Determination of attaching parameters of inhomogeneous beams in the presence of damping

A. O. Vatulyan ; L. V. Vasiliev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 4, pp. 449-456

Résumé

Characterization of solids by additional data on displacements amplitudes or resonance frequencies have been increasingly attracting attention of researchers in recent years. Among the tasks of this type, the problems associated with definition of parameters describing boundary conditions and characterizing an interaction of the body studied with the surrounding bodies are of particular interest. In this paper, we investigate the problem of determining the parameters of the boundary conditions in a beam. We propose a new approach to solve the inverse problem of a reconstruction of the bearing parameters of an inhomogeneous viscoelastic beam with viscoelastic bonds on the right end and being fixed at the left end based on the analysis of the amplitude and shift phase of the displacement at two points at a fixed frequency. We have used the principle of conformity to derive the differential equation of oscillations based on the standard model of viscoelastic body. We present a way of reduction of the original problem to the canonical form. We have formulated the auxiliary Cauchy problems for a numerical solution of both direct and inverse problems by the false position method. Within the framework of the present model, we have performed the computational experiments to restore 4 parameters characterizing the viscoelastic bonds in the boundary conditions. We have analyzed the influence of changes in the parameters on the resonant frequency and on the displacements amplitude. The influence of the input data noise on the reconstruction of the desired parameters is investigated. It is revealed that the method proposed for the reconstruction of the unknown parameters can be employed in order to their retrieval in the boundary conditions with high accuracy.