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Lyapunov Controller for the System with MR Damper
For MR damper, a Lyapunov Based Controller is developed as Mows. The equation (3) can be rewritten
in absolute coordinates for a rigid superstructure as follows
= AX a (i) + B/(t) 4- Bu + Bk b u a = g(X a ,/, u g ) (8)
i a = the absolute displacement of rigid
mass m fc , / = the aonlinear force at the isolation level, u = c(t) u bl c(i) is the tune varying damping
coefficient of the MR damper, u b — relative displacement of mass with respect to the ground, and u g =
ground displacement. Based on a appropriate Lyapunov function V and V being negative or minimum the
following switching control algorithm is derived.
C (+\
( t ) _ j
^
— \ o or
where C min is the minimum damping coefficient for one volt, C max is the maximum damping coefficient
for 4 volts, pi and p> are constants.
Figure 1 shows a smart sliding isolated two-story building model with MR Damper, a restoring spring,
and four sliding bearings, considered in this analytical study. A corresponding scaled model has been tested
by authors (Sahasrabudhe et al. 2000,2001). The 1:5 scale model is 1.47m in length and the height is
1.48m, with 0.74m height of each floor. The base mass is 5.54 N-s 2 /cm, and the mass of the first and second
floor is 5.92 N-s 2 /cm each. The total stiffness of recentering springs connected between the base of the
building and the shaking table is 720 N/cm. The condensed superstructure stiffness matrix in the fixed base
condition (including joint rotations), and corresponding clamping matrix are considered. The fundamental
period of the superstructure in the fixed base condition (2DOF) is 0.15 sec. The fundamental period of the
model sliding base isolated structure (3DOF) is nearly 1.0 sec (2.24 sec at prototype scale). Simulations
are performed at both the prototype and model scale with proper consideration being given to the scaling
issues. For the superstructure (2DOF), in the fixed base condition, the damping coefficients are & = 2.56%
and g> = 1.48%. Simulations are performed with MR clamper off—constant zero volts—low damping, MR
damper on—constant four volts—high damping, and controlled cases where the voltage is switched between
one and four volts.
Results
The response of the prototype smart sliding isolated rigid structure (SDOF) with an isolation period of 2
sec to a cosine pulse with a period of 2 sec, representative of the Rinaldi fault normal earthquake, is shown
in Figure 3. It is evident from the response in Figure 3 that in the controlled case the base displacement as
well as the total shear force at the isolation level are reduced further than the passive high damping case;
the switching of the MR damper dissipation force, from one volt level to the four volt level, in the controlled
case is clearly evident. It is clearly evident that the smart isolation system performs better than the passive
high damping case.
The response of the prototype smart sliding isolated rigid structure (3DOF) with an isolation period of
4 sec to Newhall fault normal earthquake, is shown in Figure 4. It is evident from the response in Figure 4
that the controlled case clearly reduces the base displacement as well as the total shear force at the isolation
level. In the total isolation force-base displacement loops shown in Figure 4 the difference in loops of the
three cases is primarily due to changes in MR clamper forces—switching of the MR damper, between one
and four volt levels, in the controlled case is clearly evident in the force-displacement loops. The ability of
the smart MR damper to reduce the response further than the passive high damping case is clearly evident.
Next the simulations of the smart sliding isolated model structure (SDOF) are performed for the following
earthquakes: (1) El-Centro SOOE Earthquake (May 18,1940), peak acceleration: 0.34 g (100%); (2) Newhall
Channel 1 90 Deg. Fault Parallel (Jan. 17, 1994), peak acceleration: 0.608 (105%)g; (3) Newhall Channel