Keywords: non-linear vibrations, curved beams, p-version finite element.

The large amplitude oscillations of curved beams are represented by differential equations with quadratic and cubic non-linearities, due to the combined effects of the initial curvature and of the large displacements. Moreover, transverse excitations of curved beams cause longitudinal motions and possibly buckling phenomena. Therefore, one expects quite interesting dynamics in this problem.

In this paper, the -version, hierarchical finite element method [1] is employed to investigate the large amplitude motions of isotropic, linear elastic, curved beams. Timoshenko's model, where it is assumed that the transverse shear deformation is constant along the cross section, is followed. Equating the sum of the virtual works of all forces - including the inertia forces - to zero, and assuming stiffness proportional damping, one derives equations of motion of the form:

Matrices of the type and are constant, matrices of the types and depend linearly on the unknown generalised transverse displacements, and matrix depends quadratically on these displacements. The superscripts , , , , represent, respectively, longitudinal, bending, rotational, initial curvature and shear effects. The in-plane and rotational inertia are included in the model.

As an example, a curved beam is analysed with some detail. The time domain, non-linear, second order differential equations of motion are solved by the shooting method [2] or by Newmark's scheme [3]. The motions due to the harmonic excitations applied at the beam's centre, are always periodic, but at some excitation frequencies they are far from harmonic.

By inspecting the shapes assumed by the beam along each vibration period, for several excitation frequencies close to the first and second natural frequencies, it was verified that at low vibration amplitudes, the shape is similar to the second mode of vibration (first symmetric mode, with respect to a vertical axis); however, the importance of the fourth mode (second symmetric mode) grows with the vibration amplitude, and it becomes essential at some frequencies of excitation. In this case, unstable, far from harmonic motions may appear.

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, 591-624, 1999. doi:10.1006/jsvi.1999.2193

2

A.H. Nayfeh, B. Balachandram, "Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods". John Wiley and Sons, New York, 1995.

3

M. Petyt, "Introduction to Finite Element Vibration Analysis". Cambridge University Press, Cambridge, 1990.

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