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Journal of Mathematical Sciences

, Volume 243, Issue 1, pp 63–72 | Cite as

Generalization of the Cauchy–Poisson Method and the Construction of Timoshenko-Type Equations

  • I. T. Selezov
Article
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We consider a generalization of the Cauchy–Poisson method to the n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. The presented investigation generalizes and supplements the results obtained earlier. In the Euclidean space, we introduce certain restrictions for the derivatives. The principle of hyperbolic degeneracy in terms of parameters is formulated and its realization in the form of necessary and sufficient conditions is presented. In a special case of fourdimensional space (in which the operators are preserved up to the sixth order), we obtain a generalized hyperbolic equation for the bending vibrations of plates with coefficients that depend only on the Poisson ratio. This equation includes, as special cases, the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As the development of Maxwell's and Einstein's investigations of the propagation of perturbations with finite velocity in continuous media, we can mention the nontrivial construction of Timoshenko’s equation for the bending vibrations of a beam.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • I. T. Selezov
    • 1
  1. 1.Institute of Hydromechanics, Ukrainian National Academy of SciencesKievUkraine

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