Modal analysis of circular plates with radial side cracks and in contact with water on one side based on the Rayleigh–Ritz method

2012 ◽  
Vol 331 (1) ◽  
pp. 231-251 ◽  
Author(s):  
X.H. Si ◽  
W.X. Lu ◽  
F.L. Chu
Keyword(s):  
Modal Analysis ◽  
Ritz Method ◽  
Circular Plates ◽  
Radial Side

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.

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.

scholarly journals Modal Analysis of Circular Symmetrical Plates by Means of Generalized Finite Difference Method

2019 ◽  
Vol 17 (3) ◽  
pp. 88-98
Author(s):  
A. E. Mansour
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Modal Analysis ◽  
Difference Method ◽  
Numerical Solutions ◽  
Analysis Procedure ◽  
Machine Design ◽  
Polar Coordinates ◽  
Circular Plates ◽  
Classical Formulation

In this paper, a simplified modal analysis procedure of circular plates procedures (on polar domains) through generalized (modernized) finite difference method (abbreviated next as – FDM) is developed.Generally, circular plates are widely used for a plenty of modern civilian and industrial utilities, machine design and many other purposes. They form a spectrum of elements starting with trains’ bogies along with engine pistons, dampers and up to slabs and roofs over circular-shaped buildings, train stationsand other transportation facilities. Nowadays, FDM predominates the numerical solutions of partial differential equations (abbreviated next as – PDE) not less than the method of finite elements (abbreviated next as – FEM). This is wide-famous mathematical-discretization method that is economic to compute and simple to code, less regarding to computation tools in hands and how powerful/less powerful they are, since it bases on replacing each derivative by a difference algebraic quotient in a classical formulation. In a sense, a finite difference formulation offers a more direct approach to the numerical solution of the PDE especially in polar coordinates domain problems considering curvilinear dimensions that even FEM does not.The generalized approach of FDM considers many parameters less regarded by the classical one.  Consequently, the use of classical approach negatively affects the accuracy of calculation (convergence to the exact solution values) and the tendency of results, the thing been healed by the generalized approach.

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.

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.

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