On the analysis of large amplitude vibrations of non-uniform rectangular plates

An analysis of the large amplitude vibrations of non-uniform rectangular plates is presented. The method is based on Berger's assumption of neglecting the second invariant of the middle surface strain in the expression corresponding to the total potential energy of the system, in conjunction with a Galerkin procedure.     

Large amplitude vibrations and damage detection of rectangular plates

In this work, geometrically nonlinear vibrations of fully clamped rectangular plates are used to study the sensitivity of some nonlinear vibration response parameters to the presence of damage. The geometrically nonlinear version of the Mindlin plate theory is used to model the plate behaviour. Damage is represented as a stiffness reduction in a small area of the plate. The plate is subjected to harmonic loading with a frequency of excitation close to the first natural frequency leading to large amplitude vibrations. The plate vibration response is obtained by a pseudo-load mode superposition method. The main results are focussed on establishing the influence of damage on the vibration response of the plate and the change in the time-history diagrams and the Poincaré maps caused by the damage. Finally, a criterion and a damage index for detecting the presence and the location of the damage is proposed. The criterion is based on analysing the points in the Poincaré sections of the damaged and healthy plate. Numerical results for large amplitude vibrations of damaged and healthy rectangular and square plates are presented and the proposed damage index for the considered cases is calculated. The criterion demonstrates quite good abilities to detect and localize damage.     

Large amplitude vibrations of circular plates with varying thickness

A simple finite element formulation is presented to evaluate the large amplitude vibration frequencies of orthotropic circular plates with linearly varying thicknesses. Period ratios are presented in tables and figures for different values of the orthotropy and taper parameters.     

Vibration analysis of rectangular plates with general elastic boundary supports

In this investigation, the Rayleigh-Ritz method is used to determine the modal characteristics of a rectangular plate with general elastic supports alone its edges. Each of the admissible functions here is composed of a trigonometric function and an arbitrary continuous function that is introduced to ensure the sufficient smoothness of the so-called residual displacement function at the edges. As a result, a drastic improvement of the convergence can be expected of the solution expressed as a series expansion in terms of the admissible functions. Perhaps more importantly, this study has developed a general approach for deriving a complete set of admissible functions that can be universally applied to various boundary conditions. Several numerical examples are given to demonstrate the accuracy and convergence of the current solution.     

Transverse vibrations of circular plates of varying thickness with non-uniform edge constraints

Free vibrations of circular plates varying in thickness and with flexible edge supports have been studied by several investigators for the restricted case when the supports are represented by uniformly distributed springs of constant stiffness.In the present study an approximate method is presented for dealing with supports possessing rotational flexibility which varies arbitrarily around the boundary.The method consists in representing the varying stiffness in terms of a Fourier expansion in the polar angle and approximately expressing the displacement function using a summation of polynomial co-ordinate functions which exactly satisfies only the essential boundary condition. The Ritz method is then applied in order to obtain the frequency determinant. The method can be easily extended to the forced vibrations case.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.     

Transverse vibrations of a non-uniform rectangular plate on an elastic foundation

Free transverse vibrations of an isotropic rectangular plate of variable thickness resting on an elastic foundation has been studied on the basis of classical plate theory. The fourth-order differential equation governing the motion is solved by using the quintic spline interpolation technique. Characteristic equations for plates of exponentially varying thickness have been obtained for three combinations of boundary conditions at the edges. Frequencies, mode shapes and moments have been computed for different values of the taper constant and the foundation moduli for the first three modes of vibration.     

Large amplitude axisymmetric vibrations of geometrically imperfect circular plates

This paper deals with the effects of geometric imperfections on the large amplitude vibrations of circular plates. It is found that geometric imperfections of the order of a fraction of the plate thickness may significantly raise the linear vibration frequencies. Furthermore, such imperfections may even change the inherent non-linear hard-spring character of the circular plates and cause them to exhibit soft-spring behavior. The effects of various boundary conditions are examined.     

Large amplitude free vibrations of annular plates of varying thickness

The non-linear (i.e., large deflection) free vibrations of thick, orthotropic annular plates with varying thickness are calculated. The formulation is based on the more general Reissner plate equations as well as the von Kármán plate equations for variable thickness annular plates. Numerical results for the ratio of the non-linear period to the linear period of natural vibration are compared with those existing in the literature. New results are also included for future comparisons.     

Large amplitude vibrations of circular plates carrying a concentrated mass

A finite element analysis for the large amplitude vibrations of circular plates carrying a concentrated mass is presented in this note. The variation of the fundamental frequency with the central amplitude is obtained for simply supported and clamped plates for various positions and various magnitudes of the concentrated mass.     

Free vibration analysis by the superposition method of rectangular Mindlin plates with internal columns resting on uniform elastic edge supports

Free vibration analysis of rectangular Mindlin plates is carried out via the superposition method and is shown to produce accurate results. The analytical method proposed by the authors in this paper solves the problem for the case where the plate has simultaneous elastic edge and internal supports. The conditions of edge supports are uniform lateral, rotational and torsional elastic supports, whereas, the internal supports are column supports with finite area. Compatibility between the plate and column is achieved by requiring that the column and plate rotations be equal. The results presented herein are verified through comparison with results presented by others. Numerical examples presented in this study confirm that the analytical method is able to model the property of the elastic edge supports while simultaneously considering the effect of column restraint, an effect which increases with column number and area.     

Large amplitude vibrations of clamped circular plates of linearly varying thickness

The large amplitude vibrations of clamped circular plates of linearly varying thickness are investigated, by applying the procedure proposed by Banerjee and Datta [1].     

Forced vibrations of a rectangular plate with non-uniform thickness

The dependence on frequency of the maximum deflection and surface stresses of a simply supported rectangular plate subjected to a uniformly distributed sinusoidal excitation is discussed and simple formulae are proposed for estimating the deflection and surface stresses. The thickness of the plate varies linearly in one direction parallel to a side of the plate.     

Large amplitude axisymmetric vibrations of orthotropic circular plates elastically restrained against rotation

A finite element formulation is employed to obtain the linear and non-linear frequencies of orthotropic circular plates with elastically restrained edges. Results are presented in the form of linear frequency parameters and ratios of non-linear to linear periods for several values of the spring constants, orthotropy parameter and central deflections.