Study of cable vibration by large deformations using Rayleigh and Runge-Kutta methods.

Resumo

The hypothesis of large deformations is a working condition assumed for restricted structural systems, and it finds specific applications in engineering practice. Problems involving cables are found in this context. The study of cables is a relevant topic for the Brazilian and global industry because they are essential elements for modern life. In particular, the study of their vibration arouses special interest because cables are used, among numerous applications, for the transmission of electrical energy. In this condition, they are subject to environmental excitations, which can represent a source of resonance. The vibration of a cable always occurs in the deformed configuration of the system, therefore under the influence of the loadings. Two are the loadings to which cables are usually subjected. The first is its own weight, the second is an axial force that allows the system to exist and keeps the cable tensioned in its working condition. Even with an applied tensile force, systems composed of cables can exhibit large deformations depending on the geometric and materials conditions to which they are subjected to. The tensile force modifies the stiffness of the system and, consequently, its natural vibration frequencies. In the context of what has been described, the present work aims to study the vibration of a structural system formed by a cable, assuming the hypothesis of large deformations. The adopted model will be that of a simply supported beam at its ends and subjected to both loads, one uniformly distributed due to the self-weight of the structural element, and a tensile force, which will be applied in conjunction with the first one, under the condition assumed for the deformed state of the cable. The analytical method to be employed for determining the vibration frequency is based on the Rayleigh method, for which the shape function will be obtained numerically by solving the exact elastic line of the beam model. For this, the differential equations that characterize that hypothesis will be solved using the Runge-Kutta method for the boundary conditions of the problem, in the Mathcad environment.