Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements.

Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

This article presents a simple and efficient procedure for the natural vibration analysis of rectangular plates and stiffened panels in contact with fluid on one side. The assumed mode method is applied, where the natural frequencies and mode shapes are obtained by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equation of motion. The Mindlin thick plate theory is applied for a plate, and in the case of stiffened panels, the effect of framing is taken into account by adding its strain and kinetic energies to the corresponding plate energies. Potential flow theory assumptions are adopted for the fluid, and free surface waves are ignored. The fluid velocity potential is derived from the boundary conditions for the fluid and structure and is utilized for the calculation of added mass using the assumed modes. The applicability and accuracy of the developed procedure are illustrated with several numerical examples using a developed in-house code. A comparison of the results with those obtained by general purpose finite element analysis software is provided, where very good agreement is achieved.

Geometrically nonlinear transverse steady-state periodic forced vibration of multi-degree-of-freedom discrete systems with a distributed nonlinearity

A nine-node isoparametric plate element in conjunction with first-order shear deformation theory is used for free vibration analysis of rectangular plates with central cutouts. Both thick and thin plate problems are solved for various aspect ratios and boundary conditions. In this article, primary focus is given to the effect of rotary inertia on natural frequencies of perforated rectangular plates. It is found that rotary inertia has significant effect on thick plates, while for thin plates the rotary inertia term can be ignored. It is seen that the numerical convergence is very rapid and based on comparison with experimental and analytical data from literature, it is proposed that the present formulation is capable of yielding highly accurate results. Finally, some new numerical solutions are provided here, which may serve as benchmark for future research on similar problems.

Performance of similitude methods for structural vibration analyses of rectangular plates

Several sets of static beam functions have been established by the authors, which are the static solutions of beams under arbitrary static loads. The beam is a strip taken from the rectangular plate to be considered, and the static load acting on the beam is expanded into a series which may be a series of point loads or a series of sinusoid loads or a series of polynomial loads. These sets of static beam functions have been successfully applied to the vibration analysis of rectangular plates with varying complexity by the Rayleigh-Ritz method and finite strip method. Some numerical results are given. By comparing the results with those available from literature it can be shown that the present method is extremely effective for solving the aforementioned problems and usually a few static beam functions will be sufficient to give acceptable results.

Three-dimensional vibration solutions are presented for rectangular plates with mixed boundary conditions, based on the small strain linear elasticity theory. The analysis is focused on two kinds of rectangular plates, the boundaries of which are partially fixed while the others are free. One of those studied is a rectangular plate with partially fixed boundaries symmetrically arranged around four corners and the other one is a rectangular plate with partially fixed boundaries around one corner only. A global analysis approach is developed. The Ritz method is applied to derive the governing eigenvalue equation by minimizing the energy functional of the plate. The admissible functions for all displacement components are taken as a product of a characteristic boundary function and the triplicate Chebyshev polynomial series defined in the plate domain. The characteristic boundary functions are composed of a product of four components of which each corresponds to one edge of the plate. The R-function method is applied to construct the characteristic boundary function components for the edges with mixed boundary conditions. The convergence and comparison studies demonstrate the accuracy and correctness of the present method. The influence of the length of the fixed boundaries and the plate thickness on frequency parameters of square plates has been studied in detail. Some valuable results are given in the form of tables and figures, which can serve as the benchmark for the further research.

Free vibrations of rectangular plates with interior point supports

There are several papers concerned with vibration analysis of rectangular plates without stiffeners subject to inplane loads. In this paper the authers studied the vibration of reinforced rectangular plates subject to inplane loads. And they found that the frequency of a reinforced rectangular plate under uniform compression, pure bending or pure uniform shear is less than that under no in-plane loads as well as in case of plates without stiffeners. The analysis of plates with reinforcement under any boundary conditions can be treated simply and generally by this method.

A new type of admissible function is used to study free, flexural vibration problems for rectangular plates with arbitrary interior elastic point supports. The one-dimensional deformation function of a beam with point load is applied in forming the admissible deformation shape function of a plate with a corresponding boundary condition. This study provides an extension of the classical solution in the vibration analysis of a plate with interior elastic point supports. The extended solution allows one to solve this type of problem easily when compared to other solutions available in the literature. Numerical results are presented for a number of specific problems, including convergence and accuracy of the approach, which include natural frequencies and mode shapes of some typical plates with interior point supports.

The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges

In this paper an examination of methods for vibration analysis of moderately thick rectangular plates has been presented. First, the state-of-the art in thick plate vibration theories and analysis methods is described and it is followed with basic equations of the original Mindlin theory, which represents a starting point for the development of all other mathematical models. Then, the problem of analytical solving of equilibrium equations is considered based on the modified Mindlin theory of thick plate vibrations, which has been published in the literature recently. Further, energy methods that can be applied to arbitrary boundary conditions are discussed and outline of the assumed mode method is presented. Finally, in the context of numerical methods a new quadrilateral finite element, based on the above mentioned advanced thick plate theory, is included. It should be emphasized that it doesn’t suffers of shear locking problem associated with finite elements, due to natural relation among bending and shear polynomials, and moreover, it gives very accurate results. Application of the presented methods is illustrated by several numerical examples which include natural vibration analyses of rectangular plates with various thicknesses and different combinations of boundary conditions (simply supported, clamped, free and elastically restrained). Comparisons of natural frequencies and mode shapes with results available in the relevant literature and with those obtained by the commercial finite element software are also provided.

Free Vibration Analysis of Rectangular Plates With Step-Wise Discontinuities in Rotational Stiffness Along the Edges

The coupled thermomechanical dynamic behavior of a heterogeneous solid is investigated by an example of laminated inelastic rectangular plates subjected to forced vibration and dissipative heating. The problem is solved in two formulations: complete and approximate. The complete formulation uses generalized flow theory to describe the inelastic properties of the material. The approximate formulation employs a model based on the idea of complex moduli. Special distributions of the field variables and amplitude–frequency responses of the rectangular plate are analyzed. The effect of the loading level on the mode shape and stress distribution is studied as well. A good agreement between the results obtained in the complete and approximate formulations is pointed out.

A method based on Hamilton’s principle and spectral analysis has been applied recently to nonlinear transverse vibrations of discrete systems with cubic nonlinearities, leading to calculation of the nonlinear free modes of transverse vibration and their associated nonlinear frequencies. The objective of the present work was the extension of this method to the nonlinear forced transverse steady-state periodic response of 2-dof system leading to nonlinear frequency response function in the neighbourhood of the two modes