Steady state rocking response of rigid blocks part 1: Analysis

Abstract The result of a theoretical study on the rocking response of rigid blocks subjected to sinusoidal base motion is presented. The study indicates that, for a given excitation amplitude and frequency, a rigid block can respond in several different ways. Based on analysis, the regions of different classes of steady state symmetric response solutions are mapped on the excitation amplitude‐frequency parameter space. The steady state response solutions (both harmonic and subharmonic) are classified into two classes, out‐of‐phase and in‐phase with respect to the excitation. Only out‐of‐phase solutions are found to be stable. A parametric study shows that steady rocking response amplitude is highly sensitive to the size of the block and the excitation frequency in the low frequency range. It is relatively insensitive to the excitation amplitude and the system's coefficient of restitution of impact. For two blocks of the same aspect ratio and coefficient of restitution subjected to the same excitation, the larger block always responds in smaller amplitude than the smaller block. Computer simulation is carried out to study the stability of the symmetric steady state response solutions obtained from analysis. It is found that as the excitation frequency is decreased beyond the boundary of stable symmetric response, the response becomes unsymmetric where the mean amplitude of oscillation is non‐zero. Further decrease in excitation frequency beyond the stable unsymmetric response boundary causes instability in the form of overturning.