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Conditional stability of solitary waves propagating in elastic rods. (English) Zbl 0899.73282
Summary: The problem of dynamic stability of solitary wave solutions of the equations describing bending oscillations of inextensible elastic rods is treated. The governing equations describe sufficiently large displacements (though we are restricted to small strains). We show that under the condition of well-posedness of the initial value problem (in some specific sense) the family of solitary wave solutions of the mentioned equations is stable for the planar elastic rod. The framework of the analysis is largely based on the spectral properties of the ”linearized Hamiltonian” H. We show that for planar oscillations H is positively semidefinite subject to a certain constraint, which implies the stability.

MSC:
74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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