Vibration of Nonhomogeneous Orthotropic Elliptic and Circular Plates With Variable Thickness

2006 ◽  
Vol 129 (2) ◽  
pp. 256-259 ◽  
Author(s):  
S. Chakraverty ◽  
Ragini Jindal ◽  
V. K. Agarwal
Keyword(s):  
Variable Thickness ◽  
Ritz Method ◽  
Thickness Variation ◽  
Circular Plates ◽  
Plate Material ◽  
Orthotropic Plates ◽  
Various Boundary Conditions ◽  
Vibrational Characteristics ◽  
Schmidt Orthogonalization ◽  
Orthogonalization Procedure

In this paper, study of nonhomogeneity as well as variable thickness in elliptic and circular orthotropic plates is undertaken. Nonhomogeneity of plate material is assumed to be a quadratic variation of Young’s modulii and density whereas shear modulus, is considered to vary linearly along both the axes. The quadratic thickness variation in orthotropic nonhomogeneous plates is also considered. Effect of variation of these parameters on vibrational characteristics are analyzed for various boundary conditions at the edges. Results are obtained using boundary characteristic orthogonal polynomials generated by using Gram-Schmidt orthogonalization procedure in Rayleigh-Ritz method.

LINEAR BENDING ANALYSIS OF INHOMOGENEOUS VARIABLE-THICKNESS ORTHOTROPIC PLATES UNDER VARIOUS BOUNDARY CONDITIONS

2007 ◽  
Vol 04 (03) ◽  
pp. 417-438 ◽  
Author(s):  
A. M. ZENKOUR ◽  
M. N. M. ALLAM ◽  
D. S. MASHAT
Keyword(s):  
Boundary Conditions ◽  
Variable Thickness ◽  
Flexural Rigidity ◽  
Approximate Solutions ◽  
Rectangular Plates ◽  
Orthotropic Plates ◽  
Simply Supported ◽  
Various Boundary Conditions ◽  
Thickness Parameter ◽  
Space Concept

An exact solution to the bending of variable-thickness orthotropic plates is developed for a variety of boundary conditions. The procedure, based on a Lévy-type solution considered in conjunction with the state-space concept, is applicable to inhomogeneous variable-thickness rectangular plates with two opposite edges simply supported. The remaining ones are subjected to a combination of clamped, simply supported, and free boundary conditions, and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for inhomogeneous variable-thickness plates as for isotropic homogeneous plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Benchmark numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of inhomogeneity, aspect ratio, thickness parameter, and the degree of nonuniformity on the deflections and stresses are investigated.

Splines in vibration analysis of non-homogeneous circular plates of quadratic thickness

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Robin Singh ◽  
Neeraj Dhiman ◽  
Mohammad Tamsir
Keyword(s):  
Radial Direction ◽  
Outer Edge ◽  
Thickness Variation ◽  
Circular Plates ◽  
Plate Material ◽  
Quintic Spline ◽  
Simply Supported ◽  
Edge Conditions ◽  
Modes Of Vibration ◽  
First Time

Abstract Mathematical model to account for non-homogeneity of plate material is designed, keeping in mind all the physical aspects, and analyzed by applying quintic spline technique for the first time. This method has been applied earlier for other geometry of plates which shows its utility. Accuracy and versatility of the technique are established by comparing with the well-known existing results. Effect of quadratic thickness variation, an exponential variation of non-homogeneity in the radial direction, and variation in density; for the three different outer edge conditions namely clamped, simply supported and free have been computed using MATLAB for the first three modes of vibration. For all the three edge conditions, normalized transverse displacements for a specific plate have been presented which shows the shiftness of nodal radii with the effect of taperness.

scholarly journals Asymmetric Vibration of Polar Orthotropic Annular Circular Plates of Quadratically Varying Thickness with Same Boundary Conditions

2008 ◽  
Vol 15 (6) ◽  
pp. 599-617 ◽  
Author(s):  
N. Bhardwaj ◽  
A.P. Gupta ◽  
K.K. Choong
Keyword(s):  
Ritz Method ◽  
Plate Thickness ◽  
Thickness Variation ◽  
Lateral Vibration ◽  
Circular Plates ◽  
Thickness Profile ◽  
Varying Thickness ◽  
Winkler Elastic Foundation ◽  
Polar Orthotropic ◽  
Asymmetric Vibration

In the present paper, asymmetric vibration of polar orthotropic annular circular plates of quadratically varying thickness resting on Winkler elastic foundation is studied by using boundary characteristic orthonormal polynomials in Rayleigh-Ritz method. Convergence of the results is tested and comparison is made with results already available in the existing literature. Numerical results for the first ten frequencies for various values of parameters describing width of annular plate, thickness profile, material orthotropy and foundation constant for all three possible combinations of clamped, simply supported and free edge conditions are shown and discussed. It is found that (a) higher elastic property in circumferential direction leads to higher stiffness against lateral vibration; (b) Lateral vibration characteristics ofF-Fplatesis more sensitive towards parametric changes in material orthotropy and foundation stiffness thanC-CandS-Splates; (c) Effect of quadratical thickness variation on fundamental frequency is more significant in cases ofC-CandS-S platesthan that ofF-Fplates. Thickness profile which is convex relative to plate center-line tends to result in higher stiffness of annular plates against lateral vibration than the one which is concave and (d) Fundamental mode of vibration ofC-CandS-Splatesis axisymmetrical while that ofF-Fplatesis asymmetrical.

scholarly journals Transverse Vibrations of Clamped and Simply-Supported Circular Plates with Two Dimensional Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 273-285 ◽  
Author(s):  
N. Bhardwaj ◽  
A.P. Gupta ◽  
K.K. Choong ◽  
C.M. Wang ◽  
Hiroshi Ohmori
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Two Dimensional ◽  
Circular Plates ◽  
Simply Supported ◽  
Modes Of Vibration ◽  
Dimensional Boundary ◽  
Special Case

Two dimensional boundary characteristic orthonormal polynomials are used in the Ritz method for the vibration analysis of clamped and simply-supported circular plates of varying thickness. The thickness variation in the radial direction is linear whereas in the circumferential direction the thickness varies according to coskθ, wherekis an integer. In order to verify the validity, convergence and accuracy of the results, comparison studies are made against existing results for the special case of linearly tapered thickness plates. Variations in frequencies for the first six normal modes of vibration and mode shapes for various taper parameters are presented.

Axi-Symmetric Buckling of Orthotopic Circular Plates with Variable Thickness

1967 ◽  
Vol 71 (675) ◽  
pp. 218-223 ◽  
Author(s):  
Sharad A. Patel ◽  
Franklin J. Broth
Keyword(s):  
Circular Plate ◽  
Variable Thickness ◽  
Material Properties ◽  
Radial Direction ◽  
Constant Thickness ◽  
Thickness Variation ◽  
Circular Plates ◽  
Plate Edge ◽  
Simply Supported

Axi-symmetric buckling of a circular plate having different material properties in the radial and circumferential directions was analysed in ref. 1. A plate with constant thickness and subjected to a uniform edge compression was considered. The plate edge was assumed clamped or simply-supported. The analysis of ref. 1 is extended to include plates with thickness variation in the radial direction.

scholarly journals Modelling of fibre steered plates with coupled thickness variation from overlapping continuous tows

2020 ◽  
Author(s):  
Lander Vertonghen ◽  
Saullo G. P. Castro
Keyword(s):  
Variable Thickness ◽  
Plane Surface ◽  
Thickness Distribution ◽  
Ritz Method ◽  
Variable Stiffness ◽  
Element Model ◽  
Analytical Models ◽  
Thickness Variation ◽  
Wide Range ◽  
Modelling Strategies

Previous research has hinted on further improvements of the buckling behaviour of variable-stiffness laminates by incorporating overlaps, resulting in a variable thickness profile that is non-linearly coupled to the steering angles. The present study compares two modelling strategies to consider the variable thickness distribution: 1) as manufactured with discrete thicknesses; and 2) smoothed with a continuous thickness distribution. The as-manufactured discrete thickness created by overlapping tows is obtained by means of virtually manufactured laminates. The smeared approximation is much simpler to implement, whereby the local thickness is a non-linear function of the local steering angle. Linear buckling analyses are performed by means of fast semi-analytical models based on the Ritz method using hierarchical polynomials and classical plate formulation. By assuming a smooth manufacturing mould on one side, a one-sided thickness variation is produced, resulting in non-symmetric laminates for which the mid-plane surface is varied accordingly. Modelling guidelines are provided regarding the use of the smeared model in a study covering a wide range of geometries, loading and boundary conditions. With these guidelines, one can apply the smeared thickness technique in semi-analytical models to reach a correlation within ±5% compared to a costly discrete-thickness finite element model.

scholarly journals Vibration analysis of functionally graded circular plates of variable thickness under thermal environment by generalized differential quadrature method

2019 ◽  
Vol 26 (1-2) ◽  
pp. 73-87 ◽  
Author(s):  
Roshan Lal ◽  
Rahul Saini
Keyword(s):  
Variable Thickness ◽  
Thermal Environment ◽  
Differential Quadrature Method ◽  
Functionally Graded ◽  
Quadrature Method ◽  
Circular Plates ◽  
Plate Material ◽  
Generalized Differential ◽  
Taper Constant ◽  
Plates Of Variable Thickness

The vibration of functionally graded circular plates of variable thickness under a thermal environment is analyzed when the nodal lines are concentric circles by using the generalized differential quadrature method for the nonlinear temperature distribution in the thickness direction. The parabolic variation in thickness along the radial direction is controlled by a taper constant. The plate material is graded in the transverse direction and its mechanical properties are temperature-dependent. The thermal environment over the top and bottom surfaces of the plate is assumed to be uniform. Hamilton's principle has been used in obtaining the governing differential equations for thermo-elastic equilibrium and axisymmetric motion for such a plate model employing Kirchhoff plate theory. Numerical results for thermal displacements and natural frequencies of clamped and simply supported plates have been obtained using MATLAB. The effect of the taper constant, volume fraction index, and temperature difference on the vibration characteristics has been analyzed for the lowest three modes of vibration. A study in which the plate material has temperature-independent properties has also been performed. The accuracy of the present technique is verified by comparing the results with those available in the literature.

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