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 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.

Uniformly Loaded Circular Plates With a Central Hole and Both Edges Supported

1957 ◽  
Vol 24 (2) ◽  
pp. 306-310
Author(s):  
J. C. Georgian
Keyword(s):  
Outer Edge ◽  
Central Hole ◽  
Circular Plates ◽  
Simply Supported ◽  
Edge Conditions

Abstract Several catalogs giving results of circular plates with a central hole with various loading and edge conditions have been published. However, these do not cover the case where each edge is supported, either simply or clamped. The usual recommendation is to solve these cases by superposition from the known elementary cases. This is not easily done. The results of the following two cases are given in this paper: (a) Inner and outer edge clamped; (b) inner and outer edge simply supported.

Bending of Circular Plates Under a Uniform Load on a Concentric Circle—Part II

1968 ◽  
Vol 90 (2) ◽  
pp. 279-293
Author(s):  
J. C. Heap
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Concentric Circle ◽  
Outer Edge ◽  
Circular Plates ◽  
Uniform Load ◽  
Simply Supported ◽  
Basic Equations

The basic equations of deflection, slope, and moments for a thin, flat, circular plate subjected to a uniform load on a concentric circle were derived for four generalized cases. From these generalized cases, six simplified cases were deduced. The four generalized cases have the uniform load acting on a concentric circle of the plate between the inner and outer edges, with the following boundary conditions: (a) Outer edge supported and fixed, inner edge fixed; (b) outer edge simply supported, inner edge free; (c) outer edge simply supported, inner edge fixed; and (d) outer edge supported and fixed, inner edge free.

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.

Effect of Pasternak Foundation on Axisymmetric Vibration of Polar Orthotropic Annular Plates of Varying Thickness

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Seema Sharma ◽  
U. S. Gupta ◽  
R. Lal
Keyword(s):  
Plate Theory ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Axisymmetric Vibration ◽  
Classical Plate Theory ◽  
Energy Principle ◽  
Annular Plates ◽  
Edge Conditions ◽  
Modes Of Vibration ◽  
Polar Orthotropic

Free axisymmetric vibrations of polar orthotropic annular plates of variable thickness resting on a Pasternak-type elastic foundation have been studied based on the classical plate theory. Hamilton’s energy principle has been used to derive the governing differential equation of motion. Frequency equations for an annular plate for two different combinations of edge conditions have been obtained employing Chebyshev collocation technique. Numerical results thus obtained have been presented in the form of tables and graphs. The effect of foundation parameter and thickness variation together with various plate parameters such as rigidity ratio, radius ratio, and taper parameter on natural frequencies has been investigated for the first three modes of vibration. Mode shapes for specified plates have been presented. A close agreement of results with those available in the literature shows the versatility of the present technique.

Bending of Circular Plates Under a Variable Symmetrical Load—Part I

1968 ◽  
Vol 90 (2) ◽  
pp. 268-278 ◽  
Author(s):  
J. C. Heap
Keyword(s):  
Radial Distance ◽  
Outer Edge ◽  
Variable Load ◽  
Constant Force ◽  
Circular Plates ◽  
Simply Supported ◽  
Inner Radius ◽  
Solid Plate ◽  
Basic Equations ◽  
Symmetrical Load

The basic equations of deflection, slope, and moment for a thin, flat, circular plate, under a symmetrical variable load, for a constant force divided by the square of the radial distance, have been developed. Six cases have been derived. The first four cases cover the variable load acting over the entire plate, viz., (a) fixed, supported, outer edge and fixed, inner edge, (b) simply supported, outer edge and free, inner edge, (c) simply supported, outer edge and fixed, inner edge, (d) fixed, supported, outer edge and free, inner edge; and the final two cases are for a solid plate having the acting variable load bounded by circles of an inner radius and the outer support-load-radius; i.e., (e) fixed, supported, outer edge and (f) simply supported, outer edge.

scholarly journals Modeling and Simulation of Vibrations of Non-Homogeneous Annular Plate of Quadratic Thickness Resting on Elastic Foundation

2020 ◽  
Vol 8 (10S2) ◽  
pp. 61-65
Keyword(s):  
Mathematical Modeling ◽  
Numerical Simulation ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Annular Plate ◽  
Thickness Variation ◽  
Quintic Spline ◽  
Form Accuracy ◽  
Varying Thickness ◽  
Edge Conditions

Mathematical modeling is presented to analyze natural frequencies of vibrations of an isotropic annular plate of quadratic varying thickness resting on Winkler type elastic foundation where numerical simulation is carried out using quintic spline technique for three different combinations of edge conditions. Effect of elastic foundation, together with nonhomogeneity variation, on the natural frequencies of vibration is illustrated for variety of thickness variation for the first three modes. To compare parametric effect on a specific plate, transverse displacements are presented in normalized form. Accuracy of the results and validity of numerical method is demonstrated by comparing the existing results in the literature.

scholarly journals Influence of boundary constraints on the appearance of asymmetrical equilibrium states in circular plates under normal pressure

2020 ◽  
pp. 38-46
Author(s):  
Svetlana M. Bauer ◽  
Eva B. Voronkova
Keyword(s):  
Radial Direction ◽  
Elasticity Modulus ◽  
Normal Pressure ◽  
Outer Edge ◽  
Buckling Load ◽  
Circular Plates ◽  
Boundary Constraints ◽  
Angular Coordinate ◽  
Elastically Restrained Edge ◽  
Elastically Restrained

Unsymmetrical buckling of nonuniform circular plates with elastically restrained edge and subjected to normal pressure is studied in this paper. The unsymmetric part of the solution is sought in terms of multiples of the harmonics of the angular coordinate. A numerical method is employed to obtain the lowest load value, which leads to the appearance of waves in the circumferential direction. The effect of material heterogeneity and boundary on the buckling load is examined. It is shown that if the outer edge of a plate is elastically restrained against radial deflection, the buckling load for unsymmetrical buckling is larger than for a plate with a movable edge. The elasticity modulus decrease away from the center of a plate leads to sufficient lowering of the buckling pressure if the outer edge can move freely in the radial direction.

The Bending of the Cylindrically Aeolotropic Plate

1944 ◽  
Vol 11 (3) ◽  
pp. A129-A133
Author(s):  
G. F. Carrier
Keyword(s):  
Radial Symmetry ◽  
Outer Edge ◽  
Circular Plates ◽  
Concentrated Force ◽  
Design Problems ◽  
Simply Supported ◽  
Uniform Shear ◽  
Small Deflection ◽  
Degree Of Anisotropy ◽  
Deflection Theory

Abstract In this paper, the small-deflection theory, applicable to plates of cylindrically aeolotropic material, is presented, and expressions are obtained for the moments and deflections produced by the following combinations of loading and boundary conditions: The disk clamped along its circumference and loaded by a uniform lateral pressure; the clamped disk loaded by a central concentrated force; the simply supported disk loaded by uniform edge moment; the disk with a rigid core clamped along its circumference and loaded by a central concentrated force; the ring clamped along its outer edge and loaded by a uniform shear distribution along the inner edge; the simply supported disk with an elastic isotropic core loaded by a uniform edge moment; and the disk under the loading given by p = p0r cos θ. The six symmetrical problems of this group were chosen for evaluation since they and their linear combinations comprise a large part of the total set of such problems. Curves are included showing, for three of the foregoing problems, the effect of the degree of anisotropy on the stresses. The practical application of solutions obtained by the following theory might lie, for example, in design problems involving circular plates which are built up of laminas of radial symmetry with different tangential and radial stiffnesses. The theory could also be applied to the design of circular concrete slabs with different amounts of reinforcing steel in the radial and tangential directions.

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