Low-frequency vibrations of coated elastic structures

Abstract

The thesis deals with 1D and 2D scalar equations governing the dynamic behaviour of coated elastic structures. Low-frequency vibration of a composite rod, beam, rectangular plate and circular plate are studied. The main focus is on physical effects that occur in composite elastic structures with a thin coating. We start with two auxiliary 1D problems for two-component rods and beams.
Then elastic waves localised near the edge of a semi infinite plate reinforced by a strip plate are considered within the framework of the 2D classical Kirchhoff theory for plate bending. The boundary value problem for the strip plate is subject to an asymptotic analysis assuming that a typical wave length is much greater the strip thickness. As a result, effective conditions along the interface, corresponding to a plate reinforced by a beam with a narrow rectangular cross-section, are established. They support an approximate dispersion relation perturbed from that for the homogeneous plate with a free edge. The accuracy of the approximate dispersion relation is tested by comparison with the numerical data obtained from the 'exact' matrix relation for a composite plate. The effect of the problem parameters on the localisation rate is studied.
In addition, edge bending waves on a thin isotropic semi infinite plate reinforced by a beam are considered within the framework of the classical plate and beam theories. The boundary conditions at the plate edge incorporate both dynamic bending and twisting of the beam. A dispersion relation is derived along with its long-wave approximation. The effect of the problem parameters on the cut-off frequencies of the wave in question is studied asymptotically. The obtained results are compared with calculations for the case when the reinforcement takes the form of a plate strip.
Finally, a circular plate reinforced by a thin annular strip of the same thickness is considered. Asymptotic treatment of a strip circular plate with a free outer edge and its inner edge subject to prescribed de ection and rotation is presented. The effective boundary conditions are derived, and approximate dispersion relation is deduced.