Eventos Anais de eventos
COBEM 2023
27th International Congress of Mechanical Engineering
ON THE EVALUATION OF NATURAL FREQUENCIES AND MODE SHAPES OF BEAMS UNDER TENSILE AXIAL LOAD
Submission Author:
Kevin Mauricio Menon Ribeiro , PR , Brazil
Co-Authors:
Kevin Mauricio Menon Ribeiro, Gabriel do Amaral, Eduardo Márcio de Oliveira Lopes, José Manoel Balthazar, Alexandre de Macêdo Wahrhaftig
Presenter: Kevin Mauricio Menon Ribeiro
doi://10.26678/ABCM.COBEM2023.COB2023-0142
Abstract
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. Analytical solutions are classical mathematical procedures. Traditionally, the path taken in analytical solutions starts by admitting an acceptable kinematic hypothesis for a given problem, so that, in the end, a closed mathematical equation can be obtained. Sometimes the problem can be implemented and solved numerically. A practical example of its application can be found in problems involving overhead transmission line cables, for which the corresponding Euler-Bernoulli beam model is broadly employed. Although the analytical formulation for calculating natural frequencies and mode shapes of vibration for beams under tensile load 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 general, in the vast majority of studies, 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 their supports. On the other hand, numerical simulations indicate that more accurate and reliable solutions can be obtained when the analytical solution is expressed by a linear combination of trigonometric and exponential terms. In this context, the present work aims to compare the two aforementioned ways of describing the analytical solution for calculating natural frequencies and their respective mode shapes of vibration, regarding overhead transmission line cables. 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 presented formulation, especially regarding a more accurate prediction of the mode shapes for the type of structure of concern.
Keywords
Analytical Solution, axial loading, Euler-Bernoulli beam, Mode shapes, overhead cables
