ON THE NONLINEAR TIMOSHENKO-KIRCHHOFF BEAM EQUATION

Abstract

When an elastic string with fixed ends is subjected to transverse vibrations, its length varies with the time: this introduces changes of the tension in the string. This induced Kirchhoff to propose a nonlinear correction of the classical D'Alembert equation. Later on, Woinowsky-Krieger (Nash & Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends. Here a new equation for the small transverse vibrations of a simply supported beam is proposed. Such equation takes into account Kirchhoff's correction, as well as the correction for rotary inertia of the cross section of the beam and the influence of shearing strains, already present in the Timoshenko beam equation (cf. the Mindlin-Timoshenko equation for the plate). The model is inspired by a remark of Rayleigh, and by a joint paper with Panizzi & Paoli. It looks more complicated than the one proposed by Sapir & Reiss, but as a matter of fact it is easier to study, if a suitable change of variables is performed. The author proves the local well-posedness of the initial-boundary value problem in Sobolev spaces of order ≥ 2.5. The technique is abstract, i.e. the equation is rewritten as a fourth order evolution equation in Hilbert space (thus the results could be applied also to the formally analogous equation for the plate).