Vibration of extremely flexible beam axially tensioned by incremental force

Resumo

Cables can be associated with extremely flexible beams. Particularly, electrical system cables supported on transmission towers are subject to climatic conditions whose effects are capable of mobilizing their vibration modes. The first mode is considered to be of special importance due to the shape assumed by the cable when subjected to the Earth's gravitational field. Under these conditions, cable deformation is influenced by its geometric and material properties, including viscoelastic behavior, when considered, making these systems intrinsically nonlinear in terms of geometry and material. The balance of these structural parts can only be found in the deformed configuration of the system. As a rule, cables need to be pulled for proper and correct use in transmission lines. The use of a traction force modifies the rigidity of the system, causing an increase in the natural frequencies of vibration. This occurs because the force that pulls the cable mobilizes the geometric stiffness portion of the total system stiffness. In solving the vibration problem, numerical methods can be employed. To this end, the model adopted is analogous to a double-supported beam, for which the successive integration of the bending moment differential equation leads to displacements of the axis, or the elastic line, for each stage of the iterative process. These iterations are repeated until the sum of the portions equals the total axial force to be applied. The smaller the force portion in each iteration, the better the result obtained. However, the number of iterations required will be greater, making processing potentially costly from a computational point of view. To solve the proposed problem, a programming routine was developed in the Python Jupyter Notebook language that allows calculating the deformation of the structural system and the natural frequency of vibration in each iteration. The process begins with the analytical solution of the approximate elastic line under the effect of gravity and the subsequent tension in the cable, with the definition of the natural frequency of vibration following the Rayleigh method. In the end, the first natural vibration frequency of the cable was raised non-linearly from 0.358 Hz to 5.944 Hz.