1
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T ansien esponse
o magne o heological luid
on apid change o magne ic ield
in shea mode
Michal Kubík1*, Jose Válek1, Jiří Žáček1, Filip Jeniš1, Dmi y Bo in2, Zbyněk S ecke 1 &
I an Mazů ek1
The ansien beha iou o magne o heological (MR) de ices is an impo an pa ame e o mode n
semi-ac i ely con olled suspension sys ems. A signi ican pa o he MR de ice esponse ime is
he MR luid esponse ime i sel . A signi ican ac o is he so-called heological esponse ime. The
heological esponse ime is connec ed wi h he s uc u ing pa icle’s ime and he de elopmen
o shea s ess in MR luid du ing he de o ma ion. The main aim o his pape is o expe imen ally
de e mine he heological esponse ime o MR luid and e alua ed he e ec o shea a e, magne ic
ield le el, and ca ie luid iscosi y. The unique design o he heome e , which allows he apid
change o a magne ic ield, is p esen ed. The heological esponse ime o MRF 132-DG and MRC-
C1L is in he ange o 0.8–1.4 ms, depending on he shea a e. The highe he shea a e, he sho e
he esponse ime. I can be s a ed ha he highe he magne iza ion o he MR luid, he lowe he
esponse ime. The highe he iscosi y, he highe he heological esponse ime. The measu ed da a
o heological esponse ime was gene alized and one mas e cu e was de e mined.
Magne o heological (MR) luid is he suspension o ine, non-colloidal, low-coe ci i y, high-magne izable pa -
icles in a ca ie luid. These pa icles a e usually made o ca bonyl i on and ha e a sphe ical shape due o
hei du abili y and ibological p ope ies. The con inuous phase o MR luids is ypically silicon o syn he ic
hyd oca bon oils1. The lowes possible iscosi y o he con inuous phase is equi ed, bu his signi ican ly a ec s
he sedimen a ion s abili y2. MR luid also con ains se e al addi i es ha a ec heological3, ibological4, o
sedimen a ion s abili y5. When he MR luid is ene gized by he magne ic ield, he e omagne ic pa icles a e
magne ized and o m chain-like s uc u es in he di ec ion o he magne ic ield6. The heology o MR luid in
ac i a ed s a e is cha ac e ized by p e-yield and pos -yield egime. In he p e-yield egime, he MR luid exhibi s
iscoelas ic beha iou . The complex modulus G is a magne ic ield H and pa icle concen a ion dependen . The
shea s ess τ in he luid can be desc ibed by he equa ion below
whe e
γ
is shea s ain,
˙γ
is shea a e and τ0(H) is MR luid yield s ess. The pos -yield egime is usually desc ibed
by Bingham model as ollows:
whe e τ(H) is shea s ess,
η
is Bingham iscosi y, and H magne ic lux in ensi y. I is he simples model ha
can desc ibed his beha iou . The MR dampe s7,8, clu ches/b akes9, o seals10,11 ake ad an age o he unique
beha io o MR luid.
The ansien beha iou ( ansien esponse) o MR luid is an impo an pa ame e o mode n magne o -
heological de ices wo king wi h eal- ime con ol12,13. The MR luid esponse ime is composed o o he pa ial
esponse imes which a e di e en ly impo an depending on he ope a ing condi ions and he me hod o MR
luid loading. The esponse ime o MR luid can be di ided in o (1) hyd odynamic esponse ime, (2) pa icle
s uc u e de elopmen esponse ime,and (3) heological esponse ime.
τ
=
Gγ,τ<τ
0(H)and
˙
γ
=
0
τ(H)=τ0(H)+η˙γ
OPEN
1Facul y o Mechanical Enginee ing, B no Uni e si y o Technology, B no, Czech Republic. 2Chai o
Magne o luiddynamics, Ins i u e o Mecha onic Enginee ing, Technische Uni e si a D esden, D esden,
Ge many. *email: Michal.kubik@ u b .cz
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The esea ch s udies o She man14 o Goldasz e al.15 show ha MR al e p essu e d op due o MR luid
yield s ess dec eases wi h he inc easing gap eloci y. A high eloci ies, his p essu e d op is app oaching o be
ze o. This s a emen is based on CFD (compu a ional luid dynamics) simula ions. This phenomenon is ela ed
o ansien heology connec ed wi h he de elopmen o he eloci y p o ile in he gap and is o en e e ed o
as he hyd odynamic luid esponse ime. Goncal es e al.16 expe imen ally de e mined ha he hyd odynamic
esponse ime is 0.73ms o magne ic ield 100kA/m and 0.53ms o magne ic ield 200kA/m. The comme cial
MRF-132LD (Lo d Co p., USA) was used in his s udy. Kubík e al.17 published simila s udy. This eam measu ed
he hyd odynamic esponse ime o MR luid MRF-132DG (Lo d Co p., USA) and anges om 0.4 o 1ms o a
selec ed gap size and a ange o magne ic ield s imuli. The eloci y p o ile de elopmen mechanism is simila o
MR luid and elec o heological (ER) luid18. Howe e , ER luid show as e esponse ime han MR al e. ER luid
is he suspension o ine elec ically ac i e pa icles in luid. This luid exhibi s a apid inc ease o luid yield s ess
unde he applica ion o an elec ic ield. Ga in e al.19 modelled he ansi ion om a ully de eloped Bingham
p o ile o a New onian low o ER luid. The yield s ess o ER luid was assumed o d op o ze o quicke han
he dissipa ion ene gy due o he de elopmen o he eloci y p o ile19. I can be s a ed ha his hyd odynamic
esponse ime is connec ed wi h high shea a es o as changes o he magne ic ield in al e mode.
The pa icle s uc u e de elopmen esponse ime is ela ed o he ime needed o he s uc u ing o pa icles
in he di ec ion o he magne ic ield wi hou he low condi ions o he MR luid. Jolly e al.20 p oposed an expe i-
men al me hod ha mic os uc u e o ma ion ime can be deduced om he ansien changes in he ela i e
magne ic pe meabili y o he MR luid. The chained pa icles a e assumed o ha e a highe magne ic pe meabili y
han he dispe sed. Two- ime esponses we e obse ed20. The i s a ibu es he connec ion wi h he ans e o
pa icles in o di e se chains (pai o ma ion) and he second (an o de o magni ude slowe ) connec ion wi h
he mig a ion o hese ini ial chains in o longe and s onge s uc u es. The esponse ime was be ween 5 and
10ms. A simila measu emen me hod was also published by Ho á h e al.21. Pei e al.22 s a ed ha he esponse
ime o d y MR luid was in he o de o µs by he model. This s a emen is based on simula ion esul s.
The heological esponse ime is connec ed wi h he s uc u ing pa icle’s ime and he de elopmen o shea
s ess in MR luid du ing he de o ma ion ( low). She man e al.23 c ea e a chain model o MR luid. This model
is based on one million pa icles. One esul o his pape is he shea s ess ime his o y on he s ep change o a
magne ic ield. Fo his da a, he heological esponse ime can be de e mined as oughly 0.4ms. The MR luid
had a olume pa icle ac ion o 25% and was unde he shea a e o 500 s−1. Laun and Gab iel24 de e mined
he esponse ime o MR luid o 2.8ms. They used sinusoidal exci a ion and he de e mined ime lag be ween
magne ic lux densi y and shea s ess. Kikuchi e al.25 examined he esponse ime o a s ep elec ic cu en and
in oduce non-dimensional esponse ime pa ame e . I can be expec ed ha he mechanism o chain o ma ion
in Elec o- heological (ER) luids and MR luids is simila . Koyanagi e al.26 de eloped a me hod o a measu e-
men esponse ime o ER luid. This eam expe imen ally de e mined he esponse ime as 0.95ms.
The in o ma ion abou he ansien beha iou o MR luid is limi ed. This issue is becoming mo e impo an
due o he de elopmen o MR de ices wi h a sho esponse ime7,27, whe e he limi ing pa is now he MR luid
i sel . The cu en design o he MR dampe achie ed a esponse ime o abou 1.2ms. In he cu en s a e o he
a , mo e s udies can be ound dealing wi h he esponse ime o MR luid7,12 han is p esen ed abo e. In hese
se e al cases, he au ho s measu ed he ime cons an o measu ing de ices ins ead o he ime cons an o MR
luid14. The heological esponse ime o MR o ER luid was jus expe imen ally de e mined in s udies24,26. Bo h
s udies p esen ed esponse ime jus o one expe imen al condi ion. The main aim o ou pape is o expe i-
men ally de e mine he heological esponse ime o MR luid and e alua ed he e ec o shea a e, magne ic
ield le el, and ca ie luid iscosi y. Ou esul s will be compa ed wi h he published analy ical app oach14.
Ma e ials and me hods
Desc ip ion o he measu ed phenomenon and measu ing me hods. The aim o he measu e-
men is o expe imen ally de e mine he ime cons an o MR luid in he shea mode ( om he inc ease in shea
s ess τ) on a apid change in he magne ic ield B. The p ocedu e o he expe imen is desc ibed in Fig.1. A
ime T1, he MR luid is loaded by gi en shea a es and he magne ic ield is o . A ime 0, he magne ic ield is
ac i a ed and, a ime T2, he magne ic ield is al eady a he maximum alue. Howe e , un il ime T3, he shea
s ess emains a he same le el as a ime T1. In he au ho ’s opinion, his delay is associa ed wi h pa icle s uc-
u e o ma ion in he MR luid. In eali y, he e a e no sepa a e single chains. Tha is jus a en a i e simpli ica-
ion. A ime T4, he e is a apid inc ease in shea s ess in he MR luid due o he de o ma ion o he pa icle
s uc u e. This is shown as il ing chains in he shea di ec ion bu he mechanisms o s uc u e ac u e a e mo e
complex. Gene ally, he simples dynamic sys em, ha can se e as an app oxima ion o he ansien beha io o
MR luid is a i s -o de sys em. The ansien esponse is exp essed by he ime cons an T63 (p ima y esponse
ime), which de e mines he ime when moni o ed o que (calcula ed shea s ess) achie ed 63.2% o he inal
con olled alue (s eady-s a e). This app oxima ion can be used o he desc ip ion o he dynamic beha iou o
MR ac ua o s28. In he case o heology measu emen , he MR luid can be desc ibed by a simple Maxwell model
and by Bingham cons i u i e equa ion. Fo s ep change on magne ic ield, he excep ed shea s ess esponse τ( )
would be:
whe e is ime. Mo e names o a a iable T63 can be ound in he li e a u e as swi ching ime24, esponse ime17
o heological esponse ime14. Howe e , he ansien esponse o MR luid exhibi s di e en beha iou han
he i s -o de sys em, see Fig.5. The e o e, we decided o de e mine hose ime cons an s in ou pape : (1) i s -
o de ime cons an T63 (0–63.2%) and (2) ise ime T90 (0–90%), see Fig.1. This esponse ime we e so-called
(1)
τ
( )=τ0

1−e−
T63

+η
˙γ
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heological esponse ime because i is connec ed wi h changes in he heology o MR luid. Those ime cons an s
we e selec ed due o a sui able compa ison o ou expe imen al da a wi h esul s om published pape s.
Expe imen al es ig. The expe imen al es ig is composed o an elec ic mo o wi h encode (1), de el-
oped heome e (2), load ine ia (3), le e (4), and o ce senso (5), see Fig.2A. The load ine ia accumula ed
ene gy o s abilize o a ion du ing he ac i a ion o MR luid using he magne ic ield (an inc ease mo o load).
The load ine ia had 4.6kg (momen ine ia 5900kg/mm2). The whole sys em is moun ed o a ionally and he
o que is measu ed by a o ce senso on he le e (52mm). The homemade heome e is composed o he o o
(a), s a o (b), and MR luid sample (c), see Fig.2. The elec omagne ic coil c ea es he magne ic lux (d) in he
magne ic ci cui (show in g ey). The gap size was 0.6mm, see Fig.2. The ansien beha iou o he heome e
is undamen al o he p ecise measu emen o MR luid esponse ime. The esponse ime o ha dwa e ( heom-
e e ) has o be as sho as possible and wo main sou ces we e iden i ied in he li e a u e: (1) eddy cu en s in
he magne ic ci cui 7, and (2) induc ance o he heome e elec omagne ic coil7. In ou heome e , we used so
magne ic composi e (SMC) ma e ial ( adema k Sin ex) o he magne ic ci cui o elimina e eddy cu en s.
SMC ma e ial is magne ic conduc i e and elec ic non-conduc i e ( esis i i y 280µΩm). The sui able design o
a magne ic ci cui wi h ou pa en ed cu en con olle allows a apid inc ease o elec ic cu en on he elec o-
magne ic coil (T63I = 0.21ms).
Me hodology measu emen . The aim o he expe imen s was o de e mine shea s ess in MR luid and
magne ic ield o e ime. The shea s ess τ was calcula ed om a ea and o que which was measu ed indi ec ly
based on da a om he o ce senso (MEG20) on he le e , see Fig.2. The o ce senso measu ing ange was
0–200N. The o ce ange (de o ma ion) was chosen o maximize sys em igidi y and only he i s 10% o he
ange was used o measu emen . The magne ic ield in he gap co esponds wi h he elec ic cu en cou se and
was measu ed by Fluke i30 cu en clamps. These wo signals we e eco ded and condi ioned wi h a sampling
equency o 200kHz by he Dewe on USB-50 analyze . The MFG-2120MA signal gene a o gene a es a squa e
wa e ol age signal which inpu s o he cu en con olle a a equency o 1Hz. Ou de eloped cu en con-
olle gene a es an elec ic cu en on he elec omagne ic coil wi h o e - ol age up o 100V. The measu emen
p ocedu e was as ollows: (1) 10s measu emen wi hou magne ic ield, and (2) 10s measu emen wi h he appli-
ca ion o he magne ic ield. This p ocedu e was necessa y o he elimina ion o non-cons an ic ion o ces in
he heome e and iscous o ces. Those phenomena can signi ican ly complica e he subsequen e alua ion o
esponse ime. The expe imen s we e conduc ed 5 imes unde he same condi ions. The da a was no il e ed bu
a e aged om aw da a. Then, he amp da a was no malized. All measu emen s we e pe o med a 25°C ± 1°C.
Me hodology e alua ion o esponse ime. The measu ed esponse ime o he magne ic ield (elec ic
cu en ) achie ed a alue o τ63I = 0.21ms and τ90I = 0.33ms, see Fig.4. In se e al cases o he ansien beha iou
o MR ac ua o s, his ime can be expec ed as a s ep change. In ou case, we canno make his simpli ica ion
because he expec ed esponse ime o MR luid om published models14 is in he same ime scale ( oughly
1.5ms). The e o e, i was necessa y o de e mine he ans e unc ion be ween he measu ed magne ic ield and
shea s ess in MR luid. We used a p ocess model o desc ibing he MR luid ansien esponse. The p ocess
Figu e1. Demons a ing measu ed me hod and de e mina ion o ime cons an s.
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model is popula o desc ibing sys em dynamics in many indus ial applica ions29. We used he so-called simple
SISO (Single Inpu , Single Ou pu ) p ocess model which is desc ibed by his ans e unc ion:
whe e
Kp
is he p opo ional gain,
Tp
is he ime cons an , and
Td
is dead ime. A simila app oach was used in
s udy26. The Ma lab Sys em iden i ica ion oolbox was used o he iden i ica ion o cons an s. The leng h o he
e alua ed sec ion was 20ms.
Magne o heological luid samples. The comme cial MR luid MRF-132DG supplied by Lo d Co p.,
MR luid MRHCCS4-A and MRHCCS4-B supplied by Liquids Resea ch, and MRC-C1L supplied by CK Ma e i-
als we e chosen as he samples, see Table1. These luids we e chosen because hey ha e a simila pa icle size and
a di e en iscosi y o he ca ie luid.
The iscosi y lis ed in he able was measu ed by he Haake Ro o isco 1 heome e , and de e mined as a slope
be ween 400 and 800 s−1. I should be no ed ha ca ie luid o MR luids exhibi s New onian beha iou bu MR
(2)
sys
=
K
p
1+T
p
se−Td
s
Figu e2. (A) Rheome e design wi h impo an dimensions (g ey, so magne ic composi e ma e ial;
o ange, coppe ; ligh blue, aluminium; yellow, MR luid sample, g een, s eel) and (B) magne ic lux densi y
measu emen in he gap.
Table 1. MR luid samples.
MRF-132DG MRHCCS4-A MRHCCS4-B MRC-C1L
Solid con en by weigh (%) 80.98 70 80 80
MR luid iscosi y a 40°C (Pas) 0.114 0.167 0.237 0.108
Ca ie luid iscosi y a 40°C/25°C (Pas) 0.011/0.018 0.03/0.051 0.03/0.051 0.008/0.011
A e age pa icle size (µm) Sphe ical 2.1 Sphe ical 1.8 Sphe ical 1.8 Sphe ical 1–5
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luids a e in gene al non-New onian. The pa icle sizes we e measu ed by a scanning elec on mic oscope, FEG
SEM ZEISS Ul a Plus, and analysed by sc ip using ools o pic u e analysis in Ma lab. Howe e , he in o ma-
ion abou pa icle size o MRC-C1L was aken om s udy30.
Magne ic model and expe imen al alida ion se ings. The da a om he magne ic model a e nec-
essa y o he gene aliza ion o measu ed esponse ime da a. The magne ic model was c ea ed in Ansys Elec-
onics Desk op 19.2. The geome y o he magne ic ci cui was simpli ied. The magne iza ion cu e o he mag-
ne ic ci cui ma e ial (SMC ma e ial) was ex ac ed om he da ashee o he supplie . The elec omagne ic coil
(70 u ns) ca ie was made o plas ic wi h ela i e pe meabili y 1. The lids we e made o aluminium also wi h
a ela i e pe meabili y o 1. The magne iza ion cu e o MR luid was aken om a MR luid supplie da ashee .
This model was necessa y o he calcula ion o MR luid magne iza ion M, which is an impo an inpu o he
calcula ion o Mason numbe Mn. The Mason numbe Mn is he a io o magne ic o ces o iscous o ces and is
usually used o he desc ip ion o MR luid’s beha iou a he mic oscopic le el31. The magne ome e F.W. Bell
5180 wi h an ul a hin ans e se p obe (STB1X-0201) was used o magne ic measu emen . The Fluke i30 cu -
en clamps we e used o elec ic cu en measu emen . These wo signals a e eco ded and condi ioned wi h
a sampling equency o 100Hz by a on -end Dewe on USB-50-USB2-8 connec ed o he lap op, see Fig.2B.
Resul s and discussion
Magne ic model alida ion. The compa ison o esul s o magne ic lux densi y B o e he elec ic cu en
om he magne ic model and expe imen wi h ai in he gap can be seen in Fig.3, le . Magne ic lux densi y
measu emen s in he gap we e pe o med o wo posi ions ha a e pe pendicula , see Fig.2B. The ag eemen
be ween model and expe imen is accep able. This expe imen ally e i ied model was used o he calcula ion o
magne iza ion M in he gap wi h MR luid. The esul s can be seen in Fig.3 igh . This da a is necessa y o he
calcula ion o Mason numbe Mn.
Con ol elec ic cu en signal. Fi s o all, i was necessa y o p ecisely desc ibe he exci a ion o MR
luid. I can be assumed ha he cou se o magne ic lux densi y in he MR luid copies he cou se o an elec ic
cu en due o he elimina ion o eddy cu en in he magne ic ci cui . This is ensu ed by a special design o he
heome e . The cou se o he elec ic cu en I in ime can be seen in Fig.4 o wo le els o elec ic cu en
Figu e3. The esul s om he magne ic model and expe imen o ai gap (le ), The calcula ed magne iza ion
M dependency o elec ic cu en I on he coil o di e en MR luids ( igh ).
Figu e4. The cou se o he elec ic cu en I in he ime o he inal alue o elec ic cu en 1 A (le ) and 2
A ( igh ).

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I. A as ise in he elec ic cu en I is achie ed by connec ing a highe ol age han esul s om he Ohm law
(o e - ol age me hod). When he equi ed elec ic cu en alue is achie ed, he cu en con olle s a s o
egula e a a equency o 8kHz. The e o e, he elec ic cu en exhibi s oscilla ions in ime a e 0.5ms. Nex ,
educing he elec ic cu en esponse ime was no possible due o he a ailable cu en con olle (maximum
100V) and heome e design (coil induc ance). The esponse ime (90%) o he elec ic cu en achie ed a alue
o
T90I=0.335
ms o elec ic cu en 1A and a alue o
T90I=0.365
ms o elec ic cu en 2 A, see Fig.4.
The selec ed cou ses o MR luid shea s ess in ime. The Fig.5A show he cou se o shea s ess
τ
and elec ic cu en I o e ime . The cou se o shea s ess
τ
exhibi s oscilla ions wi h a cons an equency
o 360Hz, which is connec ed wi h he na u al equency o some pa o he heome e . This hypo hesis was
e i ied by measu emen s using an accele ome e and e alua ion based on FFT (Fas Fou ie ans o m). The
measu ed equency was 337Hz ± 4.88Hz. The igh o Fig.5B he e ec o shea a e
˙γ
on he cou se o shea
s ess τ. Wi h he inc ease o shea a es
˙γ
, he esponse ime dec eases. An ini ial dead ime o 0.4ms can also
be seen, which is independen o he shea a e le el, see Fig.5B. I should be no ed ha his phenomenon may
be associa ed wi h an inc ease in he magne ic ield. The p e ious s udy26 measu ed a dead ime o 0.5ms o
ER luids and dead ime o 0.6ms o MR luid7, which is consis en wi h ou expe imen s. We assume ha he
measu ed dead ime o 0.4ms is ela ed o he chaining o e omagne ic pa icles (mic os uc u e o ma ion)
in he MR luid.
The e ec o shea a e on he heological esponse ime. The esponse imes shown in Fig.5C we e
de e mined om he expe imen al da a and e alua ed acco ding o a p ocess model. The ela ionship be ween
esponse ime and shea a e
˙γ
is nonlinea . The measu ed da a o esponse ime can be i ed by a powe -law
unc ion, see Fig.5C. The highe he shea a e, he sho e he esponse ime. The da a we e measu ed o he
same elec ic cu en (2 A), bu he magne iza ion o he luid sample was di e en . The esponse ime
T90
anges
om 5.5 o 1.9ms o shea a e
˙γ
om 11 o 218 s−1 (MRHCCS4-A and MRHCCS4-B). Wi hin he measu e-
men and e alua ion e o , i can be s a ed ha bo h luids a e iden ical in e ms o ansien esponse. The
e ec o pa icle concen a ion is he e o e nonsigni ican . MRF 132-DG and MRC-C1L luids exhibi a sho e
esponse ime
T90
han LR luids in he ange om 1.4 o 0.8ms. This is p obably due o he lowe iscosi y o he
ca ie liquid, which is abou 3 imes lowe . The Fig.5D shows he esponse ime
T63
dependen on shea a e
˙γ
.
The e can be seen he same end as in he case o 90%. Koyanagi e al.26 expe imen ally de e mined he esponse
ime
τ90
o ER luid as 0.95ms (dead ime + ime cons an ) which is nea o ou esul s. Lee e al.32 measu ed
Figu e5. (A) The selec ed shea s ess and elec ic cu en cou se o e ime o shea a e 39 s−1 and MRF 132-
DG, (B) he e ec o shea a e on he no malized cou se o shea s ess in MRF 132-DG; The e ec o shea a e
on esponse ime 90% (C), 63% (D) o di e en MR luids a he same elec ic cu en exci a ion o 2 A.
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esponse ime
τ63
as 5.1ms and
τ90
as 6.1ms o ER luid in shea mode (da a es ima ed om he publica ion
g aph). These alues a e sligh ly highe han he p esen ed da a in his pape . Laun and Gab iel24 measu ed he
MR luid esponse ime based on sinusoidal exci a ion. The expe imen de e mined he esponse ime
τ63
o MR
luid o 2.8ms ± 0.5ms a a shea a e o 100 s−1 a a magne ic lux densi y o 0.9T. This measu ed alue is abou
3 imes highe han he p esen ed esponse ime o Liquids Resea ch luids. Howe e , i can be s a ed ha he
di ec compa ison o esul s is complica ed because he measu ing sys ems a e no compa able. The s udies24,26
used pla e-pla e con igu a ion o s udy32 used o a ing cylinde .
E ec o magne iza ion and ca ie luid iscosi y on he heological esponse ime. The e ec
o luid magne iza ion M on he esponse ime T90 was demons a ed on MRHCCS4-B luid because he e ec o
magne iza ion M was mos no iceable. The luid was measu ed a h ee le els o magne iza ion M, see Fig.6, le .
Fo all h ee le els o magne iza ion M, he dependence on he shea a e
˙γ
is exponen ial. I can be s a ed ha he
highe he magne iza ion M o he MR luid, he lowe he esponse ime
T90
. This is consis en wi h he heo y.
The e ec o ca ie luid iscosi y η on esponse ime
T90
will be demons a ed on wo selec ed MR luids
(MRF 132-DG and MRHCCS4-B). These luids ha e simila pa icle concen a ions and di e en ca ie luid
iscosi ies η. The iscosi y o he MRHCCS4-B ca ie luid is app oxima ely 2.8 imes highe han he ca ie
luid o MRF 132-DG. The igh o Fig.6 shows ha an MR luid wi h a highe ca ie luid iscosi y η shows a
signi ican ly highe esponse ime
τ90
. I should be no ed ha he MR luids had he same magne iza ions M o
230 kA/m, bu di e en elec ic cu en exci a ion (MRF 132-DG elec ic cu en o 1.5 A; MRHCCS4-B elec ic
cu en o 2A). The e ec o addi i es o ca ie luid iscosi y was no conside ed he e.
Gene aliza ion o measu ed da a. She man14 s a ed ha MR luid esponse ime da a in shea mode can
be gene alized using non-dimensional esponse ime
T∗
and Mason numbe
Mn
. This s udy p o ided he equa-
ion o he calcula ion o non-dimensional esponse ime as:
T90
is he heological esponse ime (90%),
η
is he iscosi y o ca ie luid,
M
is MR luid magne iza ion and
µ0
is acuum pe meabili y. The Mason numbe can be calcula ed as ollow:
whe e
˙γ
is shea a e. The Non-dimensional esponse ime
T∗
and Mason numbe
Mn
we e calcula ed om meas-
u ed da a, see Fig.7. The mas e cu e can be de e mined om measu ed da a, see Fig.7— ed line. The esul s
show a signi ican di e ence be ween he published model14 and ou expe imen o Mn alues highe han 0.005.
The
T∗
and Mn was also e alua ed (es ima ed) om pape s24,26. This da a is ou o ange o ou measu emen .
Howe e , i should be no ed ha he da a ob ained om he expe imen a e only om s udy24. The di e ence in
he esul s may be due o (1) he model simpli ica ion and (2) inaccu acies in he measu emen and e alua ion o
he measu ed da a. I has been hypo hesized ha he di e ence may be due o he de o ma ion o he measu ing
de ice ( heome e ), which is no included in he model. This would esul in a signi ican inc ease in esponse
ime
T90
a low shea a es
˙γ
compa ed o he model.
Figu e8 shows a compa ison o he esponse ime
T90
cou se on shea a e
˙γ
om he She man model, p o-
posed model (Fig.7 ed) and om he expe imen o MRHCCS4-B. The ca ie luid iscosi y
η
, magne iza ion
M, shea a es
˙γ
a e he same o expe imen s and also o he model. I can be seen ha he esponse ime
T90
om expe imen s is signi ican ly lowe han ha om he model. Thus, i can be s a ed ha he possible de o -
ma ion o he measu ing de ice is no he sou ce o he di e ence be ween he expe imen and he model. The
di e ence can be explained by ce ain simpli ica ions o he model. Howe e , bo h cu es ha e an exponen ial
(3)
T
∗=
T
90
144η
M
2
µ0
(4)
M
n=
144η˙γ
M
2
µ0
Figu e6. The e ec o magne iza ion M (le ) and ca ie luid iscosi y η ( igh ) on esponse ime
T90
.
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cha ac e and he e o e he model desc ibes ends e y well. Ano he signi ican di e ence is ha measu ed
MR luid con ains addi i es ha a e no included in he model. The ques ion is how signi ican a di e ence can
c ea e his simpli ica ion. The su ace oughness can also a ec MR luid dynamics33. This is also no included
in he model, and can also play an impo an ole.
Conclusion
This pape deals wi h he expe imen al de e mina ion o magne o heological luid ansien esponse ( heological
esponse ime) on he apid change o a magne ic ield in shea load mode. A unique heome e was p esen ed
ha allows almos uni s ep o magne ic ields and also allows he measu ing o he de elopmen o MR luid
shea s ess o e ime. The ansien esponse was de e mined on ou MR luids ha di e in supplie , pa icle
concen a ion, o ca ie luid iscosi y. The pape also includes a magne ic model and i s expe imen al e i ica-
ion. The mos impo an conclusions o he pape a e he ollowing:
• The esponse ime o he magne ic ield is
T90I=0.335
ms and sligh ly inc eases wi h an inc easing maximum
alue o elec ic cu en .
• The ise o shea s ess exhibi s an ini ial dead ime o 0.4ms, which is independen o he shea a e le el.
• The alue o he shea a e signi ican ly in luences he heological esponse ime a low shea a es. The
highe he shea a e, he sho e he esponse ime. The measu ed da a o he esponse ime can be i ed by
a powe -law unc ion. The esponse ime
T90
anges om 5.5 o 1.9ms o shea a e
˙γ
om 11 o 218 s−1 o
MR luid MRHCCS4-A and MRHCCS4-B.
• The luid magne iza ion M signi ican ly a ec s he heological esponse ime. The highe he magne iza ion
M o he MR luid, he lowe he esponse ime
T90
.
• The ca ie luid iscosi y also a ec s he heological esponse ime. The MR luid wi h a highe ca ie luid
iscosi y
η
shows a signi ican ly highe esponse ime
T90
.
Figu e7. Dependency non-dimensional esponse ime
T∗
on Mason numbe
Mn
; Da a om o he publica ions
ha e been es ima ed acco ding o a ailable in o ma ion.
Figu e8. The compa ison o model and expe imen o he same inpu s (MRHCCS4-B, elec ic cu en 2 A).
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• All measu ed da a was gene alized in he e m o non-dimensional esponse ime
T∗
and Mason numbe
Mn
.
One mas e cu e (T* = 4.1939Mn−0.35) can be de e mined om measu ed da a independen o magne iza ion
M, ca ie luid iscosi y
η
, shea a es
˙γ
, e c. This is an impo an conclusion because he mas e cu e allows
he de e mina ion o heological ime esponse o a gi en MR luid and gi en load (shea a es).
I should be no ed ha he ou expe imen ally de e mined mas e cu e shows a de ia ion om he model14.
MR luids used in he expe imen and model di e in he ype o concen a ion o addi i es ( he model does
no include addi i es), which may also a ec he ansien esponse. Fo his eason, a plan o u he esea ch
in his a ea is o de e mine he heological esponse ime o homemade MR luid ( ull con ol o addi i es) and
measu emen o a highe ange o Mason numbe s. We also see he po en ial o u u e esea ch in he a ea o
a pa icle chaining model ha allows he showing o pa icle mo ion du ing he s ep change o a magne ic ield.
Da a a ailabili y
The da a p esen ed in his s udy a e a ailable on eques om he co esponding au ho .
Recei ed: 29 Ap il 2022; Accep ed: 10 June 2022
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