Dimensional analysis of the earthquake-induced pounding between adjacent structures

In this paper the dynamic response of two and three pounding oscillators subjected to pulse-type excitations is revisited with dimensional analysis. Using Buckingham's II-theorem the number of variables that govern the response of the system is reduced by three. When the response is presented in the dimensionless II-terms remarkable order emerges. It is shown that regardless of the acceleration level and duration of the pulse all response spectra become self-similar and follow a single master curve. This is true despite the realization of finite duration contacts with increasing durations as the excitation level increases. All physically realizable contacts (impacts, continuous contacts, and detachments) are captured via a linear complementarity approach. The study confirms the existence of three spectral regions. The response of the most flexible among the two oscillators amplifies in the low range of the frequency spectrum (flexible structures); whereas, the response of the most stiff among the two oscillators amplifies at the upper range of the frequency spectrum (stiff structures). Most importantly, the study shows that pounding structures such as colliding buildings or interacting bridge segments may be most vulnerable for excitations with frequencies very different from their natural eigenfrequencies. Finally, by applying the concept of intermediate asymptotics, the study unveils that the dimensionless response of two pounding oscillators follows a scaling law with respect to the mass ratio, or in mathematical terms, that the response exhibits an incomplete self-similarity or self-similarity of the second kind with respect to the mass ratio.