ИСТИНА

A projection approach to modelling of natural vibrations for 3D elastic beams is proposed based on polynomial semi-discretization of displacement and stress fields. This projection technique is developed as modification of the Petrov-Galerkin method in the frame of the method of integro-differential relations (MIDR), in which the local constitutive laws are replaced by an integral equation. As a result, the original eigenvalue problem in partial differential equations is approximated by a system of ordinary differential equations (ODEs). The approximations include a polynomial expansion of finite dimension over two coordinate components and unknown functions over one remaining component. As an example of thick beams, a homogeneous isotropic elastic body occupying the domain of a right prism with a quadratic cross section is under study. The lateral faces of the prism are free of external loads, whereas the displacements are fixed and equal to zero on its bases. Due to the problem symmetry, natural vibrations of the prismatic beam are decomposed into six independent groups of eigenmodes, namely, two types of “breathing”' motions, two types of lateral vibrations, torsions, and longitudinal vibrations. Spectrum characteristics of the beam and their specific properties caused by the domain symmetry and beam thickness are discussed.