Analysis of lower modes of vibration of rectangular plates of linearly varying thickness

The title problem is solved assuming that the thickness varies symmetrically with respect to the x-axis. The edges defined by $\text{x = ± a/2}$ are elastically restrained against rotation while the remaining edges are clamped or simply supported.Approximate expressions for four of the lower natural frequencies of vibration (including the fundamental) are given.A forced vibrations situation is also dealt with.     

Rayleigh-Ritz vibration analysis of Mindlin plates

The Rayleigh-Ritz method is applied to the prediction of the natural frequencies of flexural vibration of square plates having general boundary conditions. The analysis is based on the use of Mindlin plate theory so that the effects of shear deformation and rotary inertia are included. The spatial variations of the plate deflection and the two rotations over the plate middle surface are assumed to be series of products of appropriate Timoshenko beam functions. Results are presented for a number of types of plate and these demonstrate the manner of convergence of the method as the number of terms in the assumed series increases.     

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 vibration of rectangular plates of arbitrary thickness

Free vibration of thick rectangular plates is investigated by using the “method of initial functions” proposed by Vlasov. The governing equations are derived from the three-dimensional elastodynamic equations. They are obtained in the form of series and theories of any desired order can be constructed by deleting higher terms in the infinite order differential equations. The numerical results are compared with those of classical, Mindlin, and Lee and Reismann solutions.     

Free vibration of some circular plates of arbitrary thickness

In this paper, in order to be self-contained, a brief account of the construction technique of the method of initial functions which has been developed for circular plates by the author is given. Then the new method is applied to investigate the free vibration of two circular plates: i.e., simply supported and completely free plates. Numerical results are obtained and compared with those of the classical, Reissner and Mindlin theories.     

Rayleigh-Ritz vibration analysis of rectangular Mindlin plates subjected to membrane stresses

A previously developed analysis of the flexural vibration of isotropic rectangular plates is extended to include the presence of a membrane stress system. The method of analysis is the Rayleigh-Ritz method and Mindlin plate theory is used which takes into account effects which are disregarded in the classical plate theory. As in the aforementioned earlier analysis the spatial variations of the deflection and two rotations over the plate middle surface are based on the use of Timoshenko beam functions. The membrane stress system comprises biaxial direct stress plus in-plane shearing stress and is uniform throughout the plate. Numerical results are presented for a number of types of plate and of applied stress which show the manner of variation of the frequencies of vibration as the intensity of stress changes. This manner of variation is similar in form to that demonstrated elsewhere by analyses based on the use of the classical plate theory but the magnitudes of the present calculated frequencies are considerably reduced for moderately thick plates.     

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.     

Free vibration analysis of continuous rectangular plates

Vibration characteristics of rectangular plates continuous over full range line supports or partial line supports have been studied by using a discrete method. Concentrated loads with Heaviside unit functions and Dirac delta functions are used to simulate the line supports. The fundamental differential equations are established for the bending problem of the continuous plate. By transforming these differential equations into integral equations and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the line support, the variable thickness and aspect ratio on the frequencies and mode shapes are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.     

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].     

Free vibration of rectangular plates of arbitrary thickness with one or more edges clamped

Frequencies of free vibration of rectangular plates of arbitrary thickness, with different support conditions, are calculated by using the Method of Initial Functions (MIF), proposed by Vlasov. Sixth and fourth order MIF theories are used for the solution. Numerical results are presented for three square plates for three thickness ratios. The support conditions considered are (i) three sides simply supported and one side clamped, (ii) two opposite sides simply supported and the other two sides clamped and (iii) all sides clamped. It is found that the results produced by the MIF method are in fair agreement with those obtained by using other methods. The classical theory gives overestimates of the frequencies and the departures from the MIF results increase for higher modes and larger thickness ratios.     

Free vibration of super elliptical plates with constant and variable thickness by Ritz method

In this study free vibration of simply supported and clamped super elliptical plates is investigated. This class of plates includes a wide range of external boundaries varying from an ellipse to a rectangle. Although studies on the upper and lower bounds of these plate geometries, namely circle and rectangle, are quite extensive, contributions on the mid-shapes, especially for simply supported boundary edges are very limited. The Kirchhoff plate model with isotropic and homogeneous material is studied. The super elliptical powers are chosen from 1 to 10. The Ritz method is employed for the solution of the energy equations of the plates. The effects of Poisson's ratio, which should not be neglected for simply supported plates with curved boundaries, and the aspect ratio of the plate are both examined in detail. The effect of thickness variation is also considered in this study. In order to avoid long computational run times, physically pertinent trial functions are utilized. The frequency parameters obtained are presented and compared with published results for plate shapes that match the current cases.     

On vibration of plates with varying stiffener length

Fabrication considerations sometimes make it necessary to stop a stiffening member at a short distance away from the supporting edge of a panel. The effect of this gap between the stiffener tip and the supporting edge on the natural frequencies is investigated here by using a finite element approach.     

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.     

Free vibration of skew Mindlin plates by p-version of F.E.M.

Accurate natural frequencies and mode shapes of skew plates with and without cutouts are determined by p-version finite element method using integrals of Legendre polynomials for p=1-14. The hierarchical plate element is formulated based on Mindlin's plate theory including rotatory inertia effects and based on a skew co-ordinate system. Non-dimensional frequency parameter and mode shapes are presented for a range of skew angle (β), aspect ratio (a/b), thickness-width ratio (h/b), cutout dimensions and different boundary conditions. The results were verified by comparison with those available in the open literature.