Axisymmetric vibrations of a circular plate of linearly varying thickness on an elastic foundation according to Mindlin theory

Free axisymmetric vibrations of an elastic circular plate of linearly varying thickness on an elastic foundation have been studied on the basis of shear theory [1,2]. The transverse displacement and local rotation are expressed as an infinite series. The frequencies corresponding to the first two modes of vibrations are obtained for a circular plate with clamped and simply supported edge conditions for various values of the taper constant and the foundation modulus. The results have been compared with those of reference [3].     

Axisymmetric vibrations of annular sandwich plates of linearly varying thickness

Free axisymmetric transverse vibrations of annular sandwich plates of linearly varying thickness are considered. Equations of motion are derived by Hamilton's principle and solved by a cubic spline technique. Numerical results are obtained for the first four normal modes of vibration.     

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].     

Free vibrations of an infinite plate of parabolically varying thickness on elastic foundation

A simple model is presented for use by research workers in seismology, soil mechanics and earthquake vibrations in problems where the resistance offered by the soil foundation is very close to that of Winkler's assumption. In earth vibrations one earth layer can be treated as an infinite plate of variable thickness while layers below it may approximately serve as an elastic foundation. Here, the transverse displacement of a plate of parabolically varying thickness on an elastic foundation is expressed as a power series and the frequencies, deflections and moments corresponding to the first two modes of vibration are computed for various values of foundation modulus and taper constant, and for two combinations of boundary conditions. A problem discussed in previous work [1] can be considered as a particular case of the present problem if the foundation modulus is taken to be zero.     

On flexural vibrations of isotropic elastic circular plate according to Mindlin's theory

This paper presents a numerical solution for an isotropic elastic circular plate. Effects of shear and rotatory inertia are included in a manner analogus to Timoshenko's one dimensional theory of bars. Matrix method has been employed to solve the set of equations obtained by using finite difference approximations. The use of IBM 1620 Computer was made for the computation of results showing thereby that even small machines could be used to handle the problem by the suggested approach.Notation ² the Laplace two dimensional operator - h thickness of plate - a diameter of plate - v Poisson's ratio - E Young's modulus of elasticity - G shear modulus - G K ² E/2(1 +v) - K a constant - density of the plate material - q superimposed normal load per unit plate area - D Eh ³/12(1–v²) - deflection of a point normal to the plane of plate - W amplitude of harmonic deflection - M amplitude of harmonic moment     

Transverse vibrations of circular plates of linearly varying thicknesses

The title problem is solved using very simple polynomial co-ordinate functions and a variational approach.Rather general boundary conditions are assumed at the edge support. It is shown that the approach is valid for axi- and antisymmetric modal configurations.     

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.     

Axisymmetric vibrations of linearly tapered annular plates under an in-plane force

Analysis and numerical results are presented for the axisymmetric vibrations of circular annular plates with linear variation in thickness under the action of a hydrostatic in-plane force on the basis of the classical theory of plates. The governing differential equation with variable coefficients has been solved by Chebyshev collocation technique. The effect of inplane force on the natural frequencies of vibration has been investigated for two different boundary conditions and for different radii ratio and taper constant. Transverse displacements, moments and the critical buckling loads in compression with thickness variation have also been computed for the first two modes.     

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.     

Antisymmetric vibrations of circular plates with thickness varying in a bilinear fashion

The title problem is tackled using simple polynomial co-ordinate functions and the Ritz method. The algorithmic procedure is quite straightforward and yields very accurate results in the case of circular plates of uniform thickness. It can also be easily implemented by a desk computer.     

Vibration and stability of a circular plate of unidirectionally varying thickness

An analysis is presented for the vibration and stability of an elastically restrained circular plate of unidirectionally varying thickness subjected to an in-plane force. For this purpose, the transverse deflection of a circular plate of variable thickness is written in a series of the deflection functions of a uniform circular plate without the action of a force. The dynamical energies of the plate are evaluated analytically, and the frequency equation of the plate is derived by the Ritz method. The analysis is applied to circular plates of unidirectionally tapered or stepped thickness; the natural frequencies and the divergence loads are calculated numerically, and the effects of the varying thickness and edge conditions on the vibration and stability are studied.     

The acoustic field excited by flexural vibrations of an elastic plate with a circular inclusion

The acoustic field excited by flexural vibrations of a thin elastic plate and the perturbations of this field caused by a homogeneous circular inclusion with other elastic properties are considered. Because the density of air widely differs from that of a metal, this problem can be solved with fair accuracy in two steps: first, by considering the vibrations of the plate in a free space, and, then, by calculating the acoustic field excited by the field of plate’s vertical deflections. The main results of this work are the asymptotic expressions for the far acoustic field excited by each of the Fourier components F _m (r)cosmφ of the flexural wave scattered by the inclusion.     

Axisymmetric non-linear oscillations of isotropic layered circular plates

Non-linear equations of motion for isotropic layered circular plates are presented for axisymmetric motion. Further simplification is made by ignoring the in-plane and rotatory inertia terms. Explicit solutions are obtained for the forced and free oscillations. In this case it is found that the non-linearity is of hardening type. Numerical results are presented for the case of a two layered plate of aluminium and steel.