Robust Modeling of the Rocking Problem

Abstract

The rocking motion of a solid block on a moving deformable base is a dynamic problem that, despite its apparent simplicity, involves a number of complex dynamic phenomena such as impacts, sliding, geometric and material nonlinearities and, under some circumstances, chaotic behavior. For this reason, since the first model proposed by G.W. Housner in 1963, a number of alternative models have been proposed for its mathematical simulation. In this work, two new models are developed for the simulation of a rigid body experiencing a 2D rocking motion on a moving deformable base. The first model, the concentrated springs model, simulates the ground as tensionless vertical springs with vertical dampers placed at each of the two bottom corners of the body, whereas the second, the Winkler model, simulates the ground as a continuous medium of tensionless vertical springs with vertical dampers. Both models take into consideration sliding (with the use of both a penalty method and an analytical formulation for friction) and uplift and both are geometrically nonlinear. The models are used for simple free vibrational problems in which the effects of the ground deformability, sliding, and uplift are noted. In addition, the stability diagram for various parameters of the system, under excitation by ground motions that correspond to one full cycle sine pulses with varying amplitude and frequency, is created. The behavior of the two models is discussed and compared with the classic theory proposed by Housner.