Vibrations of Rectangular Plates With Nonuniform Elastic Edge Supports

1980 ◽  
Vol 47 (4) ◽  
pp. 891-895 ◽  
Author(s):  
A. W. Leissa ◽  
P. A. A. Laura ◽  
R. H. Gutierrez
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Exact Solutions ◽  
Ritz Method ◽  
Equation Of Motion ◽  
The Other ◽  
Rectangular Plates ◽  
Governing Differential Equation ◽  
Vibration Problems ◽  
Elastic Edge

Two methods are introduced for the solution of free vibration problems of rectangular plates having nonuniform, elastic edge constraints, a class of problems having no previous solutions in the literature. One method uses exact solutions to the governing differential equation of motion, and the other is an extension of the Ritz method. Numerical results are presented for problems having parabolically varying rotational constraints.

Heat Induced Vibration of a Rectangular Plate

1974 ◽  
Vol 96 (3) ◽  
pp. 1015-1021 ◽  
Author(s):  
N. D. Jadeja ◽  
Ta-Cheng Loo
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Rectangular Plate ◽  
Equation Of Motion ◽  
The Other ◽  
Double Trigonometric Series ◽  
Deflection Curve ◽  
Thermally Induced ◽  
Simply Supported ◽  
Governing Differential Equation

The purpose of this paper is to investigate thermally induced vibration of a rectangular plate with one edge fixed and other three edges simply supported. The plate was subjected to a sinusoidal heat input, which varied with respect to time, on one face while the other face of the plate was insulated. An approximate solution to the governing differential equation of motion of the plate was assumed in the form of a double trigonometric series which satisfied all the boundary conditions. Galerkin’s method was then used to obtain the deflection curve for the plate and corresponding stresses at various points in the plate. Certain interesting phenomena indicate the possibility of predicting early fatigue failure. Results are presented in graph forms and discussed.

scholarly journals Analysis on free vibration and critical buckling load of a FGM porous rectangular plate

2021 ◽  
Vol 39 (2) ◽  
pp. 317-325
Author(s):  
Zhaochun Teng ◽  
Pengfei Xi
Keyword(s):  
Differential Equation ◽  
Elastic Modulus ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Natural Frequencies ◽  
Gradient Index ◽  
Rectangular Plates ◽  
Buckling Loads ◽  
Governing Differential Equation

The properties of functionally gradient materials (FGM) are closely related to porosity, which has effect on FGM's elastic modulus, Poisson's ratio, density, etc. Based on the classical theory of thin plates and Hamilton principle, the mathematical model of free vibration and buckling of FGM porous rectangular plates with compression on four sides is established. Then the dimensionless form of the governing differential equation is also obtained. The dimensionless governing differential equation and its boundary conditions are transformed by differential transformation method (DTM). After iterative convergence, the dimensionless natural frequencies and critical buckling loads of the FGM porous rectangular plate are obtained. The problem is reduced to the free vibration of FGM rectangular plate with zero porosity and compared with its exact solution. It is found that DTM gives high accuracy result. The validity of the method is verified in solving the free vibration and buckling problems of the porous FGM rectangular plates with compression on four sides. The results show that the elastic modulus of FGM porous rectangular plate decreases with the increase of gradient index and porosity. Furthermore, the effects of gradient index and porosity on dimensionless natural frequencies and critical buckling loads are further analyzed under different boundary conditions with constant aspect ratio, and the effects of aspect ratio and load on dimensionless natural frequencies under different boundary conditions.

Free-Vibration Analysis of Rectangular Plates With Clamped-Simply Supported Edge Conditions by the Method of Superposition

1977 ◽  
Vol 44 (4) ◽  
pp. 743-749 ◽  
Author(s):  
D. J. Gorman
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Formidable Challenge ◽  
Edge Conditions ◽  
Governing Differential Equation

In this paper attention is focused on the free-vibration analysis of rectangular plates with combinations of clamped and simply supported edge conditions. Plates with at least two opposite edges simply supported are not considered as they have been analyzed in a separate paper. It is well known that the family of problems considered here have presented researchers with a formidable challenge over the years. This is because they are not directly amenable to Le´vy-type solutions. It has been pointed out in the literature that most of the existing solutions are approximate in that they either do not satisfy exactly the governing differential equation or the boundary conditions, or both. In a new approach taken by the author the method of superposition is exploited for handling these dynamic problems. It is found that solutions of any degree of exactitude are easily obtained. The governing differential equation is completely satisfied and the boundary conditions are satisfied to any degree of exactitude by merely increasing the number of terms in the series. Convergence is shown to be remarkably rapid and tabulated results are provided for a large range of parameters. The immediate applicability of the method to problems involving elastic restraint or inertia forces along the plate edges has been discussed in an earlier publication.

Exact Solutions for Free-Vibration Analysis of Rectangular Plates Using Bessel Functions

2005 ◽  
Vol 74 (6) ◽  
pp. 1247-1251 ◽  
Author(s):  
Jiu Hui Wu ◽  
A. Q. Liu ◽  
H. L. Chen
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Exact Solutions ◽  
Thin Plate ◽  
Vibration Analysis ◽  
Bessel Functions ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Edge Conditions

A novel Bessel function method is proposed to obtain the exact solutions for the free-vibration analysis of rectangular thin plates with three edge conditions: (i) fully simply supported; (ii) fully clamped, and (iii) two opposite edges simply supported and the other two edges clamped. Because Bessel functions satisfy the biharmonic differential equation of solid thin plate, the basic idea of the method is to superpose different Bessel functions to satisfy the edge conditions such that the governing differential equation and the boundary conditions of the thin plate are exactly satisfied. It is shown that the proposed method provides simple, direct, and highly accurate solutions for this family of problems. Examples are demonstrated by calculating the natural frequencies and the vibration modes for a square plate with all edges simply supported and clamped.

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.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

A Novel and Accurate Ritz Formulation for Free Vibration of Rectangular and Skew Plates

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
S. A. Eftekhari ◽  
A. A. Jafari
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Rectangular Plates ◽  
Natural Boundary ◽  
Boundary Points ◽  
Natural Boundary Conditions ◽  
Free Edges

One of the major limitations of the conventional Ritz method is its difficulty in implementation to the differential equations with natural boundary conditions at the boundary points/lines. Plates involving free edges/corners and irregularly shaped plates are two historical and classical examples which show that their solutions cannot be accurately approximated by the conventional Ritz method. To solve this difficulty, a simple, novel, and accurate Ritz formulation is introduced in this paper. It is revealed that the proposed methodology can produce much better accuracy than the conventional Ritz method for rectangular plates involving free edges/corners and skew plates.

Free Vibration of a Spinning Stepped Timoshenko Beam

2000 ◽  
Vol 67 (4) ◽  
pp. 839-841 ◽  
Author(s):  
S. D. Yu ◽  
W. L. Cleghorn
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Linear Transformations ◽  
Stepped Beam ◽  
Uniform Cross Section ◽  
Vibration Problems

The finite element method is employed in this paper to investigate free-vibration problems of a spinning stepped Timoshenko beam consisting of a series of uniform segments. Each uniform segment is considered a substructure which may be modeled using beam finite elements of uniform cross section. Assembly of global equation of motion of the entire beam is achieved using Lagrange’s multiplier method. The natural frequencies and mode shapes are subsequently reduced with the help of linear transformations to a standard eigenvalue problem for which a set of natural frequencies and mode shapes may be easily obtained. Numerical results for an overhung stepped beam consisting of three uniform segments are obtained and presented as an illustrative example. [S00021-8936(01)00101-5]

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