Rocking of rigid blocks: Analytical solution and exact energy-based overturning criteria

Abstract: 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 regime 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. Additionally, pulse-type overturning spectra are constructed easily by numerical integration of the equation of motion only during the pulse duration. If the block does not topple during the forced regime, the angle and angular velocity at the end of the excitations are employed to assess immediately the safety or unsafety of the block using the exact overturning nonlinear criterion. Finally, new insights into the stability regions of the overturning spectra are illuminated and explained in detail using a color mapping.