Research ArticleDynamic Model of MR Dampers Based ona Hysteretic Magnetic Circuit

Pengfei Guo 1 Jing Xie1 and Xinchun Guan 2

1School of Civil Engineering Xirsquoan University of Architecture and Technology Xirsquoan China2Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education Harbin Institute of Technology Harbin China

Correspondence should be addressed to Pengfei Guo pengfei guo126com

Received 29 December 2017 Revised 21 February 2018 Accepted 28 February 2018 Published 3 April 2018

Academic Editor Simone Cinquemani

Copyright copy 2018 Pengfei Guo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a key to understand dynamic performances ofMRdampers a comprehensive dynamicmagnetic circuitmodel is proposed in thiswork on the basis of Amperersquos and Gaussrsquos laws It takes into account not only the magnetic saturation which many existing studieshave focused on but also the magnetic hysteresis and eddy currents in a MR damper The hysteresis of steel parts of MR dampersis described by Jiles-Atherton (J-A) models and the eddy current is included based on the field separation Compared with theFEM results the proposedmodel is validated in low- and high-frequency studies for the predictions of the magnetic saturation thehysteresis and the effect of eddy currents A simple multiphysics model is developed to demonstrate how to combine the proposedmagnetic circuit model with the commonly used Bingham fluid model The damping force in the high-frequency case obviouslylags behind the coil current which exhibits a hysteresis loop in the current-force plane The lag of damping force even exists in alow-frequency varying magnetic field and becomes more severe in the presence of eddy currents

1 Introduction

The dynamic magnetic behavior of MR dampers cannot befully understood yet due to the lack of a comprehensiveelectromagneticmodel in which themagnetic saturation theferromagnetic hysteresis and the eddy current in the steelparts of MR dampers should be fully considered under atime-varying magnetic field

The finite element method (FEM) has been the maintool for the design and analysis of the magnetic circuit ofMR dampers with many previous studies focusing on themagnetic saturation analysis [1ndash5] In contrast with FEMthe electric circuit analogy method based on Amperersquos andGaussrsquos laws is much easier to implement flexible and lesstime consuming and proved to be particularly effective in theoptimization of magnetic circuits of MR devices [6 7]

Due to the hysteresis of ferromagnetic materials mag-netic induction follows different paths with increasing anddecreasing magnetic field and this ambiguous characteristicfurther complicates the predicting of the performances ofMR devices So far there have been limited studies onthe magnetic hysteresis modeling of MR devices And not

surprisingly few commercial FEM software tools support thehysteresis modeling except that ANSYS Maxwell starts toinclude such capability until its recent release [8] Severalresearch efforts have been made to integrate numeroushysteresis models such as Preisach model [9] and J-A model[10] into FEM for a general electromagneticmodeling [11 12]In this way Jedryczka et al [13] showed that the residualmagnetic flux density had a significant influence on thetorque of a MR clutch resulting in an increase of thefriction torque in the disengagement state An and Kwon[14] developed a nonlinear model of MR actuator using theHodgdon hysteresis model [15] for the magnetic circuit andBinghammodel for theMRfluidThey successfully simulatedthe electric current-torque hysteresis Instead of starting fromthe hysteresis of materials Deur et al [16] described theelectric current-torque hysteresis in a phenomenological waybased on the experimental data and showed that themodelingaccuracy was significantly improved after including the effectof hysteresis

From an energy point of view the effect of eddy currentcan be taken into account by incorporating in a physicalmodel the classical (macroscopic) and excess (microscopic)

HindawiShock and VibrationVolume 2018 Article ID 2784950 13 pageshttpsdoiorg10115520182784950

2 Shock and Vibration

eddy current losses with the total loss (119882) decomposed intohysteresis loss (119882ℎ) classical loss (119882119888) and excess loss (119882119890)that is

119882 = 119882ℎ +119882119888 +119882119890 (1)

The hysteresis loss (119882ℎ) increases linearly with frequencywhile the eddy current loss (119882119888) increases with the frequencysquaredThere is usually a discrepancy between themeasuredloss and the loss expected from the sum of hysteresis andeddy current losses and this is usually referred to as theanomalousexcess loss (119882119890) [17]

Thus under a high-frequency changing magnetic fieldthe eddy current in an electrically conducting materialparticularly in a solid electrically conducting material (egthe steel piston in aMR damper) is significant and delays themagnetic responses of devices Replacing the high conductiv-ity aluminum alloy by an insulating material Takesue et al[18] sharply improved the magnetic response of a magneticactuator by reducing the eddy current The existence of eddycurrent changed the inductance dynamics of a 200 kN large-scale MR damper as shown by Jiang and Christenson [19]and such effect should be considered in order to fully capturethe time-varying effective current that magnetizes the MRfluid Nam and Park [20] qualitatively analyzed the magneticresponse characteristic of a MR damper and concluded thatthe magnetic reluctance should be increased as much aspossible in order to improve the magnetic response Underthe assumptions of ignorance of the hysteresis Farjoud andBagherpour [21] recently performed a more detailed analysison the eddy current in MR devices The flux leakage andmagnetic saturation are taken into account by integratingthe static FEM into the whole Simulink model as an 119878-function However it is worth noting that the hysteresis lossin steel sheet accounts formore than 70of the total loss evenunder a 50Hz high-frequency varying magnetic field [22] sothe assumption of negligible hysteresis effect should be usedcarefully

In this paper a general dynamic magnetic circuit modelof MR dampers will be proposed It is expected to have thefollowing advantages compared to the available studies

(1)The proposed model is based on Amperersquos and Gaussrsquoslaws and is essentially a simple lumped parameter modelwith high computational efficiency achieved All the modelparameters have clear physical meanings and the resultsare easy to interpret Additionally the proposed magneticcircuit model can be easily extended by coupling it withmechanical models (eg Bingham model for MR fluid) torealize a multiphysics analysis

(2) By using J-A models to describe the ferromagneticmaterials (eg the piston and house cylinder of a MRdamper) the proposed model is a fully dynamic magneticcircuit model with the magnetic saturation hysteresis andeddy current all included This makes it possible to conductmore comprehensive studies on the dynamic performancesof a MR damper Moreover the proposed model can be alsoapplied to the permanent magnets based hybrid-magnetic-circuit [23] self-sensing [24] and energy-harvesting [25]MRdampers which receive more and more research interests

because permanent magnets [26 27] can also be accuratelycharacterized by J-A models

2 J-A Model for Ferromagnetic Materials

The J-A model [10] is based on physical premises concerningdomain wall movement in soft magnetic materials It isinteresting from both the theoretical and practical points ofview It offers insights into the details of the magnetizationprocessesThe formof J-Amodel has evolved over the last fewdecades and more and more physical phenomena have beentaken into account such as the effects frommechanical stressanisotropy temperature eddy current and dependency offrequency

J-A model consists of algebra-differential equations inwhich the magnetization (119872) is either calculated from themagnetic induction (119861) or from themagnetic field (119867)Thesetwo forms 119872(119867) and 119872(119861) are equivalent and the modelparameters share the same set of values The 119872(119867) form ismore convenient to characterize a material while the 119872(119861)form referred to as the inverse J-A model is preferred in amagnetic circuit analysis in which magnetic induction (119861)is often used as the independent variable Hereafter the J-Amodel refers to the119872(119867) form unless otherwise stated

As shown by Zirka et al [28] the dynamic J-A modelcan be expressed in a convenient form after some algebraicmanipulations

119889119872119889119867 = 119873119896120575 + 119867119889 minus 120572119873 (2)

where

119873 = 120575119872 (119872an minus119872) + 119896120575119888119889119872an119889119867119890 (3)

and the anhysteretic magnetization is given by the Langevinfunction

119872an = 119872119904 [coth(119867119890119886 ) minus 119886119867119890 ] (4)

the effective field

119867119890 = 119867 + 120572119872 (5)

the dynamic field

119867119889 = 119903119888 119889119861119889119905 + 119903119890120575 100381610038161003816100381610038161003816100381610038161198891198611198891199051003816100381610038161003816100381610038161003816100381612

(6)

and the directional parameter 120575 = sign(119889119867119889119905) The param-eter 120575119872 is introduced to eliminate the unphysical negativesusceptibility and calculated by

120575119872 = 0 (119872an minus119872) 119889119867119889119905 lt 01 otherwise

(7)

Model parameters 119872119904 120572 119886 119888 119896 119903119888 119903119890 are obtained frommeasurements with their physical meanings listed in Table 1

Shock and Vibration 3

J-A model

Nonlinear least squares method

Criterion function

Initial parameters

Measured H-B

SimulatedH-B Output

J-A parameters

x ＬＬＩＬ gt

ＬＬＩＬ lt (x0)

(xlowast)

Figure 1 Block diagram of the identifications of J-A model parameters

Table 1 Physical meanings and estimated value ranges of J-A model parameters

Model parameter Physical meaning Value range [31]119872119904 Saturation magnetization 119872tipsim15119872tip120572 Domain coupling 05(119867119888119872tip)sim07(119867119888119872tip)119886 Domain density 05119867119888sim5119867119888119888 Domain wall bowing 0sim1119896 Primarily determining the coercivity 05119867119888sim5119867119888119903119888 Determining classical eddy current loss mdashmdash119903119890 Determining excess eddy current loss mdashmdash119872tip magnetization of loop tip119867119888 coercivity

The corresponding inverse J-A model showed a similarform [28] except that

119889119872119889119861 = 1198731205830 [119896120575 + 119867119889 + (1 minus 120572)119873] (8)

and 119889119861119889119905 rather than 119889119867119889119905 is used in parameters 120575 and120575119872 1205830 is the permeability of free spaceIn addition efforts also have been devoted to improving

the accuracy and convergence of J-A model For exampleZirka et al [28] suggested that 119903119890 = 1199031198901(1 + 11990311989021198612) inorder to refine the accuracy of J-A model under high-frequency magnetic fields Wilson et al [29] proposed thatthe convergence can be improved by making parameter 119896dependent on the applied field in a formofGaussian function

119896 = 1198960119890minus(119867221205902) (9)

where 1198960 is the default value of the 119896 parameter in the originalJ-A model 119867 is the applied field and 120590 is the standarddeviation of the Gaussian function

Before using J-A model in the magnetic circuit of MRdampers it is safe to validate its effectiveness of charac-terizing magnetic hysteresis Kis and Ivanyi [30] tested themagnetic hysteresis of C19 structural steel by using a two-coil measurement system in which a prescribed magneticfield was generated by the primary coil and the magneticflux density was calculated from the induced voltage on thesecondary coilMore details of themeasurement can be foundin the references therein Here their experimental data isused for investigating the effectiveness of J-A model and a

simple least-squares method is used to determine the modelparameters with the block scheme of identification shownin Figure 1 The ranges of parameters proposed by Chwastekand Szczygłowski [31] in Table 1 are used here and the initialparameters (x0 in Figure 1) are chosen as the medians of theranges

The following criterion function is used to generate anerror by comparing the experimental and the simulatedhysteresis

Error = 1119899radic119899sum119894=1

(119861exp119894 minus 119861sim119894119861max)2 (10)

where 119899 is the number ofmeasurement points119861exp119894 and119861sim119894are the 119894th measured and simulated magnetic induction and119861max is the maximum of the measured magnetic induction

The detailed identification process of the model parame-ters is as below

(a) Input the initial guesses of model parameters(b) Using the adopted parameter values solve the J-

A model and obtain the simulated magnetic flux densities(119861sim119894) at different magnetic fields points specified by mea-surements

(c) Compare the simulated results (119861sim119894) with the mea-sured ones (119861exp119894) and calculate the error using (10)

(d) If the error is larger than the tolerance (120576) thenonlinear least-squares method automatically modifies theinput values of model parameters If the error is below thetolerance the corresponding parameter values are acceptedand the identification process is finished

4 Shock and Vibration

MeasuredSimulated

minus4000 minus2000 0 2000 4000

02 Hzminor loop

H (Am)

02 Hz major loop

50 Hz

minus1

0

1

B (T

)

minus2

minus1

0

1

2

B (T

)

minus2

minus1

0

1

2

B (T

)

minus4000 minus2000 0 2000 4000

minus4000 minus2000 0 2000 4000

Figure 2 The simulated and measured [30] hysteresis curves fordifferent frequencies

After about 50 iterations the parameters are identified as120572 = 21502 times 10minus3 119872119904 = 14334 times 106 119886 = 11143 times 103119888 = 33493 times 10minus1 1198960 = 63306 times 102 120590 = 14149 times 102119903119888 = 10505 times 101 1199031198901 = 13827 times 10minus1 1199031198902 = 11271 times 10minus1As shown in Figure 2 the model results agree well with theexperimental data so J-A model is effective and will be usedin the following sections to describe the magnetic hysteresisof the steel parts in MR dampers

3 A Dynamic Magnetic Circuit Model ofMR Dampers

As shown in Figure 3 the magnetic circuit of a MR damperis composed of a gap and three steel parts (denoted by AB and C) An independent current source is consideredhere because current drivers are preferred for faster magneticresponses [32]

For simplicity a static J-A model in which the dynamicfield (119867119889) vanishes will be used to describe the steel parts ofMR dampers and the effect of eddy current will be includedlater in the magnetic circuit modeling Another main reasonfor choosing a static J-Amodel is that the hysteresis algorithmin Maxwell software which will be compared with themagnetic circuit model proposed in this paper is still ablack box with little information available However such atemporary simplificationwill not undermine the generality ofthe proposed model because the dynamic hysteresis modelwould be recovered by simply restoring the dynamic field(119867119889)

The steels are described by the following three inverse J-Amodels which constitute three differential algebraic equations(DAEs) in terms of the magnetizations (119872119894)

119889119872119894119889119861119894 = 1198731198941205830 [119896120575 + (1 minus 120572)119873119894] 119894 = 1 2 3 (11)

g

III

II

IRd

Rp

Rb

Lp

Lb

i

①

③

③

③

③

②

②

Figure 3 Typical magnetic circuit of a MR damper

Similarly the magnetic saturationhysteresis of MR fluid canbe described by introducing another J-Amodel However themagnetic permeability of MR fluid is very low in comparisonwith that of steels so it is neglected here and regarded as airWithout loss of generality the piston and house cylinder areassumed to bemade of the same steel so the three J-Amodelsshare the same set of parameters

If the effects of magnetic flux fringing and leakage areignored according to Gaussrsquos law the magnetic inductions inthe different parts of the circuit are related by

119861119892119860119892 = 119861119894119860 119894 = Φ 119894 = 1 2 3 (12)

where 119861119892 and 119861119894 are the average magnetic inductions in theair gap and steels Φ is the magnetic flux in the circuit loopas shown in Figure 31198601 and1198602 are the cross-sectional areasof the piston core and the house cylinder 1198603 is the averagelateral area of the piston head (ie the lateral area at positionIII in Figure 3) and 119860119892 is the average lateral area of theannular gap These four areas are given by

1198601 = 12058711987721198871198602 = 120587 (1198772119889 minus (119877119901 + 119892)2) 1198603 = 2120587(119877119887 + 119877119901 minus 1198771198872 ) (119871119901 minus 119871119887) 119860119892 = 2120587(119877119901 + 1198922 ) (119871119901 minus 119871119887)

(13)

with the meanings of all the variables shown in Figure 3The flux leakage fringing flux and the nonuniformity

of magnetic fieldinduction as shown in Figure 4 are stillchallenging to be accurately considered and their combinedeffect is reflected here by introducing parameter 119896119891119886 for thegap and 119896119891119894 for the steels that is

119896119891119886119861119892119860119892 = 119896119891119894119861119894119860 119894 = Φ 119894 = 1 2 3 (14)

Shock and Vibration 5

Nonuniformmagnetic field

House cylinder

Fringingflux Piston

head

Leakage flux

Figure 4The fringing flux leakage flux and nonuniformmagneticfield near the working gap of a MR damper

The governing equation of the wholemagnetic circuit canbe obtained by Amperersquo law as

3sum119894=1

119867119894119871 119894 + 119867119892119892 = 119868tot (15)

where 119868tot is the total current in the source 119871 119894 is the magneticcircuit length of the steel parts 1198711 = 1198712 = (119871119901 + 119871119887)2 1198713 =119877119901 minus 119877119887 119892 is the gap width The magnetic field in differentsteel parts (119867119894) are coupled to the corresponding J-A modelby

119867119894 = 1198611198941205830 minus119872119894 119894 = 1 2 3 (16)

And the magnetic field in the gap is simply calculated by

119867119892 = Φ1198601198921205830 (17)

In a magnetic circuit analysis the effect of either classicalor excess eddy current can be taken into account based on thefollowing field separation [28 33 34]

119867tot = 119867 +119867119888 + 119867119890 (18)

where the total magnetic field (119867tot) is the sum of thehysteretic magnetic field (119867) the dynamic field due toclassical eddy current (119867119888) and the dynamic field concerningexcess eddy current (119867119890) In the cases of low frequenciesandor magnetic circuits with small cross-sections the mag-netic field and induction can be assumed to be uniformConsequently the field separation is equivalent to the afore-mentioned loss separation (1) and the dynamic fields can beestimated by [35 36]

119867119888 = 11988922120588120573 119889119861119889119905 119867119890 = (1198661198891199081198670120588 ) sign(119889119861119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198891199051003816100381610038161003816100381610038161003816100381612

(19)

where 120588 is the resistivity 119889 is the cross-sectional dimen-sion (thickness for laminations diameter for cylinders andspheres) and 120573 is a geometrical factor which varies form120573 = 6 in laminations to 120573 = 16 in cylinders and 120573 = 20 inspheres

In order to keep the generality of the proposedmodel thegeneral form (6) is used for the dynamic magnetic field thatis

119867119888119894 = 119903119888119894 119889119861119894119889119905 119894 = 1 2 3119867119890119894 = 119903119890119894 sign(119889119861119894119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198941198891199051003816100381610038161003816100381610038161003816100381612 119894 = 1 2 3 (20)

Only the dynamic field due to classical eddy current will beconsidered for simplicity Accordingly (15) is extended toinclude the effect of classical eddy current

3sum119894=1

(119867119894 + 119867119888119894) 119871 119894 + 119867119892119892 = 119868tot (21)

Finally (11) and (21) expressed in terms of four inde-pendent variables (119872111987221198723 Φ) constitute a completemagnetic circuit model of MR dampers This model isessentially a set of DAEs and can be numerically solved bytheMatlab built-in ODEDAE solver ode15i with the relativeerror tolerance and absolute tolerance respectively set to1e minus 5 and 1e minus 64 Model Validation and Analysis

The effectiveness of the proposed magnetic circuit modelis investigated for describing the low- and high-frequencymagnetic performances of a MR damper with the mainstructural parameters chosen as 119877119889 = 346mm 119877119901 = 30mm119877119887 = 20mm 119892 = 06mm 119871119887 = 25mm and 119871119901 = 30mmThecoil has 284 turns The current driver supplies a sinusoidalcurrent with an amplitude of 2A and a frequency of 02Hzin the low-frequency study and 10Hz in the high-frequencystudy

For the validation of the proposed magnetic circuitmodel it should be noted that although comparison withexperimental data is always the most direct way it is often(eg in the predesign stage of a MR damper) not a preferredway for validation because the magnetic response of a MRdamper indispensable for identifying the parameters of alumped parameter model is not available until the damperis finally fabricated and then tested In this case a FEMsimulation although coming with drawbacks such as beingtime consuming and hard-to-interpret is often a practicaland reliable alternative

More importantly although validation by an indirect testof damping forces is always much easier it should not be apreferredway for validating amagnetic circuitmodel becausethe damping force itself is greatly dependent on many factorssuch as the compressibility ofMR fluid and the compliance ofinstrument systemMoreover the effect of stress onmagnetichysteresis is often unavoidable which further makes the

6 Shock and Vibration

Part ① Part ② Part ③

FEMProposed model

FEMProposed model

FEMProposed model

B(T

)

B(T

)

B(T

)

06

04

02

00

minus02

minus04

minus06

06

04

02

00

minus02

minus04

minus06

H (Am) H (Am)H (Am)

09

06

03

00

minus03

minus06

minus0910000minus1000 10000minus1000 10000minus1000

Figure 5 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under a low-frequency (02Hz) sinusoidal electric current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute errortolerance 1e minus 6 nonlinear residual of FEM solver 1e minus 5)

damping force an unsuitable indicator to validate a magneticcircuit model

Since the FEM simulation will be used to validate theproposedmodel the FEMresults will act as the ldquoexperimentaldatardquo in (10) and the similar error will be calculated as

Error = 1119899radic119899sum119894=1

(119861lum119894 minus 119861fem119894119861FEMmax)2 (22)

where 119861fem119894 is the 119894th magnetic induction obtained byFEM 119861lum119894 is the 119894th magnetic induction obtained by theproposed model 119861FEMmax is the maximum FEM magneticinduction and 119899 is the number FEM results It should benoted that the summation index 119894 implicitly includes allthe average FEM results at three parts (lines I II and III inFigure 3)

ANSYS Maxwell employs a vector hysteresis model topredict the hysteresis loops and losses for soft and hardmagnetic materials and permanent magnets Hysteresis isautomatically activated by properly setting the material dataand the detailed material settings in ANSYS Maxwell aregiven below

(a) Specify the steel parts of the damper as materials withnonlinear relative permeabilities and edit the 119861119867 curves

(b) Input the descending branch of the saturation hys-teresis loop It should be noted that the first point and thelast point must mirror each other otherwise the curve isconsidered invalid

(c) After finishing the 119861119867 curve input a coercivity valueis automatically calculated and listed

(d) Set all three components of the ldquoUnit vectorrdquo to zerootherwise it acts like a permanent magnet

41 Low-Frequency StudyMagnetic Saturation andHysteresisUnder a low-frequency (02Hz) current excitation the effectof eddy current is insignificant and the magnetic hystere-sissaturation predicting capability of the proposed model isexamined in this case Accordingly the three parameters 11990311988811199031198882 1199031198883 related to the dynamic field vanish

The magnetic responses in the three steel parts are sim-ulated by the proposed model with the parameters identifiedas 120572 = 11440 times 10minus3 119872119904 = 11527 times 106 119886 = 63750 times 102119888 = 22428 times 10minus1 119896 = 63750 times 102 119896119891119886 = 13284 and1198961198911 = 1198961198912 = 1198961198913 = 10000 The three average cross-sectionalmagnetic responses at positions I II and III (Figure 3) arealso obtained inMaxwell (v160) for comparison As shown inFigure 5 the results predicted by the proposed model are in asatisfactory agreement with the FEM results so the proposedmagnetic circuit model is reliable for describing the staticmagnetic hysteretic behavior of MR dampers

It can be seen from values of the parameters (1198961198911 = 1198961198912 =1198961198913 = 10000) that the combined effects of magnetic fringingleakage and nonuniformity of magnetic fieldinduction arenegligible in all the three steel parts and will be ignored inthe following high-frequency study

42 High-Frequency Study Effect of Eddy Current The high-frequency study is performed under a sinusoidal current of10Hz and the maximum of the eddy current density is oneorder ofmagnitude larger than that under a current of 02Hzas shown in Figure 6

Because the five J-A material parameters have beenalready identified in the low-frequency study only fourparameters (1199031198881 1199031198882 1199031198883 119896119891119886) are left to be determined Using

Shock and Vibration 7

J (AＧ2)75E + 004

70E + 004

65E + 004

60E + 004

56E + 004

51E + 004

46E + 004

41E + 004

36E + 004

31E + 004

26E + 004

21E + 004

16E + 004

11E + 004

57E + 003

75E + 002

(a) Low-frequency study 02Hz

J (AＧ2)27E + 006

25E + 006

23E + 006

21E + 006

19E + 006

18E + 006

16E + 006

14E + 006

12E + 006

10E + 006

85E + 005

67E + 005

49E + 005

31E + 005

12E + 005

minus58E + 004

(b) High-frequency study 10Hz

Figure 6 Eddy currents at half-period instants (ie time 119905 = 1198792)

Part ① Part ② Part ③

H (Am)H (Am) H (Am)10000minus1000 10000minus1000 10000minus1000

minus050

minus025

000

025

050

B(T

)

B(T

)

B(T

)

minus070

minus035

000

035

070

minus050

minus025

000

025

050

FEMProposed model

FEMProposed model

FEMProposed model

Figure 7 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under high-frequency (10Hz) sinusoidal current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute error tolerance1e minus 6 nonlinear residual of FEM solver 1e minus 5)

an identification method similar to that shown in the abovesection they are obtained as 1199031198881 = 16650 times 102 1199031198882 =27785 times 101 1199031198882 = 48578 times 101 and 119896119891119886 = 10593 As shownin Figure 7 the predicted results in the steel parts show anoverall satisfactory agreement with the FEM results althougha larger discrepancy observed as compared to the results inthe low-frequency study

The numerical errors in low- and high-frequency studiescalculated by (22) are compared in Figure 8 The agree-ment between FEM and the proposed model results in the

low-frequency study is much better than that in the high-frequency study with the largest error occurring in steel partA

The larger discrepancy or error especially that in steelpart A is believed to be the consequence of the largernonuniformity ofmagnetic induction due to the effect of eddycurrent The hysteresis curves along lines I II and III (inFigure 3) are shown in Figure 9 with obvious nonuniformitiesof magnetic fields observed under high frequency Comparedto the nonuniformities of magnetic fields in parts B and C

8 Shock and Vibration

Different parts 02 HzDifferent parts 10 Hz

Whole damper 02 HzWhole damper 10 Hz

Part 3Part 2Part 1

002

003

004

005

Erro

r

006

007

008

Figure 8 Comparison of numerical errors under low- and high-frequency studies

the nonuniformity in partA is much larger so using averagevalues in the lumped parameter model produces largererror

Moreover the comparison between the two groups ofhysteresis curves in Figures 8 and 9 shows that the magneticinduction which directly determines the yields stress of MRfluid is apparently reduced under high-frequency changingfields due to the effect of eddy currents

5 A Demonstration of How to Usethe Proposed Magnetic Circuit Model

For the purpose of the demonstration of how to use theproposed magnetic circuit model a simple multiphysicsmodel will be developed in this section The flow field in aMR damper can be controlled by the magnetic field due tochanges of the shear yield-strength of MR fluid In contrastthe magnetic field is generally assumed to be independentof the fluid flow field so the analyses of these two fieldscan be conducted separately in series The experimentalyield stresses (120591119910) of the MR fluid are shown in Figure 10under different magnetic inductions (119861) and their relation-ship which is required for the damping force modelingcan be obtained by a least square curve fitting methodas

120591119910 = 5483033 [1 minus exp (minus314 |119861|203)] (23)

As shown in Figure 10 the fitted curve agrees well with theexperimental data

As shown in Figure 11 after validating the proposed mag-netic circuit model the damping force (119865) and the flow field(119906) of aMR damper can be analyzed by substituting the yield-strength history (obtained from the magnetic circuit model)into any extensively studied fluid model such as the Bingham

model theHerschel-Bulkleymodel and the biviscousmodelThe simple Bingham fluid model is adopted here for thedemonstration purpose Since the proposed magnetic circuitmodel has been validated in previous sections the resultsfrom the combinations of these two reliable models shouldbe also reliable

Describing the flow of MR fluid by the Bingham flowequations (ie combination of the Navier-Stokes equationsand the Bingham constitutive equation) in a rectangular ductthe damping force (119865) of a MR damper can be related to thepiston velocity (V119901) by the following mechanical model [37]

P (T) = 23 (1 + 3T)sdot [cos(13 arccos(1 minus 54 ( T1 + 3T)3)) + 12]

(24)

with the dimensionless pressure gradient (P) and yield stress(T) defined by

P = 1199081198923Δ11990112119876120578119871 T = 119908119892212059111991012119876120578

(25)

where 119908 is the mean circumference of the annular flow pathΔ119901 is the pressure drop responding to the volumetric flowrate 119876 = V119901119860119901 119860119901 is the effective area of the piston headthe damping force 119865 = Δ119901119860119901 and 120578 is the postyield plasticviscosity and taken as 13 Pasdots

Using the time history of the shear yield stress (23) as thebridge connecting the proposed magnetic circuit model andthe Binghamfluid basedmechanicalmodel (24) the dampingforces corresponding to the low- and high-frequency studiesin the above section are examined with the piston velocitychosen as 10mms

As shown in Figure 12 compared to the results of thelow-frequency study an obvious damping force lag withrespect to the coil current is observed in the high-frequencystudy due to the effect of eddy currents Such lag exhibitsa hysteresis loop in the current-force plane as shown inFigure 13 which even exists in a low-frequency varyingmagnetic field because of the magnetic hysteresis Thishysteresis behavior produces a variation in the damping forcebetween an increasing and decreasing coil current Thus in adynamic environment such as oscillatory current excitationuncertainty is introduced in any calculated torque Worsestill the eddy currents make such hysteresis more severe Asexpected an apparent decrease in the maximum dampingforce is also observed in the high-frequency study becauseof the decreasing maximummagnetic induction as shown inFigures 5 and 7

For better understanding of dynamic performances ofMR dampers it is often desirable to gain an insight intothe flow analysis of MR fluid in the working gap and this

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

Shock and Vibration 3

J-A model

Nonlinear least squares method

Criterion function

Initial parameters

Measured H-B

SimulatedH-B Output

J-A parameters

x ＬＬＩＬ gt

ＬＬＩＬ lt (x0)

(xlowast)

Figure 1 Block diagram of the identifications of J-A model parameters

Table 1 Physical meanings and estimated value ranges of J-A model parameters

Model parameter Physical meaning Value range [31]119872119904 Saturation magnetization 119872tipsim15119872tip120572 Domain coupling 05(119867119888119872tip)sim07(119867119888119872tip)119886 Domain density 05119867119888sim5119867119888119888 Domain wall bowing 0sim1119896 Primarily determining the coercivity 05119867119888sim5119867119888119903119888 Determining classical eddy current loss mdashmdash119903119890 Determining excess eddy current loss mdashmdash119872tip magnetization of loop tip119867119888 coercivity

The corresponding inverse J-A model showed a similarform [28] except that

119889119872119889119861 = 1198731205830 [119896120575 + 119867119889 + (1 minus 120572)119873] (8)

and 119889119861119889119905 rather than 119889119867119889119905 is used in parameters 120575 and120575119872 1205830 is the permeability of free spaceIn addition efforts also have been devoted to improving

the accuracy and convergence of J-A model For exampleZirka et al [28] suggested that 119903119890 = 1199031198901(1 + 11990311989021198612) inorder to refine the accuracy of J-A model under high-frequency magnetic fields Wilson et al [29] proposed thatthe convergence can be improved by making parameter 119896dependent on the applied field in a formofGaussian function

119896 = 1198960119890minus(119867221205902) (9)

where 1198960 is the default value of the 119896 parameter in the originalJ-A model 119867 is the applied field and 120590 is the standarddeviation of the Gaussian function

Before using J-A model in the magnetic circuit of MRdampers it is safe to validate its effectiveness of charac-terizing magnetic hysteresis Kis and Ivanyi [30] tested themagnetic hysteresis of C19 structural steel by using a two-coil measurement system in which a prescribed magneticfield was generated by the primary coil and the magneticflux density was calculated from the induced voltage on thesecondary coilMore details of themeasurement can be foundin the references therein Here their experimental data isused for investigating the effectiveness of J-A model and a

simple least-squares method is used to determine the modelparameters with the block scheme of identification shownin Figure 1 The ranges of parameters proposed by Chwastekand Szczygłowski [31] in Table 1 are used here and the initialparameters (x0 in Figure 1) are chosen as the medians of theranges

The following criterion function is used to generate anerror by comparing the experimental and the simulatedhysteresis

Error = 1119899radic119899sum119894=1

(119861exp119894 minus 119861sim119894119861max)2 (10)

where 119899 is the number ofmeasurement points119861exp119894 and119861sim119894are the 119894th measured and simulated magnetic induction and119861max is the maximum of the measured magnetic induction

The detailed identification process of the model parame-ters is as below

(a) Input the initial guesses of model parameters(b) Using the adopted parameter values solve the J-

A model and obtain the simulated magnetic flux densities(119861sim119894) at different magnetic fields points specified by mea-surements

(c) Compare the simulated results (119861sim119894) with the mea-sured ones (119861exp119894) and calculate the error using (10)

(d) If the error is larger than the tolerance (120576) thenonlinear least-squares method automatically modifies theinput values of model parameters If the error is below thetolerance the corresponding parameter values are acceptedand the identification process is finished

4 Shock and Vibration

MeasuredSimulated

minus4000 minus2000 0 2000 4000

02 Hzminor loop

H (Am)

02 Hz major loop

50 Hz

minus1

0

1

B (T

)

minus2

minus1

0

1

2

B (T

)

minus2

minus1

0

1

2

B (T

)

minus4000 minus2000 0 2000 4000

minus4000 minus2000 0 2000 4000

Figure 2 The simulated and measured [30] hysteresis curves fordifferent frequencies

After about 50 iterations the parameters are identified as120572 = 21502 times 10minus3 119872119904 = 14334 times 106 119886 = 11143 times 103119888 = 33493 times 10minus1 1198960 = 63306 times 102 120590 = 14149 times 102119903119888 = 10505 times 101 1199031198901 = 13827 times 10minus1 1199031198902 = 11271 times 10minus1As shown in Figure 2 the model results agree well with theexperimental data so J-A model is effective and will be usedin the following sections to describe the magnetic hysteresisof the steel parts in MR dampers

3 A Dynamic Magnetic Circuit Model ofMR Dampers

As shown in Figure 3 the magnetic circuit of a MR damperis composed of a gap and three steel parts (denoted by AB and C) An independent current source is consideredhere because current drivers are preferred for faster magneticresponses [32]

For simplicity a static J-A model in which the dynamicfield (119867119889) vanishes will be used to describe the steel parts ofMR dampers and the effect of eddy current will be includedlater in the magnetic circuit modeling Another main reasonfor choosing a static J-Amodel is that the hysteresis algorithmin Maxwell software which will be compared with themagnetic circuit model proposed in this paper is still ablack box with little information available However such atemporary simplificationwill not undermine the generality ofthe proposed model because the dynamic hysteresis modelwould be recovered by simply restoring the dynamic field(119867119889)

The steels are described by the following three inverse J-Amodels which constitute three differential algebraic equations(DAEs) in terms of the magnetizations (119872119894)

119889119872119894119889119861119894 = 1198731198941205830 [119896120575 + (1 minus 120572)119873119894] 119894 = 1 2 3 (11)

g

III

II

IRd

Rp

Rb

Lp

Lb

i

①

③

③

③

③

②

②

Figure 3 Typical magnetic circuit of a MR damper

Similarly the magnetic saturationhysteresis of MR fluid canbe described by introducing another J-Amodel However themagnetic permeability of MR fluid is very low in comparisonwith that of steels so it is neglected here and regarded as airWithout loss of generality the piston and house cylinder areassumed to bemade of the same steel so the three J-Amodelsshare the same set of parameters

If the effects of magnetic flux fringing and leakage areignored according to Gaussrsquos law the magnetic inductions inthe different parts of the circuit are related by

119861119892119860119892 = 119861119894119860 119894 = Φ 119894 = 1 2 3 (12)

where 119861119892 and 119861119894 are the average magnetic inductions in theair gap and steels Φ is the magnetic flux in the circuit loopas shown in Figure 31198601 and1198602 are the cross-sectional areasof the piston core and the house cylinder 1198603 is the averagelateral area of the piston head (ie the lateral area at positionIII in Figure 3) and 119860119892 is the average lateral area of theannular gap These four areas are given by

1198601 = 12058711987721198871198602 = 120587 (1198772119889 minus (119877119901 + 119892)2) 1198603 = 2120587(119877119887 + 119877119901 minus 1198771198872 ) (119871119901 minus 119871119887) 119860119892 = 2120587(119877119901 + 1198922 ) (119871119901 minus 119871119887)

(13)

with the meanings of all the variables shown in Figure 3The flux leakage fringing flux and the nonuniformity

of magnetic fieldinduction as shown in Figure 4 are stillchallenging to be accurately considered and their combinedeffect is reflected here by introducing parameter 119896119891119886 for thegap and 119896119891119894 for the steels that is

119896119891119886119861119892119860119892 = 119896119891119894119861119894119860 119894 = Φ 119894 = 1 2 3 (14)

Shock and Vibration 5

Nonuniformmagnetic field

House cylinder

Fringingflux Piston

head

Leakage flux

Figure 4The fringing flux leakage flux and nonuniformmagneticfield near the working gap of a MR damper

The governing equation of the wholemagnetic circuit canbe obtained by Amperersquo law as

3sum119894=1

119867119894119871 119894 + 119867119892119892 = 119868tot (15)

where 119868tot is the total current in the source 119871 119894 is the magneticcircuit length of the steel parts 1198711 = 1198712 = (119871119901 + 119871119887)2 1198713 =119877119901 minus 119877119887 119892 is the gap width The magnetic field in differentsteel parts (119867119894) are coupled to the corresponding J-A modelby

119867119894 = 1198611198941205830 minus119872119894 119894 = 1 2 3 (16)

And the magnetic field in the gap is simply calculated by

119867119892 = Φ1198601198921205830 (17)

In a magnetic circuit analysis the effect of either classicalor excess eddy current can be taken into account based on thefollowing field separation [28 33 34]

119867tot = 119867 +119867119888 + 119867119890 (18)

where the total magnetic field (119867tot) is the sum of thehysteretic magnetic field (119867) the dynamic field due toclassical eddy current (119867119888) and the dynamic field concerningexcess eddy current (119867119890) In the cases of low frequenciesandor magnetic circuits with small cross-sections the mag-netic field and induction can be assumed to be uniformConsequently the field separation is equivalent to the afore-mentioned loss separation (1) and the dynamic fields can beestimated by [35 36]

119867119888 = 11988922120588120573 119889119861119889119905 119867119890 = (1198661198891199081198670120588 ) sign(119889119861119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198891199051003816100381610038161003816100381610038161003816100381612

(19)

where 120588 is the resistivity 119889 is the cross-sectional dimen-sion (thickness for laminations diameter for cylinders andspheres) and 120573 is a geometrical factor which varies form120573 = 6 in laminations to 120573 = 16 in cylinders and 120573 = 20 inspheres

In order to keep the generality of the proposedmodel thegeneral form (6) is used for the dynamic magnetic field thatis

119867119888119894 = 119903119888119894 119889119861119894119889119905 119894 = 1 2 3119867119890119894 = 119903119890119894 sign(119889119861119894119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198941198891199051003816100381610038161003816100381610038161003816100381612 119894 = 1 2 3 (20)

Only the dynamic field due to classical eddy current will beconsidered for simplicity Accordingly (15) is extended toinclude the effect of classical eddy current

3sum119894=1

(119867119894 + 119867119888119894) 119871 119894 + 119867119892119892 = 119868tot (21)

Finally (11) and (21) expressed in terms of four inde-pendent variables (119872111987221198723 Φ) constitute a completemagnetic circuit model of MR dampers This model isessentially a set of DAEs and can be numerically solved bytheMatlab built-in ODEDAE solver ode15i with the relativeerror tolerance and absolute tolerance respectively set to1e minus 5 and 1e minus 64 Model Validation and Analysis

The effectiveness of the proposed magnetic circuit modelis investigated for describing the low- and high-frequencymagnetic performances of a MR damper with the mainstructural parameters chosen as 119877119889 = 346mm 119877119901 = 30mm119877119887 = 20mm 119892 = 06mm 119871119887 = 25mm and 119871119901 = 30mmThecoil has 284 turns The current driver supplies a sinusoidalcurrent with an amplitude of 2A and a frequency of 02Hzin the low-frequency study and 10Hz in the high-frequencystudy

For the validation of the proposed magnetic circuitmodel it should be noted that although comparison withexperimental data is always the most direct way it is often(eg in the predesign stage of a MR damper) not a preferredway for validation because the magnetic response of a MRdamper indispensable for identifying the parameters of alumped parameter model is not available until the damperis finally fabricated and then tested In this case a FEMsimulation although coming with drawbacks such as beingtime consuming and hard-to-interpret is often a practicaland reliable alternative

More importantly although validation by an indirect testof damping forces is always much easier it should not be apreferredway for validating amagnetic circuitmodel becausethe damping force itself is greatly dependent on many factorssuch as the compressibility ofMR fluid and the compliance ofinstrument systemMoreover the effect of stress onmagnetichysteresis is often unavoidable which further makes the

6 Shock and Vibration

Part ① Part ② Part ③

FEMProposed model

FEMProposed model

FEMProposed model

B(T

)

B(T

)

B(T

)

06

04

02

00

minus02

minus04

minus06

06

04

02

00

minus02

minus04

minus06

H (Am) H (Am)H (Am)

09

06

03

00

minus03

minus06

minus0910000minus1000 10000minus1000 10000minus1000

Figure 5 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under a low-frequency (02Hz) sinusoidal electric current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute errortolerance 1e minus 6 nonlinear residual of FEM solver 1e minus 5)

damping force an unsuitable indicator to validate a magneticcircuit model

Since the FEM simulation will be used to validate theproposedmodel the FEMresults will act as the ldquoexperimentaldatardquo in (10) and the similar error will be calculated as

Error = 1119899radic119899sum119894=1

(119861lum119894 minus 119861fem119894119861FEMmax)2 (22)

where 119861fem119894 is the 119894th magnetic induction obtained byFEM 119861lum119894 is the 119894th magnetic induction obtained by theproposed model 119861FEMmax is the maximum FEM magneticinduction and 119899 is the number FEM results It should benoted that the summation index 119894 implicitly includes allthe average FEM results at three parts (lines I II and III inFigure 3)

ANSYS Maxwell employs a vector hysteresis model topredict the hysteresis loops and losses for soft and hardmagnetic materials and permanent magnets Hysteresis isautomatically activated by properly setting the material dataand the detailed material settings in ANSYS Maxwell aregiven below

(a) Specify the steel parts of the damper as materials withnonlinear relative permeabilities and edit the 119861119867 curves

(b) Input the descending branch of the saturation hys-teresis loop It should be noted that the first point and thelast point must mirror each other otherwise the curve isconsidered invalid

(c) After finishing the 119861119867 curve input a coercivity valueis automatically calculated and listed

(d) Set all three components of the ldquoUnit vectorrdquo to zerootherwise it acts like a permanent magnet

41 Low-Frequency StudyMagnetic Saturation andHysteresisUnder a low-frequency (02Hz) current excitation the effectof eddy current is insignificant and the magnetic hystere-sissaturation predicting capability of the proposed model isexamined in this case Accordingly the three parameters 11990311988811199031198882 1199031198883 related to the dynamic field vanish

The magnetic responses in the three steel parts are sim-ulated by the proposed model with the parameters identifiedas 120572 = 11440 times 10minus3 119872119904 = 11527 times 106 119886 = 63750 times 102119888 = 22428 times 10minus1 119896 = 63750 times 102 119896119891119886 = 13284 and1198961198911 = 1198961198912 = 1198961198913 = 10000 The three average cross-sectionalmagnetic responses at positions I II and III (Figure 3) arealso obtained inMaxwell (v160) for comparison As shown inFigure 5 the results predicted by the proposed model are in asatisfactory agreement with the FEM results so the proposedmagnetic circuit model is reliable for describing the staticmagnetic hysteretic behavior of MR dampers

It can be seen from values of the parameters (1198961198911 = 1198961198912 =1198961198913 = 10000) that the combined effects of magnetic fringingleakage and nonuniformity of magnetic fieldinduction arenegligible in all the three steel parts and will be ignored inthe following high-frequency study

42 High-Frequency Study Effect of Eddy Current The high-frequency study is performed under a sinusoidal current of10Hz and the maximum of the eddy current density is oneorder ofmagnitude larger than that under a current of 02Hzas shown in Figure 6

Because the five J-A material parameters have beenalready identified in the low-frequency study only fourparameters (1199031198881 1199031198882 1199031198883 119896119891119886) are left to be determined Using

Shock and Vibration 7

J (AＧ2)75E + 004

70E + 004

65E + 004

60E + 004

56E + 004

51E + 004

46E + 004

41E + 004

36E + 004

31E + 004

26E + 004

21E + 004

16E + 004

11E + 004

57E + 003

75E + 002

(a) Low-frequency study 02Hz

J (AＧ2)27E + 006

25E + 006

23E + 006

21E + 006

19E + 006

18E + 006

16E + 006

14E + 006

12E + 006

10E + 006

85E + 005

67E + 005

49E + 005

31E + 005

12E + 005

minus58E + 004

(b) High-frequency study 10Hz

Figure 6 Eddy currents at half-period instants (ie time 119905 = 1198792)

Part ① Part ② Part ③

H (Am)H (Am) H (Am)10000minus1000 10000minus1000 10000minus1000

minus050

minus025

000

025

050

B(T

)

B(T

)

B(T

)

minus070

minus035

000

035

070

minus050

minus025

000

025

050

FEMProposed model

FEMProposed model

FEMProposed model

Figure 7 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under high-frequency (10Hz) sinusoidal current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute error tolerance1e minus 6 nonlinear residual of FEM solver 1e minus 5)

an identification method similar to that shown in the abovesection they are obtained as 1199031198881 = 16650 times 102 1199031198882 =27785 times 101 1199031198882 = 48578 times 101 and 119896119891119886 = 10593 As shownin Figure 7 the predicted results in the steel parts show anoverall satisfactory agreement with the FEM results althougha larger discrepancy observed as compared to the results inthe low-frequency study

The numerical errors in low- and high-frequency studiescalculated by (22) are compared in Figure 8 The agree-ment between FEM and the proposed model results in the

low-frequency study is much better than that in the high-frequency study with the largest error occurring in steel partA

The larger discrepancy or error especially that in steelpart A is believed to be the consequence of the largernonuniformity ofmagnetic induction due to the effect of eddycurrent The hysteresis curves along lines I II and III (inFigure 3) are shown in Figure 9 with obvious nonuniformitiesof magnetic fields observed under high frequency Comparedto the nonuniformities of magnetic fields in parts B and C

8 Shock and Vibration

Different parts 02 HzDifferent parts 10 Hz

Whole damper 02 HzWhole damper 10 Hz

Part 3Part 2Part 1

002

003

004

005

Erro

r

006

007

008

Figure 8 Comparison of numerical errors under low- and high-frequency studies

the nonuniformity in partA is much larger so using averagevalues in the lumped parameter model produces largererror

Moreover the comparison between the two groups ofhysteresis curves in Figures 8 and 9 shows that the magneticinduction which directly determines the yields stress of MRfluid is apparently reduced under high-frequency changingfields due to the effect of eddy currents

5 A Demonstration of How to Usethe Proposed Magnetic Circuit Model

For the purpose of the demonstration of how to use theproposed magnetic circuit model a simple multiphysicsmodel will be developed in this section The flow field in aMR damper can be controlled by the magnetic field due tochanges of the shear yield-strength of MR fluid In contrastthe magnetic field is generally assumed to be independentof the fluid flow field so the analyses of these two fieldscan be conducted separately in series The experimentalyield stresses (120591119910) of the MR fluid are shown in Figure 10under different magnetic inductions (119861) and their relation-ship which is required for the damping force modelingcan be obtained by a least square curve fitting methodas

120591119910 = 5483033 [1 minus exp (minus314 |119861|203)] (23)

As shown in Figure 10 the fitted curve agrees well with theexperimental data

As shown in Figure 11 after validating the proposed mag-netic circuit model the damping force (119865) and the flow field(119906) of aMR damper can be analyzed by substituting the yield-strength history (obtained from the magnetic circuit model)into any extensively studied fluid model such as the Bingham

model theHerschel-Bulkleymodel and the biviscousmodelThe simple Bingham fluid model is adopted here for thedemonstration purpose Since the proposed magnetic circuitmodel has been validated in previous sections the resultsfrom the combinations of these two reliable models shouldbe also reliable

Describing the flow of MR fluid by the Bingham flowequations (ie combination of the Navier-Stokes equationsand the Bingham constitutive equation) in a rectangular ductthe damping force (119865) of a MR damper can be related to thepiston velocity (V119901) by the following mechanical model [37]

P (T) = 23 (1 + 3T)sdot [cos(13 arccos(1 minus 54 ( T1 + 3T)3)) + 12]

(24)

with the dimensionless pressure gradient (P) and yield stress(T) defined by

P = 1199081198923Δ11990112119876120578119871 T = 119908119892212059111991012119876120578

(25)

where 119908 is the mean circumference of the annular flow pathΔ119901 is the pressure drop responding to the volumetric flowrate 119876 = V119901119860119901 119860119901 is the effective area of the piston headthe damping force 119865 = Δ119901119860119901 and 120578 is the postyield plasticviscosity and taken as 13 Pasdots

Using the time history of the shear yield stress (23) as thebridge connecting the proposed magnetic circuit model andthe Binghamfluid basedmechanicalmodel (24) the dampingforces corresponding to the low- and high-frequency studiesin the above section are examined with the piston velocitychosen as 10mms

As shown in Figure 12 compared to the results of thelow-frequency study an obvious damping force lag withrespect to the coil current is observed in the high-frequencystudy due to the effect of eddy currents Such lag exhibitsa hysteresis loop in the current-force plane as shown inFigure 13 which even exists in a low-frequency varyingmagnetic field because of the magnetic hysteresis Thishysteresis behavior produces a variation in the damping forcebetween an increasing and decreasing coil current Thus in adynamic environment such as oscillatory current excitationuncertainty is introduced in any calculated torque Worsestill the eddy currents make such hysteresis more severe Asexpected an apparent decrease in the maximum dampingforce is also observed in the high-frequency study becauseof the decreasing maximummagnetic induction as shown inFigures 5 and 7

For better understanding of dynamic performances ofMR dampers it is often desirable to gain an insight intothe flow analysis of MR fluid in the working gap and this

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

4 Shock and Vibration

MeasuredSimulated

minus4000 minus2000 0 2000 4000

02 Hzminor loop

H (Am)

02 Hz major loop

50 Hz

minus1

0

1

B (T

)

minus2

minus1

0

1

2

B (T

)

minus2

minus1

0

1

2

B (T

)

minus4000 minus2000 0 2000 4000

minus4000 minus2000 0 2000 4000

Figure 2 The simulated and measured [30] hysteresis curves fordifferent frequencies

After about 50 iterations the parameters are identified as120572 = 21502 times 10minus3 119872119904 = 14334 times 106 119886 = 11143 times 103119888 = 33493 times 10minus1 1198960 = 63306 times 102 120590 = 14149 times 102119903119888 = 10505 times 101 1199031198901 = 13827 times 10minus1 1199031198902 = 11271 times 10minus1As shown in Figure 2 the model results agree well with theexperimental data so J-A model is effective and will be usedin the following sections to describe the magnetic hysteresisof the steel parts in MR dampers

3 A Dynamic Magnetic Circuit Model ofMR Dampers

As shown in Figure 3 the magnetic circuit of a MR damperis composed of a gap and three steel parts (denoted by AB and C) An independent current source is consideredhere because current drivers are preferred for faster magneticresponses [32]

For simplicity a static J-A model in which the dynamicfield (119867119889) vanishes will be used to describe the steel parts ofMR dampers and the effect of eddy current will be includedlater in the magnetic circuit modeling Another main reasonfor choosing a static J-Amodel is that the hysteresis algorithmin Maxwell software which will be compared with themagnetic circuit model proposed in this paper is still ablack box with little information available However such atemporary simplificationwill not undermine the generality ofthe proposed model because the dynamic hysteresis modelwould be recovered by simply restoring the dynamic field(119867119889)

The steels are described by the following three inverse J-Amodels which constitute three differential algebraic equations(DAEs) in terms of the magnetizations (119872119894)

119889119872119894119889119861119894 = 1198731198941205830 [119896120575 + (1 minus 120572)119873119894] 119894 = 1 2 3 (11)

g

III

II

IRd

Rp

Rb

Lp

Lb

i

①

③

③

③

③

②

②

Figure 3 Typical magnetic circuit of a MR damper

Similarly the magnetic saturationhysteresis of MR fluid canbe described by introducing another J-Amodel However themagnetic permeability of MR fluid is very low in comparisonwith that of steels so it is neglected here and regarded as airWithout loss of generality the piston and house cylinder areassumed to bemade of the same steel so the three J-Amodelsshare the same set of parameters

If the effects of magnetic flux fringing and leakage areignored according to Gaussrsquos law the magnetic inductions inthe different parts of the circuit are related by

119861119892119860119892 = 119861119894119860 119894 = Φ 119894 = 1 2 3 (12)

where 119861119892 and 119861119894 are the average magnetic inductions in theair gap and steels Φ is the magnetic flux in the circuit loopas shown in Figure 31198601 and1198602 are the cross-sectional areasof the piston core and the house cylinder 1198603 is the averagelateral area of the piston head (ie the lateral area at positionIII in Figure 3) and 119860119892 is the average lateral area of theannular gap These four areas are given by

1198601 = 12058711987721198871198602 = 120587 (1198772119889 minus (119877119901 + 119892)2) 1198603 = 2120587(119877119887 + 119877119901 minus 1198771198872 ) (119871119901 minus 119871119887) 119860119892 = 2120587(119877119901 + 1198922 ) (119871119901 minus 119871119887)

(13)

with the meanings of all the variables shown in Figure 3The flux leakage fringing flux and the nonuniformity

of magnetic fieldinduction as shown in Figure 4 are stillchallenging to be accurately considered and their combinedeffect is reflected here by introducing parameter 119896119891119886 for thegap and 119896119891119894 for the steels that is

119896119891119886119861119892119860119892 = 119896119891119894119861119894119860 119894 = Φ 119894 = 1 2 3 (14)

Shock and Vibration 5

Nonuniformmagnetic field

House cylinder

Fringingflux Piston

head

Leakage flux

Figure 4The fringing flux leakage flux and nonuniformmagneticfield near the working gap of a MR damper

The governing equation of the wholemagnetic circuit canbe obtained by Amperersquo law as

3sum119894=1

119867119894119871 119894 + 119867119892119892 = 119868tot (15)

where 119868tot is the total current in the source 119871 119894 is the magneticcircuit length of the steel parts 1198711 = 1198712 = (119871119901 + 119871119887)2 1198713 =119877119901 minus 119877119887 119892 is the gap width The magnetic field in differentsteel parts (119867119894) are coupled to the corresponding J-A modelby

119867119894 = 1198611198941205830 minus119872119894 119894 = 1 2 3 (16)

And the magnetic field in the gap is simply calculated by

119867119892 = Φ1198601198921205830 (17)

In a magnetic circuit analysis the effect of either classicalor excess eddy current can be taken into account based on thefollowing field separation [28 33 34]

119867tot = 119867 +119867119888 + 119867119890 (18)

where the total magnetic field (119867tot) is the sum of thehysteretic magnetic field (119867) the dynamic field due toclassical eddy current (119867119888) and the dynamic field concerningexcess eddy current (119867119890) In the cases of low frequenciesandor magnetic circuits with small cross-sections the mag-netic field and induction can be assumed to be uniformConsequently the field separation is equivalent to the afore-mentioned loss separation (1) and the dynamic fields can beestimated by [35 36]

119867119888 = 11988922120588120573 119889119861119889119905 119867119890 = (1198661198891199081198670120588 ) sign(119889119861119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198891199051003816100381610038161003816100381610038161003816100381612

(19)

where 120588 is the resistivity 119889 is the cross-sectional dimen-sion (thickness for laminations diameter for cylinders andspheres) and 120573 is a geometrical factor which varies form120573 = 6 in laminations to 120573 = 16 in cylinders and 120573 = 20 inspheres

In order to keep the generality of the proposedmodel thegeneral form (6) is used for the dynamic magnetic field thatis

119867119888119894 = 119903119888119894 119889119861119894119889119905 119894 = 1 2 3119867119890119894 = 119903119890119894 sign(119889119861119894119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198941198891199051003816100381610038161003816100381610038161003816100381612 119894 = 1 2 3 (20)

Only the dynamic field due to classical eddy current will beconsidered for simplicity Accordingly (15) is extended toinclude the effect of classical eddy current

3sum119894=1

(119867119894 + 119867119888119894) 119871 119894 + 119867119892119892 = 119868tot (21)

Finally (11) and (21) expressed in terms of four inde-pendent variables (119872111987221198723 Φ) constitute a completemagnetic circuit model of MR dampers This model isessentially a set of DAEs and can be numerically solved bytheMatlab built-in ODEDAE solver ode15i with the relativeerror tolerance and absolute tolerance respectively set to1e minus 5 and 1e minus 64 Model Validation and Analysis

The effectiveness of the proposed magnetic circuit modelis investigated for describing the low- and high-frequencymagnetic performances of a MR damper with the mainstructural parameters chosen as 119877119889 = 346mm 119877119901 = 30mm119877119887 = 20mm 119892 = 06mm 119871119887 = 25mm and 119871119901 = 30mmThecoil has 284 turns The current driver supplies a sinusoidalcurrent with an amplitude of 2A and a frequency of 02Hzin the low-frequency study and 10Hz in the high-frequencystudy

For the validation of the proposed magnetic circuitmodel it should be noted that although comparison withexperimental data is always the most direct way it is often(eg in the predesign stage of a MR damper) not a preferredway for validation because the magnetic response of a MRdamper indispensable for identifying the parameters of alumped parameter model is not available until the damperis finally fabricated and then tested In this case a FEMsimulation although coming with drawbacks such as beingtime consuming and hard-to-interpret is often a practicaland reliable alternative

More importantly although validation by an indirect testof damping forces is always much easier it should not be apreferredway for validating amagnetic circuitmodel becausethe damping force itself is greatly dependent on many factorssuch as the compressibility ofMR fluid and the compliance ofinstrument systemMoreover the effect of stress onmagnetichysteresis is often unavoidable which further makes the

6 Shock and Vibration

Part ① Part ② Part ③

FEMProposed model

FEMProposed model

FEMProposed model

B(T

)

B(T

)

B(T

)

06

04

02

00

minus02

minus04

minus06

06

04

02

00

minus02

minus04

minus06

H (Am) H (Am)H (Am)

09

06

03

00

minus03

minus06

minus0910000minus1000 10000minus1000 10000minus1000

Figure 5 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under a low-frequency (02Hz) sinusoidal electric current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute errortolerance 1e minus 6 nonlinear residual of FEM solver 1e minus 5)

damping force an unsuitable indicator to validate a magneticcircuit model

Since the FEM simulation will be used to validate theproposedmodel the FEMresults will act as the ldquoexperimentaldatardquo in (10) and the similar error will be calculated as

Error = 1119899radic119899sum119894=1

(119861lum119894 minus 119861fem119894119861FEMmax)2 (22)

where 119861fem119894 is the 119894th magnetic induction obtained byFEM 119861lum119894 is the 119894th magnetic induction obtained by theproposed model 119861FEMmax is the maximum FEM magneticinduction and 119899 is the number FEM results It should benoted that the summation index 119894 implicitly includes allthe average FEM results at three parts (lines I II and III inFigure 3)

ANSYS Maxwell employs a vector hysteresis model topredict the hysteresis loops and losses for soft and hardmagnetic materials and permanent magnets Hysteresis isautomatically activated by properly setting the material dataand the detailed material settings in ANSYS Maxwell aregiven below

(a) Specify the steel parts of the damper as materials withnonlinear relative permeabilities and edit the 119861119867 curves

(b) Input the descending branch of the saturation hys-teresis loop It should be noted that the first point and thelast point must mirror each other otherwise the curve isconsidered invalid

(c) After finishing the 119861119867 curve input a coercivity valueis automatically calculated and listed

(d) Set all three components of the ldquoUnit vectorrdquo to zerootherwise it acts like a permanent magnet

41 Low-Frequency StudyMagnetic Saturation andHysteresisUnder a low-frequency (02Hz) current excitation the effectof eddy current is insignificant and the magnetic hystere-sissaturation predicting capability of the proposed model isexamined in this case Accordingly the three parameters 11990311988811199031198882 1199031198883 related to the dynamic field vanish

The magnetic responses in the three steel parts are sim-ulated by the proposed model with the parameters identifiedas 120572 = 11440 times 10minus3 119872119904 = 11527 times 106 119886 = 63750 times 102119888 = 22428 times 10minus1 119896 = 63750 times 102 119896119891119886 = 13284 and1198961198911 = 1198961198912 = 1198961198913 = 10000 The three average cross-sectionalmagnetic responses at positions I II and III (Figure 3) arealso obtained inMaxwell (v160) for comparison As shown inFigure 5 the results predicted by the proposed model are in asatisfactory agreement with the FEM results so the proposedmagnetic circuit model is reliable for describing the staticmagnetic hysteretic behavior of MR dampers

It can be seen from values of the parameters (1198961198911 = 1198961198912 =1198961198913 = 10000) that the combined effects of magnetic fringingleakage and nonuniformity of magnetic fieldinduction arenegligible in all the three steel parts and will be ignored inthe following high-frequency study

42 High-Frequency Study Effect of Eddy Current The high-frequency study is performed under a sinusoidal current of10Hz and the maximum of the eddy current density is oneorder ofmagnitude larger than that under a current of 02Hzas shown in Figure 6

Because the five J-A material parameters have beenalready identified in the low-frequency study only fourparameters (1199031198881 1199031198882 1199031198883 119896119891119886) are left to be determined Using

Shock and Vibration 7

J (AＧ2)75E + 004

70E + 004

65E + 004

60E + 004

56E + 004

51E + 004

46E + 004

41E + 004

36E + 004

31E + 004

26E + 004

21E + 004

16E + 004

11E + 004

57E + 003

75E + 002

(a) Low-frequency study 02Hz

J (AＧ2)27E + 006

25E + 006

23E + 006

21E + 006

19E + 006

18E + 006

16E + 006

14E + 006

12E + 006

10E + 006

85E + 005

67E + 005

49E + 005

31E + 005

12E + 005

minus58E + 004

(b) High-frequency study 10Hz

Figure 6 Eddy currents at half-period instants (ie time 119905 = 1198792)

Part ① Part ② Part ③

H (Am)H (Am) H (Am)10000minus1000 10000minus1000 10000minus1000

minus050

minus025

000

025

050

B(T

)

B(T

)

B(T

)

minus070

minus035

000

035

070

minus050

minus025

000

025

050

FEMProposed model

FEMProposed model

FEMProposed model

Figure 7 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under high-frequency (10Hz) sinusoidal current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute error tolerance1e minus 6 nonlinear residual of FEM solver 1e minus 5)

an identification method similar to that shown in the abovesection they are obtained as 1199031198881 = 16650 times 102 1199031198882 =27785 times 101 1199031198882 = 48578 times 101 and 119896119891119886 = 10593 As shownin Figure 7 the predicted results in the steel parts show anoverall satisfactory agreement with the FEM results althougha larger discrepancy observed as compared to the results inthe low-frequency study

The numerical errors in low- and high-frequency studiescalculated by (22) are compared in Figure 8 The agree-ment between FEM and the proposed model results in the

low-frequency study is much better than that in the high-frequency study with the largest error occurring in steel partA

The larger discrepancy or error especially that in steelpart A is believed to be the consequence of the largernonuniformity ofmagnetic induction due to the effect of eddycurrent The hysteresis curves along lines I II and III (inFigure 3) are shown in Figure 9 with obvious nonuniformitiesof magnetic fields observed under high frequency Comparedto the nonuniformities of magnetic fields in parts B and C

8 Shock and Vibration

Different parts 02 HzDifferent parts 10 Hz

Whole damper 02 HzWhole damper 10 Hz

Part 3Part 2Part 1

002

003

004

005

Erro

r

006

007

008

Figure 8 Comparison of numerical errors under low- and high-frequency studies

the nonuniformity in partA is much larger so using averagevalues in the lumped parameter model produces largererror

Moreover the comparison between the two groups ofhysteresis curves in Figures 8 and 9 shows that the magneticinduction which directly determines the yields stress of MRfluid is apparently reduced under high-frequency changingfields due to the effect of eddy currents

5 A Demonstration of How to Usethe Proposed Magnetic Circuit Model

For the purpose of the demonstration of how to use theproposed magnetic circuit model a simple multiphysicsmodel will be developed in this section The flow field in aMR damper can be controlled by the magnetic field due tochanges of the shear yield-strength of MR fluid In contrastthe magnetic field is generally assumed to be independentof the fluid flow field so the analyses of these two fieldscan be conducted separately in series The experimentalyield stresses (120591119910) of the MR fluid are shown in Figure 10under different magnetic inductions (119861) and their relation-ship which is required for the damping force modelingcan be obtained by a least square curve fitting methodas

120591119910 = 5483033 [1 minus exp (minus314 |119861|203)] (23)

As shown in Figure 10 the fitted curve agrees well with theexperimental data

As shown in Figure 11 after validating the proposed mag-netic circuit model the damping force (119865) and the flow field(119906) of aMR damper can be analyzed by substituting the yield-strength history (obtained from the magnetic circuit model)into any extensively studied fluid model such as the Bingham

model theHerschel-Bulkleymodel and the biviscousmodelThe simple Bingham fluid model is adopted here for thedemonstration purpose Since the proposed magnetic circuitmodel has been validated in previous sections the resultsfrom the combinations of these two reliable models shouldbe also reliable

Describing the flow of MR fluid by the Bingham flowequations (ie combination of the Navier-Stokes equationsand the Bingham constitutive equation) in a rectangular ductthe damping force (119865) of a MR damper can be related to thepiston velocity (V119901) by the following mechanical model [37]

P (T) = 23 (1 + 3T)sdot [cos(13 arccos(1 minus 54 ( T1 + 3T)3)) + 12]

(24)

with the dimensionless pressure gradient (P) and yield stress(T) defined by

P = 1199081198923Δ11990112119876120578119871 T = 119908119892212059111991012119876120578

(25)

where 119908 is the mean circumference of the annular flow pathΔ119901 is the pressure drop responding to the volumetric flowrate 119876 = V119901119860119901 119860119901 is the effective area of the piston headthe damping force 119865 = Δ119901119860119901 and 120578 is the postyield plasticviscosity and taken as 13 Pasdots

Using the time history of the shear yield stress (23) as thebridge connecting the proposed magnetic circuit model andthe Binghamfluid basedmechanicalmodel (24) the dampingforces corresponding to the low- and high-frequency studiesin the above section are examined with the piston velocitychosen as 10mms

As shown in Figure 12 compared to the results of thelow-frequency study an obvious damping force lag withrespect to the coil current is observed in the high-frequencystudy due to the effect of eddy currents Such lag exhibitsa hysteresis loop in the current-force plane as shown inFigure 13 which even exists in a low-frequency varyingmagnetic field because of the magnetic hysteresis Thishysteresis behavior produces a variation in the damping forcebetween an increasing and decreasing coil current Thus in adynamic environment such as oscillatory current excitationuncertainty is introduced in any calculated torque Worsestill the eddy currents make such hysteresis more severe Asexpected an apparent decrease in the maximum dampingforce is also observed in the high-frequency study becauseof the decreasing maximummagnetic induction as shown inFigures 5 and 7

For better understanding of dynamic performances ofMR dampers it is often desirable to gain an insight intothe flow analysis of MR fluid in the working gap and this

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

Shock and Vibration 5

Nonuniformmagnetic field

House cylinder

Fringingflux Piston

head

Leakage flux

Figure 4The fringing flux leakage flux and nonuniformmagneticfield near the working gap of a MR damper

The governing equation of the wholemagnetic circuit canbe obtained by Amperersquo law as

3sum119894=1

119867119894119871 119894 + 119867119892119892 = 119868tot (15)

where 119868tot is the total current in the source 119871 119894 is the magneticcircuit length of the steel parts 1198711 = 1198712 = (119871119901 + 119871119887)2 1198713 =119877119901 minus 119877119887 119892 is the gap width The magnetic field in differentsteel parts (119867119894) are coupled to the corresponding J-A modelby

119867119894 = 1198611198941205830 minus119872119894 119894 = 1 2 3 (16)

And the magnetic field in the gap is simply calculated by

119867119892 = Φ1198601198921205830 (17)

In a magnetic circuit analysis the effect of either classicalor excess eddy current can be taken into account based on thefollowing field separation [28 33 34]

119867tot = 119867 +119867119888 + 119867119890 (18)

where the total magnetic field (119867tot) is the sum of thehysteretic magnetic field (119867) the dynamic field due toclassical eddy current (119867119888) and the dynamic field concerningexcess eddy current (119867119890) In the cases of low frequenciesandor magnetic circuits with small cross-sections the mag-netic field and induction can be assumed to be uniformConsequently the field separation is equivalent to the afore-mentioned loss separation (1) and the dynamic fields can beestimated by [35 36]

119867119888 = 11988922120588120573 119889119861119889119905 119867119890 = (1198661198891199081198670120588 ) sign(119889119861119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198891199051003816100381610038161003816100381610038161003816100381612

(19)

where 120588 is the resistivity 119889 is the cross-sectional dimen-sion (thickness for laminations diameter for cylinders andspheres) and 120573 is a geometrical factor which varies form120573 = 6 in laminations to 120573 = 16 in cylinders and 120573 = 20 inspheres

In order to keep the generality of the proposedmodel thegeneral form (6) is used for the dynamic magnetic field thatis

119867119888119894 = 119903119888119894 119889119861119894119889119905 119894 = 1 2 3119867119890119894 = 119903119890119894 sign(119889119861119894119889119905 )

100381610038161003816100381610038161003816100381610038161198891198611198941198891199051003816100381610038161003816100381610038161003816100381612 119894 = 1 2 3 (20)

Only the dynamic field due to classical eddy current will beconsidered for simplicity Accordingly (15) is extended toinclude the effect of classical eddy current

3sum119894=1

(119867119894 + 119867119888119894) 119871 119894 + 119867119892119892 = 119868tot (21)

Finally (11) and (21) expressed in terms of four inde-pendent variables (119872111987221198723 Φ) constitute a completemagnetic circuit model of MR dampers This model isessentially a set of DAEs and can be numerically solved bytheMatlab built-in ODEDAE solver ode15i with the relativeerror tolerance and absolute tolerance respectively set to1e minus 5 and 1e minus 64 Model Validation and Analysis

The effectiveness of the proposed magnetic circuit modelis investigated for describing the low- and high-frequencymagnetic performances of a MR damper with the mainstructural parameters chosen as 119877119889 = 346mm 119877119901 = 30mm119877119887 = 20mm 119892 = 06mm 119871119887 = 25mm and 119871119901 = 30mmThecoil has 284 turns The current driver supplies a sinusoidalcurrent with an amplitude of 2A and a frequency of 02Hzin the low-frequency study and 10Hz in the high-frequencystudy

For the validation of the proposed magnetic circuitmodel it should be noted that although comparison withexperimental data is always the most direct way it is often(eg in the predesign stage of a MR damper) not a preferredway for validation because the magnetic response of a MRdamper indispensable for identifying the parameters of alumped parameter model is not available until the damperis finally fabricated and then tested In this case a FEMsimulation although coming with drawbacks such as beingtime consuming and hard-to-interpret is often a practicaland reliable alternative

More importantly although validation by an indirect testof damping forces is always much easier it should not be apreferredway for validating amagnetic circuitmodel becausethe damping force itself is greatly dependent on many factorssuch as the compressibility ofMR fluid and the compliance ofinstrument systemMoreover the effect of stress onmagnetichysteresis is often unavoidable which further makes the

6 Shock and Vibration

Part ① Part ② Part ③

FEMProposed model

FEMProposed model

FEMProposed model

B(T

)

B(T

)

B(T

)

06

04

02

00

minus02

minus04

minus06

06

04

02

00

minus02

minus04

minus06

H (Am) H (Am)H (Am)

09

06

03

00

minus03

minus06

minus0910000minus1000 10000minus1000 10000minus1000

Figure 5 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under a low-frequency (02Hz) sinusoidal electric current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute errortolerance 1e minus 6 nonlinear residual of FEM solver 1e minus 5)

damping force an unsuitable indicator to validate a magneticcircuit model

Since the FEM simulation will be used to validate theproposedmodel the FEMresults will act as the ldquoexperimentaldatardquo in (10) and the similar error will be calculated as

Error = 1119899radic119899sum119894=1

(119861lum119894 minus 119861fem119894119861FEMmax)2 (22)

where 119861fem119894 is the 119894th magnetic induction obtained byFEM 119861lum119894 is the 119894th magnetic induction obtained by theproposed model 119861FEMmax is the maximum FEM magneticinduction and 119899 is the number FEM results It should benoted that the summation index 119894 implicitly includes allthe average FEM results at three parts (lines I II and III inFigure 3)

ANSYS Maxwell employs a vector hysteresis model topredict the hysteresis loops and losses for soft and hardmagnetic materials and permanent magnets Hysteresis isautomatically activated by properly setting the material dataand the detailed material settings in ANSYS Maxwell aregiven below

(a) Specify the steel parts of the damper as materials withnonlinear relative permeabilities and edit the 119861119867 curves

(b) Input the descending branch of the saturation hys-teresis loop It should be noted that the first point and thelast point must mirror each other otherwise the curve isconsidered invalid

(c) After finishing the 119861119867 curve input a coercivity valueis automatically calculated and listed

(d) Set all three components of the ldquoUnit vectorrdquo to zerootherwise it acts like a permanent magnet

41 Low-Frequency StudyMagnetic Saturation andHysteresisUnder a low-frequency (02Hz) current excitation the effectof eddy current is insignificant and the magnetic hystere-sissaturation predicting capability of the proposed model isexamined in this case Accordingly the three parameters 11990311988811199031198882 1199031198883 related to the dynamic field vanish

The magnetic responses in the three steel parts are sim-ulated by the proposed model with the parameters identifiedas 120572 = 11440 times 10minus3 119872119904 = 11527 times 106 119886 = 63750 times 102119888 = 22428 times 10minus1 119896 = 63750 times 102 119896119891119886 = 13284 and1198961198911 = 1198961198912 = 1198961198913 = 10000 The three average cross-sectionalmagnetic responses at positions I II and III (Figure 3) arealso obtained inMaxwell (v160) for comparison As shown inFigure 5 the results predicted by the proposed model are in asatisfactory agreement with the FEM results so the proposedmagnetic circuit model is reliable for describing the staticmagnetic hysteretic behavior of MR dampers

It can be seen from values of the parameters (1198961198911 = 1198961198912 =1198961198913 = 10000) that the combined effects of magnetic fringingleakage and nonuniformity of magnetic fieldinduction arenegligible in all the three steel parts and will be ignored inthe following high-frequency study

42 High-Frequency Study Effect of Eddy Current The high-frequency study is performed under a sinusoidal current of10Hz and the maximum of the eddy current density is oneorder ofmagnitude larger than that under a current of 02Hzas shown in Figure 6

Because the five J-A material parameters have beenalready identified in the low-frequency study only fourparameters (1199031198881 1199031198882 1199031198883 119896119891119886) are left to be determined Using

Shock and Vibration 7

J (AＧ2)75E + 004

70E + 004

65E + 004

60E + 004

56E + 004

51E + 004

46E + 004

41E + 004

36E + 004

31E + 004

26E + 004

21E + 004

16E + 004

11E + 004

57E + 003

75E + 002

(a) Low-frequency study 02Hz

J (AＧ2)27E + 006

25E + 006

23E + 006

21E + 006

19E + 006

18E + 006

16E + 006

14E + 006

12E + 006

10E + 006

85E + 005

67E + 005

49E + 005

31E + 005

12E + 005

minus58E + 004

(b) High-frequency study 10Hz

Figure 6 Eddy currents at half-period instants (ie time 119905 = 1198792)

Part ① Part ② Part ③

H (Am)H (Am) H (Am)10000minus1000 10000minus1000 10000minus1000

minus050

minus025

000

025

050

B(T

)

B(T

)

B(T

)

minus070

minus035

000

035

070

minus050

minus025

000

025

050

FEMProposed model

FEMProposed model

FEMProposed model

Figure 7 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under high-frequency (10Hz) sinusoidal current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute error tolerance1e minus 6 nonlinear residual of FEM solver 1e minus 5)

an identification method similar to that shown in the abovesection they are obtained as 1199031198881 = 16650 times 102 1199031198882 =27785 times 101 1199031198882 = 48578 times 101 and 119896119891119886 = 10593 As shownin Figure 7 the predicted results in the steel parts show anoverall satisfactory agreement with the FEM results althougha larger discrepancy observed as compared to the results inthe low-frequency study

The numerical errors in low- and high-frequency studiescalculated by (22) are compared in Figure 8 The agree-ment between FEM and the proposed model results in the

low-frequency study is much better than that in the high-frequency study with the largest error occurring in steel partA

The larger discrepancy or error especially that in steelpart A is believed to be the consequence of the largernonuniformity ofmagnetic induction due to the effect of eddycurrent The hysteresis curves along lines I II and III (inFigure 3) are shown in Figure 9 with obvious nonuniformitiesof magnetic fields observed under high frequency Comparedto the nonuniformities of magnetic fields in parts B and C

8 Shock and Vibration

Different parts 02 HzDifferent parts 10 Hz

Whole damper 02 HzWhole damper 10 Hz

Part 3Part 2Part 1

002

003

004

005

Erro

r

006

007

008

Figure 8 Comparison of numerical errors under low- and high-frequency studies

the nonuniformity in partA is much larger so using averagevalues in the lumped parameter model produces largererror

Moreover the comparison between the two groups ofhysteresis curves in Figures 8 and 9 shows that the magneticinduction which directly determines the yields stress of MRfluid is apparently reduced under high-frequency changingfields due to the effect of eddy currents

5 A Demonstration of How to Usethe Proposed Magnetic Circuit Model

For the purpose of the demonstration of how to use theproposed magnetic circuit model a simple multiphysicsmodel will be developed in this section The flow field in aMR damper can be controlled by the magnetic field due tochanges of the shear yield-strength of MR fluid In contrastthe magnetic field is generally assumed to be independentof the fluid flow field so the analyses of these two fieldscan be conducted separately in series The experimentalyield stresses (120591119910) of the MR fluid are shown in Figure 10under different magnetic inductions (119861) and their relation-ship which is required for the damping force modelingcan be obtained by a least square curve fitting methodas

120591119910 = 5483033 [1 minus exp (minus314 |119861|203)] (23)

As shown in Figure 10 the fitted curve agrees well with theexperimental data

As shown in Figure 11 after validating the proposed mag-netic circuit model the damping force (119865) and the flow field(119906) of aMR damper can be analyzed by substituting the yield-strength history (obtained from the magnetic circuit model)into any extensively studied fluid model such as the Bingham

model theHerschel-Bulkleymodel and the biviscousmodelThe simple Bingham fluid model is adopted here for thedemonstration purpose Since the proposed magnetic circuitmodel has been validated in previous sections the resultsfrom the combinations of these two reliable models shouldbe also reliable

Describing the flow of MR fluid by the Bingham flowequations (ie combination of the Navier-Stokes equationsand the Bingham constitutive equation) in a rectangular ductthe damping force (119865) of a MR damper can be related to thepiston velocity (V119901) by the following mechanical model [37]

P (T) = 23 (1 + 3T)sdot [cos(13 arccos(1 minus 54 ( T1 + 3T)3)) + 12]

(24)

with the dimensionless pressure gradient (P) and yield stress(T) defined by

P = 1199081198923Δ11990112119876120578119871 T = 119908119892212059111991012119876120578

(25)

where 119908 is the mean circumference of the annular flow pathΔ119901 is the pressure drop responding to the volumetric flowrate 119876 = V119901119860119901 119860119901 is the effective area of the piston headthe damping force 119865 = Δ119901119860119901 and 120578 is the postyield plasticviscosity and taken as 13 Pasdots

Using the time history of the shear yield stress (23) as thebridge connecting the proposed magnetic circuit model andthe Binghamfluid basedmechanicalmodel (24) the dampingforces corresponding to the low- and high-frequency studiesin the above section are examined with the piston velocitychosen as 10mms

As shown in Figure 12 compared to the results of thelow-frequency study an obvious damping force lag withrespect to the coil current is observed in the high-frequencystudy due to the effect of eddy currents Such lag exhibitsa hysteresis loop in the current-force plane as shown inFigure 13 which even exists in a low-frequency varyingmagnetic field because of the magnetic hysteresis Thishysteresis behavior produces a variation in the damping forcebetween an increasing and decreasing coil current Thus in adynamic environment such as oscillatory current excitationuncertainty is introduced in any calculated torque Worsestill the eddy currents make such hysteresis more severe Asexpected an apparent decrease in the maximum dampingforce is also observed in the high-frequency study becauseof the decreasing maximummagnetic induction as shown inFigures 5 and 7

For better understanding of dynamic performances ofMR dampers it is often desirable to gain an insight intothe flow analysis of MR fluid in the working gap and this

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

6 Shock and Vibration

Part ① Part ② Part ③

FEMProposed model

FEMProposed model

FEMProposed model

B(T

)

B(T

)

B(T

)

06

04

02

00

minus02

minus04

minus06

06

04

02

00

minus02

minus04

minus06

H (Am) H (Am)H (Am)

09

06

03

00

minus03

minus06

minus0910000minus1000 10000minus1000 10000minus1000

Figure 5 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under a low-frequency (02Hz) sinusoidal electric current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute errortolerance 1e minus 6 nonlinear residual of FEM solver 1e minus 5)

damping force an unsuitable indicator to validate a magneticcircuit model

Since the FEM simulation will be used to validate theproposedmodel the FEMresults will act as the ldquoexperimentaldatardquo in (10) and the similar error will be calculated as

Error = 1119899radic119899sum119894=1

(119861lum119894 minus 119861fem119894119861FEMmax)2 (22)

where 119861fem119894 is the 119894th magnetic induction obtained byFEM 119861lum119894 is the 119894th magnetic induction obtained by theproposed model 119861FEMmax is the maximum FEM magneticinduction and 119899 is the number FEM results It should benoted that the summation index 119894 implicitly includes allthe average FEM results at three parts (lines I II and III inFigure 3)

ANSYS Maxwell employs a vector hysteresis model topredict the hysteresis loops and losses for soft and hardmagnetic materials and permanent magnets Hysteresis isautomatically activated by properly setting the material dataand the detailed material settings in ANSYS Maxwell aregiven below

(a) Specify the steel parts of the damper as materials withnonlinear relative permeabilities and edit the 119861119867 curves

(b) Input the descending branch of the saturation hys-teresis loop It should be noted that the first point and thelast point must mirror each other otherwise the curve isconsidered invalid

(c) After finishing the 119861119867 curve input a coercivity valueis automatically calculated and listed

(d) Set all three components of the ldquoUnit vectorrdquo to zerootherwise it acts like a permanent magnet

41 Low-Frequency StudyMagnetic Saturation andHysteresisUnder a low-frequency (02Hz) current excitation the effectof eddy current is insignificant and the magnetic hystere-sissaturation predicting capability of the proposed model isexamined in this case Accordingly the three parameters 11990311988811199031198882 1199031198883 related to the dynamic field vanish

The magnetic responses in the three steel parts are sim-ulated by the proposed model with the parameters identifiedas 120572 = 11440 times 10minus3 119872119904 = 11527 times 106 119886 = 63750 times 102119888 = 22428 times 10minus1 119896 = 63750 times 102 119896119891119886 = 13284 and1198961198911 = 1198961198912 = 1198961198913 = 10000 The three average cross-sectionalmagnetic responses at positions I II and III (Figure 3) arealso obtained inMaxwell (v160) for comparison As shown inFigure 5 the results predicted by the proposed model are in asatisfactory agreement with the FEM results so the proposedmagnetic circuit model is reliable for describing the staticmagnetic hysteretic behavior of MR dampers

It can be seen from values of the parameters (1198961198911 = 1198961198912 =1198961198913 = 10000) that the combined effects of magnetic fringingleakage and nonuniformity of magnetic fieldinduction arenegligible in all the three steel parts and will be ignored inthe following high-frequency study

42 High-Frequency Study Effect of Eddy Current The high-frequency study is performed under a sinusoidal current of10Hz and the maximum of the eddy current density is oneorder ofmagnitude larger than that under a current of 02Hzas shown in Figure 6

Because the five J-A material parameters have beenalready identified in the low-frequency study only fourparameters (1199031198881 1199031198882 1199031198883 119896119891119886) are left to be determined Using

Shock and Vibration 7

J (AＧ2)75E + 004

70E + 004

65E + 004

60E + 004

56E + 004

51E + 004

46E + 004

41E + 004

36E + 004

31E + 004

26E + 004

21E + 004

16E + 004

11E + 004

57E + 003

75E + 002

(a) Low-frequency study 02Hz

J (AＧ2)27E + 006

25E + 006

23E + 006

21E + 006

19E + 006

18E + 006

16E + 006

14E + 006

12E + 006

10E + 006

85E + 005

67E + 005

49E + 005

31E + 005

12E + 005

minus58E + 004

(b) High-frequency study 10Hz

Figure 6 Eddy currents at half-period instants (ie time 119905 = 1198792)

Part ① Part ② Part ③

H (Am)H (Am) H (Am)10000minus1000 10000minus1000 10000minus1000

minus050

minus025

000

025

050

B(T

)

B(T

)

B(T

)

minus070

minus035

000

035

070

minus050

minus025

000

025

050

FEMProposed model

FEMProposed model

FEMProposed model

Figure 7 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under high-frequency (10Hz) sinusoidal current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute error tolerance1e minus 6 nonlinear residual of FEM solver 1e minus 5)

an identification method similar to that shown in the abovesection they are obtained as 1199031198881 = 16650 times 102 1199031198882 =27785 times 101 1199031198882 = 48578 times 101 and 119896119891119886 = 10593 As shownin Figure 7 the predicted results in the steel parts show anoverall satisfactory agreement with the FEM results althougha larger discrepancy observed as compared to the results inthe low-frequency study

The numerical errors in low- and high-frequency studiescalculated by (22) are compared in Figure 8 The agree-ment between FEM and the proposed model results in the

low-frequency study is much better than that in the high-frequency study with the largest error occurring in steel partA

The larger discrepancy or error especially that in steelpart A is believed to be the consequence of the largernonuniformity ofmagnetic induction due to the effect of eddycurrent The hysteresis curves along lines I II and III (inFigure 3) are shown in Figure 9 with obvious nonuniformitiesof magnetic fields observed under high frequency Comparedto the nonuniformities of magnetic fields in parts B and C

8 Shock and Vibration

Different parts 02 HzDifferent parts 10 Hz

Whole damper 02 HzWhole damper 10 Hz

Part 3Part 2Part 1

002

003

004

005

Erro

r

006

007

008

Figure 8 Comparison of numerical errors under low- and high-frequency studies

the nonuniformity in partA is much larger so using averagevalues in the lumped parameter model produces largererror

Moreover the comparison between the two groups ofhysteresis curves in Figures 8 and 9 shows that the magneticinduction which directly determines the yields stress of MRfluid is apparently reduced under high-frequency changingfields due to the effect of eddy currents

5 A Demonstration of How to Usethe Proposed Magnetic Circuit Model

For the purpose of the demonstration of how to use theproposed magnetic circuit model a simple multiphysicsmodel will be developed in this section The flow field in aMR damper can be controlled by the magnetic field due tochanges of the shear yield-strength of MR fluid In contrastthe magnetic field is generally assumed to be independentof the fluid flow field so the analyses of these two fieldscan be conducted separately in series The experimentalyield stresses (120591119910) of the MR fluid are shown in Figure 10under different magnetic inductions (119861) and their relation-ship which is required for the damping force modelingcan be obtained by a least square curve fitting methodas

120591119910 = 5483033 [1 minus exp (minus314 |119861|203)] (23)

As shown in Figure 10 the fitted curve agrees well with theexperimental data

As shown in Figure 11 after validating the proposed mag-netic circuit model the damping force (119865) and the flow field(119906) of aMR damper can be analyzed by substituting the yield-strength history (obtained from the magnetic circuit model)into any extensively studied fluid model such as the Bingham

model theHerschel-Bulkleymodel and the biviscousmodelThe simple Bingham fluid model is adopted here for thedemonstration purpose Since the proposed magnetic circuitmodel has been validated in previous sections the resultsfrom the combinations of these two reliable models shouldbe also reliable

Describing the flow of MR fluid by the Bingham flowequations (ie combination of the Navier-Stokes equationsand the Bingham constitutive equation) in a rectangular ductthe damping force (119865) of a MR damper can be related to thepiston velocity (V119901) by the following mechanical model [37]

P (T) = 23 (1 + 3T)sdot [cos(13 arccos(1 minus 54 ( T1 + 3T)3)) + 12]

(24)

with the dimensionless pressure gradient (P) and yield stress(T) defined by

P = 1199081198923Δ11990112119876120578119871 T = 119908119892212059111991012119876120578

(25)

where 119908 is the mean circumference of the annular flow pathΔ119901 is the pressure drop responding to the volumetric flowrate 119876 = V119901119860119901 119860119901 is the effective area of the piston headthe damping force 119865 = Δ119901119860119901 and 120578 is the postyield plasticviscosity and taken as 13 Pasdots

Using the time history of the shear yield stress (23) as thebridge connecting the proposed magnetic circuit model andthe Binghamfluid basedmechanicalmodel (24) the dampingforces corresponding to the low- and high-frequency studiesin the above section are examined with the piston velocitychosen as 10mms

As shown in Figure 12 compared to the results of thelow-frequency study an obvious damping force lag withrespect to the coil current is observed in the high-frequencystudy due to the effect of eddy currents Such lag exhibitsa hysteresis loop in the current-force plane as shown inFigure 13 which even exists in a low-frequency varyingmagnetic field because of the magnetic hysteresis Thishysteresis behavior produces a variation in the damping forcebetween an increasing and decreasing coil current Thus in adynamic environment such as oscillatory current excitationuncertainty is introduced in any calculated torque Worsestill the eddy currents make such hysteresis more severe Asexpected an apparent decrease in the maximum dampingforce is also observed in the high-frequency study becauseof the decreasing maximummagnetic induction as shown inFigures 5 and 7

For better understanding of dynamic performances ofMR dampers it is often desirable to gain an insight intothe flow analysis of MR fluid in the working gap and this

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

Shock and Vibration 7

J (AＧ2)75E + 004

70E + 004

65E + 004

60E + 004

56E + 004

51E + 004

46E + 004

41E + 004

36E + 004

31E + 004

26E + 004

21E + 004

16E + 004

11E + 004

57E + 003

75E + 002

(a) Low-frequency study 02Hz

J (AＧ2)27E + 006

25E + 006

23E + 006

21E + 006

19E + 006

18E + 006

16E + 006

14E + 006

12E + 006

10E + 006

85E + 005

67E + 005

49E + 005

31E + 005

12E + 005

minus58E + 004

(b) High-frequency study 10Hz

Figure 6 Eddy currents at half-period instants (ie time 119905 = 1198792)

Part ① Part ② Part ③

H (Am)H (Am) H (Am)10000minus1000 10000minus1000 10000minus1000

minus050

minus025

000

025

050

B(T

)

B(T

)

B(T

)

minus070

minus035

000

035

070

minus050

minus025

000

025

050

FEMProposed model

FEMProposed model

FEMProposed model

Figure 7 Hysteresis curves of steel parts in the MR damper predicted by the proposed model (lines) and FEM (symbols) under high-frequency (10Hz) sinusoidal current excitation (lumped parameter model solver relative error tolerance 1e minus 5 absolute error tolerance1e minus 6 nonlinear residual of FEM solver 1e minus 5)

an identification method similar to that shown in the abovesection they are obtained as 1199031198881 = 16650 times 102 1199031198882 =27785 times 101 1199031198882 = 48578 times 101 and 119896119891119886 = 10593 As shownin Figure 7 the predicted results in the steel parts show anoverall satisfactory agreement with the FEM results althougha larger discrepancy observed as compared to the results inthe low-frequency study

The numerical errors in low- and high-frequency studiescalculated by (22) are compared in Figure 8 The agree-ment between FEM and the proposed model results in the

low-frequency study is much better than that in the high-frequency study with the largest error occurring in steel partA

The larger discrepancy or error especially that in steelpart A is believed to be the consequence of the largernonuniformity ofmagnetic induction due to the effect of eddycurrent The hysteresis curves along lines I II and III (inFigure 3) are shown in Figure 9 with obvious nonuniformitiesof magnetic fields observed under high frequency Comparedto the nonuniformities of magnetic fields in parts B and C

8 Shock and Vibration

Different parts 02 HzDifferent parts 10 Hz

Whole damper 02 HzWhole damper 10 Hz

Part 3Part 2Part 1

002

003

004

005

Erro

r

006

007

008

Figure 8 Comparison of numerical errors under low- and high-frequency studies

the nonuniformity in partA is much larger so using averagevalues in the lumped parameter model produces largererror

Moreover the comparison between the two groups ofhysteresis curves in Figures 8 and 9 shows that the magneticinduction which directly determines the yields stress of MRfluid is apparently reduced under high-frequency changingfields due to the effect of eddy currents

5 A Demonstration of How to Usethe Proposed Magnetic Circuit Model

For the purpose of the demonstration of how to use theproposed magnetic circuit model a simple multiphysicsmodel will be developed in this section The flow field in aMR damper can be controlled by the magnetic field due tochanges of the shear yield-strength of MR fluid In contrastthe magnetic field is generally assumed to be independentof the fluid flow field so the analyses of these two fieldscan be conducted separately in series The experimentalyield stresses (120591119910) of the MR fluid are shown in Figure 10under different magnetic inductions (119861) and their relation-ship which is required for the damping force modelingcan be obtained by a least square curve fitting methodas

120591119910 = 5483033 [1 minus exp (minus314 |119861|203)] (23)

As shown in Figure 10 the fitted curve agrees well with theexperimental data

As shown in Figure 11 after validating the proposed mag-netic circuit model the damping force (119865) and the flow field(119906) of aMR damper can be analyzed by substituting the yield-strength history (obtained from the magnetic circuit model)into any extensively studied fluid model such as the Bingham

model theHerschel-Bulkleymodel and the biviscousmodelThe simple Bingham fluid model is adopted here for thedemonstration purpose Since the proposed magnetic circuitmodel has been validated in previous sections the resultsfrom the combinations of these two reliable models shouldbe also reliable

Describing the flow of MR fluid by the Bingham flowequations (ie combination of the Navier-Stokes equationsand the Bingham constitutive equation) in a rectangular ductthe damping force (119865) of a MR damper can be related to thepiston velocity (V119901) by the following mechanical model [37]

P (T) = 23 (1 + 3T)sdot [cos(13 arccos(1 minus 54 ( T1 + 3T)3)) + 12]

(24)

with the dimensionless pressure gradient (P) and yield stress(T) defined by

P = 1199081198923Δ11990112119876120578119871 T = 119908119892212059111991012119876120578

(25)

where 119908 is the mean circumference of the annular flow pathΔ119901 is the pressure drop responding to the volumetric flowrate 119876 = V119901119860119901 119860119901 is the effective area of the piston headthe damping force 119865 = Δ119901119860119901 and 120578 is the postyield plasticviscosity and taken as 13 Pasdots

Using the time history of the shear yield stress (23) as thebridge connecting the proposed magnetic circuit model andthe Binghamfluid basedmechanicalmodel (24) the dampingforces corresponding to the low- and high-frequency studiesin the above section are examined with the piston velocitychosen as 10mms

As shown in Figure 12 compared to the results of thelow-frequency study an obvious damping force lag withrespect to the coil current is observed in the high-frequencystudy due to the effect of eddy currents Such lag exhibitsa hysteresis loop in the current-force plane as shown inFigure 13 which even exists in a low-frequency varyingmagnetic field because of the magnetic hysteresis Thishysteresis behavior produces a variation in the damping forcebetween an increasing and decreasing coil current Thus in adynamic environment such as oscillatory current excitationuncertainty is introduced in any calculated torque Worsestill the eddy currents make such hysteresis more severe Asexpected an apparent decrease in the maximum dampingforce is also observed in the high-frequency study becauseof the decreasing maximummagnetic induction as shown inFigures 5 and 7

For better understanding of dynamic performances ofMR dampers it is often desirable to gain an insight intothe flow analysis of MR fluid in the working gap and this

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

8 Shock and Vibration

Different parts 02 HzDifferent parts 10 Hz

Whole damper 02 HzWhole damper 10 Hz

Part 3Part 2Part 1

002

003

004

005

Erro

r

006

007

008

Figure 8 Comparison of numerical errors under low- and high-frequency studies

the nonuniformity in partA is much larger so using averagevalues in the lumped parameter model produces largererror

Moreover the comparison between the two groups ofhysteresis curves in Figures 8 and 9 shows that the magneticinduction which directly determines the yields stress of MRfluid is apparently reduced under high-frequency changingfields due to the effect of eddy currents

5 A Demonstration of How to Usethe Proposed Magnetic Circuit Model

For the purpose of the demonstration of how to use theproposed magnetic circuit model a simple multiphysicsmodel will be developed in this section The flow field in aMR damper can be controlled by the magnetic field due tochanges of the shear yield-strength of MR fluid In contrastthe magnetic field is generally assumed to be independentof the fluid flow field so the analyses of these two fieldscan be conducted separately in series The experimentalyield stresses (120591119910) of the MR fluid are shown in Figure 10under different magnetic inductions (119861) and their relation-ship which is required for the damping force modelingcan be obtained by a least square curve fitting methodas

120591119910 = 5483033 [1 minus exp (minus314 |119861|203)] (23)

As shown in Figure 10 the fitted curve agrees well with theexperimental data

As shown in Figure 11 after validating the proposed mag-netic circuit model the damping force (119865) and the flow field(119906) of aMR damper can be analyzed by substituting the yield-strength history (obtained from the magnetic circuit model)into any extensively studied fluid model such as the Bingham

model theHerschel-Bulkleymodel and the biviscousmodelThe simple Bingham fluid model is adopted here for thedemonstration purpose Since the proposed magnetic circuitmodel has been validated in previous sections the resultsfrom the combinations of these two reliable models shouldbe also reliable

Describing the flow of MR fluid by the Bingham flowequations (ie combination of the Navier-Stokes equationsand the Bingham constitutive equation) in a rectangular ductthe damping force (119865) of a MR damper can be related to thepiston velocity (V119901) by the following mechanical model [37]

P (T) = 23 (1 + 3T)sdot [cos(13 arccos(1 minus 54 ( T1 + 3T)3)) + 12]

(24)

with the dimensionless pressure gradient (P) and yield stress(T) defined by

P = 1199081198923Δ11990112119876120578119871 T = 119908119892212059111991012119876120578

(25)

where 119908 is the mean circumference of the annular flow pathΔ119901 is the pressure drop responding to the volumetric flowrate 119876 = V119901119860119901 119860119901 is the effective area of the piston headthe damping force 119865 = Δ119901119860119901 and 120578 is the postyield plasticviscosity and taken as 13 Pasdots

Using the time history of the shear yield stress (23) as thebridge connecting the proposed magnetic circuit model andthe Binghamfluid basedmechanicalmodel (24) the dampingforces corresponding to the low- and high-frequency studiesin the above section are examined with the piston velocitychosen as 10mms

As shown in Figure 12 compared to the results of thelow-frequency study an obvious damping force lag withrespect to the coil current is observed in the high-frequencystudy due to the effect of eddy currents Such lag exhibitsa hysteresis loop in the current-force plane as shown inFigure 13 which even exists in a low-frequency varyingmagnetic field because of the magnetic hysteresis Thishysteresis behavior produces a variation in the damping forcebetween an increasing and decreasing coil current Thus in adynamic environment such as oscillatory current excitationuncertainty is introduced in any calculated torque Worsestill the eddy currents make such hysteresis more severe Asexpected an apparent decrease in the maximum dampingforce is also observed in the high-frequency study becauseof the decreasing maximummagnetic induction as shown inFigures 5 and 7

For better understanding of dynamic performances ofMR dampers it is often desirable to gain an insight intothe flow analysis of MR fluid in the working gap and this

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

Shock and Vibration 9

10

05

00

minus05

minus10

B(T

)

0

510

15

20

Line I (mm) minus3000minus2000

minus10000

10002000

3000

H(Am)

02 Hz10 Hz

(a) Line I

0 2000

12

3

4

Line II (mm)

10

05

00

minus05

minus10

B(T

)

minus3000minus2000

minus10000

1000

3000

H(Am)

02 Hz10 Hz

(b) Line II

B(T

)

0

minus1000minus500

0500

1000

H(Am)

08

06

04

02

00

minus02

minus04

minus06

minus08

12

34

5

Line III (mm)

02 Hz10 Hz

(c) Line IIIFigure 9 Hysteresis curves at points of lines (in Figure 3) under low- and high-frequency current excitation (nonlinear residual of FEMsolver 1e minus 5)becomes straightforward once the pressure drop (Δ119901) hasbeen obtained by solving (24) The flow velocity (119906) can bedirectly calculated by [38]

119906 (119910)

=

Δ1199012120578119871 (119892 minus 1205752 )2 10038161003816100381610038161199101003816100381610038161003816 le 1205752Δ1199012120578119871 [(119892 minus 1205752 )2 minus (10038161003816100381610038161199101003816100381610038161003816 minus 1205752)2] 10038161003816100381610038161199101003816100381610038161003816 gt 1205752

120575 = 2120591119910119871Δ119901

(26)

where 119910 is the gap coordinate and 120575 is the plug thickness asshown in Figure 14

The velocity profiles corresponding to the low- and high-frequency studies are shown in Figure 15 in which the mag-nitudes of the coil currents are indicated by the surface colorsThe parabolic velocity profiles occurring at the instants ofzero current and flat velocity profiles at other instants appearalternately Typically for a flow of MR fluids the larger thecurrents are the flatter the flow velocity profiles are

6 Conclusion

In this research work a novel dynamic magnetic circuitmodel of MR dampers is proposed on the basis of Amperersquos

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

10 Shock and Vibration

Experimental dataFitted curve

0

10

20

30

40

50

60

Shea

r yie

ld st

ress

y

(kPa

)

02 04 06 08 10 1200

Magnetic induction B (T)

Figure 10 Magnetic induction dependence of the shear yield-strength of MR fluid

Fluid model

Rheology of MR fluid Magnetic model

Powersupply

Mechanical modelPiston

velocityNavierndashstokes

equations

A simple multiphysics model of MR dampers

Q flow rate

Ap piston area

u flow velocityF damping force

p piston velocity

Δp pressure drop shear rate

shear stressy shear yield-strengthB magnetic inductionic c coil currentvoltage

ic c

B

y(B)

y

B (ic) B (c)

(y )

p F(p) = 0

F = ΔpAp

f(u p ) = 0

F u

= uyp

Qlarrrarr u

Figure 11 A multiphysics model of MR dampers

and Gaussrsquos laws With the steel parts of MR dampersdescribed by J-A models the magnetic saturation hysteresisand eddy currents are all taken into account A low-frequencystudy is conducted to validate the modelrsquos capability ofmodeling the magnetic saturation and hysteresis The effectof eddy current is investigated in a high-frequency studywith an obvious reduction of magnetic induction observedBy combining the proposed magnetic circuit model with theBingham fluid model a simple multiphysics model is devel-oped Compared to the results in the low-frequency studythe damping force in the high-frequency case apparently lags

behind the coil current due to the effect of eddy currentsSuch lag exhibits a hysteresis loop in the current-force planewhich even appears in a low-frequency varyingmagnetic fieldbecause of the magnetic hysteresis The eddy currents makesuch hysteresis more severe Apparent decrease is observedin themaximumdamping force as that found in themagneticinduction

Conflicts of InterestThe authors declared that they have no conflicts of interest tothis work

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

Shock and Vibration 11

CurrentDamping force proposed modelDamping force FEM

Nor

mal

ized

curr

ent o

r for

ce 02 Hz

10 Hz

010005000

00

05

10

00

05

10

2 4 60

Time (s)

Figure 12 Time histories of normalized forces and currents

02 Hz

10 Hz

Lines proposed modelSymbols FEM

0

1000

2000

3000

4000

5000

Dam

ping

forc

e (N

)

minus1 0 1 2minus2

Coil current (A)

Figure 13 Force-current in low- and high-frequency current excitation

Postyield region

Preyieldregion

Postyield region

g

y

xO u(y t)

Figure 14 Velocity profile across the gap in a MR rectangular duct

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

12 Shock and Vibration

0402

0minus02

minus04

Gap coordinates (mm)0

24

68

10

Time (s)

0

01

02

03

04

Flow

velo

city

u(m

s) 02 Hz

15

1

05

0

(a)

0005

01015

0204

020

minus02minus04

Gap coordinates (mm)Time (s)

0

01

02

03

04

Flow

vel

ocity

u(m

s) 10 Hz

0 1015

020

15

1

05

0

(b)

Figure 15 Evolution of velocity profiles under sinusoidal coil currents (a) 02Hz (b) 10Hz piston velocity 1 cms (Surface color indicatesthe magnitude of excitation current)

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (no 51308450) Natural Science Foun-dation of Shaanxi Provincial Department of Science andTechnology (no 2016JQ5002) Research Foundation of XirsquoanUniversity of Architecture and Technology (no RC1368)and Foundation of Key Laboratory of Structures DynamicBehavior and Control (Ministry of Education) in HarbinInstitute of Technology (no HITCE201708)

References

[1] D Case B Taheri and E Richer ldquoDynamical modelingand experimental study of a small-scale magnetorheologicaldamperrdquo IEEEASME Transactions on Mechatronics vol 19 no3 pp 1015ndash1024 2014

[2] Z Parlak T Engin and I Calli ldquoOptimal design of MR dampervia finite element analyses of fluid dynamic andmagnetic fieldrdquoMechatronics vol 22 no 6 pp 890ndash903 2012

[3] B Sapinski and S Krupa ldquoEfficiency improvement in a vibra-tion power generator for a linearMR damper numerical studyrdquoSmart Materials and Structures vol 22 no 4 Article ID 0450112013

[4] H H Zhang C R Liao W M Chen and S L Huang ldquoA mag-netic design method of MR fluid dampers and FEM analysis onmagnetic saturationrdquo Journal of Intelligent Material Systems andStructures vol 17 no 8-9 pp 813ndash818 2006

[5] J Zheng Z Li J H Koo and J Wang ldquoMagnetic circuit designandmultiphysics analysis of a novelMRdamper for applicationsunder high velocityrdquoAdvances inMechanical Engineering vol 6Article ID 402501 2014

[6] P-B Nguyen and S-B Choi ldquoA new approach to magneticcircuit analysis and its application to the optimal design of abi-directional magnetorheological brakerdquo Smart Materials andStructures vol 20 no 12 Article ID 125003 2011

[7] QHNguyen S B Choi Y S Lee andM S Han ldquoAn analyticalmethod for optimal design of MR valve structuresrdquo SmartMaterials and Structures vol 18 no 9 p 095032 2009

[8] ANSYSMaxwell Features 2015 httpwwwansyscomPro-ductsSimulation+TechnologyElectronicsElectromechanicalANSYS+MaxwellFeatures

[9] F Preisach ldquoUber die magnetische Nachwirkungrdquo Zeitschriftfur Physik vol 94 no 5-6 pp 277ndash302 1935

[10] D C Jiles and D L Atherton ldquoTheory of ferromagnetic hys-teresisrdquo Journal of Magnetism and Magnetic Materials vol 61no 1-2 pp 48ndash60 1986

[11] D Marcsa and M Kuczmann ldquoAnalysis of ferromagnetic corecombining preisach hysteresis modeling and finite elementtechniquesrdquo Journal of Advanced Research in Physics vol 1 no1 Article ID 011004 2010

[12] N Sadowski N J Batistela J P A Bastos and M Lajoie-Mazenc ldquoAn inverse Jiles-Atherton model to take into accounthysteresis in time-stepping finite-element calculationsrdquo IEEETransactions on Magnetics vol 38 no 2 I pp 797ndash800 2002

[13] C Jedryczka P Sujka andW Szelag ldquoThe influence ofmagnetichysteresis onmagnetorheological fluid clutch operationrdquoCOM-PEL -The International Journal for Computation andMathemat-ics in Electrical and Electronic Engineering vol 28 no 3 pp 711ndash721 2009

[14] J An and D-S Kwon ldquoModeling of a magnetorheologicalactuator including magnetic hysteresisrdquo Journal of IntelligentMaterial Systems and Structures vol 14 no 9 pp 541ndash550 2003

[15] M L Hodgdon ldquoApplications of a theory of ferromagnetichysteresisrdquo IEEE Transactions on Magnetics vol 24 no 1 pp218ndash221 1988

[16] J K Deur Z Herold and M Kostelac ldquoModeling of elec-tromagnetic circuit of a magnetorheological fluid clutchrdquo inProceedings of the Control Applications (CCA) and IntelligentControl IEEE Petersburg Russia 2009

[17] D Jiles Introduction to magnetism and magnetic materialsSpringer London UK 2nd edition 1991

[18] N Takesue J Furusho and Y Kiyota ldquoAnalytic and experimen-tal study on fast response MR-fluid actuatorrdquo in Proceedings ofthe International Conference on Robotics and Automation ICRArsquo03 IEEE Taipei Taiwan 2003

[19] Z Jiang and R E Christenson ldquoA fully dynamic magneto-rheological fluid damper modelrdquo Smart Materials and Struc-tures vol 21 no 6 Article ID 065002 2012

[20] Y-J Nam and M-K Park ldquoElectromagnetic design of a mag-netorheological damperrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 2 pp 181ndash191 2009

[21] A Farjoud and E A Bagherpour ldquoElectromagnet design formagneto-rheological devicesrdquo Journal of Intelligent MaterialSystems and Structures vol 27 no 1 pp 51ndash70 2014

Shock and Vibration 13

[22] K Chwastek ldquoModelling of dynamic hysteresis loops usingthe Jiles-Atherton approachrdquoMathematical andComputerMod-elling of Dynamical Systems vol 15 no 1 pp 95ndash105 2009

[23] H Bose and J Ehrlich ldquoMagnetorheological dampers with var-ious designs of hybrid magnetic circuitsrdquo Journal of IntelligentMaterial Systems and Structures vol 23 no 9 pp 979ndash987 2012

[24] C Chen and W-H Liao ldquoA self-sensing magnetorheologicaldamperwith power generationrdquo SmartMaterials and Structuresvol 21 no 2 Article ID 025014 2012

[25] B Sapinski ldquoEnergy-harvesting linearMRdamper prototypingand testingrdquo Smart Materials and Structures vol 23 no 3Article ID 035021 2014

[26] L H Lewis J Gao D C Jiles and D O Welch ldquoModelingof permanent magnets Interpretation of parameters obtainedfrom the Jiles-Atherton hysteresis modelrdquo Journal of AppliedPhysics vol 79 no 8 pp 6470ndash6472 1996

[27] D Zhang Z Ren and C-S Koh ldquoHysteresis modeling ofanisotropic and isotropic bonded NdFeB PM utilizing the jiles-atherton hysteresis modelrdquo in Proceedings of the InternationalConference on Electrical Machines and Systems ICEMS rsquo13 pBusan South Korea IEEE October 2013

[28] S E Zirka Y I Moroz R G Harrison and K Chwastek ldquoOnphysical aspects of the Jiles-Atherton hysteresismodelsrdquo Journalof Applied Physics vol 112 no 4 Article ID 043916 2012

[29] P R Wilson J N Ross and A D Brown ldquoOptimizing theJiles-Atherton model of hysteresis by a genetic algorithmrdquo IEEETransactions on Magnetics vol 37 no 2 pp 989ndash993 2001

[30] P Kis and A Ivanyi ldquoParameter identification of Jiles-Athertonmodel with nonlinear least-square methodrdquo Physica B Con-densed Matter vol 343 no 1-4 pp 59ndash64 2004

[31] K Chwastek and J Szczygłowski ldquoEstimation methods for theJiles-Atherton model parameters - a reviewrdquo in Proceedings ofthe 2nd Symposium on Applied Electromagnetics 2008

[32] G Yang B F Spencer Jr J D Carlson and M K Sain ldquoLarge-scale MR fluid dampers modeling and dynamic performanceconsiderationsrdquo Engineering Structures vol 24 no 3 pp 309ndash323 2002

[33] S Zirka Y Moroz P Marketos A Moses and D Jiles ldquoMea-surement and modeling study of B-H loops and losses of highsilicon non oriented steelsrdquo in Proceedings of the InternationalMagnetics Conference INTERMAG rsquo06 IEEE California CalifUSA May 2006

[34] S E Zirka Y I Moroz P Marketos A J Moses D C Jilesand T Matsuo ldquoGeneralization of the classical method forcalculating dynamic hysteresis loops in grain-oriented electricalsteelsrdquo IEEE Transactions on Magnetics vol 44 no 9 pp 2113ndash2126 2008

[35] G Bertotti ldquoGeneral properties of power losses in soft ferro-magnetic materialsrdquo IEEE Transactions on Magnetics vol 24no 1 pp 621ndash630 1988

[36] D C Jiles ldquoFrequency dependence of hysteresis curves in con-ducting magnetic materialsrdquo Journal of Applied Physics vol 76no 10 pp 5849ndash5855 1994

[37] H P Gavin R D Hanson and F E Filisko ldquoElectrorheologicaldampers part i analysis anddesignrdquo Journal of AppliedMechan-ics vol 63 no 3 pp 669ndash675 1996

[38] G A Dimock J-H Yoo and N M Wereley ldquoQuasi-steadyBingham biplastic analysis of electrorheological and magne-torheological dampersrdquo Journal of Intelligent Material Systemsand Structures vol 13 no 9 pp 549ndash559 2002

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom