This paper presents the investigation of the nonlinear eigenvalue problem describing free vibrations of plates with elastically attached masses. We study properties of eigenvalues and eigenfunctions and prove the existence theorem. Theoretical results are illustrated by numerical experiments.

This paper presents the investigation of the nonlinear eigenvalue problem describing free vibrations of plates with elastically attached masses. We study properties of eigenvalues and eigenfunctions and prove the existence theorem. Theoretical results are illustrated by numerical experiments.

[pt] Neste trabalho é apresentado um método numérico de resolução para a equação diferencial de placas: o método de Galerkin Iterativo. O método é utilizado para obtenção das cargas críticas de flambagem e das freqüências naturais para placas retangulares com condições de contorno arbitrárias. São determinados ainda os modos de vibração de placas para diversas condições de contorno. É também apresentada uma análise do comportamento estático e dinâmico de placas planas retangulares. Utilizando-se dos resultados obtidos nesta análise e do método de Galerkin Iterativo, analisa- se a influência dos carregamentos axiais sobre as propriedades de vibração de uma placa com diversas condições de contorno, como proposta de um meio de controle passivo de vibrações em placas retangulares. Realiza-se uma análise linear para o carregamento no plano médio da placa e outra não-linear no caso de placas submetidas a carregamentos excêntricos, ou seja, fora do plano médio da placa. Mostra-se que o método de Galerkin Iterativo permite a obtenção de modos de vibração ortogonais possibilitando a resolução de problemas dinâmicos através do método de superposição de modos. Além disso, mostra-se que o método de controle passivo de vibrações em placas, através da aplicação de forças de compressão no plano, reduz a amplitude da resposta na região de ressonância. / [en] The aim of this work is to present a procedure for the solution of differential equations for plates: the Iterative Galerkin method. With the aid of this method, the buckling loads and natural frequencies of plates are obtained for plates with arbitrary sets of boundary conditions. The vibration modes of plates with various boundary conditions are obtained and compared with results found in literature. An analysis of the static and dynamic behavior of unloaded and in-plane loaded rectangular plates is presented. The use of in-plane loads as a passive vibration control technique for rectangular plates is investigated using the results obtained by the Iterative Galerkin`s method. A linear analysis is conducted for loads applied on the plate mid-surface and a non-linear one for plates with in-plane eccentric loads. Moreover, it is shown that the Iterative Galerkin method leads to a set of orthogonal vibration modes allowing the use of superposition methods in the solution of dynamic problems. Furthermore, the results show that the proposed passive vibration control through the use of in- plane compression loads, decrease the response in the resonance region.

In this Thesis, the problems of the Free Flexural (or Bending) Vibrations of Composite, Orthotropic Plates and/or Panels with Various Bonded Joints are formulated and investigated in detail. The composite bonded plate system is composed of Plate Adherends adhesively bonded by relatively very thin adhesive layers. The general problem is considered in terms of the three Main PROBLEMS, namely Main PROBLEM I, Main PROBLEM II and Main PROBLEM III. The theoretical formulation of the Main PROBLEMS is based on Mindlin Plate Theory which is a First Order Shear Deformation Plate Theory (FSDPT). Thus, the transverse shear deformations, the transverse and the rotary moments of inertia of the plates are included in the formulation. Very thin, elastic deformable adhesive layers are considered as continua with transverse normal and shear stresses. The damping effects in the plates and the adhesive layers are neglected. The entire composite bonded joint assembly is assumed to be simple supported along the two opposite edges, so that the Classical Levy&amp / #8217 / s Solutions can be applied in this direction. The dynamic equations of the Bonded Joint System which combines together the Mindlin Plate dynamic equations with the adhesive layer equations are reduced to a system of First Order Ordinary Differential Equations in the state vector form. This special form of the Governing System of the First Order Ordinary Differential Equations are numerically integrated by means of the Modified Transfer Matrix Method which is a combination of the Classical Levy&amp / #8217 / s Method, the Transfer Matrix Method and the Integrating Matrix Method (with Interpolation Polynomials and/or Chebyshev Polynomials). The Main PROBLEMS are investigated and presented in terms of the mode shapes and the corresponding natural frequencies for various sets of boundary conditions. The significant effects of the hard and the soft adhesive layer elastic constants on the mode shapes and on the natural frequencies are demonstrated. Some important parametric studies such as the influences of the Joint Length Ratio, the Joint Position Ratio, the Bending Stiffness Ratio, etc. on the natural frequencies are computed and plotted for the hard and soft adhesive cases for several support conditions.