VIBRATIONS OF COMPOSITE CIRCULAR AND ANNULAR PLATES OF VARYING THICKNESS

Abstract

In the present days, sandwich constructions are found to have enormous applications in the field of modern science and technology. Their use in automobile, shipbuilding and transportation industries, geotextile infrastructures, aeronautics and astronautics, marine and offshore oil structures, wind industry and sport goods is growing rapidly due to their specific stiffness, light weight and maximum fatigue resistance. In construction, sandwich structures have been applied as structural elements in foot and vehicular bridges, in the rehabilitation as replacement of concrete bridge, cladding, roofing and also as partition wall elements, sometimes with translucent properties. Employment of sandwich structural elements of varying thickness further helps the designer not only in enhancing the performance but also reduce the weight and size of the structure. To use them efficiently, a good knowledge of their constructional and dynamic behavior is essential with a fair amount of accuracy. In the present thesis, an attempt has been made to study the vibration characteristics of non-uniform circular and annular sandwich plates comprise with relatively stiff core and isotropic / orthotropic facings. It consists of ten chapters. Chapter I provide an up-to-date survey of the literature on the dynamic behavior of plate-type sandwich constructions with their usage and significance in various technological situations. In Chapters II-IX, the analysis of such plates with different considerations has been presented employing three different numerical techniques, namely; differential quadrature method, harmonic differential quadrature method and Chebyshev collocation technique. Numerical computations have been made using the software MATLAB. The detailed analysis of the results in the form of conclusions together with the future scope of the present work is given in chapter X. The chapter wise summary is given as below: Chapter II deals with the free axisymmetric vibrations of circular sandwich plates with relatively stiff core of linearly varying thickness. The governing differential equations of motion for such a model of the plate have been derived using Hamilton’s principle and solved by employing differential quadrature method for three different boundary conditions. The effect of various plate parameters such as taper parameter, thickness of the core at the center, face thickness on the natural frequencies for the first three modes of vibration has been studied. Three dimensional mode shapes for a specified plate have been plotted. In chapter III, an analysis for radially symmetric vibrations of annular sandwich plates with relatively stiff core of linearly varying thickness has been presented. The governing differential equations of motion obtained for circular plate in chapter II have been extended for the annular sandwich plates of linearly varying thickness. The frequency equations for three ii different combinations of boundary conditions at the two edges, namely; clamped at the inner edge and clamped or simply supported or free at the outer edge are obtained employing differential quadrature method. The lowest three roots of these frequency equations have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters such as taper parameter, thickness of the core at the centre, face thickness and radii ratio on the natural frequencies has been analyzed. Three dimensional mode shapes for a specified plate have been illustrated. Chapter IV deals with the free axisymmetric vibrations of circular sandwich plates with relatively stiff core of parabolically varying thickness. The facings are taken of the same material and of the same thickness. Due to parabolic variation in thickness of the core, facings take the shape of paraboloid of revolution and membrane forces of facings contribute to the transverse shear and bending of the core. The equations of motion have been derived using Hamilton’s energy principle. The first three natural frequencies for clamped, simply supported and free edge conditions have been obtained using differential quadrature method, taking the grid points as zeros of Chebyshev polynomials. The effect of various plate parameters such as taper parameter, facing thickness and core thickness on the frequency parameter has been investigated. Mode shapes for a specified plate have been plotted. In chapter V, an analysis has been presented for the axisymmetric vibrations of annular sandwich plates with relatively stiff core of parabolically varying thickness using a refined theory. The governing differential equations of motion derived in chapter IV has been used for annular sandwich plates. The frequency equations for three different combinations of boundary conditions, namely; clamped at the inner and outer edges, clamped at inner and simply supported or free at the outer edge are obtained employing differential quadrature method. The lowest three roots of these frequency equations have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters such as radii ratio, taper parameter, thickness of the facings and core at the centre on the natural frequencies has been studied. Three dimensional mode shapes for the specified plates have been illustrated. In chapter VI, harmonic differentia quadrature (HDQ) method has been applied to analyze the axisymmetric vibrations of composites circular sandwich plates with isotropic core and orthotropic face sheets. The thickness of the core is assumed to vary quadratically in the radial direction and the face sheets are treated as membranes of constant thickness. The effect of shear deformation and rotatory inertia in the core has been taken into account and the mathematical model is based upon first-order shear deformation theory. The governing differential equations of motion for this model have been established using Hamilton’s energy iii principle. Frequency equations have been obtained employing HDQ method in the resulting equations for the three edge conditions, namely; clamped, simply supported and free. The lowest three roots of these equations have been reported as the frequencies for the first three modes of vibration for all the three plates. The effect of various plate parameters such as taper parameter, core thickness at the centre and facing thickness on the natural frequencies has been studied. Three-dimensional mode shapes for the specified plates have been illustrated. Comparison of results has been presented in special cases. In chapter VII, the analysis presented in chapter VI has been extended to study the radially symmetric vibrations of composite annular sandwich plates using HDQ for the three different sets of boundary conditions, as mentioned in earlier chapters of annular plate. The lowest three roots of these equations have been reported as the frequencies for the first three modes of vibration. The effect of various plate parameters on the natural frequencies has been studied. Three-dimensional mode shapes for the specified plates have been illustrated. To check the validity of the present considerations and the technique, frequency parameter has been compared in particular cases. Chapter VIII presents the free axisymmetric vibrations of composite circular sandwich plates with isotropic core and orthotropic facings using first-order shear deformation theory. The thickness of the core is assumed to vary exponentially in the radial direction and the face sheets is treated as membrane of constant thickness. The governing differential equations of the present model have been obtained by applying Hamilton’s principle. Chebyshev collocation technique has been used to obtain the frequency equations for the plate when it is clamped or simply supported or free at the edge. The first three roots of these equations have been reported as the natural frequencies for the first three modes of vibration. The effect of various plate parameters on the frequency parameter has been investigated. Mode shapes for a specified plate for all the three boundary conditions have been plotted. In chapter IX, mode shapes and frequencies for axisymmetric vibrations of composite annular sandwich plates with isotropic core of exponentially varying thickness and orthotropic face sheets have been obtained by extending the analysis of chapter VIII for the same three edge conditions as considered before for annular plate. The lowest three roots of these equations have been obtained employing Chebyshev collocation technique and reported as the frequencies for the first three modes of vibration. The effect of various plate parameters on the natural frequencies has been studied. In Chapter X, conclusions and future scope of the present work have been presented