Dimensional Analysis of the Earthquake Response of a Pounding Oscillator

Abstract

In this paper, the dynamic response of a pounding oscillator subjected to pulse type excitations is revisited with dimensional analysis. The study adopts the concept of the energetic length scale which is a measure of the persistence of the distinguishable pulse of strong ground motions and subsequently presents the dimensionless $\Pi$ products that govern the response of the pounding oscillator. The introduction of Buckingham’s $\Pi$ theorem reduces the number of variables that govern the response of the system from 7 to 5. The proposed dimensionless $\Pi$ products are liberated from the response of an oscillator without impact and most importantly reveal remarkable order in the response. It is shown that, regardless the acceleration level and duration of the pulse, all response spectra become self-similar and, when expressed in the dimensionless $\Pi$ products, follow a single master curve. This is true despite the realization of contacts with increasing durations as the excitation level increases. All physically realizable contacts (impacts, continuous contacts, and detachment) are captured via a linear complementarity approach. The proposed analysis stresses the appreciable differences in the response due to the directivity of the excitation (toward or away the stationary wall) and confirms the existence of three spectral regions where the response of the pounding oscillator amplifies, deamplifies, and equals the response of the oscillator without pounding.