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].     

Vibrations of a square plate with parabolically varying thickness

Vibration results for a square plate with a parabolically varying thickness distribution and built-in edges are presented. Frequency and mode shape predictions obtained from a finite element analysis are compared with measurements made with real-time laser holography. In general, the agreement between the two is very good except for a few of the lower modes where the predicted frequencies are about 15% high. So far, a satisfactory explanation for this has not been found. The vibration mode shapes for the plate exhibit a striking blend of radial and square symmetries that results from the axisymmetric thickness distribution and the square symmetry of the boundary frame.     

Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness

Free axisymmetric vibrations of an isotropic, elastic, non-homogeneous circular plate of linearly varying thickness have been studied on the classical theory of plates. The non-homogeneity of the material of the plate is assumed to arise due to the variation of Young's modulus and density with the radius vector whereas Poisson's ratio is assumed to remain constant. The governing differential equation of motion is solved by the method of Frobenius. The transverse displacement of the plate has been expressed as a power series in terms of the radial co-ordinate. The frequency parameters corresponding to the first two modes of vibration have been computed for different values of the non-homogeneity parameter and taper constant and for clamped and simply supported edge conditions of the plate. A comparison between the numerical results for homogeneous and non-homogeneous material of the plate is also made.     

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.     

Effect of an elastic foundation on axisymmetric vibrations of polar orthotropic annular plates of variable thickness

Free axisymmetric vibrations of a polar orthotropic annular plate of linearly varying thickness resting on an elastic foundation of Winkler type are studied on the basis of classical theory of plates. The fourth order linear differential equation with variable coefficients governing the motion is solved by using the quintic spline interpolation technique for three different combinations of boundary conditions. The effect of the elastic foundation together with the orthotropy on the natural frequencies of vibration is illustrated for different values of the radii ratio and the thickness variation parameter for the first three modes of vibration. Transverse displacements and moments are presented for a specified plate. The validity of the spline technique is demonstrated by presenting a comparison of present results with those available in the literature.     

On the vibration of an elastic plate on an elastic foundation

The forced vibration of an elastic plate under a time harmonic point force is studied. The plate is infinite in extent and supported by an elastic foundation. This study is made on the basis of the improved (Timoshenko) plate theory. The mathematical problem is to seek a fundamental solution (the Green's function) of the time-reduced plate equation of the improved plate theory. Such a fundamental solution is constructed by the distributional Fourier transform method. From the explicit expressions of the fundamental solution, the behavior of the fundamental singularity as a function of the vibration frequency and the foundation stiffness is examined. Conditions under which plate resonance occurs are also determined.     

Effects of edge restraints on the non-linear flexural vibrations of an imperfect cross-ply laminated plate resting on an elastic foundation

A solution is presented, for the non-linear Marguerre dynamic equilibrium and compatability equations, for the large amplitude free flexural vibrations of an imperfect, cross-ply, laminated plate, having elastically restrained edges and resting on an elastic foundation. The analysis is used to study the effects of edge restraints and elastic foundation constants on the frequency ratios of isotropic and CFRP plates. Numerical results are presented graphically.     

Dynamic analysis of an elastic plate on a thin,elastic foundation

An improved theory for the motion of an elastic foundation is developed. Included is the interaction of two or more plates resting on a common foundation. A simplified model is deduced which appears useful for some applications.     

Free vibrations of a plate with an inner support

The title problem is tackled by using a simple polynomial co-ordinate function and the Rayleigh-Schmidt method. It is assumed that the inner support is parallel to the free edge. When the support coincides with the free edge the frequency equation degenerates properly into the case of a simply supported edge. Numerical results are presented for the situation where two opposite edges are simply supported and the edge parallel to the free edge is either clamped or simply supported.     

Free vibration of a shearwall-frame building on an elastic foundation

The frequencies and associated mode shapes of a shearwall-frame building interacting with an elastic foundation are determined by using an iterative technique iith the Rayleigh-Ritz method. The eigenfunctions of a clamped-free prismatic shear-flexure beam are suitably modified to generate the required shape functions. The influence of the soil on the frequency spectrum of a shearwall-frame building is investigated.     

Random vibration of an initially stressed thick plate on an elastic foundation

The mean-square bending moment of a thick rectangular plate excited by a uniform distribution of stationary random forces that are uncorrelated in space is calculated. The plate has in-plane compressive or tensile stresses. In addition, the plate is mounted on an elastic foundation. Numerical results are given for plates with uniform initial stress when the temporal correlation function of the excitation possesses an exponential decay. In general it can be said that the position on the plate where the mean-square moment takes on a maximum value depends upon the relative values of the initial stress, the stiffness of the foundation and the aspect ratio of the plate. The mean-square response amplitude of the plate on a foundation never exceeds that of the plate without a foundation, regardless of the intensity of the initial stress or the geometrical configuration of the plate.     

Note on the forced vibration of an orthotropic plate on an elastic foundation

An analysis is presented of the response of an infinite, linearly elastic, orthotropic plate, resting on an elastic foundation, to a forcing function which may vary in both time and space. Some numerical results are given for a case of concentrated blast loading.     

Free vibrations of elastic circular toroidal shells

In this paper, the free vibrations of elastic in vacuo circular toroidal shells under different boundary conditions are studied using the linear Sanders thin shell theory. Beam functions are used to describe the motion along the meridional direction whilst trigonometric functions are used to represent the deformation of the cross section. It is shown that both the natural frequencies and the mode shapes can be accurately predicted as long as the employed beam functions satisfy the boundary conditions at the ends of the shells. The dependence of the free vibration characteristics of an elastic toroidal shell upon boundary conditions and toroidal to cross-sectional radius ratio is also illustrated and explained in this paper.     

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.