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Riemann’s Formulae and Their Application to the Timoshenko Beam Model

  • J. T. Schmidt
Part of the Lecture Notes in Engineering book series (LNENG, volume 28)

Abstract

In this paper we consider the transverse vibrations of an infinite TIMOSHENKO beam. The hyperbolic type of the governing partial differential equations allows the application of a theory first developed by RIEMANN and later enhanced by HAACK and HELLWIG. In this theory, the solution of arbitrary initial value problems for hyperbolic systems is reduced to simple quadratures involving the initial data and a set of so-called adjoint functions, not dependening on the particular initial data. In the case of the string, this theory yields. the well-known d’ALEMBERT formula, and an appliation to the string with elastic bedding is given in /4/. In our paper, we determine the adjoint functions for the TIMOSHENKO beam and give a first application of the formulae to the study of wave propagation.

Keywords

Initial Data Phase Velocity Wave Mode Travel Wave Solution TIMOSHENKO Beam 
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References

  1. /1/.
    Hawk, W. and Hellwig, G., Ober Systeme hyperbolischer Differentialgleichungen erster Ordnung. I, Mathematische Zeitschrift, Band 53, Heft 3, 1950, S. 244–266.Google Scholar
  2. /2/.
    Haack, W. and Hellwig, G., Ober Systeme hyperbolischer Differentialgleichungen erster Ordnung. II, Mathematische Zeitschrift, Band 53, Heft 4, 1950, S. 340–356Google Scholar
  3. /3/.
    / Hellwig, G., Ober Systeme hyperbolischer Differentialgleichungen erster Ordnung III, Mathematische Zeitschrift, Band 68, 1958, S. 325–337Google Scholar
  4. /4/.
    Hellwig, G., Partial Differential Equations, B.G. Teubner, Stuttgart, 1977Google Scholar
  5. /5/.
    Pontryagin, L.S., Ordinary Differential Equations, Addison-Weseley Publishing Company Inc.,1962Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1987

Authors and Affiliations

  • J. T. Schmidt
    • 1
  1. 1.TH DarmstadtInstitut für Mechanik IIDarmstadtDeutschland

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