New insights on rocking of rigid blocks: Analytical solutions and exact energy-based overturning criteria

In this paper, the underlying energy flow in rocking dynamics of rigid blocks is utilized to elucidate their response details and overturning conditions. Based merely on the law of the conservation of energy, the exact criteria for a rigid block to overturn in the free vibration regime are established for the first time in the context of both nonlinear and linear theory. That is, if the initial conditions of the free oscillation fall within the prescribed stable libration region, then it is a priori determined that the block will not overturn and vice versa. The analytical solution to the nonlinear free rocking problem is also derived, utilizing integration techniques originating from the solution of the nonlinear pendulum. Regarding the forced rocking problem, an approximate analytical solution is presented based on a Taylor series expansion of the equation of motion. Moreover, employing the newly derived overturning criterion richer pulse-type overturning spectra are constructed, revealing stability and instability regions that had never been shown before in the literature. The new characteristics of the overturning spectra are elucidated using color maps. Finally, some new results concerning the design of solid blocks against rectangular and half-sine acceleration pulses are presented.

Automated summary

This paper utilizes the energy flow in rocking dynamics of rigid blocks to explain their response details and overturning conditions. Based on the conservation of energy, exact criteria for a rigid block to overturn in the free vibration regime are established for the first time in both nonlinear and linear theory. Analytical solutions for the nonlinear free rocking problem and an approximate analytical solution for the forced rocking problem are derived. By employing the newly derived overturning criterion, richer overturning spectra are constructed, and the new characteristics are elucidated using color maps. Finally, new results regarding the design of solid blocks against different acceleration pulses are presented.