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Homoclinic and heteroclinic bifurcations in the non-linear dynamics of a beam resting on an elastic substrate. (English) Zbl 1342.74017
Summary: The non-linear dynamics of a slender “elastica”, fixed at its base and free at the top, resting on an elastic substrate, axially loaded and subjected to periodic excitation, has been analyzed. Taking into account the non-linear inertial terms, the single-mode dynamics of the systems is governed by a Duffing equation with fifth-order non-linearities. In the considered range of parameters, two qualitatively different phase portraits exist. When the axial load p is less than the Eulerian critical value, there are three centers and two saddles (with the related stable and unstable manifolds). After the pitchfork bifurcation, the two saddles and the middle center coalesce in an unique new saddle which has a pair of symmetric homoclinic solutions. Melnikov criteria on the chaotic dynamics of the system are derived on the basis of analytical expressions for the homoclinic and the heteroclinic orbits. They involve transverse intersections of the stable and unstable manifolds that represent the starting point for a subsequent route to a chaotic dynamics. Numerical simulations which aim to show some effects of the global bifurcations on the actual dynamics are presented.

74B20 Nonlinear elasticity
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
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