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combines the best features of both active and
passive control systems. Smart control devices
include variable-orifice dampers, variable-
stiffness devices, variable-friction dampers,
controllable-fluid dampers, shape memory alloy
actuators, piezoelectrics, etc. (Hurlebaus & Gaul
2006). In particular, one of the controllable-fluid
dampers, magnetorheological (MR) damper has
attracted considerable attention in recent years
due to its appealing characteristics. In general,
an MR damper consists of a hydraulic cylinder,
magnetic coils, and MR fluid comprising micron-
sized magnetically polarizable particles floating
within oil-type fluid. The MR damper is operated
as a passive damper; however, when a magnetic
field is applied to the MR fluid, it is changed into
a semi-solid state in a few milliseconds. This is
one of the most unique aspects of the MR damp-
ers compared to active systems: the malfunction
of the active control system might occur if some
control feedback components, e.g., wires and sen-
sors, are broken for some reasons during severe
earthquake events, while a semiactive system
operates as a passive damping system although
the control feedback components are not function-
ing properly. Its characteristics are summarized
in Kim et al. (2009)
To fully use the best features of the MR
damper, a mathematical model that portrays the
nonlinear behavior of the MR damper has to be
developed first. However, this is challenging
because the MR damper is a highly nonlinear
hysteretic device. In this study, a modified Bouc-
Wen model is used to predict the dynamic behav-
ior of the MR damper force because it accurately
predicts the dynamic behaviors at both low and
high velocity regions as shown in figure 1 (Spen-
cer et al. 1997), where the MR damper force
f
1
l
−
l
z
= −
γ
x
−
y
z
z
−
β
(
x
−
y
)
z
BW
BW
BW BW BW
BW
BW
BW
A x
(
y
),
+
−
BW BW
BW
(20)
1
{
}
y
=
α
z
+
c
0
x
+
k
0
(
x
−
y
) ,
BW
BW
BW BW
BW BW
BW
(
c
0
+
c
1
)
BW
BW
(21)
α
=
α
+
α
b B
u
,
(22)
a
c
1
=
c
1
a
+
c
1
b
u
,
(23)
BW
BW
BW BW
0
0
a
0
b
c
=
c
+
c
u
,
(24)
BW
BW
BW BW
u
= −
η
(
u
−
v
),
(25)
BW
BW
BW
where zBW and α, called the evolutionary vari-
ables, describe the hysteretic behavior of the MR
damper;
c
B
0
is the viscous damping parameter at
high velocities;
c
B
1
is the viscous damping pa-
rameter for the force roll-off at low velocities; αa,
αb,
c
BW
1
, and
c
B
1
are parameters that
account for the dependence of the MR damper
force on the voltage applied to the current driver;
k
BW
0
,
c
BW
0
,
c
BW
0
1
controls the stiffness at large velocities;
k
BW
represents the accumulator stiffness;
x
B
0
and
x
BW
are the initial and arbitrary displacements of the
spring stiffness
k
B
1
, respectively; γ, β, l and
A
BW
are adjustable shape parameters of the hysteresis
loops, i.e., the linearity in the unloading and the
transition between pre-yielding and post-yielding
regions;
v
BW
and
u
BW
are input and output volt-
ages of a first-order filter, respectively; and
η
is
the time constant of the first-order filter. Note that
nonlinear phenomena occur when the highly
nonlinear MR dampers are applied to structural
systems for effective energy dissipation. Such an
integrated structure-MR damper system behaves
MR
( )
predicted by the modified Bouc-Wen
model is governed by the following differential
equations
t
1
1
0
f
=
c
y
+
k
(
x
−
x
),
(19)
MR
BW BW
BW BW
BW
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