Dynamic Response of a MDOF System subjected to Harmonic and Impulsive Loadings and Free Vibration: An Analytical Approach

Abstract

The influence of different kinds of dynamic forces on multistory structures is notable. This paper studies the effect of harmonic and impulsive loading and free-vibration on a three-story shear frame structure. A persistent pulse loading was considered at the upper level of the structure, and the resulted mathematical model of the time-dependent multi-level displacements were derived. For harmonic force, normalized response amplitudes under the applied harmonic loading is plotted against the frequency ratio 𝜔/𝜔1. These frequency-response curves show three resonance conditions at 𝜔=𝜔1, 𝜔=𝜔2 and 𝜔=𝜔3; at these exciting frequencies, the steady-state response is unbounded. At other exciting frequencies, the vibration is finite and could be calculated from the derived equations. When the structure is excited with harmonic loading over a range of frequencies, the structure experiences resonance at some frequency. Resonance occurs when the frequency of the excitation is equal to the natural frequency of the structure. At the resonant frequency, the structure experiences its largest response as compared to any other frequency of loading. For rectangular pulse force, the time-history of the multi-level displacements were presented. The results indicate that the extreme displacement occurs at the top-level. Additionally, when the persistent pulse loading had been expired, the maximum top-level displacement response was obtained. As the impact of the preliminary conditions is essential, the final response of the structure to the pulse loading is not being “steady-state. For free vibration, floor displacements versus t /T1 was plotted and observed the relative contributions of the three vibration modes to the response that was produced by the given initial displacement. Although all three modes contribute to the response, the dominant response is due to the first mode since the shape of the given initial displacement is similar to the configuration of the first mode.