In this paper we study eigenvalue problems for fourth-order ordinary differential equations. These boundary problems usually describe the bending vibrations of a homogeneous beam. Our aim here is to present mixed variational forms depending on a wide class of boundary conditions. In particular, we show when the symmetry in variational formulations is available. This property ensures real spectrum of the corresponding problem. The effect of the theoretical results is illustrated by some realistic examples.
Author: Andreev A.
THEORETICAL FOUNDATIONS AND SPECIFICITY OF MATHEMATICAL MODELLING
ON THE USE OF CONFORMING AND NONCONFORMING RECTANGULAR FINITE ELEMENTS FOR EIGENVALUE APPROXIMATIONS
The paper deals with some combinations of conforming and nonconforming rectangular finite elements in order to obtain twosided bounds of eigenvalues, applied to second-order elliptic opertor. The aim is to use the lowest possible order finite elements. Namely, the combination of serendipity conforming and rotated bilinear nonconforming elements is considered in details. This work continues some recent researches of the authors concerning eigenvalue approximations. Computational aspects of the used algorithm are also discussed. Finally, results from numerical experiments are presented.
In this paper a new algorithm and numerical approach which gives two-sided approximations of eigenvalues for second-order problem is presented. Conforming finite element methods are used in combination with an appropriate nonconforming interpolation. Numerical Aspects are discussed and also experiments which demonstrate the proposed algorithm are given.
TECHNOLOGICAL BASIS OF “INDUSTRY 4.0”
The paper deals with a shaft subjected to an axial pressure with force P and relatively high moment of rotation M. In order to determne the lines of deflection we propose various approaches. The assessment of critical values of P and M is also discussed when loss of stability is available. Thus two fourth-order eigenvalue problems are considered and their mixed variational models are proposed. Finally, some numerical experimental results are presented.