Vibration analysis of an extremely flexible beam with concentrated load axially stressed by incremental force

Resumo

Throughout human history, cables have been essential for the development of various tools. In prehistoric times, they helped with hunting, fishing and harvesting; during the Portuguese caravel period, they were fundamental in the structural design of ships; in cable-stayed bridges, they became exceptional elements for overcoming large spans; and in prestressed beams, they made it possible to reduce the cross-section. The modern world is driven by information and communication, and overhead cables are responsible for conducting electricity and the internet, making them indispensable in the current globalized world. Overhead cables can be calculated in a manner analogous to an extremely flexible two-supported beam using the approximate elastic line, where the cables only offer structural resistance when subjected to axial tensile forces. Therefore, there is a mobilization of geometric rigidity in the total rigidity of the system, causing a modification in the natural frequency of vibration, in which structural equilibrium can only be achieved in the deformed configuration of the system. Therefore, numerical methods are desirable, since the cable tensioning process involves the incremental application of force until the total load is reached. It is essential to determine the natural vibration frequency of the first mode, since it represents the shape of the transmission line when subjected to the Earth's gravitational field. The overhead cable is subjected to various weather conditions, together with the characteristic of being a slender structure by nature. As a consequence of these aspects, the cables are vulnerable to dynamic problems. A viable alternative to solve this issue is transmission lines with concentrated loads along the cable, dividing the transmission line into excerpt. This helps to avoid structural resonance, a phenomenon that can occur when the frequency of the external load coincides with the natural vibration frequency of the structure. To implement a numerical solution, a computer program in Python was developed that allows determining the shape of the transmission line at each increment of axial load, using three concentrated loads along the span, in comparison to a transmission line without concentrated loads. The approximate elastic line equation was used to define the respective shapes of each iteration. Furthermore, the Rayleigh method was used to determine the natural frequency of the structure.