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

Abstract

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.

Geometric Recursive Dynamic Modelling and Simulation of Soft Robotic Systems

“…Multiple extensions on FEM exist to help expedite solutions and integrate machine learning but they are still more computationally expensive than FDM [118]. Perturbation theory has been used to extend the solutions from known PDEs to those for a Cosserat rod, but they are limited to small perturbations which nullify the nonlinear model to capture finite deformations [119]. Data driven models developed using machine learning offer an alternative to the derived models, but require significant training and cannot operate well in a new unknown environment if it was trained on different case [27,28].…”