5. On the Evaluation of Natural Frequencies and Mode Shapes of Beams Under Tensile Axial Loading

The accurate prediction of lateral vibration in beams subjected to axial loads is of great practical interest due to its wide use in describing the dynamic behavior of structures in the field of civil, mechanical, and aerospace engineering. A distinct example of its application can be found in problems involving overhead cables of transmission lines, for which the corresponding Euler–Bernoulli beam model is widely used. Although the analytical formulation for calculating natural frequencies and mode shapes for beams under tensile axial loading is well known and developed, it is worth noting that in certain applications, such as in the case of overhead cables, the results obtained may exhibit numerical instabilities for some boundary conditions. In the vast majority of studies, the mode shapes are described in terms of trigonometric and hyperbolic functions, despite the fact that hyperbolic functions tend to grow rapidly to infinity as their argument increases, which can cause numerical errors, especially in the vicinity of the beam supports. On the other hand, numerical simulations indicate that more accurate and reliable results can be obtained when the analytical solutions are expressed by linear combinations of trigonometric and exponential terms. In this context, the present work aims to compare the two aforementioned ways of describing the analytical solutions for calculating natural frequencies and their respective mode shapes of vibration with respect to overhead cables of transmission lines. In this effort solutions are developed for two classical boundary conditions (fixed–fixed and fixed-simply supported) and one non-classical boundary condition (simply supported with torsional springs). Numerical simulations are carried out in the MATLAB® environment. A significant gain is observed in the use of the preferred formulation, especially in terms of a more accurate prediction of mode shapes for the type of structure under consideration.