The non-dimensional response spectra of impact oscillators subjected to pulse-type base excitation

The oscillation due to vibration results in impact of the structures if the at-rest separation is insufficient to accommodate the relative displacements. The impact between the floors of the two adjacent buildings or the impact between adjacent decks and abutments is modeled by an impact oscillator subjected to closed-form mathematical formulations of near-fault earthquake motions as pulse-type excitation, such as cosine (±) and sinusoidal pulses (±). This study investigates the impacting responses of an impact oscillator against a barrier from an analytical perspective. The momentum-based stereo-mechanical method is used as a contact force-based model to perform the impact analysis. The equation of the motion of the impact oscillator under lateral pulse-type base excitation is non-dimensionalized by two dimensionless parameters using the standard process of non-dimensionalization of a differential equation. The system has been analyzed for the complete forced vibration part as it was excited by a single cycle of cosine and sinusoidal pulses. However, the transient solutions are also derived for the analysis. Therefore, the free vibration phase is determined until the impactor comes to a non-impacting state. The steady-state with the transient solutions for the complete impacting responses are provided for the exact solutions of the dynamic responses of the impact oscillator, such as displacement response spectra, velocity response spectra. The governing system parameters, such as damping ratio, frequency ratio of the impact oscillator, the coefficient of restitution, and the direction of the base excitation ( ±) are the governing system parameters to control the impact between the oscillator and barrier. A frequency point under resonance region from each response spectra of impact oscillator subjected to cosine ( ±) and sinusoidal pulses ( ±) is subtracted to define the non-dimensional time domain responses. The exact impact determines in out-of-phase state and the dynamic responses of impact oscillator are attenuated. The non-dimensional relative distance between the impacting oscillator and the barrier varies from 0.5 to 3.0. However, for δ~>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\delta } > 3$$\end{document}, no successive impact obtains. Accordingly, the impacting oscillator’s dynamic response spectra closely match the linear oscillator’s dynamic response spectra. All the results are mathematically derived and accurate for practical applications of a damped impact oscillator (i.e., precisely for single-degree-of-freedom systems) to determine the non-dimensional dynamic responses.