EFFECTS OF LINEARIZING ON ROCKING-BLOCK TOPPLING

We report on the free and harmonically forced vibration responses of a rigid block rotating on a rigid surface. For free vibration, we compa re the approximate quarter period for an initially open block with the exact quarter period, which we derive to be an elliptic integral of t he first kind. The difference between the two periods increases as a b lock becomes less slender; however, the approximate solution improves as a function of an increasing initial rotation. The maximum error is 2.0% for a single segment of rotation for practical geometries. For ha rmonic excitation, we compare ensembles of time histories for linear a nd nonlinear responses. The poorly conditioned nature of the system em erges prominently when very small amplitude (<10(-4)) rocking is predi cted and when the forcing amplitude to slenderness angle ratio is betw een 1.2 and 1.8. Parameter studies delineating between surviving and t oppling reveal that models are least sensitive to slenderness angle an d peak amplitude, more sensitive to rebound velocity coefficient, and most sensitive to block diagonal. This sensitivity is generally reduce d when using the nonlinear model. We conclude the nonlinear model prov ides more robust results, even for slender blocks.