1 Copyright © 2003 by ASME Proceedings of IMECE’03 2003 ASME International Mechanical Engineering Congress & Exposition Washington, D.C., November 16-21, 2003 IMECE2003 – 44600 OVERTURNING THRESHOLDS OF A ROCKING BLOCK SUBJECTED TO HARMONIC EXCITATION: COMPUTER SIMULATIONS AND ANALYTICAL TREATMENT Stefano Lenci (Lenci@univpm.it) Istituto di Scienza e Tecnica delle Costruzioni, Università Politecnica delle Marche, Via Brecce Bianche, 60131, Ancona, Italy Giuseppe Rega (Giuseppe.Rega@uniroma1.it) Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma "La Sapienza", via A. Gramsci 53, 00197, Roma, Italy ABSTRACT When a rigid block resting on a horizontal rigid foundation is shaken by a periodic excitation it can topples if the excitation amplitude is sufficiently high. This question is addressed in this work by the combined use of numerical and analytical tools. Numerical computations are first performed aimed at understanding the overall overturning behavior, and how it is modified by varying relevant parameters, such as the excitation phase and the coefficient of restitution at impacts. Then the problem is addressed from a theoretical point of view, with the objective of analytically detecting the most relevant features highlighted by previous numerical simulations. We succeed in obtaining closed form and manageable criteria for overturning. KEYWORDS: Rigid block dynamics, harmonic excitation, overturning, invariant manifolds. 1. INTRODUCTION In this work we investigate the overturning of a rigid block subjected to harmonic base excitation, an issue which deserves interest in many branches of mechanical and civil engineering. It is an old topic tracing back to the nineteen century (Milne, 1881; Perry, 1881), which has been largely investigated in the past from theoretical (Housner, 1963; Hogan, 1989), numerical and experimental points of view (Wong & Tso, 1989; Fielder et al., 1997). Several questions have been addressed, such as the existence and stability of periodic solutions, multistability, chaotic dynamics, impulsive, periodic and stochastic excitations, overturning diagrams, Melnikov analysis of invariant manifolds, and control. From a practical point of view, one of the most interesting problems is the possible overturning of the rest position, which (i) actually motivated early works, (ii) can arise in many applications and (iii) is the principal subject of the present work. In studying overturning from an engineering point of view, two points are of basic interest. The first is the detection of analytical criteria permitting reliable estimations of the occurrence of overturning, and the second is the construction of overturning charts characterizing the system response in parameters space. The first criterion for overturning seems to be the static one proposed by West (Milne & Omori, 1893) which compares the momentum of the weight with the momentum of the acceleration peak. Other criteria are given in the famous paper of Housner (1963): in the case of constant acceleration, he assumes that the (energetic) condition for overturning is that the total work done by the inertial force is equal to the difference in potential energy between the rest position and the hilltop saddles. In the case of sine-waved pulse the (kinematic) criterion is instead that the actual position reaches one of the hilltop saddles at the end of the pulse: this leads to the same results previously obtained by Kirkpatrick (1927). Energetic considerations are also used in the case of earthquake (i.e., stochastic) excitations. Criteria for overturning have been reviewed by Ishiyama (1980, 1982, 1984), where also sinusoidal (harmonic) excitations are considered. Numerical overturning charts under harmonic excitations are obtained, among others, by Spanos & Koh (1984), which consider both piecewise linear and strongly nonlinear models but produce diagrams with a scarce resolution. More refined charts are constructed, e.g., by Fielder et al. (1997), which also report on experimental overturning charts, for both symmetric and unsymmetric blocks, and which stress the influence of the restitution coefficient. In both these works, however, viscous damping during rocking is not considered. In recent years, the problem has been addressed also from a dynamical system point of view, and it has been suggested that invariant manifolds should play an important role in the complicated toppling mechanism. In fact, it is just the penetration of tongues of the basin of the overturned attractor into the safe basin, driven and bounded by the stable manifolds