Pa ame e iden i ica ion o la ge-scale
magne o heological dampe s in a benchma k building✩
A ash Baha a, F ancesc Pozo∗,b, Leona do Achob, Jos´e Rodella a, Alex
Ba ba c
aCoDAlab, Depa amen de Ma em`a ica Aplicada III, Escola T`ecnica Supe io
d’Enginye s de Camins, Canals i Po s de Ba celona (ETSECCPB), Uni e si a
Poli `ecnica de Ca alunya (UPC), Jo di Gi ona, 1-3, 08034 Ba celona. Spain.
bCoDAlab, Depa amen de Ma em`a ica Aplicada III, Escola Uni e si `a ia d’Enginye ia
T`ecnica Indus ial de Ba celona (EUETIB), Uni e si a Poli `ecnica de Ca alunya,
Com e d’U gell, 187, 08036 Ba celona, Spain.
cCIMNE, Depa amen de Resis `encia de Ma e ials i Es uc u es a l’Enginye ia, Escola
T`ecnica Supe io d’Enginye s de Camins, Canals i Po s de Ba celona, Uni e si a
Poli `ecnica de Ca alunya, Jo di Gi ona, 1-3, 08034 Ba celona, Spain.
Abs ac
Magne o heological (MR) dampe s a e de ices ha can be used o ib a ion
educ ion in s uc u es. Howe e , o use hese de ices in an e ec i e way,
a p ecise modeling is equi ed. In his sense, in his pape we conside a
modi ied pa ame e iden i ica ion me hod o la ge-scale magne o heological
dampe s which a e ep esen ed using he no malized Bouc-Wen model. The
main bene i o he p oposed iden i ica ion algo i hm is he accu acy o he
pa ame e es ima ion. The alida ion o he pa ame e iden i ica ion me hod
has been ca ied ou using a black box model o an MR dampe in a sma
base-isola ed benchma k building. Magne o heological dampe s a e used in
his nume ical pla o m bo h as isola ion bea ings as well as semiac i e con-
ol de ices.
✩Suppo ed by CICYT (Spanish Minis y o Science and Inno a ion) h ough g an
DPI2008-06463-C02-01.
∗Co esponding au ho
Email add esses: [email p o ec ed] (A ash Baha ), [email p o ec ed]
(F ancesc Pozo), leona [email protected] (Leona do Acho), [email p o ec ed]
(Jos´e Rodella ), [email p o ec ed] (Alex Ba ba )
URL: h p://www-ma3.upc.es/codalab (F ancesc Pozo),
h p://www.cimne.upc.es (Alex Ba ba )
P ep in submi ed o Else ie July 30, 2009
Key wo ds: MR dampe , pa ame e iden i ica ion, benchma k building
1. In oduc ion
Magne o heological (MR) dampe s a e de ices ha change hei mechan-
ical p ope ies when hey a e exposed o a magne ic ield. The magne o he-
ological luid o hese ac ua o s is cha ac e ized by a g ea abili y o a y,
in a e e sible way, om a ee- lowing linea iscous liquid o a semi-solid
one wi hin milliseconds [1]. Mo eo e , MR dampe s ha e a low cos , low
powe equi emen s, la ge o ce capaci y, obus ness and can be con olled
wi h a low ol age a he coils [1]. All hese ea u es make MR dampe s
e y a ac i e and p omising as ac ua o s con olled by he ol age ha can
be used in di e en enginee ing ields, such as dampe s and shock abso be s
(p essu e d i en low mode de ices), as well as clu ches, b akes, chucking,
and locking de ices (di ec -shea mode de ices) [9]. F om a s uc u al con-
ol poin o iew, MR dampe s a e usually employed as ac ua o s ope a ed
by low ol ages. In his espec , semi-ac i e con ol sys ems seem o com-
bine he bes comp omise be ween passi e and ac i e con ol: hey o e he
eliabili y o passi e de ices oge he wi h he e sa ili y and adap abili y
o ac i e sys ems [3, 5, 20]. Howe e , he i s s ep in he design o a semi-
ac i e con ol s a egy is he de elopmen o an accu a e model o he MR
de ice. I is wo h no ing ha he sys em-iden i ica ion issue plays a key
ole in his con ol p oblem [19]. High-accu acy models can be designed us-
ing wo di e en model amilies: semi-physical models [16, 20], and black box
models [10, 23]. Some o he mos known semi-physical models o desc ibe
he hys e e ic beha iou o MR dampe s a e he Bingham model and i s ex-
ended e sions, he Bouc-Wen model, he Dahl model, he modi ied LuG e
model and some o he non-pa ame ic models [12, 17]. I is impo an o e-
ma k ha hese models a e no linea -in-pa ame e s and, he e o e, classical
pa ame e iden i ica ion me hods, such as he g adien o he mean squa e
algo i hms, canno be applied.
Using a no malized e sion o he Bouc-Wen model, Ikhouane e al. [7]
p esen an iden i ica ion algo i hm which is di ec ly used o MR dampe s
in shea -mode [17]. Howe e , his me hodology can p oduce la ge pa ame-
e iden i ica ion e o s i he iscous ic ion is much smalle han he d y
ic ion [17]. To cope wi h his d awback, a modi ied s ep was p oposed by
Rod ´ıguez e al. [17] and es ed in a small-scale MR dampe . When he
2
iden i ica ion is applied o a la ge-scale MR dampe , he pa ame e iden i i-
ca ion e o s inc ease [18]. The aim o his pape is o imp o e he accu acy
o he iden i ica ion algo i hm. This is based on augmen ing he no malized
Bouc-Wen model wi h an addi ional e m. The alida ion o his modi ied
pa ame e iden i ica ion me hod has been ca ied ou using a black box model
o an MR dampe in a sma base-isola ed benchma k building [14]. This
model is unknown o he use o his benchma k and o he designe o
con ol sys ems. The benchma k pla o m is hen conside ed as a i ual
labo a o y expe imen . The nume ical esul s show ha he p oposed mod-
i ied me hod is able o imp o e signi ican ly he accu acy o he pa ame e
iden i ica ion.
The pape is o ganized as ollows. In Sec ion 2, he magne o heological
dampe model is p esen ed. In Sec ion 3, he key poin s o he modi ied
iden i ica ion me hod a e discussed. In Sec ion 4, he applica ion o he
p oposed iden i ica ion me hod o a la ge-scale MR dampe in a benchma k
building is conside ed. Finally, some concluding ema ks a e s a ed in Sec ion
5.
2. The magne o heological dampe model
The no malized e sion o he Bouc-Wen model [7] is an equi alen ep-
esen a ion o he o iginal Bouc-Wen model [23]. Fo MR dampe s in shea
mode i akes he o m:
Φn( ˙x, w)( ) = κ˙x( ) ˙x( ) + κw( )w( ),(1)
˙w( ) = ρ( ˙x( )−σ|˙x( )||w( )|n−1w( ) + (σ−1) ˙x( )|w( )|n),(2)
whe e Φn( ˙x, w) is he ou pu o ce o he MR dampe , ˙x( ) and a e he
eloci y and ol age inpu s, espec i ely. The ol age inpu is he applied
ol age a he coil o he MR dampe . The sys em pa ame e s, which a e
ol age-dependen , a e κ˙x( )>0, κw( )>0, ρ > 0, σ > 1/2, and n≥1.
These pa ame e s con ol he shape o he hys e esis loop and hei meaning
can be ound in [6]. The s a e a iable w( ) has no a physical meaning so
ha i is no accessible o measu emen s.
Since he no malized Bouc-Wen ep esen a ion desc ibed in equa ions
(1)-(2) is no a linea -in-pa ame e model, classical pa ame e iden i ica ion
me hods canno be applied. In his ega d, a new pa ame e iden i ica ion
algo i hm has been p oposed in [7, p. 38], which is based on a physical un-
de s anding o he de ice along wi h a black box desc ip ion. Rod ´ıguez e
3
al. [18] used his me hodology o a la ge-scale MR dampe . The me hod is
based on applying a pe iodic inpu eloci y ˙x( ) a a cons an ol age coil
and obse ing he pe iodic s eady-s a e o ce esponse o he MR dampe .
None heless, la ge ela i e e o s in he iden i ica ion p ocess can be obse ed
when he MR dampe has a iscous ic ion (κ˙x( ) ˙x( )) small enough wi h
espec o he d y ic ion (κw( )w( )). To cope wi h his d awback, when
he displacemen is la ge enough, an al e na i e me hod based on he plas ic
egion o he o ce- eloci y diag am o he MR dampe has been p oposed in
[17]. Howe e , he model in equa ions (1)-(2) may no gi e an accu a e ep-
esen a ion o la ge-scale MR dampe s which do no belong o he shea - ype
ca ego y. Fo ins ance, conside he black box model o an MR dampe in
he sma base-isola ed benchma k building [14]. Figu e 1 con ains he o ce-
displacemen and o ce- eloci y diag ams when his MR dampe is exci ed
by a sinusoidal displacemen and eloci y. In o de o es he goodness o
he Bouc–Wen model, his igu e also con ains he esponse o he dynamic
model in equa ions (1)-(2) wi h some app op ia e pa ame e s. I can be ob-
se ed ha he esul ing plas ic b anch in he o ce- eloci y diag am is wide
ha he co esponding b anch o he Bouc–Wen model. In he li e a u e, he
same ype o cycles ha e been expe imen ally epo ed, o ins ance, in [4,
Figu e 4(b)], [11, Figu e 9] and [13, Figu e 7]. The e o e, i can be de i ed
ha his kind o hys e e ic loops a e unable o be ep oduced wi h he o igi-
nal Bouc–Wen model. To imp o e he accu acy o he model ep esen a ion
and, consequen ly, he accu acy o he pa ame e iden i ica ion, we use he
ollowing ex ended Bouc-Wen model:
Φe(x, ˙x, w)( ) = κx( )x( ) + κ˙x( ) ˙x( ) + κww( ),(3)
˙w( ) = ρ( ˙x( )−σ|˙x( )||w( )|n−1w( ) + (σ−1) ˙x( )|w( )|n),(4)
whe e he e m κx( )x( ), which ep esen s a linea elas ic o ce, has been
added. We conside ha he coe icien κxis ol age-dependen , as he o he
pa ame e s. The e ec o his e m can be obse ed no only in he o ce-
eloci y diag am, bu also in he o ce-displacemen : he esul ing plo is
inclined. This ea u e has also been expe imen ally epo ed in, o ins ance,
[2, Figu e 8], [11, Figu e 9] and [13, Figu e 7].
3. Model pa ame e iden i ica ion
This sec ion is conce ned wi h he compu a ion o he pa ame e s o he
ex ended model in equa ions (3)-(4). The p oposed algo i hm will be di ided
4
−200
−150
−100
−50
0
50
100
150
200
250
Fo ce (kN)
−2 −1 0 1 2
−400
−300
−200
−100
0
100
200
300
400
Fo ce (kN)
−250
Displacemen (m) Veloci y (m/s)
0−0.2−0.4 0.40.2
plas ic b anch
Figu e 1: Fo ce-displacemen (le ) and o ce- eloci y ( igh ) diag ams o he black box
model o he MR dampe when i is exci ed by a sinusoidal displacemen and eloci y
(dashed). The esponse o he dynamic Bouc–Wen model in equa ions (1)-(2) unde he
same exci a ion is ep esen ed by a solid line.
in wo s eps: (a) he es ima ion o he alue o κxand (b) he es ima ion o
he es o he pa ame e s based on he iden i ica ion algo i hm in [18].
A cons an ol age, he compu a ion o he pa ame e κx( ) can be pe -
o med g aphically by conside ing he o ce-displacemen diag am o he MR
dampe . When his de ice is exci ed by a sinusoidal displacemen wi h a
la ge enough ampli ude, he a e age inclina ion o he esul ing plo gi es an
es ima ion o his pa ame e . As an example, conside he black box model
o an MR dampe in he sma base-isola ed benchma k building. When
his MR dampe is d i en wi h ze o coil command ol age, we ob ain he
o ce-displacemen diag am in Figu e 2. The es ima ed alue o κxis hen
compu ed as κx=82.8
0.4= 207 kN.
To es ima e he es o he pa ame e s, we use he knowledge o he
pa ame e κxand he ac ha
Φn( ˙x, w)( ) = Φe(x, ˙x, w)( )−κx( )x( ).
Figu e 3 depic s (in solid line) he o ce- eloci y diag am o he esul ing
ou pu o ce Φeo he MR dampe minus he linea elas ic o ce κx( )x( ),
when his de ice is exci ed by a sinusoidal displacemen and eloci y. I can
be ecognized om his igu e ha his new cycle has he same shape as
5

he o ce- eloci y cycle o he model in equa ions (1)-(2), which is plo ed in
Figu e 1 igh . As a esul , o he iden i ica ion o he es o he pa ame e s,
ha is, he pa ame e s o he model
Φn( ˙x, w)( ) = κ˙x( ) ˙x( ) + κw( )w( ),
˙w( ) = ρ( ˙x( )−σ|˙x( )||w( )|n−1w( ) + (σ−1) ˙x( )|w( )|n),
we can ollow basically he same idea as in [18]. The de ails o his me hod
a e omi ed he e bu can be ound in he Appendix.
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−400
−300
−200
−100
0
100
200
300
400
Displacemen (m)
Fo ce (kN)
82.8 kN
Figu e 2: The a e age inclina ion o he o ce-displacemen diag am, when he MR dampe
is exci ed by a sinusoidal displacemen , gi es an es ima ion o he pa ame e κx.
4. Applica ion o a benchma k building
This sec ion is conce ned wi h he applica ion o he p oposed me hod on
a i ual MR dampe . Mo e p ecisely, he p oposed iden i ica ion algo i hm
is es ed using a black box model o an MR dampe , which is a pa o a
sma base-isola ed benchma k building p oblem [14]. Consequen ly, we use
his nume ical pla o m as a i ual labo a o y es (Figu e 4). To alida e
6
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−250
−200
−150
−100
−50
0
50
100
150
200
250
Veloci y (m/s)
Fo ce (kN)
Figu e 3: Fo ce- eloci y diag am o he esul ing ou pu o ce Φeo he MR dampe
(dashed) and he o ce Φn= Φe−κx( )x( ) (solid).
he esul s, he ou pu o ces o he i ual de ice and he iden i ied one will
be compa ed using se en p ede ined ea hquake eco ds o he benchma k
p oblem wi h hei co esponding luc ua ing ol age du ing ull simula ions.
The MR dampe is used in his con ex as a semi-ac i e de ice o educe he
s uc u al esponse o he building.
4.1. Sma base-isola ed benchma k building
The sma base-isola ed benchma k building [14] is employed as an in e -
es ing and mo e ealis ic example o u he in es iga e he e ec i eness o
he p oposed design app oach. This benchma k p oblem is ecognized by he
Ame ican Socie y o Ci il Enginee s (ASCE) S uc u al Con ol Commi ee
as a s a e-o - he-a model de eloped o p o ide a compu a ional pla o m
o nume ical expe imen s o seismic con ol a enua ion [15, 21].
The benchma k s uc u e is an eigh -s o ey ame building wi h s eel-
b aces, 82.4 m long and 54.3 m wide, simila o exis ing buildings in Los
7
MR dampe
x
˙x
Φ
Figu e 4: Inpu -ou pu a iables o he i ual MR dampe .
Angeles, Cali o nia. S o ies one o six ha e an L-shaped plan while he
highe loo s ha e a ec angula plan. The supe s uc u e es s on a igid
conc e e base, which is isola ed om he g ound by an isola o laye , and
consis s o linea beam, column and b acing elemen s and igid slabs. Below
he base, he isola ion laye consis s o a a ie y o 92 isola ion bea ings. The
isola o s a e connec ed be ween he d op panels and he oo ings below, as
shown in Figu e 5.
Figu e 5: Ele a ion iew wi h de ices
4.2. Iden i ica ion esul s
In o de o implemen he iden i ica ion p ocedu e in Sec ion 2 i is nec-
essa y o apply a pe iodic exci a ion displacemen and obse e he co e-
8
sponding MR dampe o ce. Figu e 6 illus a es hese wo signals o a ze o
ol age. A se o expe imen s ha e been pe o med o di e en ol ages in
he ange [0,1] ol s.
0 2 4 6 8 10
−80
−60
−40
−20
0
20
40
60
80
Time (s)
Fo ce (kN)
0 2 4 6 8 10
−0.03
−0.02
−0.01
0
0.01
0.02
0.03
Time (s)
Displacemen (m)
Figu e 6: Response o he MR dampe model in he benchma k building pla o m.
The esul ing alues o he pa ame e s o he model in equa ions (3)-(4)
a e lis ed in Table 1. Figu e 8 plo s hese pa ame e s as a unc ion o he
ol age. To ind an accu a e ol age-dependen ela ion o hese pa ame e s,
and acco ding wi h he unc ional dependence in Figu e 8, we conside ha
κx( ) is cons an , κ˙x( ) is linea and n( ), ρ( ) and σ( ) a e exponen ial:
κx( ) = κx(5)
κ˙x( ) = κ˙x,a +κ˙x,b (6)
n( ) = na+nbexp(−13 ) (7)
ρ( ) = ρa+ρbexp(−14 ) (8)
σ( ) = σa+σbexp(−14 ) (9)
Because o he impo ance o he pa ame e κwdue o i s g ea in luence in
he esul ed o ce ( he ange o i s magni ude is, app oxima ely, om 50 kN
o 1000 kN, as can be seen in Table 1), i s ol age dependence unc ion has
been es ima ed in h ee di e en egions
κw( ) = 


κw1+κw2 1.15, ≤0.3
κw3+κw4sin(π( −0.3)
0.8) + κw5sin(3π( −0.3)
0.8),0.3≤ ≤0.7
κw6+κw7 +κw8 3+κw9 5,0.7≤
,
(10)
based on he a ia ion o he esul ed alues (Figu e 7).
9
0 5 10 15 20 25 30
−1000
−500
0
500
1000
Time (s)
Dampe Fo ce E o (kN)
0 5 10 15 20 25 30
−1000
−500
0
500
1000
Time (kN)
Dampe Fo ce E o (kN)
Figu e 11: Gene a ed dampe o ce e o s o p oposed model (abo e), and o iginal me hod
[17] (below) , unde Kobe g ound mo ion (FP-y).
iden i ica ion me hod has been es ed using he MR dampe as a semi-ac i e
de ice unde ime- a ying ol age and ea hquake exci a ion.
A. Appendix
The pa ame e iden i ica ion in [18] depa s om he nex shea -mode
model:
Φn( ˙x)( ) = κ˙x( ) ˙x( ) + κw( )w( ) (13)
˙w( ) = ρ( ˙x( )−σ|˙x( )||w( )|n−1w( ) + (σ−1) ˙x( )|w( )|n) (14)
whe e κ˙x>0, κw>0, ρ > 0, σ > 1/2, and n≥1. All o hese pa ame e s
can be ol age o cu en dependen (he e he case o ol age dependency
is unde conside a ion, as emphasized o κ˙xand κw). I has been shown in
[7] ha his model is meaning ul in he sense ha he limi cycle depends
16

di ec ly on he pa ame e s ha appea in he no malized o m, and hus
depends only indi ec ly on he pa ame e s o he s anda d o m as
ρ=A
Dz0
>0,
σ=β
β+γ≥0,
κ˙x=αk > 0,
κw= (1 −α)Dkz0>0,
whe e A, D, α, β, γ and kcomes om he s anda d Bouc–Wen model
ΦBW(x)( ) = αkx( ) + (1 −α)Dkz( ),
˙z=D−1A˙x−β|˙x|z|z|n−1−γ˙x|z|n.
Fo pa ame e iden i ica ion, a T-pe iodic inpu ˙x( ) (see Figu e 12) is ap-
plied o he Bouc-Wen sys em unde cons an ol age . I has been p o ed
[7] ha he ou pu o ce o he Bouc-Wen model goes asymp o ically o a
pe iodic s eady-s a e so ha a limi cycle is ob ained. The iden i ica ion
me hod assumes he knowledge o he ela ion ¯w(x) ha desc ibes his cycle
as illus a ed in Figu e 13. The whole iden i ica ion p ocess can be summa-
ized as ollows.
The pa ame e κ˙xis i s de e mined using he plas ic egion ( ¯w≈1) o
he hys e esis loop by a linea eg ession o each cons an ol age:
¯
F(τ) = κ˙x( ) ˙x(τ) + κw( ).
To con inue wi h pa ame ic es ima ion, a unc ion θis compu ed as:
θ(x(τ)) = ¯
F(x(τ)) −κ˙x
dx(τ)
dτ , τ ∈[0, T+],(15)
which has a unique ze o, i.e, he e exis s a ime ins an τ∗∈[0, T+], and a
co esponding alue x∗=x(τ∗)∈[Xmin, Xmax], such ha he unc ion θis
ze o. Because θis known, hen ˙x∗is also known. De ine he quan i y
a=dθ(x)
dx x=x∗
.(16)
17
0T+TmT mT +T+(m+ 1)T
Xmin
Xmax
Inpu signal x
Time
Figu e 12: A sample T-wa e pe iodic signal
Then, he pa ame e nis de e mined as:
n=
log (dθ(x)
dx )x=x∗2
−a
(dθ(x)
dx )x=x∗1
−a
log θx=x∗2
θx=x∗1(17)
whe e x∗2> x∗1> x∗a e design pa ame e s. De ine
b=
a−dθ(x)
dx x=x∗2
θ(x∗2)n.(18)
Then, he pa ame e s κwand ρa e compu ed as ollows:
κw=n
a
b,(19)
ρ=a
κw
.(20)
18
Figu e 13: Symme y p ope y o he hys e esis loop o he no malized Bouc–Wen model.
The unc ion ¯w(x) can be compu ed as:
¯w(x) = θ(x)
κw
.(21)
Finally, he emaining pa ame e σis de e mined as:
σ=1
2


(d¯w(x)
dx )x=x∗3
ρ−1
(−¯w(x∗3)n)+ 1

(22)
whe e x∗3is a design pa ame e such ha x∗3< x∗.
Re e ences
[1] G. Bossis , P. Khuzi , S. Lacis, and O. Volko a, Yield beha io o mag-
ne o heological suspensions, Jou nal o Magne ism and Magne ic Ma e-
ials,258-259:456-458, 2003.
19
[2] W.W. Chooi, S.O. Oyadiji, Design, modelling and es ing o magne-
o heological (MR) dampe s using analy ical low solu ions, Compu e s
& S uc u es,86(3-5):473–482, 2008.
[3] A. Dominguez, R. Sedagha i, and I. S iha u, Modeling and applica ion
o MR dampe s in semi-adap i e s uc u es, Compu e s & S uc u es,
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