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.

Buckling and Vibration of Orthotropic Nonhomogeneous Rectangular Plates With Bilinear Thickness Variation

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Yajuvindra Kumar ◽  
R. Lal
Keyword(s):  
Orthogonal Polynomials ◽  
Plate Theory ◽  
Ritz Method ◽  
Cartesian Product ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Two Dimensional ◽  
Classical Plate Theory ◽  
Simply Supported

An analysis and numerical results are presented for buckling and transverse vibration of orthotropic nonhomogeneous rectangular plates of variable thickness using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method on the basis of classical plate theory when uniformly distributed in-plane loading is acting at two opposite edges clamped/simply supported. The Gram–Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of various plate parameters such as nonhomogeneity parameters, aspect ratio together with thickness variation, and in-plane load on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported, and free edges correct to four decimal places. Three dimensional mode shapes for a specified plate for all the four boundary conditions have been plotted. By allowing the frequency to approach zero, the critical buckling loads in compression for various values of plate parameters have been computed correct to six significant digits. A comparison of results with those available in the literature has been presented.

AXISYMMETRIC VIBRATION OF POLAR ORTHOTROPIC CIRCULAR PLATES OF QUADRATICALLY VARYING THICKNESS RESTING ON ELASTIC FOUNDATION

2005 ◽  
Vol 05 (03) ◽  
pp. 387-408 ◽  
Author(s):  
N. BHARDWAJ ◽  
A. P. GUPTA
Keyword(s):  
Elastic Foundation ◽  
Ritz Method ◽  
Three Dimensional ◽  
Normal Modes ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Circular Plates ◽  
Varying Thickness ◽  
Edge Conditions ◽  
Polar Orthotropic

This paper is concerned with the axisymmetric vibration problem of polar orthotropic circular plates of quadratically varying thickness and resting on an elastic foundation. The problem is solved by using the Rayleigh–Ritz method with boundary characteristic orthonormal polynomials for approximating the deflection function. Numerical results are computed for frequencies, nodal radii and mode shapes. Three-dimensional graphs are also plotted for the first four normal modes of axisymmetric vibration of plates with free, simply-supported and clamped edge conditions for various values of taper, orthotropy and foundation parameters.

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 Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

2012 ◽  
Vol 19 (3) ◽  
pp. 349-364 ◽  
Author(s):  
R. Lal ◽  
Yajuvindra Kumar
Keyword(s):  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Cartesian Product ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Transverse Vibrations ◽  
Modes Of Vibration ◽  
Boundary Characteristic

The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

Vibration of Rectangular Plates by the Ritz Method

1950 ◽  
Vol 17 (4) ◽  
pp. 448-453 ◽  
Author(s):  
Dana Young
Keyword(s):  
Ritz Method ◽  
Normal Modes ◽  
Approximate Solutions ◽  
The Other ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Modes Of Vibration ◽  
Ritz’S Method ◽  
Square Plate ◽  
Set Up

Abstract Ritz’s method is one of several possible procedures for obtaining approximate solutions for the frequencies and modes of vibration of thin elastic plates. The accuracy of the results and the practicability of the computations depend to a great extent upon the set of functions that is chosen to represent the plate deflection. In this investigation, use is made of the functions which define the normal modes of vibration of a uniform beam. Tables of values of these functions have been computed as well as values of different integrals of the functions and their derivatives. With the aid of these data, the necessary equations can be set up and solved with reasonable effort. Solutions are obtained for three specific plate problems, namely, (a) square plate clamped at all four edges, (b) square plate clamped along two adjacent edges and free along the other two edges, and (c) square plate clamped along one edge and free along the other three edges.

Vibration of Clamped Circular Symmetric Laminates

1994 ◽  
Vol 116 (2) ◽  
pp. 141-145 ◽  
Author(s):  
K. M. Liew
Keyword(s):  
Ritz Method ◽  
Numerical Procedure ◽  
Flexural Vibration ◽  
Boundary Function ◽  
Energy Functional ◽  
Laminated Plates ◽  
Mode Shapes ◽  
Circular Plates ◽  
Present Solution ◽  
Contour Plots

Treated in this paper is the free-flexural vibration analysis of symmetrically laminated thin circular plates. The total energy functional for the laminated plates is formulated where the pb-2 Ritz method is applied for the solution. The assumed displacement is defined as the product of (1) a two-dimensional complete polynomial function and (2) a basic boundary function. The simplicity and accuracy of the numerical procedure will be demonstrated by solving some plate examples. In the present study, the effects of material properties, number of layers and fiber stacking sequences upon the vibration frequency parameters are investigated. Selected mode shapes by means of contour plots for several 16-ply laminated plates with different fiber stacking sequences and composite materials are presented. This study may provide valuable information for researchers and engineers in design applications. In addition, the present solution plays an important role in increasing the existing data base for future references.

Vibration Analysis of Anisotropic Simply Supported Plates by Using Variable Kinematic and Rayleigh-Ritz Method

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Erasmo Carrera ◽  
Fiorenzo Adolfo Fazzolari ◽  
Luciano Demasi
Keyword(s):  
Vibration Analysis ◽  
Ritz Method ◽  
Single Layer ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Virtual Displacement ◽  
Simply Supported ◽  
Order Expansion ◽  
Thickness Layer ◽  
Made In

This work deals with accurate free-vibration analysis of anisotropic, simply supported plates of square planform. Refined plate theories, which include layer-wise, equivalent single layer and zig-zag models, with increasing number of displacement variables are take into account. Linear up to fourth N-order expansion, in the thickness layer-plate direction have been implemented for the introduced displacement field. Rayleigh-Ritz method based on principle of virtual displacement is derived in the framework of Carrera’s unified formulation. Regular symmetric angle-ply and cross-ply laminates are addressed. Convergence studies are made in order to demonstrate that accurate results are obtained by using a set of trigonometric functions. The effects of the various parameters (material, number of layers, and fiber orientation) upon the frequencies and mode shapes are discussed. Numerical results are compared with available results in literature.

Accurate Vibration Analysis of Simply Supported Rhombic Plates by Considering Stress Singularities

1995 ◽  
Vol 117 (3A) ◽  
pp. 245-251 ◽  
Author(s):  
C. S. Huang ◽  
O. G. McGee ◽  
A. W. Leissa ◽  
J. W. Kim
Keyword(s):  
Ritz Method ◽  
Mode Shapes ◽  
Transverse Displacement ◽  
Algebraic Polynomials ◽  
Stress Singularities ◽  
Simply Supported ◽  
Free Transverse Vibration ◽  
Skewed Plates ◽  
Convergence Of Solution ◽  
Skew Angles

This is the first known work which explicitly considers the bending stress singularities that occur in the two opposite, obtuse corner angles of simply supported rhombic plates undergoing free, transverse vibration. The importance of these singularities increases as the rhombic plate becomes highly skewed (i.e., the obtuse angles increase). The analysis is carried out by the Ritz method using a hybrid set consisting of two types of displacement functions, e.g., (1) algebraic polynomials and (2) corner functions accounting for the singularities in the obtuse corners. It is shown that the corner functions accelerate the convergence of solution, and that these functions are required if accurate solutions are to be obtained for highly skewed plates. Accurate nondimensional frequencies and normalized contours of the vibratory transverse displacement are presented for simply supported rhombic plates with skew angles ranging to 75 deg. (i.e., obtuse angles of 165 deg.). Frequency and mode shapes of isosceles and right triangular plates with all edges simply supported are also available from the data presented.

scholarly journals Almost Classically Damped Continuous Linear Systems

1998 ◽  
Vol 65 (4) ◽  
pp. 1022-1031 ◽  
Author(s):  
S. Natsiavas ◽  
J. L. Beck
Keyword(s):  
Boundary Conditions ◽  
Normal Modes ◽  
Mode Shapes ◽  
Continuous Linear ◽  
Perturbation Expansions ◽  
Field Equations ◽  
Regular Perturbation ◽  
Modes Of Vibration ◽  
Proper Splitting ◽  
Damped Systems

The dynamic response of a general class of continuous linear vibrating systems is analyzed which possess damping properties close to those resulting in classical (uncoupled) normal modes. First, conditions are given for the existence of classical modes of vibration in a continuous linear system, with special attention being paid to the boundary conditions. Regular perturbation expansions in terms of undamped modeshapes are then utilized for analyzing the eigenproblem as well as the vibration response of almost classically damped systems. The analysis is based on a proper splitting of the damping operators in both the field equations and the boundary conditions. The main advantage of this approach is that it allows application of standard modal analysis methodologies so that the problem is reduced to that of finding the frequencies and mode shapes of the corresponding undamped system. The approach is illustrated by two simple examples involving rod and beam vibrations.

scholarly journals FREE VIBRATION OF ISOTROPIC HALF-ELLIPTIC PLATES OF LINEARLY VARYING THICKNESS WITH CLAMPED CURVED BOUNDARY

2010 ◽  
Vol 7 (2) ◽  
pp. 1-15
Author(s):  
A.P Gupta ◽  
N. Bhardwaj ◽  
K.K. Choong
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Mode Shapes ◽  
Chebyshev Collocation Method ◽  
Integral Equation Methods ◽  
Aspect Ratios ◽  
Dimensional Boundary ◽  
Title Problem ◽  
Significant Figures

Two-dimensional boundary characteristic orthonormal polynomials are used in Rayleigh-Ritz method to study the title problem. In general, it is found that this method gives better results than the other traditional method such as boundary integral equation methods, Spline methods, Chebyshev collocation method, Frobenius method etc. The thickness is taken to be linearly varying in two orthogonal directions. Comparisons in particular cases have been made with the existing results in the literature. Convergence of frequencies of at least up to five significant figures is obtained. Results showing the variation in frequencies with taper parameters and aspect ratios are presented in tabular form. Mode shapes are shown using three-dimensional graphs of plates in displaced configurations.

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