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In this study, free axisymmetric vibrations of composite circular sandwich plates with isotropic core and orthotropic facings have been studied 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 effect of shear deformation and rotatory inertia in the core has been taken into account. 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 taper parameter, facing thickness and core thickness on the frequency parameter has been investigated. Comparison of results for some special cases with published results obtained by other approximate methods has been presented which shows an excellent agreement. Mode shapes for a specified plate for all the three boundary conditions have been plotted. © 2019 Elsevier Ltd

Department of Mathematics

School of Advanced Sciences

Vellore Campus

World University root node

Composites Part B: Engineering

Free vibrations of composite sandwich plates by Chebyshev collocation technique

The present work deals with the free axisymmetric vibrations of circular sandwich plates with core of linearly varying thickness. The face sheets are treated as membranes of equal thickness and the core is assumed to be solid as well as moderately thick. The plate is assumed to be symmetric about the middle surface. Due to linear thickness variation in the core, the face sheets take the shape of conical shell and because of this the face sheet membrane forces contribute to the bending and transverse shear of the core of sandwich plate. The effect of transverse shear and rotatory inertia has been retained in the core. The equations of motion for such a plate have been derived by Hamilton's energy principle. The frequency equations for three different boundary conditions have been obtained by employing differential quadrature method. The effect of various plate parameters such as taper parameter, thickness of the core at the center, face thickness on the natural frequencies has been studied. Comparison of results has been made. © The Author(s) 2014.

JVC/Journal of Vibration and Control

On the use of differential quadrature method in the study of free axisymmetric vibrations of circular sandwich plates of linearly varying thickness

In the present paper, analysis and numerical results are presented for free axisymmetric vibrations of circular sandwich plate with isotropic core and polar orthotropic face sheets employing harmonic differential quadrature (HDQ) method. 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 analysis is based upon Mindlin’s theory/first-order shear deformation theory. The governing differential equations of motion of the model have been obtained by using Hamilton’s energy principle. In the resulting equation HDQ method is employed to obtain the frequency equations for the three edges, 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. To check the validity of the present considerations and the technique, frequency parameter has been compared in particular cases. © 2015, Springer Science+Business Media Dordrecht.

Meccanica

On the radially symmetric vibrations of circular sandwich plates with polar orthotropic facings and isotropic core of quadratically varying thickness by harmonic differential quadrature method

Analysis and numerical results for the axisymmetric vibrations of annular sandwich plates with relatively stiff core of non-uniform thickness have been investigated using a refined theory. The face sheets are treated as membranes of constant thickness and the core is assumed to be solid as well as moderately thick of parabolically varying thickness. Due to this-type of variation in the thickness of the core, the face sheets take the shape of paraboloid of revolution and because of this, the face sheets membrane forces contribute to the bending and transverse shear stresses of the core of the plate. The equations of motion for such a plate have been derived using Hamilton's energy principle. The frequency equations for three different combinations of boundary conditions; 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 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. A comparison of the results with published work has been made. © 2015 Elsevier Ltd. All rights reserved.

Thin-Walled Structures

On radially symmetric vibrations of non-uniform annular sandwich plates

In the present paper, axisymmetric vibrations of annular sandwich plates with isotropic core and orthotropic face sheets have been studied using the harmonic differential quadrature (HDQ) method. 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 analysis of the model is based upon first-order shear deformation theory. The governing differential equations of motion have been obtained using Hamilton’s energy principle. In the resulting equations, the HDQ method is employed to obtain the frequency equations for the three edge conditions, namely CC, i.e., clamped at both the inner and outer edges, CS, i.e., clamped at the inner edge and simply supported at the outer edge, and CF, i.e., clamped at the inner edge and free at the outer edge. 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 such as taper parameter, core thickness at the center, facing thickness and radii ratio 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, a frequency parameter has been compared in particular cases. © 2014, Springer-Verlag Wien.

Acta Mechanica

Axisymmetric vibrations of composite annular sandwich plates of quadratically varying thickness by harmonic differential quadrature method

In the present paper, axisymmetric vibrations of a circular sandwich plate with relatively stiff core of exponentially varying thickness have been investigated. The face sheets are treated as membranes of constant thickness and the core is assumed to be solid as well as moderately thick. The equations of motion have been derived using Hamilton’s energy principle. The frequency equation for clamped boundary condition is obtained by employing harmonic differential quadrature method. The lowest three roots of this equation 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 a specified sandwich plate have been illustrated. A comparison of the results with published work has been made. © Springer India 2015.

Springer Proceedings in Mathematics and Statistics

Radially symmetric vibrations of exponentially tapered clamped circular sandwich plate using harmonic differential quadrature method

Abstract In this paper, the free axisymmetric vibrations of circular sandwich plates with relatively stiff core of parabolically varying thickness have been presented on the basis of first order shear deformation theory. For the present model, the plate is assumed to be symmetric with respect to the middle surface. 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 of sandwich plate. The equations of motion have been derived by Hamilton's energy principle. The first three natural frequencies for clamped, simply supported and free edge conditions have been obtained using a 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. Comparison of results for some special cases with published results obtained from other approximate methods has been presented which shows an excellent agreement. Mode shapes for specified plates have been plotted. © 2015 Elsevier Ltd.

International Journal of Mechanical Sciences

On radially symmetric vibrations of circular sandwich plates of non-uniform thickness

An analysis and numerical results of radially symmetric vibrations of annular sandwich plates with core of linearly varying thickness are presented. The face sheets are treated as membranes of constant thickness, and the core is assumed to be solid as well as moderately thick. Due to linear thickness variation in the core, the face sheets take the shape of a truncated conical shell and because of this the face sheets membrane forces contribute to the bending and transverse shear of the core of the sandwich plate. Keeping this in view, the equations of motion for such a plate are developed by Hamilton's energy principle. The frequency equations for three different combinations of boundary conditions, namely clamped at the inner edge and clamped or simply supported or free at the outer edge, are obtained by employing the 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 center, face thickness, and radii ratio on the natural frequencies has been analyzed. Three-dimensional mode shapes for a specified plate for all the three boundary conditions are illustrated. A comparison of results is presented. © 2013 Springer-Verlag Wien.

Mode shapes and frequencies of radially symmetric vibrations of annular sandwich plates of variable thickness

In this paper axisymmetric vibration of circular sandwich plates with honeycomb core of linearly varying thickness has been studied. The facings are of the same thickness and of the same material, are treated as membranes. The facings take the shape of conical shell due to linear thickness variation in the core and hence membranes forces of the facings contribute to the transverse shear of the core. The equations of motion for such a plate have been derived by Hamilton’s energy principle. The frequency equations have been obtained by employing differential quadrature method for three different boundary conditions at the edge, namely, clamped, simply supported and free. The lowest three roots of these frequency equations have been obtained for various values of different plate parameters and reported as the frequencies for the first three modes of vibration. Three dimensional mode shapes for specified plates have been presented. Comparison of results with published work has been made. © Springer India 2014.