Overturning instability of a two‐rigid block system under ground excitation

The rocking response of a freely standing two-rigid block 2-DOF system under ground excitation is comprehensively presented. The highly nonlinear governing equations of motion, properly established, under certain conditions may be linearised leading after integration to closed form solutions. The analysis is based on the assumption that friction at the interface between the two-rigid blocks or the lower block and the ground is sufficient to prevent sliding. All possible configuration patterns exhibited by the two-rigid block system during rocking motion are examined in detail. Attention is focused on the determination of the minimum amplitude ground acceleration among all patterns which leads the system to overturning instability. The effect of week damping is included in the analysis by reducing the relative angular velocity after impact. Conditions for overturning instability with or without impact, either between the two blocks or between the lower block and the ground, associated with an escaped motion in the phase-plane portrait, are thoroughly discussed. It is found that for moderately large values of excitation frequencies overturning instability occurs after impact. Beyond these values overturning instability without impact prevails. In case of overturning instability without impact, surprisingly enough, it was also found that there are ranges of values of excitation frequencies in which a monolithic rigid block as a 1-DOF system becomes more stable when divided into two equal rigid blocks, acting as a 2-DOF system.