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Shock and Vibration
Volume 20 (2013), Issue 6, Pages 1073-1092

Influence of Axial Loads on the Nonplanar Vibrations of Cantilever Beams

Eulher C. Carvalho,1 Paulo B. Gonçalves,1 Giuseppe Rega,2 and Zenón J.G.N. Del Prado3

1Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil
2Sapienza University of Rome, Rome, Italy
3Federal University of Goiás, Goiânia, GO, Brazil

Received 8 October 2013; Accepted 8 October 2013

Copyright © 2013 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In this paper an inextensible cantilever beam subject to a concentrated axial load and a lateral harmonic excitation is investigated. Special attention is given to the effect of the axial load on the frequency-amplitude relation, bifurcations and instabilities of the beam. To this aim, the nonlinear integro-differential equations describing the flexural-flexural-torsional coupling of the beam are used, together with the Galerkin method, to obtain a set of discretized equations of motion, which are in turn solved by using the Runge-Kutta method. Both inertial and geometric nonlinearities are considered in the present analysis. Due to symmetries of the beam cross section, the beam exhibits a 1:1 internal resonance which has an important role on the nonlinear oscillations and bifurcation scenario. The results show that the axial load influences the stiffness of the beam changing its nonlinear behavior from hardening to softening. A detailed parametric analysis using several tools of nonlinear dynamics unveils the complex dynamic behavior of the beam in the parametric and external resonance regions. Bifurcations leading to multiple coexisting solutions are observed.