| A. AROSIO.[J].数学年刊A辑,1999,20(4):495~506 |
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| ON THE NONLINEAR TIMOSHENKO-KIRCHHOFF BEAM EQUATION |
| Received:February 17, 1997 Revised:October 22, 1998 |
| DOI： |
| 中文关键词: |
| 英文关键词:Timoshenko-Kirchhiff beam equation, Local well-posedness,
Fourth order evolution equation. |
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| 中文摘要: |
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| 英文摘要: |
| 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 $\geq 2.5$. 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). |
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