Occurrence phase of peak responses to symmetric pulse loads

This study fills a challenging gap in the field of structural dynamics. A potential rule is theoretically proved: The peak displacements of undamped single-degree-of-freedom (SDOF) systems subjected to nonnegative but symmetric pulse loads necessarily occur within the pulse loading duration if the frequency ratio $\beta <$1, and after the pulse loading duration if the frequency ratio $\beta$>1. As a special case, the first peak displacements accurately take place at the end of the pulse loading when $\beta$=1. Also, the occurrence time of the first peak displacements has a theoretic value of $t_{\varphi }$=$t_p$/2+$T$/4 in the case of $\beta$>1. Although this potential rule can be easily verified in certain cases, it has not been theoretically and systematically proved so far. A rigorous and complete proof is presented and featured by the proposed analysis based on Duhamel's integral. The analyzation circumvents the difficulties in analytically solving dynamic responses to different pulse loads in different shapes, but still reaches theoretical conclusions and yields a general law of structural dynamics. The proved law can be used to predict the occurrence phase of the first peak displacements when undamped SDOF systems subjected to nonnegative but symmetric pulse loads.