Abstract

This work explores symbolic and numeric solutions to the Lyapunov matrix equation as it applies to performance-based assessment of base-isolated structures supplemented by modified bang-bang control. Traditional studies of this type rely on numeric simulations alone. This study is the first to use symbolic analysis as a means of identifying key “cause and effect ” relationships existing between parameters of the active control problem and the underlying differential equations of motion. We show that symbolic representations are very lengthy, even for structures having a small number of degrees of freedom. However, under certain simplifying assumptions, symbolic solutions to the Lyapunov matrix equation assume a greatly simplified form (thereby avoiding the need for computational solutions).Regarding the behavior of the bang-bang control strategy, further analysis shows: (1) for a 1-DOF system, the actuator force acts very nearly in phase, but in opposite direction to the velocity (90 ◦ out of phase and in opposite direction to the displacement), and (2) for a wide range of 2-DOF nonlinear base-isolated models, bang-bang control is insensitive to nonlinear deformations in the isolator devices. Through nonlinear time-history analysis, we see that one- and two-DOF models are

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