Dynamic behaviour of rocking two‐block assemblies

TLDR

The dynamic behaviour of two‑block rocking assemblies, where one block rests atop another and can rock freely without sliding, is examined. The authors derive equations of motion for each vibration mode, establish criteria for rocking initiation and mode transitions, and show that transitions can involve impact‑induced dissipation and kinetic‑energy redistribution between the blocks. The system exhibits continuous mode switching that produces a nonlinear response, with the upper block increasingly contributing to the energy, resulting in a larger and longer vibration of the top block compared to the lower one.

Abstract

Abstract The dynamic behaviour of systems consisting of two blocks, one placed on the top of the other, and free to rock without sliding, is examined in this analysis. The equations of motion for each ‘mode’ of vibration are derived and criteria for the initiation of rocking and the transition between modes are given. During vibration, the system continuously changes from one mode to another and this makes the response non‐linear. This transition may be accompanied by impact, in which case dissipation of energy occurs, the amount of which depends on the relative velocities and the dimensions of the blocks. Also, redistribution of the kinetic energy of the system in the blocks happens. In most cases, the fractional contribution from the upper block to the system energy increases, which results in a larger and longer response of the top block, compared to the vibration of the lower one.