Free vibration analysis of plates and shells by using the Superposition Method

Abstract

This thesis is devoted to investigate the capability of the Superposition Method for obtaining the transient response of plates and the natural frequencies of thin doubly curved shallow shells. The Superposition Method gives accurate results with only a few terms and has proved to be efficient for both cases. To investigate the transient response, all supports of a thin simply supported rectangular plate under self weight are suddenly removed. The resulting motion comprises a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response. The modes and natural frequencies of the plate are obtained using the Superposition Method and the Rayleigh-Ritz method with the ordinary and degenerated free-free beam functions. The W–W algorithm is then used to delimit the natural frequencies from the frequency equation derived in a determinantal form. There is an excellent agreement between the results from both approaches but the modes based on the Superposition Method result in more accurate values with fewer terms, and have shown faster convergence. The results from the Superposition Method may serve as benchmarks for the transient response of completely free plates. The transient response is found to be dominated by the lower modes. The centre of vibration is shifted parallel from the original xy plane by the distance of the first mode of the plate (a rigid body translation) multiplied by the first transient coefficient. In the investigation of doubly curved shells, the natural frequency parameters of thin shallow shells with three different sets of boundary conditions were obtained for several different curvature ratios and two aspect ratios. The solutions to the building blocks, which are subject to simply-supported out-of-plane conditions and shear diaphragm in-plane conditions at all four edges, are represented by series of sine and cosine functions, generated using Galerkin’s method since an exact solution is not available for the doubly curved shells. Once displacement functions for the building blocks are obtained, the prescribed boundary conditions of the actual shell under investigation are then satisfied using the Superposition Method. The rate of convergence is found to be excellent and the results agree well with published results obtained using the Ritz method and those obtained using a Finite Element package, Abaqus. The computations show that it is possible to obtain the first 12 natural frequency parameters of the shallow shells on the rectangular planform with a rapid convergence rate.