ER valves would both converge to a damping ratio of unity. This analysis demonstrated that for the given constrained volume, the optimized MR valve showed significant performance benefits over the geometrically similar ER valve in all conditions. While the ER valve had a greater active volume of fluid, the greater yield stress of the MR fluid translated directly into a larger achievable damping coefficient. Moreover, the analysis provided a criterion for comparing the performances of general MR and ER valves with similar 1312

geometries. An MR valve will outperform a similar ER valve for an easily attainable case where the ratio of maximum MR yield stress to ER yield stress is greater than the ratio of total fluid volume to active MR fluid volume. This fact, along with MR fluid’s comparatively greater operating temperature range, lower power requirements, and comparable time response characteristics as compared to ER fluid, makes MR fluid a more attractive choice for use in damping applications. As an end note, the comparison between differing volumeconstrained ER and MR configurations can be revisited in the light of this study’s findings. It has been concluded that, for the same fluid volume, an MR valve will allow greater controllable damping than an ER valve. Expanding this idea, it is clear that increasing the fluid volume within the given constrained volume will allow for higher damping performance. A multiannular or disk-type valve affords more fluid volume than a single-annulus valve. A recent study of multi-annular ER dampers [24] (i.e., relatively large total fluid volumes) suggests that such configurations can achieve damping performance comparable to that of a similar-sized, single-annulus MR damper [25] (i.e., relatively small total fluid volume). This, in turn, suggests that a multi-annular or disk-type MR valve would be able to give still higher performance. Thus, one can tentatively conclude that MR dampers will provide superior damping in any cross-configuration comparison.

7. Conclusions This study investigated a means of optimizing MR and ER smart valves and dampers in cases where the volume available for the valve was constrained. A procedure for developing simple candidate valve geometries within the constrained volume was presented. Magnetic FEM analysis was used to calculate the performance of each candidate valve and two parameters, non-dimensional plug thickness and damping coefficient, were used to characterize and compare the different geometries. The 12-wrap geometry was chosen as the optimal MR valve. This optimal MR valve was then compared with a geometrically similar ER valve through both analytical and numerical calculations. The major conclusions of this study were: (1) The performance of an MR valve is highly dependent on both actuation power and active fluid volume, so a volumeconstrained valve must balance these two factors. In the optimization procedure, the 8-wrap geometry (i.e., large active volume) had a maximum damping coefficient of ∼19.5 while the 22-wrap geometry (i.e., large actuation power) had a maximum damping coefficient of ∼14. The optimum 12-wrap geometry (balanced active volume and actuation power) had a damping coefficient of ∼22.5. (2) A given MR damper will have a larger damping coefficient than a geometrically similar ER damper in the case where the ratio of the achievable maximum MR yield stress to maximum ER yield stress is greater than the ratio of the total length of the valves to the active length of the MR valve. This study’s conservative maximum yield stress ratio of 10 is larger than the optimized total-to-active length ratio of ∼2, indicating that the MR damper will have the greater damping coefficient.

Volume-constrained optimization of magnetorheological and electrorheological valves and dampers

(3) The superior performance of the MR damper over a geometrically similar ER damper was validated through numerical analysis. Calculations of damping coefficient at vp = 0.5 m s−1 gave an MR maximum damping coefficient of 3.65, over twice the ER one’s maximum damping coefficient of 1.56. A slower speed of vp = 0.05 m s−1 gave an MR maximum damping coefficient of 24.8, nearly four times greater than the ER one’s 6.25 maximum damping coefficient. This optimized MR valve provided a greater range of controllable damping than a geometrically similar ER valve. This result, along with comparable or superior range of operating conditions, power requirements, and response characteristics as compared to the ER valve, makes MR technology more attractive for volume-constrained conditions.

References [1] Gordaninejad F and Kelso S P 2000 Fail-safe magneto-rheological fluid dampers for off-highway, high-payload vehicles J. Intell. Mater. Syst. Struct. 11 395–406 [2] Park D W, Choi S B, Suh M S and Shin M J 2001 ER suspension units for vibration control of a tracked vehicle Proc. SPIE 4327 159–64 [3] Kamath G M, Wereley N M and Jolly M R 1999 Characterization of magnetorheological helicopter lag dampers J. Am. Helicopter Soc. 44 234–48 [4] Gandhi F, Wang K W and Xia L 2001 Magnetorheological fluid damper feedback linearization control for helicopter rotor application Smart Mater. Struct. 10 96–103 [5] Zhao C, Choi Y T and Wereley N M 2004 Semi-active damping of ground resonance in helicopters using magnetorheological dampers J. Am. Helicopter Soc. 49 at press [6] Dyke S J, Spencer B F Jr, Sain M K and Carlson J D 1997 An experimental study of MR dampers for seismic protection Smart Mater. Struct. 7 693–703 [7] Hiemenz G, Choi Y T and Wereley N M 2003 Seismic control of civil structures utilizing semi-active MR braces J. Comput. Aided Civil Infrastruct. Eng. 18 31–44 [8] Choi Y T, Wereley N M and Jeon Y-S 2002 Semi-active vibration isolation using magnetorheological isolators Proc. SPIE Smart Structures and Materials 2002: Damping and Isolation vol 4697 pp 284–91; AIAA J. Aircr. at press [9] Carlson J D, Catanzarite D M and St Clair K A 1996 Commercial magnetorheological fluid devices Int. J. Mod. Phys. B 10 2857–65 [10] Gavin H, Hoagg J and Dobossy M 2001 Optimal design of MR dampers Proc. US Japan Workshop on Smart Structures for

[11]

[12]

[13] [14] [15]

[16]

[17]

[18] [19] [20]

[21] [22] [23]

[24] [25]

Improved Seismic Performance in Urban Regions (Seattle, WA, Aug. 2001) pp 225–36 Choi Y T and Wereley N M 2002 Comparative analysis of the time response of electrorheological and magnetorheological dampers using nondimensional parameters J. Intell. Mater. Syst. Struct. 13 443–52 Choi S B, Hong S R, Cheong C C and Park Y K 1999 Comparison of field-controlled characteristics between ER and MR clutches J. Intell. Mater. Syst. Struct. 10 615–9 Carlson J D and Chrzan M J 1992 Magnetorheological fluid dampers US Patent Specification 5,277,281 Gordaninejad F and Breese D G 1997 Magneto-rheological fluid damper US Patent Specification 6,019,201 B¨olter R and Janocha H 1997 Design rules of MR fluid actuators in different working modes Proc. SPIE Smart Structures and Materials 1997: Passive Damping and Isolation vol 3045 (Bellingham, WA: SPIE Optical Engineering Press) pp 148–59 Wereley N M and Pang L 1997 Nondimensional analysis of semi-active electrorheological and magnetorheological dampers using approximate parallel plate models Smart Mater. Struct. 7 732–43 Kamath G M, Hurt M K and Wereley N M 1995 Analysis and testing of Bingham plastic behaviour in semi-active electrorheological fluid dampers Smart Mater. Struct. 5 576–90 Stanway R, Sproston J L and El-Wahed A K 1996 Applications of electro-rheological fluids in vibration control: a survey Smart Mater. Struct. 5 464–82 Namuduri C S, Alexandridis A A, Madak J and Rule D S 2001 Magnetorheological fluid damper with multiple annular flow gaps US Patent Specification 6,279,701 Dogruer U, Gordaninejad F and Evrensel C A 2003 A new magnetorheological fluid dampers for high-mobility multi-purpose wheeled vehicle (MNNWV) Proc. SPIE Smart Structures and Materials 2002: Damping and Isolation vol 5052 (Bellingham, WA: SPIE Optical Engineering Press) pp 198–206 Lord Corporation 2003 MR Fluid Product Bulletins Online http://www.rheonetic.com/fluid begin.htm Gavin H 2001 Annular Poiseuille flow of electrorheological and magnetorheological fluids J. Rheol. 45 983–94 Hu W and Wereley N 2003 Nondimensional equivalent damping analysis of flow-mode magnetorheological and electrorheological dampers Proc. ASME Int. Mechanical Engineering Congress and Exposition (Washington, DC, Nov. 2003) ASME paper no IMECE2003-43135 Gavin H P 1998 Design method for high force electrorheological dampers Smart Mater. Struct. 7 664–73 Carlson J D and Spencer B F Jr 1996 Magneto-rheological fluid dampers for semi-active seismic control Proc. 3rd Int. Conf. on Motion and Vibration Control (Chiba, Japan, 1996) vol 3, pp 35–40

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