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.

ACCURATE FREQUENCIES AND MODE SHAPES FOR MODERATELY THICK, CANTILEVERED, SKEW PLATES

2007 ◽  
Vol 07 (03) ◽  
pp. 425-440 ◽  
Author(s):  
A. W. LEISSA ◽  
C. S. HUANG ◽  
M. J. CHANG
Keyword(s):  
Plate Theory ◽  
Ritz Method ◽  
The Other ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Transverse Displacement ◽  
Algebraic Polynomials ◽  
Stress Singularities ◽  
Dependent Variables ◽  
Mindlin Plate Theory

Accurate free vibration frequencies and mode shapes are presented for complete sets of moderately thick, cantilevered skew plates of triangular, trapezoidal and parallelogram shape. These accurate results are obtained by using the Ritz method applied to the Mindlin plate theory. Two sets of functions are employed simultaneously for each of the three dependent variables: transverse displacement (w) and bending rotations (ϕx and ϕy). One set is the widely used algebraic polynomials. The other is the set of corner functions which provide the proper stress singularities in the reentrant clamped-free corner, and accelerates the convergence of the solutions. The extensive frequencies presented are exact to the four digits shown. Corresponding mode shapes are also shown, by means of nodal patterns, most of which are novel in the published literature.

Vibrations of Sectorial Plates Having Corner Stress Singularities

1993 ◽  
Vol 60 (1) ◽  
pp. 134-140 ◽  
Author(s):  
A. W. Leissa ◽  
O. G. McGee ◽  
C. S. Huang
Keyword(s):  
Outer Boundary ◽  
Full Range ◽  
Ritz Method ◽  
Trigonometric Polynomials ◽  
Mode Shapes ◽  
Vertex Angle ◽  
Stress Singularities ◽  
Simply Supported ◽  
Plate Vibrations ◽  
Continuous Boundary

A procedure is presented for determining the free vibration frequencies and mode shapes of sectorial plates having re-entrant corners (i.e., vertex angles exceeding 180 degrees). No correct results for such problems have been found in the vast literature of plate vibrations. The procedure is applicable to sectorial plates having arbitrary (but continuous) boundary conditions (e.g., clamped, simply supported, or free) along the two radial edges and the circular edge. It is based upon the Ritz method, but utilizes two sets of admissible functions simultaneously. One set consists of algebraic-trigonometric polynomials. The other is the set of corner functions derived by Williams (1952) to deal with the bending stress singularities which may arise at the corner when the vertex angle becomes large. The method is demonstrated for sectorial plates having all edges simply supported, which yields the strongest singularity in a re-entrant corner. Frequencies are compared with those obtained from an analytical solution involving Bessel functions. It is shown that the latter solution is invalid for re-entrant corners. Analytical solutions are also obtained for annular sectorial plates having very small ratios of inner to outer boundary radii. These solutions are found to be consistent with those using polynomials and corner functions. Accurate fundamental frequency data is presented for simply supported sectorial plates having three values of Poisson’s ratio (0, 0.3, 0.5) and the full range of vertex angles (0 < α ≤ 360 deg).

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.

Vibration and Damping Analysis of Rectangular Plate With Partially Covered Constrained Viscoelastic Layer

1987 ◽  
Vol 109 (3) ◽  
pp. 241-247 ◽  
Author(s):  
A. K. Lall ◽  
N. T. Asnani ◽  
B. C. Nakra
Keyword(s):  
Loss Factor ◽  
Ritz Method ◽  
Viscoelastic Damping ◽  
Viscoelastic Layer ◽  
Transverse Displacement ◽  
Modal System ◽  
Resonant Frequencies ◽  
Simply Supported ◽  
The Core ◽  
Single Term

The Rayleigh-Ritz method is applied for the damping analysis offlexural vibrations of a simply supported plate, partially covered with constrained viscoelastic damping treatment. The anaysis is carried out in terms of resonant frequencies and associated modal system loss factors. Single-term solutions for respective modes are assumed for the longitudinal displacements of the constraining layer and the transverse displacement of the plate. The variations of the resonant frequency and the associated modal loss factor with the coverage percentage, the coverage location, the core shear modulus and the thickness of the constrained and the constraining layers have been reported.

Three-Dimensional Vibration Analysis of Thick, Complete Conical Shells

2004 ◽  
Vol 71 (4) ◽  
pp. 502-507 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Present Method ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Conical Shells ◽  
Thin Shell Theory ◽  
Time Periodic

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur,uz, and uθ in the radial, axial, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z-directions. Potential (strain) and kinetic energies of the conical shells are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the conical shells. Novel numerical results are presented for thick, complete conical shells of revolution based upon the 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and a 2D thin shell theory.

Exact Solutions for the Free Vibrations and Buckling of Rectangular Plates With Linearly Varying In-Plane Loading

2001 ◽  
Author(s):  
Arthur W. Leissa ◽  
Jae-Hoon Kang
Keyword(s):  
Solution Procedure ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Buckling Loads ◽  
Simply Supported ◽  
Edge Conditions ◽  
Elastically Supported

Abstract An exact solution procedure is formulated for the free vibration and buckling analysis of rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying normal stresses. The other two edges may be clamped, simply supported or free, or they may be elastically supported. The transverse displacement (w) is assumed as sinusoidal in the direction of loading (x), and a power series is assumed in the lateral (y) direction (i.e., the method of Frobenius). Applying the boundary conditions yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to obtain adequate convergence for accurate vibration frequencies and buckling loads, as is demonstrated by two convergence tables. Some interesting and useful results for vibration frequencies and buckling loads, and their mode shapes, are presented for a variety of edge conditions and in-plane loadings, especially pure in-plane moments.

3D Vibration Analysis of Combined Shells of Revolution

2019 ◽  
Vol 19 (02) ◽  
pp. 1950005 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Legendre Polynomials ◽  
Thin Shell ◽  
Ritz Method ◽  
Three Dimensional ◽  
Surface Strain ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Thick Shells ◽  
Admissible Functions

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies and the mode shapes of combined hemispherical–cylindrical shells of revolution with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2D) conventional thin shell theories or higher-order thick shell theories, the present method is based upon the 3D dynamic equations of elasticity. Mathematically, minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangent to the shell mid-surface. Strain and kinetic energies of the combined shell of revolution with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for the combined shells of revolution with or without a top opening, which are completely free and fixed at the bottom of the combined shells. The frequencies from the present 3D Ritz method are compared with those from 2D thin shell theories by previous researchers. The present analysis is applicable to very thick shells as well as very thin shells.

Free Vibration and Tension Buckling of Circular Plates With Diametral Point Forces

1990 ◽  
Vol 57 (4) ◽  
pp. 995-999 ◽  
Author(s):  
E. F. Ayoub ◽  
A. W. Leissa
Keyword(s):  
Circular Plate ◽  
Tensile Force ◽  
Ritz Method ◽  
Compressive Force ◽  
Free Vibrations ◽  
Plane Elasticity ◽  
Mode Shapes ◽  
Simply Supported ◽  
Symmetry Classes ◽  
Tensile Forces

This paper presents the first known results for the free vibrations of a circular plate subjected to a pair of static, concentrated forces acting on the boundary at opposite ends of a diameter. The closed-form exact solution of the plane elasticity problem is used to provide the in-plane stress distribution for the vibration problem. A proper procedure using the Ritz method is developed for solving the latter problem for clamped, simply supported, or free boundary conditions. Numerical results are given for the vibration frequencies of a simply supported circular plate, which separate into four symmetry classes of mode shapes. Compressive buckling loads for each symmetry class are determined as a special case as the frequencies decrease to zero with increasing compressive force. Tracking the frequency versus loading data with increasing tensile forces shows that buckling due to tensile force can also occur, and the critical value of the force is found.

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.

FREE VIBRATION AND BUCKLING ANALYSIS OF HIGHLY SKEWED PLATES BY LEAST SQUARES-BASED FINITE DIFFERENCE METHOD

2010 ◽  
Vol 10 (02) ◽  
pp. 225-252 ◽  
Author(s):  
W. X. WU ◽  
C. SHU ◽  
C. M. WANG ◽  
Y. XIANG
Keyword(s):  
Finite Difference ◽  
Free Vibration ◽  
Least Squares ◽  
Stress Singularities ◽  
Skew Angle ◽  
Good Convergence ◽  
Mesh Free ◽  
Skewed Plates ◽  
Skew Angles ◽  
First Time

It is well-known that stress singularities occur at the obtuse corners of skew plates, especially when the skew angles are large. Owing to the stress singularities, accurate bending results, vibration frequencies and buckling loads of highly skewed plates are difficult to obtain accurately. In this paper, the mesh-free least squares-based finite difference (LSFD) method is proposed for solving the free vibration and buckling problems of highly skewed plates. As such vibration and buckling results are scarce in the open literature, the method was verified by comparing the LSFD solutions with existing ones having a skew angle θ ≤ 70°, or by carrying out convergence studies. The vibration and buckling results for plates with very large skew angle (θ = 80°) are presented for the first time. The close agreement observed in the comparison studies and the good convergence behavior of the LSFD solutions provide the confidence that these vibration and buckling results predicted by the LSFD method are of good accuracy.

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