Non-linear free vibration of a cable against a straight obstacle

The goal of this study was to show how the response of a cable was altered after it impacted a straight obstacle. The two ends of the cable were held at the same level, and the cable was initially vibrating at its first symmetric mode. The free vibration of a cable depends on the amplitude of vibration, the sag-to-span ratio, and the ratio of the elastic to geometric stiffness as measured by Irvine and Caugheyʹs cable parameter λ2. How these variables affected the response of a cable interacting with a straight obstacle was examined for three limiting cases: a nearly taut string, an extensible shallow-sag cable, and an inextensible deep-sag cable. For a nearly taut string interacting with a straight obstacle in the absence of gravity, the periodicity that occurs for a linear taut string was preserved even when the amplitude of vibration was large. However, the frequency of vibration was modified due to the non-linear effects of large amplitudes. The periodicity was lost if the cable was not taut both in the presence and in the absence of gravity. For the case of a nearly taut string with gravity, the periodicity could be recovered if the obstacle assumed the shape of the equilibrium configuration of the cable. This was because the obstacle no longer behaved as a straight obstacle but as a convex obstacle whose radius of curvature decreased with decreasing amplitude-to-sag ratio. The behavior approached that of a point obstacle in the limit. Shallow-sag cables were analyzed by keeping the amplitude-to-sag ratio constant so that the obstacle behaved in a consistent manner. Comparison of our findings with the results of Irvine and Caughey for the free vibrations of a cable in air showed that the relative frequency based on the time to the first contact was not affected by the presence of the obstacle for all values of λ2. However, the relative frequency based on the time between the first two contacts was slightly reduced when compared to the values corresponding to a non-interacting cable. This was attributed to the longer persistent contact that developed between the cable and the obstacle. Similar results were found for an inextensible deep-sag cable with various sag-to-span ratios. In both cases of shallow- and deep-sag cables, the ratio of the initial amplitude to the distance between the cable and the obstacle had a minimal effect on the reduction in the relative frequency.