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 compare the approximate quarter period for an initially open block with the exact quarter period, which we derive to be an elliptic integral of the first kind. The difference between the two periods increases as a block 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 harmonic excitation, we compare ensembles of time histories for linear and nonlinear responses. The poorly conditioned nature of the system emerges prominently when very small amplitude ( <10 ⁻⁴ ) rocking is predicted and when the forcing amplitude to slenderness angle ratio is between 1.2 and 1.8. Parameter studies delineating between surviving and toppling reveal that models are least sensitive to slenderness angle and peak amplitude, more sensitive to rebound velocity coefficient, and most sensitive to block diagonal. This sensitivity is generally reduced when using the nonlinear model. We conclude the nonlinear model provides more robust results, even for slender blocks.