Resea ch A icle
Mul iphysics Model o an MR Dampe including
Magne ic Hys e esis
M. Kub´
ık
1
and J. Goldasz
2
,
3
1
Facul y o Mechanical Enginee ing, B no Uni e si y o Technology, B no, Czech Republic
2
Facul y o Elec ical and Compu e Enginee ing, C acow Uni e si y o Technology, K ak´
ow, Poland
3
Technical Cen e K ak´ow, BWI G oup, K ak´ow, Poland
Co espondence should be add essed o M. Kub´
ık; [email p o ec ed]
Recei ed 12 Ap il 2019; Re ised 31 May 2019; Accep ed 9 June 2019; Published 26 June 2019
Gues Edi o : E ´
en D´
ıez-Jim´
enez
Copy igh ©2019 M. Kub´
ık and J. Goldasz. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion
License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is
p ope ly ci ed.
Hys e esis is one o key ac o s influencing he ou pu o magne o heological (MR) ac ua o s. The ac ua o s e eal wo p ima y
sou ces o hys e esis. The hyd o(mechanical) hys e esis can be ela ed o flow dynamics mechanisms and is equency- o a e-
dependen . Fo compa ison, he magne ic hys e esis is an inhe en p ope y o e omagne ic ma e ials o ming he magne ic
ci cui o he ac ua o s. The need o a good quali y hys e esis model has been ea ly ecognized in s udies on MR ac ua o s;
howe e , ew s udies ha e p o ided models which could be used in he design s age. In he pape we e eal a hyb id mul iphysics
model o a flow-mode MR ac ua o which could be used o ha pu pose. The model elies on he in o ma ion which can be
ex ac ed p ima ily om ma e ial da ashee s and enginee ing d awings. We e eal key de ails o he model and hen e i y i
agains measu ed da a. Finally, we employ i in a pa ame e sensi i i y s udy o examine he influence o magne ic hys e esis and
o he ele an ac o s on he ou pu o he ac ua o .
1. In oduc ion
Magne o heological (MR) dampe s a e ai ly well-known
de ices u ilizing MR fluids which, when subjec ed o
magne ic s imuli o sufficien s eng h, gene a e yield s ess
[1]. So a , he unique echnology has been comme cialized
in semiac i e passenge ehicle suspensions, powe ain
moun s [2], o op ical finishing [3]. Low powe consump-
ion, as and e e sible esponses, and high dynamic ange
ha e made he de ices a ac i e o use in ib a ion con ol
sys ems in pa icula [4]. As MR dampe s a e gene ally
ope a ed in eal- ime con ol sys ems, hei dynamic pe -
o mance is equally impo an as o mo e impo an han
hei s eady-s a e cha ac e is ics. S eady-s a e cha ac e is ics
only p o ide he e idence o an ac ua o o a dampe
mee ing he equi ed o ce/ o que ange (o o ce/ o que)
a ge s. Thei dynamic beha iou needs o be quan ified a
he same ime i i is used in a eal-li e con ol p ocess. Thus,
unde s anding he con ibu ions o a ious ac o s com-
plica ing he o ce o o que build-up dynamic p ocess is
c i ical o he de elopmen o a ealis ic applica ion. B iefly,
wi h MR ac ua o s, he e is ample e idence o se e al ac o s
complica ing he o ce/ o que gene a ion p ocess, namely,
mechanical/hyd aulic hys e esis, magne ic hys e esis, con-
ol ci cui dynamics such as eddy cu en s, d i e dynamics,
empe a u e, flow losses, ic ion, and nonlinea ela ionship
be ween he ma e ial’s yield s ess and he induced flux
[5–7]. These ac o s influence he de ice’s abili y o gene a e
he ou pu o ce/ o que and need o be accoun ed, o in-
s ance, o in he con ol algo i hm de elopmen p ocess.
In his s udy, we pay pa icula a en ion o modeling he
dampe ’s hys e e ic beha iou . In gene al, MR de ices e eal
wo p ima y sou ces o hys e esis. The (hyd o-)mechanical
hys e esis can be ela ed o he damped dynamics o a hea y
slug o MR fluid (MRF) bouncing agains complian col-
umns o MRF in fluid chambe s. The effec is a e- o e-
quency-dependen , and i s magni ude a ies wi h he
cu en applied, oo. I disappea s as he mechanical exci-
a ion equency app oaches ze o [8]. The magne ic hys-
e esis is diffe en . I is p esen in all elec omagne ic de ices,
e.g., elec omagne ic solenoids [9], mo o s [10], and mag-
ne o heological ac ua o s [11]. Fi s o all, i is he inhe en
Hindawi
Shock and Vib a ion
Volume 2019, A icle ID 3246915, 20 pages
h ps://doi.o g/10.1155/2019/3246915
p ope y o e omagne ic ma e ials o ming he magne ic
ci cui o he MR al e; he hys e esis o ca bonyl i on-
(CIP-) based MFRs is i ually nonexis en [12]. Nex , i does
no anish as he inducing cu en equency app oaches DC
limi . Also, empe a u e, load his o y, and mechanical
s esses ha e a nega i e influence on he hys e esis and
magne iza ion cha ac e is ics o e omagne ic ma e ials
[13]. Fo ins ance, i is a common p ac ice o subjec ma-
chined e omagne ic componen s o hea ea men o
in e nal s ess and hys e esis as well as coe ci e o ce
educ ion.
The need o a good quali y hys e esis model has been
ea ly ecognized in s udies on MR ac ua o s, and he eade
should e e , e.g., o Zheng e al. [14] o a e iew o sui able
phenomenological models as well as o he well-known s udy
o Spence e al. [15]. In gene al, he pos e io i pa ame ic
models we e ob ained by examining he o ce-posi ion and
o ce- eloci y ela ionships by fi ing by model esponse o
he ac ua o ’s ou pu . Such models a e sui able o con ol
s udies only. In he de ice’s de elopmen p ocess, o he
app oaches a e equi ed. In ha aspec , many MR- ela ed
esea ch s udies neglec ed he pa icula con ibu o ’s
p esence. The opic, howe e , has been well iden ified in he
field o con en ional solenoid ac ua o s whe e a ious
models we e de eloped o copy he hys e e ic beha iou o
he ac ua o s. Fo ins ance, Maye goyz [16] applied he
P eisach model o model he hys e e ic beha iou o a so-
lenoid ac ua o . In he P eisach model, he hys e esis is he
sum o elemen a y hys e esis loops. Nex , Coleman and
Hodgdon [17] de eloped a fi s -o de diffe en ial equa ion
ha links he field s eng h Hand he flux densi y B. One
model whose pa ame e s can be ela ed o physical p op-
e ies o e omagne ic ma e ials is he Jiles–A he on (J–A)
model [18, 19]. The model was ex ended o include bo h he
impac o eddy cu en s and empe a u e on hys e esis and
magne isa ion cu es [20, 21]. All o he abo e models can be
ec o ized. Tellinen [22] p oposed a simple scala model o
handling he hys e esis based on he limi ing hys e esis loop
om physical measu emen s o e omagne ic ma e ials.
Wi h MR ac ua o s, howe e , al hough he significance o a
good quali y hys e e ic model has been ecognized ea ly, he
opic does no seem o ha e dese ed enough a en ion.
Significan con ibu ions include Han e al. [23] who ex-
amined he field dependen hys e esis o ER fluids. The
au ho s used he amilia P eisach app oach. Mo eo e , Han
e al. [7] used he P eisach model o iden i y he hys e esis o
an MR fluid. Yadmella and Ke mani de eloped a model o
an MR clu ch in which he de eloped hys e esis model was
assessed agains he P eisach ope a o [24]. Fo compa ison,
in hei ea ly s udy using he Coleman-Hodgdon model, An
and Kwon modelled he hys e e ic beha iou o an MR
clu ch, and by examining he o que-cu en loops showed
ha he hys e esis is an impo an con ibu o o he de ice’s
ou pu [25]. The model pa ame e s we e iden ified om
physical measu emen s (o magne isa ion cha ac e is ics) o
he ma e ials o ming he magne ic ci cui o he clu ch.
Nex , Je
˛d yczka e al. p esen ed a fini e-elemen (FE) model
o an MR clu ch based on he J-A app oach [26]. Mo eo e ,
Guo e al. p esen ed a ansien mul idomain model o a
flow-mode dampe based on he J-A app oach and hen
e ified i agains he no el FE ec o hys e esis echnique
[27]. The in e se J-A model was ecen ly examined by Zheng
e al. [14] o copy he ansien beha iou o an MR flow-
mode dampe . Goldasz e al. [28] p oposed an ex ension o
he Bouc–Wen model in an a emp o sepa a e he magne ic
hys e esis om he mechanical one. The au ho s p oposed a
simple lumped pa ame e model o he ac ua o including a
hys e e ic ope a o . The model was e ified agains selec ed
sinusoidal AC exci a ion inpu s and p o ided accep able
accu acy o p ac ical pu poses. S ill, when compa ed o he
as numbe o esea ch s udies using pa ame ic phe-
nomenological hys e e ic models, he opic does no seem
in ensi ely s udied as al eady men ioned. Tha may be due o
ew exis ing comp ehensi e elec omagne ic models o such
ac ua o s.
Ano he aspec is dynamics. Clea ly, he insigh in o he
dynamics o MR ac ua o s should be p o ided h ough
ansien models. Such a comp ehensi e model would a -
emp o copy no only he dynamics o he fluid flow
h ough he al e and he flux dynamics bu accoun o he
physics ou side he con ol al e as well. Sui able lumped
pa ame e models usually u ilize a ne wo k o elemen s
ep esen ing physical domains o in e es (hyd aulic, he -
mal, elec ic, and magne ic) and connec ions (in e aces)
be ween hem [2]. Fo example, he elec ical ci cui o he
ac ua o can be ep esen ed in he o m o a esis o -non-
linea induc o ne wo k model [5]. Thei main disad an age
is he necessi y o using a ious simpli ying assump ions,
e.g., uni o m yield s ess/flux, ully de eloped flow, e c. On
he con a y, con inuum mul iphysics (magne ics and flow
dynamics) models u ilize ewe assump ions and can be
exe cised on ealis ic geome ies, howe e , a a significan
compu a ional expense [29].
As such, in he pape , we p opose a hyb id mul iphysics
model o he magne o heological dampe which sepa a es
he magne ic hys e esis o he magne ic ci cui o he ac-
ua o om ha o he mechanical ha dwa e. B iefly, he
elec omagne ic domain is modelled using he ec o hys-
e esis FE model (p esen in Ansys Maxwell) based on he
ex ension o well-known Maxwell equa ions [30], and he
hyd aulic sec ion is desc ibed h ough dimensionless
biplas ic Bingham app oach [31].
The pape is o ganized as ollows. Fi s , we p esen an
MR dampe geome y and key ma e ial p ope ies. Then, in
he ollowing sec ion, we e eal key de ails o he FE model
o he ac ua o such as magne ic hys e esis and he coupled
lumped pa ame e hyd omechanical model. Nex , we show
measu emen s o magne ic hys e esis loops and a com-
pa ison o he measu emen s agains he FE elec omagne
model. Finally, we show esul s o a pa ame ic s udy (also
in ol ing he hyb id model) in an a emp o examine he
hys e esis influence on he ou pu o he MR ac ua o and
hen d aw conclusions.
2. Magne o heological Dampe
In he s udy, an MR flow-mode dampe configu a ion ha ing
a single coil assembly in he elec omagne and one annula
2Shock and Vib a ion
flow pa h in he con ol al e is o esea ch in e es . The
dampe is p esen ed in Figu e 1. The hyd aulic ube houses (1)
he pis on (2), he pis on od (3), he floa ing pis on (4) and
he od guide assembly (5). The pis on sepa a es he MR fluid
olume in o ebound chambe olume and he comp ession
chambe olume. The floa ing pis on sepa a es he fluid om
he gas chambe . The MR al e loca ed in he pis on con ol
con ols he fluid flow be ween he ebound and comp ession
chambe and ice e sa. The MR al e is a con en ional
con ol al e by design. I consis s o he pis on co e (6), he
slee e (7), he nonmagne ic flanges o pla es (8), he coil
assembly (9), and he connec ing wi es (10) o connec ing o
an ex e nal powe supply. I is he mos common single- ube
MR dampe configu a ion.
The MR al e’s magne ic ci cui (6, 7) is manu ac u ed
ou o annealed low-ca bon s eel 11SMn30 (see he com-
ponen s in blue in Figu e 1). The b onze (yellow) flanges (8)
define he mu ual posi ion o he pis on co e and slee e. The
dis ance be ween he ou e diame e o he annulus and he
inne diame e o he slee e defines he annula gap heigh .
The coil assembly (9) inco po a es N�120 u ns o coppe
(pu ple) wi e (0.5 mm diame e ).
The connec ing wi es a e ou ed h ough he h u-hole in
he pis on od (3) made o s eel 42C Mo4 (AISI 4140). The
floa ing pis on, he od guide, and he emaining compo-
nen s a e manu ac u ed ou o s eel S235JR (g een). The MR
dampe is filled wi h he fluid MRF132-DG by Lo d Co p;
see he magne isa ion cu e in Figu e 2(c). The dampe key
dimensions, heological p ope ies o MR fluid, and gas
chambe de ails a e shown in Table 1. The magne isa ion
cu es o he annealed low-ca bon s eel 11SMn30 we e
de e mined using he measu emen sys em Remag aph
C-500. The ob ained i gin & hys e esis da a can be seen in
Figu es 2(a) and 2(b).
The coe ci i i y and emanence o he 11SMn30 ma e ial
sample we e de e mined om he measu ed hys e esis:
H
c
�209 A/m and emanence B
�1.09 T. The ma e ial’s
bulk conduc i i y was se a 5.8 MS/m. Based on simila
measu emen s o he od ma e ial (42C Mo4), we se i s
coe ci i y o H
c
�1250 A/m and he bulk conduc i i y o
4.5 MS/m.
3. Modeling
In he sec ion, we p esen modeling de ails. Specifically, we
highligh he ansien magne ic FE model o he MR al e
including hys e esis ollowed by a desc ip ion o a mono ube
dampe lumped pa ame e model. The lumped pa ame e
model is coupled wi h he ansien FE model h ough he
yield s ess-flux densi y in e ace. We conside he in eg a ed
model as illus a ed in Figu e 3. In he p esen ed layou , he
elec omagne ic ci cui (desc ibed in Sec ion 3.1) is d i en by
he ol age usupplied by he cu en d i e . The esul ing
ou pu flux densi y B
g
is hen con e ed in o he ma e ials’
(fluid) field-induced yield s ess τ
0
(ex ac ed om he ma-
e ial’s da ashee o heological measu emen s). Gi en he
inpu eloci y o displacemen and he yield s ess, we hen
calcula e he ou pu o ce acco ding o he equa ions in
Sec ion 3.2.
3.1. T ansien Magne ic Model wi h Magne ic Hys e esis.
To model he elec omagne ic ci cui o he MR al e, we
applied he ec o hys e esis modeling ea u e a ailable in
Ansys Maxwell R19. Fo iso opic ma e ial and 2D/3D
p oblems, he ec o play model was ecognized o be mo e
compu a ionally efficien han a ec o P eisach model
[30, 33]. In gene al, he play model assumes a decomposi ion
o he applied field Hin o he e e sible componen H
e
and
he i e e sible one H
i
. Then, he esul ing flux densi y
B�B e +Bi �μ0M H e
 􏼁+μ0Hi ,(1)
whe e B
e
is he e e sible componen o flux densi y and B
i
is he i e e sible componen . The magne iza ion M a ies
wi h he e e sible field componen along an anhys e e ic
cu e. The p ocess can be isualized as in Figu e 4. The
pa ame e s o he model can be iden ified om he majo
hys e esis loop. The majo hys e esis loop inco po a es wo
b anches, he ascending b anch and he descending b anch,
and hey can be calcula ed om each o he , he Maxwell
model u ilizes only one b anch o pa icula B-Hloops. The
algo i hm o cons uc ing he majo and symme ic mino
hys e esis loops is gi en in [34].
To de elop he FE model, we assumed he al e o be
axially symme ical a ound he cen e line in a cylind ical
coo dina e sys em. The geome y o he MR dampe
pis on ( al e) was simplified o he ansien simula ions
(Figu e 5(a)). The disc e ized model can be obse ed in
Figu e 5(b). As shown, he geome y was disc e ized using
iangula elemen s. The elemen leng h-based efinemen
wi h he maximum leng h o 0.5 mm was applied o he coil
co e and he slee e. Nex , he alue o 0.7 mm was applied
o he pis on od and he hyd aulic ube, and he c i e ion
o 0.2 mm was applied o he MR fluid egion in he ac i e
zone.
To accomplish ansien field simula ions, he FE
model was coupled o he ex e nal ci cui e ealed in
Figu e 6(a). The lumped ci cui documen s an ideal cu -
en sou ce (inpu ) in se ies wi h a esis o (coil winding)
and he FE al e objec (as ep esen ed by he nonlinea
induc o ). The model is subjec ed o p esc ibed cu en
wa e o ms, and he esul ing flux densi y in he annulus B
g
is ex ac ed om he simula ion esul s. The flux densi y B
g
p esen ed in he nex sec ions was calcula ed by a e aging
flux densi y o he middle o he annulus. The in o ma ion
is equi ed o coupling he FE model wi h he dampe
hyd aulics.
The figu e also shows he cu e fi o he expe imen al
da a o he s eel 11SMn30 (Figu e 6(b)). The ag eemen
is sa is ac o y excep o low magne ic field s eng h and
on he ini ial magne iza ion cu e only. The co e and
slee e componen s we e assigned he B-H p ope ies o he
11SMn30 alloy and he MRF componen ha o MRF132-
DG a ailable om Lo d Co p. Also, he od componen was
assigned he ma e ial p ope ies o he 42C Mo4 s eel alloy
as men ioned abo e. Finally, all da a in subsequen simu-
la ions we e ob ained using he fixed s ep-size sol e wi h he
ollowing se ings: cons an ime s ep was 0.05 ms, nonlinea
esidual was 1e−7, ime in eg a ion me hod was Backwa d
Eule .
Shock and Vib a ion 3
35 214
(a)
97 6 810
Lc
La
ϕDp
ϕDc
ϕd
g
(b)
Figu e 1: Magne o heological dampe . (a) MR dampe . (b) Pis on (con ol al e).
–1.8
–1.2
–0.6
0
0.6
1.2
1.8
–6000 –4000 –2000 0 2000 4000 6000
H (A/m)
B (T)
(a)
H (A/m)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 1000 2000 3000 4000 5000
B (T)
(b)
H (A/m)
B (T)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 100 200 300 400 500 600 700
(c)
Figu e 2: Vi gin and hys e esis magne iza ion cu es. (a) Hys e esis cu e o 11SMn30. (b) S a ic cu e o 11SMn30. (c) S a ic cu e o MRF
132-DG [32].
Table 1: MR dampe dimensions and ma e ial p ope ies.
Name Value Symbol Uni
Geome y and weigh
Pis on od ou e diame e 12 dmm
Pis on ou e diame e 36 D
p
mm
Annula gap heigh 0.65 hmm
Ac i e zone leng h 16 L
a
mm
Co e leng h 37 L
c
mm
Pis on s oke 150 L
s
mm
In e nal diame e o he gap 28 D
c
mm
Floa ing pis on weigh 0.07 m
kg
MR fluid p ope ies (MRF-132DG [32])
MR fluid iscosi y a 40°C 0.114 μPa·s
MR fluid iso he mal bulk modulus 1500 βMPa
MR fluid densi y 3090 ρkg·m
−3
4Shock and Vib a ion
3.2.Hyd omechanicalModel. To illus a e o e eal he effec
o fluc ua ing ( ansien ) magne ic field on he ou pu o he
ac ua o , a capable dampe model is equi ed. Modeling he
beha iou o MR dampe s has been clea ly he subjec o
in ensi e esea ch, o name only [36–38]. Howe e , we chose
o p oceed u he wi h he model o Goldasz and Sapinski in
[2]. The app oach is flexible, inco po a es mos key physical
phenomena occu ing in he MR al e and ou side o i , and
was success ully e ified agains se e al MR pis on al e
configu a ions (mono ube dampe , al e: single coil, single
annula flow pa h, magne ic flux bypass ea u e). The e o e,
in he sec ions ha ollow, we desc ibe de ails o he lumped
pa ame e model o he dampe and he coupling me hod
wi h he FE ansien model.
Elec omagne
model (FE)
Dampe model
Equa ion (1)–(9)
D i e
ic, Bg
u
icmd
icx ,
τ0Fd
Bg/τ0
Figu e 3: Block diag am o he p oposed model.
H
M
H e
H e
H e
Hi
2Hi
M
H
Figu e 4: Magne ic field decomposed in o e e sible/i e e sible componen s [35].
(a) (b)
Figu e 5: Simplified geome y o he pis on uni and mesh. (a) Geome y. (b) Mesh.
Table 1: Con inued.
Name Value Symbol Uni
Ai con en in he MR fluid 0.01 α—
Gas chambe
Gas olume (a mids oke) 46000 Vg0 mm
3
Gas empe a u e 40 T°C
Ini ial gas p essu e 30 Pg0 ba
Ini ial floa ing pis on posi ion 20 xgmm
O he s
Ini ial ebound chambe (uppe ) MR fluid olume 63333 V 0 mm
3
Ini ial comp ession (lowe ) chambe MR fluid
olume 71250 Vc0 mm
3
Coil u ns 120 N—
Coil esis ance 1.0 R
c
Ω
Nondimensional iscosi y a io (es .) 0.1 c—
Yield s ess numbe (es .) 0.5 δ—
Shock and Vib a ion 5

3.2.1. MR Dampe Model: Theo e ical Backg ound. The
schema ic geome y o he dampe is e ealed in Figu e 7. In
he p esen ed illus a ion, he pis on sepa a es he uppe
( ebound) fluid chambe om he fluid below i (com-
p ession chambe ). The p essu e in he uppe chambe is P ,
and i s (ini ial) olume is V (V 0). Acco dingly, he p essu e
in he comp ession chambe is Pc, and i s (ini ial) olume is
Vc(Vc0). The gas p essu e is Pg(Pg0), and he (ini ial) gas
olume below he floa ing pis on is e e ed o as Vg(Vg0).
A s a ic condi ions, he p essu e in each chambe is equal o
Pg0. The c oss-sec ional a ea o he pis on is A
p
and ha o
he od A
. As he pis on od mo es, i displaces he floa ing
pis on (sepa a ing he lowe fluid chambe and he p es-
su ised gas). The floa ing gas cup mass is m
g
, and i s dis-
placemen is x
g
. The ic ion o ces agains he od guide and
he floa ing pis on a e F
g
and F
, espec i ely. We assume
one annula flow pa h in he MR al e; dimensions: his he
gap heigh ; wis he ci cum e en ial wid h (a pe ime e );
A
g
�wh is he flow channel a ea. The flow a e h ough he
annulus is e e ed o as Q
a
. Finally, we e e o he dis-
placemen o he pis on od as x
and o ha o he cylinde
ube as x
(no shown).
The fluid’s beha iou is quan ified wi h he iscosi y μ,
he densi y ρ, he comp essibili y β, and he field-induced
yield s ess τ
0
. The non-New onian heology o he MR fluid
is desc ibed using he biplas ic Bingham model [31].
We assume ha he dampe model would accoun o
he ollowing phenomena: MR effec (using he biplas ic
Bingham model men ioned abo e), comp essibili y o fluid,
dynamics o he fluid elemen (“slug”) mo ion when o ced
h ough he annulus, en ance and exi losses in he annulus,
floa ing pis on mass ine ia, and seal ic ion. Elas ici y o he
cylinde ube, a ious effec s due o hea ing, and he de-
pendency o seal ic ion on he dampe in e nal p essu e a e
no accoun ed o .
Fi s , he gas p essu e in he olume below he floa ing
pis on can be modeled by assuming he adiaba ic p ocess
(n�1.4). The gas p essu e Pgis hen dependen on he
posi ion o he floa ing pis on x
g
in he ollowing manne :
Pg�Pg0
Vg0
Vg0 +Apxg
􏼠 􏼡n
.(2)
The p essu e a ia ion in he chambe s below/abo e he
pis on is modeled assuming iso he mal p ocesses and he
conse a ion o mass app oach [39]. Nex , he dynamics o
he mass o fluid is conside ed by examining he mo ion o
he fluid mass in he annula channel. The esul ing sys em
o o dina y diffe en ial equa ion in he s a e-space o m
which desc ibes he mu ual ela ionships be ween he e-
bound p essu e chambe P , he comp ession chambe
p essu e Pc, he floa ing pis on mo ion eloci y g, and he
Fd
F
F g
A , m
P , V
Ap
Qa
x ,
x
g, g
Pg, Vg
Pc, Vc
mg
Figu e 7: Dampe model schema ic layou .
A_ou A_in
Cu en _sou ce
Resis o
Magne ic_model
0
(a)
–2
–1.5
–1
–0.5
0
0.5
1
1.5
2
–6000 –4000 –2000 0 2000 4000 6000
B (T)
H (A/m)
(b)
Figu e 6: Ex e nal ci cui model and magne isa ion cu e: 11SMn30 (cu e fi s da a). (a) Ex e nal ci cui . (b) B-H plo .
6Shock and Vib a ion
olume ic flow a e h ough he annulus Qais shown
below
_
P �βAp−A
􏼐 􏼑 p−Qa
V
,(3)
_
Pc�β g− p
􏼐 􏼑Ap+Qa
Vc
,(4)
_
g�1
mg
ApPc−Pg
􏼐 􏼑−F gsign g
􏼐 􏼑􏽨 􏽩,(5)
_
Qa�Ag
ρLP −Pc−Δpa
 􏼁.(6)
As al eady men ioned, he beha iou o he ene gized
MR fluid is desc ibed by inco po a ing he field-dependen
losses in o he p essu e d op Δp
a
which is desc ibed in de ail
in Sec ion 3.2.3; he eade should e e o [2, 39] o a mo e
de ailed de i a ion o he equa ions and he expe imen al
e ifica ion me hod. The p essu e e m also inco po a es he
local flow losses Δp
e
. The local flow losses as he fluid en e /
exi s he annulus a e accoun ed o using he semiempi ical
equa ion [40]:
Δpe�KρQa
2A2Qa
􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌,(7)
KSE �Kco 1−Ag
Ap
􏼠 􏼡2
,(8)
KSC �Kco 1−Ag
Ap
􏼠 􏼡0.75
,(9)
whe e K/K
SE
is he p essu e loss coefficien o he sudden
enla gemen (exi om he gap), K/K
SC
is he p essu e loss
coefficien o he sudden con ac ion (en ance o he an-
nula gap), and K
co
is he co ec ion ac o . Finally, con-
side ing he o ces ac ing on he pis on yields he ollowing
ela ionship:
Fd�Ap−A
􏼐 􏼑P −PcAp+F +F g.(10)
The ic ion o ce in MR dampe is assumed o be he
sum o S ibeck, Coulomb, and iscous componen s [40]. As
al eady men ioned, he effec s o iscosi y change wi h
empe a u e (hea ing) a e no included.
We sol e he sys em o equa ions (2)–(10) using he
mul idomain modeling package Simscape which ex ends
Simulink wi h ools o objec o ien ed modeling and
simula ing mul iphysics sys ems [40]. Ou model as
shown in Figu e 8 consis s o mechanical, hyd aulic, and
physical signals domains. Using ha en i onmen , he
MR dampe model was de eloped wi h iso he mal hy-
d aulic double-ac ing cylinde componen s, adiaba ic gas
blocks, and ic ion componen s. The MR fluid beha iou
as copied specifically by equa ion (6) was defined by
means o a cus om componen based on he biplas ic
Bingham model app oach [31] in se ies wi h he local loss
model.
To allow simula ions o he ansien pe o mance o
he dampe , he model was coupled o he FE model in
Ansys Maxwell h ough he magne ic flux densi y s yield
s ess ela ionship, τ
0
�τ
0
(B
g
). The in e ace assumes ze o
delay be ween he elec omagne ic esponse o he ci cui
and he MRF esponse. MRF measu emen s indica e he
esponse ime o he fluid o be below 0.6 ms; he e o e,
ha pa icula con ibu ion is omi ed in he de eloped
equa ion se .
3.2.2. Magne o heological Val e Model. In his sec ion, we
desc ibe he biplas ic Bingham compu ing scheme o de-
e mining he p essu e d op ac oss he magne o heological
al e. Specifically, he ela ionship be ween he flow a e
h ough he annulus Q
a
and he p essu e d op Δp
a
is needed.
The biplas ic scheme is p e e ed a he han he con en-
ional Bingham app oach as i is mo e flexible and allows o
a mo e effec i e modeling o low eloci y ea u es in he
annulus, e.g., magne ic bypass [2]. Using he dimensionless
ep esen a ion o he scheme in e ms o he p essu e
numbe Gand he plas ici y S, we exp ess he ela ionship
be ween he flow a e Q
a
and he p essu e d op ac oss he
annulus Δp
a
as
Δpa�2τ2La
hG(S) � 2τ0La
h(1−c(1−δ))G(S),(11)
whe e
G�−hΔpa
2Laτ2
,
S�12μQa
wh2τ2
.
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩(12)
The wo addi ional pa ame e s, cand δ, a e e e ed o as
he a ificial iscosi y a io and he (nondimensional) bypass
numbe (yield s ess a io), espec i ely. As he biplas ic
model was well s udied in p io esea ch pape s, he eade
should e e he e o in-dep h de ails and he pa ame e
es ima ion me hod. B iefly, δcon ols he in e cep o ce a
he ze o pis on eloci y, and cinfluences he cu e’s slope
below he knee-poin o he o ce- eloci y cha ac e is ics [2].
The wo pa ame e s o he biplas ic model a e ela ed o he
al e’s geome y a he han ma e ial p ope ies. The es i-
ma ion p ocedu e was highligh ed, e.g., in [36]. Fo example,
based on p io knowledge, he alue o δ(0.5) was selec ed
o a al e wi h no leakage flow pa h in he annulus. Using
he model, we classi y he al e’s beha iou in o wo flow
egimes: p eyield (G≤1) and pos yield (G>1). The ansi-
ion poin coo dina es a which he beha iou o he
pseudoma e ial changes om he p eyield egime o he
pos yield egime a e equal o G= 1, S
0
=c(2 −3δ+δ
3
).
B iefly, when in he pos yield egime, he ela ionship be-
ween Gand Scan be exp essed as
Shock and Vib a ion 7
G�1
6[3(1−c(1−δ))+S]2 cos 1
3a an 2(y, x)
􏼒 􏼓+1
􏼔 􏼕,
(13)
y�12 ������������
−81b2+12ba3
√,
x�−108b+8a3,
a�3
2(1−c(1−δ))+1
2S,
b�1
21−c1−δ3
􏼐 􏼑􏼐 􏼑,
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩(14)
whe eas in he p eyield egime, he ela ionship be ween he
p essu e d op and he flow a e h ough he annulus is
go e ned by he ollowing o mula:
G�δ1
6
S
cδ+3
􏼢 􏼣2 cos 1
3a an 2 y1, x1
 􏼁􏼒 􏼓+1
􏼔 􏼕,(15)
whe e
y1�6�3
√���������������������
27 S
cδ+9S
cδ
􏼠 􏼡2
+S
cδ
􏼠 􏼡3
,
􏽳
x1�−27 +27 S
cδ+9S
cδ
􏼠 􏼡2
+S
cδ
􏼠 􏼡3
.
⎧⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩(16)
The wo model pa ame e s (c,δ) can be iden ified om
eal dampe expe imen al da a o CFD (compu a ional flow
dynamics) simula ions. Finally, equa ion (11) can be mod-
ified o include he con ibu ion o he nonene gized egion
abo e he coil o he leng h L
c
−L
a
h ough
Δpa�2τ2La
hG(S)+12μLc−La
 􏼁Qa
wh3.(17)
4. Magne ic Flux Measu emen s
Fo he specific elec omagne geome y, we pe o med a
se ies o measu emen s o ex ac ing he flux densi y in-
o ma ion wi h espec o he con ol (exci ing) cu en
inpu . The goal was o e i y he FE model. The e o e, in his
sec ion, we e eal he expe imen al p ocedu e o acqui ing
he magne ic flux ela ionship agains he exci ing cu en
and p esen he ob ained da a.
4.1. Tes Rig Configu a ion. The magne ic flux densi y was
measu ed in he middle o he ai gap wi h he ul a hin
Hall ans e se p obe (STB1X-0201) and he magne o-
me e F. W. Bell 5180 a he sampling equency o 100 Hz.
The coil cu en magni ude coil was simul aneously ac-
qui ed by means o he Fluke i30s cu en clamp. The MR
dampe coil was exci ed using wo labo a o y powe
supplies: (1) Manson SDP2603 de ice o lowe ampli ude
cu en exci a ions and (2) G. W. Ins ek PST-3202 powe
supply o highe cu en inpu s. The wo signals a e
eco ded simul aneously using he on -end Dewe on
USB-50-USB2-8 da a acquisi ion module connec ed o he
lap op (Figu e 9).
The p ocedu e was pe o med as ollows: (1) cu en
inc ease up o he maximum p esc ibed cu en I
max
, which
was ollowed by dec easing he cu en down o 0 A, (2)
inpu ol age pola i y change, (3) epea S ep 1, (4) epea
B
-T- P
Gas
AA
(x)=0
MTS ( ube)
-K-
P
-T-
R
C
C
Dampe
B
R
-T-
MTS ( od)
RC
-K-
Displacemen
( od)
Viewe
[x od]
[xbase]
[a od]
o ce
[xgcup]
S
[x od]
Flux densi y
MR al e
P
S
Displacemen
( ube)
C
1
1
2
Figu e 8: High-le el Simscape model layou .
8Shock and Vib a ion
S ep 2, and (5) epea S ep 1. Using he highligh ed p o-
cedu e, he magne ic flux densi y was measu ed o he
maximum cu en le els I
max
�{0.5, 1, 2, 3, 4, 5} A,
espec i ely.
The measu emen s o magne ic flux in he annula gap
we e pe o med wi hou he MR fluid. No e ha he ela i e
pe meabili y o he Hall senso (μ
�1) placed in he hin
annulus wi h MR fluid would dis o he accu acy o he
expe imen as he flux flows a ound he p obe as illus a ed
in Figu e 10. In he simula ions, we assume he p esence o
MR fluid would no deg ade he accu acy o he model.
4.2. Resul s. The ob ained da a a e e ealed in Figu e 11 as
plo s o flux densi y s coil cu en .
Obse a ions o he plo s o flux densi y s cu en e eal
he p esence o hys e esis and nonlinea beha iou wi h he
ac ua o app oaching he sa u a ion a he highes cu en
le el (I
max
�5 A).
5. Modelling Resul s
The se ies o modelling expe imen s was spli in o wo
s ages. Fi s , we alida e he ansien FE model agains he
expe imen al da a, and hen we s udy he beha iou o he
hyd aulic model.
5.1. FE Model Ve ifica ion: Ai Gap, No MR Fluid. He e, he
FE model o he dampe desc ibed in Sec ion 3.1 was e ified
agains he ob ained ai gap flux densi y measu emen s. The
compa ison o he ob ained da a agains he model ou pu
can be obse ed in Figu e 12 as plo s o flux densi y s coil
cu en . Due o he low cu en change a e, he eddy
cu en s we e neglec ed in he model, and only he hys e esis
con ibu ion was s udied.
Again, obse a ions o he plo s e eal sa is ac o y
ag eemen wi h he model anywhe e excep o he smalles
exci ing cu en . O e all, he plo s p o e he a ionali y o
he p oposed app oach.
5.2. Hys e esis Assessmen o he MR Val e. Due o easons
explained in Sec ion 4.1, di ec assessmen o he hys e e ic
beha iou o he MR al e wi h he fluid in he annulus was
no possible wi h he a ailable labo a o y equipmen . How-
e e , CIP- (ca bonyl powde i on-) based MR fluids show
i ually ze o hys e esis [12]. The e o e, he p esence o he
fluid in he annulus only modifies he flux densi y-cu en
ela ionship h ough i s (nonlinea ) magne isa ion cha ac-
e is ics. Hence, i is easonable o p oceed u he unde he
assump ion ha elec omagne model o he ac ua o was
alida ed, and i would be accu a e also in he scena ios in
which he MR annulus would be filled wi h he fluid. Due o
he magne ic ci cui sa u a ion abo e 2 A, we e eal he esul s
o he exci ing cu en s up o 2 A (Figu e 13). The nonlinea
con ibu ion o he fluid is e iden in he p esen ed esul s.
Nex , we examine he beha iou o he al e model o
he wo ollowing a ian s:
(i) Hys e esis (co e loss) ON, eddy cu en s ON (solid
line)
(ii) Hys e esis (co e loss) ON, eddy cu en s OFF
(dashed line)
The hys e esis model was applied o all 11SMn30
componen s (co e, slee e). As p esen ed in Figu e 14, he
calcula ed emanen flux densi y is ela i ely independen o
he p e ious magne ic his o y wi hin he examined coil
cu en ange om 0.5 A o 2 A, and he effec o eddy
cu en s is a he insignifican in he examined case as al-
eady e ealed in Figu e 14.
Fu he mo e, we epea ed he flux densi y calcula ions
o one selec ed elec ic cu en le el (I
max
�2 A) o he
ollowing h ee model a ian s:
(i) Hys e esis swi ched OFF, eddy cu en s swi ched
OFF (dashed line)
(ii) Hys e esis ON, eddy cu en s OFF (do ed line)
(iii) Hys e esis ON and eddy cu en s ON (solid line)
The esul s a e e ealed in Figu e 15. I is now appa en
ha he hys e esis has he bigges impac on he ini ial
F on -end
DEWE 50
Cu en clamp
fluke i30
Powe supply
manson SDP2603
Magne ome e
F.W. Bell 5180
Figu e 9: Tes ig configu a ion.
Shock and Vib a ion 9
5.6.3. Influence o Pis on Veloci y. Se ing he elec ical
conduc i i y o 1 MS/m and he coe ci i i y o 200 A/m, we
hen es ed he influence o he pis on eloci y on he o ce.
In he p esen ed examples, he o ce ou pu ime his o ies
we e no malized o be e compa ison. The ob ained esul s
imply ha he slowe he pis on eloci y o con ol cu en
ise, he slowe he magni ude o he o ce change a e
gene a ed. A e exceeding he pis on eloci y o 0.2 m/s, he
ac ua o esponse in he cu en ise s age is independen o
he pis on eloci y (Figu e 30). Howe e , he exac eloci y
alue will depend on he pa icula dampe design. Fo
compa ison, he ac ua o esponse in he cu en d op s age
is independen o he p esc ibed pis on eloci y.
Apa om he eddy cu en s, he main sou ce o slowe
o ce ise is he comp essibili y o he MR fluid i sel . Th ee
diffe en alues o MR fluid bulk modulus we e es ed o il-
lus a e his effec (Figu e 31).
The p ima y esponse ime o o ce (63.3% o final o ce)
was calcula ed om he simula ed da a (Figu e 30) (Figu e 32).
The p ima y esponse ime o con ol cu en ise is influ-
enced by he pis on eloci y. The lowe he pis on eloci y, he
lowe he p ima y esponse ime. Howe e , he p ima y e-
sponse ime o con ol cu en d op is independen o pis on
eloci y. I is no ewo hy ha simila ends we e expe i-
men ally de e mined in o he esea ch s udies [42, 43].
6. Conclusions and Summa y
In his pape , we p esen he esul s o a modeling s udy
in ol ing a mul iphysics model o a flow-mode MR dampe .
The model allows in eg a ing an FE elec omagne model o
he de ice wi h a hyd aulic lumped pa ame e model o he
de ice. The modeling app oach elies only on he in-
o ma ion which can be ex ac ed om enginee ing d aw-
ings (geome y), ma e ial da a shee s (ma e ial p ope ies),
and he e o e, i can be used in s udies on he pe o mance o
eal ac ua o s o i ual p o o ypes.
The elec omagne was e ified expe imen ally. Based on
he ob ained da a, we conclude ha he model is capable o
p edic ing he magne ic hys e e ic beha iou o he MR
al e. The esul s conce ning he hyd aulic model a e
simula ed; howe e , i should be no ed ha he model is
based on a well-es ablished and expe imen ally e ified
heo y [39]. To demons a e he use ulness o he model, we
applied i in a pa ame ic s udy, in which he con ibu ion o
a ious geome ic pa ame e s and ma e ial p ope ies o he
ou pu o he ac ua o was s udied and analyzed. Fo in-
s ance, we can conclude he ollowing.
(i) Residual magne ic flux is di ec ly ela ed o he
cu en his o y and he annula gap heigh
(ii) La ge annula gaps induce lowe emanen ( e-
sidual) flux densi y and hen less undesi ed o ce
inc ease (Figu es 22 and 23)
(iii) Val es wi h la ge annula gaps heigh e eal highe
u n-up a io (dynamic ange), howe e , a he
expense o maximum damping o ces (Figu es 25
and 26)
(i ) Tu n-up a io (dynamic ange) a ies wi h he
coe ci i y and gap heigh (Figu e 25)
( ) Demagne izing cu en cycles a e equi ed o e-
duce/elimina e he esidual flux (and he o ce
inc emen due o he esidual flux)
4.0
4.6
5.0
4.8 5.5
6.0
0
1
2
3
4
5
6
7
0.6 0.7 0.8 0.9 1 1.1
K (-)
h (mm)
200 A/m
120 A/m
0.3m/s
Figu e 25: Tu n-up a io a ia ion wi h gap heigh and coe ci i y,
V �0.3 m/s.
–2000
–1500
–1000
–500
0
500
1000
1500
2000
–0.3 –0.2 –0.1 0 0.1 0.2 0.3
(m/s)
Fd (N)
0.65mm
0.8mm
1mm
Figu e 26: Gap heigh influence on damping o ce ou pu ,
I
max
�2 A V �0.3 m/s.
0
0.5
1
1.5
2
2.5
0
100
200
300
400
500
600
0 10203040506070
(ms)
Bg (mT)
120 A/m
200 A/m
600 A/m
ic (A)
Figu e 27: Coe ci i y H
c
: (gap) a e aged magne ic flux densi y
ime his o y.
16 Shock and Vib a ion

( i) The elec ical conduc i i y has a majo influence on
he dynamic beha iou o he ac ua o (Figu e 29)
( ii) The pis on eloci y influences he ac ua o ’s e-
sponse ime. Low pis on eloci ies deg ade he
esponse ime in he cu en ise s age (Figu es 30
and 32). I is likely due o he comp essibili y o he
fluid (Figu e 31).
The collec ed da a enhance unde s anding he key
mechanisms go e ning he flux/ o ce ou pu o MR
0
0.5
1
1.5
2
2.5
0
500
1000
1500
2000
0 5 10 15 20 25 30
ic (A)
(ms)
120 A/m
200 A/m
600 A/m
Fd (N)
(a)
ic (A)
(ms)
120 A/m
200 A/m
600 A/m
0
0.5
1
1.5
2
2.5
0
500
1000
1500
2000
0 5 10 15 20 25 30
Fd (N)
(b)
ic (A)
(ms)
120 A/m
200 A/m
600 A/m
0
0.5
1
1.5
2
2.5
0
500
1000
1500
2000
0 5 10 15 20 25 30
Fd (N)
(c)
Figu e 28: Impac o he coe ci i y H
c
. Full line: damping o ce ime his o y; g een dashed line: cu en s ep inpu . (a) Fo ce s ime: cu en
ise (ini ial condi ion �demag.). (b) Fo ce s ime: cu en d op. (c) Fo ce s ime: cu en ise (no demag.).
0
0.5
1
1.5
2
2.5
0
500
1000
1500
2000
0 102030
(ms)
5.8 MS/m
1 MS/m
Fd (N)
ic (A)
Figu e 29: Co e ma e ial’s elec ic conduc i i y: Full line: damping o ce ime his o y; g een dashed line: cu en s ep inpu . H
c
�200 A/m,
V �0.3 m/s.
Shock and Vib a ion 17
ac ua o s. They allow o a clea sepa a ion o a ious
con ibu o s o he s a ic and dynamic beha iou o such
de ices. In ou opinion, he p oposed model can be a use ul
ool as i inco po a es majo key phenomena occu ing in
he ac ua o including magne ic hys e esis and emanence/
coe ci i y, eddy cu en s, comp essibili y, and fluid ine ia
(hyd aulic hys e esis), he MR effec . I allows sepa a ing he
magne ic hys e esis om he hyd omechanical one so ha
he wo phenomena can be examined sepa a ely.
Da a A ailabili y
The da a used o suppo he findings o his s udy a e
a ailable om he co esponding au ho upon eques .
Con lic s o In e es
The au ho s decla e ha hey ha e no conflic s o in e es .
Acknowledgmen s
The au ho s wish o acknowledge he suppo o he g an
NAWA: E-Mobili y and Sus ainable Ma e ials and Tech-
nologies (EMMAT) (numbe PPI/APM/2018/1/00027/U/
001) sponso ed by he Polish Na ional Agency o Academic
Exchange (NAWA).
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02468101214161820
ic (A)
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