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  4. Generalization and application of the Cauchy-Poisson method to elastodynamics of a layer and the Timoshenko equation

Generalization and application of the Cauchy-Poisson method to elastodynamics of a layer and the Timoshenko equation

MMC.
2018;
Volume 5, Number 1
: pp. 88-97
https://doi.org/10.23939/mmc2018.01.088
Received: March 21, 2018

Math. Model. Comput. Vol. 5, No. 1, pp. 88-97 (2018)

Authors:
  1. I. Selezov
1
Institute of Hydromechanics, NASU

The Cauchy-Poisson method is extended to n-dimensional Euclidean space so that to obtain partial differential equations (PDEs) of a higher order.   The application in the construction of hyperbolic approximations is presented, generalizing and supplementing the previous investigations.   Restrictions on derivatives in Euclidean space are introduced.  The hyperbolic degeneracy by parameters and its realization in the form of necessary and sufficient conditions are considered.  As a particular case of 4-dimensional Euclidean space, keeping operators up to the 6th order, we obtain a generalized hyperbolic equation of transverse (bending) vibrations of plates with coefficients depending only on the Poisson number.  Numerical calculations are carried out and presented.  This equation includes, as special cases, all the known equations of Bernoulli-Euler, Kirchhoff, Rayleigh, Timoshenko.   It should be noted that the refined equation of bending oscillations of a beam, firstly presented by Timoshenko, must be considered as the development of Maxwell's and Einstein's investigations on the perturbation propagation with finite velocity in media.  For the first time, the conformity with the Cosserat theory is noted.

Cauchy-Poisson method
Euclidean space
partial differential equation (PDE)
elastodynamics
layer
hyperbolic approximations
Timoshenko equation
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