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

TL;DR

This paper develops an energy-based analytical framework to determine the overturning conditions of rigid blocks during rocking motion, providing exact criteria and solutions for stability analysis in both nonlinear and linear regimes.

Contribution

It introduces the first energy conservation-based exact overturning criteria and analytical solutions for rigid block rocking, enhancing understanding of stability and overturning behavior.

Findings

Exact overturning criteria derived from energy conservation.

Analytical solutions for nonlinear free rocking motion.

Stability regions visualized through color mapping.

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.