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where gj^i = g(X fcTl , F^+^it
formulation iterations are avoided.
) (4)
.); Hence, the method is implicit, needing iteration. In the incremental
Modeling of Base Isolation Bearings and MR Dampers
Sliding bearings are modeled using the following equation:
fb (t) = pN. S (t) (5)
where N is the normal force at the sliding bearing, the coefficient of friction /x = /max — (/^ax - / mm ) •
e-aK! wj t k u b the velocity of the sliding base, / max the coefficient of sliding friction at high velocity, / m i n
is the coefficient of sliding friction at essentially zero velocity of sliding, and a is the coefficient controlling
the dependency of friction on velocity of sliding and z (t] is a Bouc-Wen hysteretic dimensionless quantity
governed by the equation:
Y-Z + T \u b \ • z \z\ n ~ l + p-u b -z n -Au b = Q (6)
where Y is the yield displacement, 7, /3, n, and A are dimensionless parameters of the model that govern
the hysteretic behavior, and lib is the base displacement.
Magnetorheological (MR) dampers are semi-active control devices that use MR fluids to produce controllable
damping. In this paper an extension to the MR damper model by Spencer et al (2000 ) is developed.
The Bouc-Wen model hysteretic element with spring and variable dashpot in parallel are used to model the
MR damper, as shown hi Figure 2.
Figure 2. MR Damper Model
The governing equation of the model shown in Figure 2 is given by
where k is the stiffness of the accumulator, the variable damping coefficient, c = c a -f c& with c a being
constant and c b — GI • hi(v), a = a a 4- a b with a a being constant and a& » aj. • h^(v}^ with &i(v), /ia(u) and
f(v) being functions of voltage, v, applied to the MR damper, f(u) accounts for the variation in the force
generated by the damper under different voltages. The hysteretic variable z of Bouc-Wen Model is governed
by equation (6), where Y is the yield displacement and 7 = 0.9, / = 0.1, n = 2, .4 - 1.
Equations (5), (6) and (7) are solved using the unconditionally stable semi-implicit Runge-Kutta method
suitable for stiff differential equations. Equations (3), (4), (5) and (6) are solved using an efficient predictorcorrector
algorithm developed by the authors.