SYMMETRY EQUIVALENCES OF BOUNDARY VALUE PROBLEMS FOR THE NON-UNIFORM BEAMS

Abstract

In this paper, the models of Euler–Bernoulli non-uniform beams with the axial loads on the Winkler foundations are considered. The non-uniform beam in the model is described by three variable parameters/coefficients: bending stiffness, foundation and beam mass per unit length. The key finding of this study is the clear demonstration of how the agreed symmetry of variable parameters affects the spectral properties of a problem. The qualitative results for the symmetric equivalence (factorisation of sets of eigenvalues and eigenfunctions) of eigenvalues of non-uniform beams for two types of fixing at the ends (clamped-clamped and hinged-hinged) have been obtained. In order to demonstrate equivalence, a hybrid algorithm has been devised, based on the qualitative spectral properties of fourth-order ordinary differential equations and axial load calculations. The results have been validated using examples on the Maple computer package and compared with the experimental measurements.