Keywords: non-linear, free vibrations, beams, torsion, bending, harmonic balance, finite element, p-version.

The finite element method is a powerful numerical tool, and, particularly in non-linear dynamics, the -version of the finite element method has important advantages over the -version [1]. There are some publications devoted to the analysis of non-linear vibrations of beams in a plane [1] and in space [2]. In these references, the efficiency of the p elements in geometrically non-linear problems was shown. However, the three dimensional study carried out in [2] was in the time domain and limited to steady-state vibrations under harmonic excitations. Here, we intend to continue the former work with a free vibration analysis in the frequency domain.

The beam, -version, hierarchical finite element employed allows one to model longitudinal, torsional, and bending vibrations in any plane. Clamped-clamped, isotropic and elastic beams of circular cross section are considered.

The theory of Bernoulli-Euler for bending and the Saint-Venant theory for torsion are followed. The geometrical non-linearity is taken into account by considering a simplified version of Green's strain tensor, which ought to be valid for small displacements in the longitudinal direction and for moderate rotation angles. The mass and stiffness matrices are derived by the principle of the virtual work, and the longitudinal inertia is neglected, thus reducing the number of degrees of freedom of the model.

The linear natural frequencies in bending and torsion of a particular beam are computed. It is shown that convergence to the exact values is achieved with a reduced number of degrees of freedom.

The harmonic balance method is employed to map the non-linear equations of motion into the frequency domain. It is assumed that the generalised displacements can be represented by a Fourier series where the constant term and the first three harmonics are considered. The ensuing algebraic non-linear system of equations is solved by a continuation, predictor-corrector procedure [1,3].

It is known that internal resonances may occur when the natural frequencies are commensurable, i.e., when they are related by an equation of the type: , where are integers. In this case, due to the non-linearity, a mode of vibration can be excited by another [4].

For large vibration amplitudes in bending and for small vibration amplitudes in torsion, internal resonances are found in this study. The variations of the bending and torsional non-linear natural frequencies and modes of vibration are investigated.

Although the equations of motion present quadratic and cubic non-linear terms, it turned out that the constant term and the second harmonic did not interfere in the motion, because no coupling was found between flexure and torsion, or between flexure in different planes.

1

P. Ribeiro, M. Petyt, "Non-Linear Vibration of Beams with Internal Resonance by the Hierarchical Finite Element Method", Journal of Sound and Vibration, 224 (4), 591-624, 1999. doi:10.1006/jsvi.1999.2193

2

J.R. Fonseca, P. Ribeiro, "Beam Hierarchical Finite Element for Geometrically Non-linear Vibrations in Space", Fifth World Congress on Computational Mechanics (WCCM V), July 7-12, 2002, Vienna, Austria, H.A. Mang, F.G. Rammerstorfer, J. Eberhardsteiner (Editors) Vienna University of Technology, Austria, http://wccm.tuwien.ac.at.

3

R. Seydel, "Pratical Bifurcation and Stability Analysis: From Equilibrium to Chaos", Springer Verlag, New York, 1994.

4

W. Szemplinska-Stupnicka, "The Behaviour of Non-Linear Vibrating Systems", Kluwer Academic Publishers, Dordretch, 1990.

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