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.

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.

scholarly journals Buckling and Vibration of Non-Homogeneous Rectangular Plates Subjected to Linearly Varying In-Plane Force

2013 ◽  
Vol 20 (5) ◽  
pp. 879-894 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Material ◽  
Classical Plate Theory ◽  
Simply Supported

The present work analyses the buckling and vibration behaviour of non-homogeneous rectangular plates of uniform thickness on the basis of classical plate theory when the two opposite edges are simply supported and are subjected to linearly varying in-plane force. For non-homogeneity of the plate material it is assumed that young's modulus and density of the plate material vary exponentially along axial direction. The governing partial differential equation of motion of such plates has been reduced to an ordinary differential equation using the sine function for mode shapes between the simply supported edges. This resulting equation has been solved numerically employing differential quadrature method for three different combinations of clamped, simply supported and free boundary conditions at the other two edges. The effect of various parameters has been studied on the natural frequencies for the first three modes of vibration. Critical buckling loads have been computed. Three dimensional mode shapes have been presented. Comparison has been made with the known results.

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 ANALYSES OF CRACKED PLATES BY THE RITZ METHOD WITH MOVING LEAST-SQUARES INTERPOLATION FUNCTIONS

2014 ◽  
Vol 14 (02) ◽  
pp. 1350060 ◽  
Author(s):  
C. S. HUANG ◽  
C. W. CHAN
Keyword(s):  
Least Squares ◽  
Plate Theory ◽  
Ritz Method ◽  
Thin Plates ◽  
Mode Shapes ◽  
Moving Least Squares ◽  
Rectangular Plates ◽  
Classical Plate Theory ◽  
Cracked Plates ◽  
Admissible Functions

The solutions for the vibrations of cracked thin plates are obtained by the Ritz method with admissible functions. Based on the classical plate theory, the basis functions comprising polynomials and crack functions are adopted to generate the admissible functions by the moving least-squares approach for a set of nodes randomly distributed in the domain. The crack functions account for the singular behaviors of stress resultants at crack tip(s), which are discontinuous in displacement and slope across the crack. The present solutions are validated through convergence tests of frequencies and by comparison with the published results for simply-supported cracked rectangular plates. The solutions are further employed to determine the natural frequencies of cantilevered skewed rhombic and isosceles triangular plates and completely free circular plates, each with a crack of varying length, location and orientation. The numerical results are tabulated and some corresponding mode shapes are also presented, by means of nodal patterns. Most of the results shown here are new to the literature.

Effect of Shear Deformations on the Bending of Rectangular Plates

1960 ◽  
Vol 27 (1) ◽  
pp. 54-58 ◽  
Author(s):  
V. L. Salerno ◽  
M. A. Goldberg
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Plate Theory ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Wide Range

The three partial differential equations derived by Dr. E. Reissner2, 3 have been reduced to a fourth-order partial differential equation resembling that of the classical plate theory and to a second-order differential equation for determining a stress function. The general solution for the two partial differential equations has been applied to a simply supported plate with a constant load p and to a plate with two opposite edges simply supported and the other two edges free. Numerical calculations have been made for the simply supported plate and the results compared with those of classical theory. The calculations for a wide range of parameters indicate that the deviation is small.

Effect of Nonhomogeneity on Vibration of Orthotropic Rectangular Plates of Varying Thickness Resting on Pasternak Foundation

2009 ◽  
Vol 131 (1) ◽  
Author(s):  
Roshan Lal ◽  
Dhanpati
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
The Other ◽  
Thickness Variation ◽  
Order Partial Differential Equation ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Varying Thickness ◽  
Aspect Ratios ◽  
Displacement Mode

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates of varying thickness with two opposite simply supported edges (y=0 and y=b) and resting on two-parameter foundation (Pasternak-type) have been studied on the basis of classical plate theory. The other two edges (x=0 and x=a) may be any combination of clamped and simply supported edge conditions. The nonhomogeneity of the plate material is assumed to arise due to the exponential variations in Young’s moduli and density along one direction. By expressing the displacement mode as a sine function of the variable between simply supported edges, the fourth order partial differential equation governing the motion of such plates of exponentially varying thickness in another direction gets reduced to an ordinary differential equation with variable coefficients. The resulting equation is then solved numerically by using the Chebyshev collocation technique for two different combinations of clamped and simply supported conditions at the other two edges. The lowest three frequencies have been computed to study the behavior of foundation parameters together with other plate parameters such as nonhomogeneity, density, and thickness variation on the frequencies of the plate with different aspect ratios. Normalized displacements are presented for a specified plate. A comparison of results with those obtained by other methods shows the computational efficiency of the present approach.

Applications of Holography to Dynamics: High-Frequency Vibrations of Plates

1970 ◽  
Vol 37 (4) ◽  
pp. 1083-1090 ◽  
Author(s):  
R. Aprahamian ◽  
D. A. Evensen
Keyword(s):  
High Frequency ◽  
Plate Theory ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Classical Plate Theory ◽  
Simply Supported ◽  
Transverse Vibrations ◽  
Shear Effects ◽  
High Frequency Vibrations ◽  
Very High

The techniques of holographic interferometry are applied to study the high-frequency transverse vibrations of a simply supported rectangular plate. Over 110 vibration modes were identified using stored beam holography, at frequencies ranging from 162 cps to 20,000 cps. In addition, three very high modes were identified at frequencies up to 76.8 kcps. The corresponding modal numbers were m = 17, n = 31, for the highest mode identified. Time average holograms were made of these and other modes, and photographs made from the holograms are included herein. The experimental mode shapes and frequencies were generally in agreement with classical plate theory, except for the three highest modes. The latter agreed with Mindlin’s plate theory, which includes rotary inertia and shear effects. Other applications of holography to dynamics are briefly discussed.

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.

Mode shapes and frequencies of thin rectangular plates with arbitrarily varying non-homogeneity along two concurrent edges

2016 ◽  
Vol 23 (17) ◽  
pp. 2841-2865 ◽  
Author(s):  
Roshan Lal ◽  
Renu Saini
Keyword(s):  
Eigenvalue Problems ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Transverse Vibrations ◽  
Modes Of Vibration ◽  
Generalized Differential

Analysis and numerical results are presented for free transverse vibrations of isotropic rectangular plates having arbitrarily varying non-homogeneity with the in-plane coordinates along the two concurrent edges on the basis of Kirchhoff plate theory. For the non-homogeneity, a general type of variation for Young’s modulus and density of the plate material has been assumed. Generalized differential quadrature method has been used to obtain the eigenvalue problem for such model of plates for four different combinations of boundary conditions at the edges namely, (i) fully clamped, (ii) two opposite edges are clamped and other two are simply supported, (iii) two opposite edges are clamped and other two are free, and (iv) two opposite edges are simply supported and other two are free. By solving these eigenvalue problems using software MATLAB, the lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration. The effect of various plate parameters on the vibration characteristics has been analysed. Three dimensional mode shapes have been plotted. A comparison of results with those available in literature has been presented.

Vibration Behavior and Simplified Design of Thick Rectangular Plates With Variable Thickness

2002 ◽  
Vol 124 (2) ◽  
pp. 302-309 ◽  
Author(s):  
M. Sasajima ◽  
T. Kakudate ◽  
Y. Narita
Keyword(s):  
Optimal Design ◽  
Boundary Conditions ◽  
Fundamental Frequency ◽  
Plate Theory ◽  
Free Vibration Analysis ◽  
Design Approach ◽  
Mindlin Plate ◽  
Thickness Variation ◽  
Rectangular Plates ◽  
Classical Plate Theory

A free vibration analysis and an optimal design approach have been presented for thick isotropic rectangular plates with varying thickness under general edge conditions. First, the analysis is developed for vibrating rectangular plates by using the Mindlin plate theory and an eigenvalue problem is formulated by extending a method of Ritz to arbitrary sets of standard boundary conditions. The classical plate theory is also used to derive the frequency equation for comparison purpose. Secondly, a simplified optimal design approach is proposed to maximize the fundamental frequency of the plates. In applying this approach, the thickness variation is assumed to be linear in one direction and a taper ratio is chosen to be a design variable that represents the whole plate design. This approach significantly reduces the process for obtaining optimal or nearly optimal design under constraint of the constant plate volume. Numerical results are presented for various sets of boundary conditions, thickness ratio and two different plate theories, and their effects on the optimal taper ratio and its corresponding maximized fundamental frequency are discussed.

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