Dimensional Analysis of Bilinear Oscillators under Pulse-Type Excitations

In this paper the response of a bilinear oscillator subjected to pulse-type motions is revisited with dimensional analysis. Using Buckingham’s Π theorem the number of variables in the response analysis is reduced from six (6) to four (4). When the response is presented in terms of dimensionless Π terms remarkable order emerges. It is shown that for a given value of dimensionless strength and dimensionless yield displacement, the response (relative dimensionless displacements and dimensionless base shears) is self-similar regardless of the intensity and duration of the pulse excitation. These self-similar solutions scale better with the peak pulse acceleration rather than with the peak pulse velocity, indicating that peak pulse acceleration is a superior intensity measure of the induced shaking. Most importantly, the paper demonstrates that for relatively small values of strength (larger values of ductility) the value of the normalized yield displacement is immaterial in the response, a finding that shows that the response of the bilinear single-degree-of-freedom oscillator exhibits a complete similarity (similarity of the first kind) in the normalized yield displacement. This finding implies that under a strong earthquake an isolated bridge will exhibit the same maximum displacement regardless if it is supported on lead-rubber bearings or friction pendulum bearings that exhibit the same strength and offer the same isolation period.