Performance of similitude methods for structural vibration analyses 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.

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.

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

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

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 objective of this work is to study the geometrically non-linear steady state periodic forced response of fully clamped rectangular plates (FCRP) with immovable in-plane conditions, taking into account the effect of the in-plane displacements. A complete formulation has been proposed first, reducing the equations of motion to a system of coupled non-linear algebraic equations, which are decoupled once the in-plane inertia is omitted. An averaging technique has then been developed, in order to simplify the first method and to develop an engineering complete theory. The forced response is given in the case of a concentrated harmonic excitation force with various intensities. The numerical results obtained here with the two formulations, using an explicit analytical solution, were compared with those obtained previously using a formulation in which the in-plane displacements have been neglected, showing an “over-stiff” effect.

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.

Part 2 of this series of two papers presents the applications of the discrete singular convolution (DSC) algorithm. The main purpose of this paper is to explore the utility, test the accuracy and examine the convergence of the proposed approach for the vibration analysis of rectangular plates with internal supports. Both partial internal line supports and complex internal supports are considered for 21 square plates of various combinations of edge support conditions. The effects of different size, shape and topology of the internal supports and different boundary conditions on the vibration response of plates are investigated. The partial internal line supports may vary from a central point support to a full range of cross or diagonal line supports. Several closed‐loop supports, such as ring, square and rhombus, and their combinations are studied for complex internal supports. Convergence and comparison studies are carried out to establish the correctness and accuracy of the DSC algorithm. The DSC results are compared with those in the available literature obtained by using other methods. Numerical results indicate that the DSC algorithm exhibits controllable accuracy for plate analysis and shows excellent flexibility in handling complex geometries, boundary conditions and support conditions. Copyright © 2002 John Wiley & Sons, Ltd.

An experimental and numerical investigation on temperature profile of underground soil in the process of heat storage

Various faults may simultaneously occur in a rotor dynamic system. It is crucial to detect and monitor these faults to ensure the safe operation of machinery. Usually, vibration response is utilized for fault diagnosis; however, the identification process becomes more challenging when multiple faults exhibit similar symptoms in the vibration response. In this paper, the vibrational transient response of a cracked rotor is analyzed in the presence of unbalance and coupling misalignment. The study focuses on investigating the possibility, advantages and limitations of using the sub-critical startup response in solving the fault identification problem. Numerical simulations using finite element modeling and experimental investigations are carried out for different individual and multi-fault conditions. Various factors are taken into consideration, including sub-critical limit speed, crack parameters, unbalance parameters and misalignment severity. Numerical and experimental investigations showed that the sub-critical transient response is suitable for the fault detection and diagnosis of the simultaneously existing faults studied. Compared to the critical response, the sub-critical response is more informative and each fault shows a unique feature that can be extracted without the need for sophisticated post signal processing. The study provides useful insights on the interaction between the various fault effects on the vibratory response and presents a simple method for condition monitoring of industrial equipment.

Interest in the free vibration frequencies and associated mode shapes of rectangular plates resting on internal point supports has arisen in connection with the design of electronic circuit boards, solar panels and other industrial problems. An analytical solution is presented for the free vibration analysis of rectangular plates with multiple internal point supports. The solution is also shown to be easily modified to account for the effects of attached masses. The basic solution for each case consists of the Levy type solution for a plate with one discrete point support. N similar solutions for N discrete point supports are then superimposed to create an eigenvalue matrix from which the plate's natural frequencies and associated mode shapes can be determined. Due to the nature of the Levy type solution, only plates with two opposite edges simply supported are considered. The remaining edges are either simply supported, clamped or free. There are therefore six possible combinations of plate boundary conditions. The objective of this thesis is to give a concise and clear description of the mathematical procedure employed and to present the results of some representative frequency and mode shape studies.