On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of the flexural vibration and buckling of isotropic and orthotropic rectangular plates

1986 ◽  
Vol 108 (1) ◽  
pp. 51-62 ◽  
Author(s):  
S.M. Dickinson ◽  
A. Di Blasio
Keyword(s):  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Flexural Vibration ◽  
Rectangular Plates

Dynamic analysis of rectangular plates crossed by distributed moving loads

2017 ◽  
Vol 23 (9) ◽  
pp. 1291-1302 ◽  
Author(s):  
S Sorrentino ◽  
G Catania
Keyword(s):  
Coupling Effect ◽  
Ritz Method ◽  
Flexural Vibration ◽  
Analytical Form ◽  
Relative Magnitude ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Moving Loads ◽  
Railway Bridges ◽  
Parallel Direction

This study investigates the dynamic behaviour of plates crossed by distributed moving gravitational and inertial loads, in the case in which the relative magnitude of the moving mass introduces a coupling effect with the structure, with possible applications to the vibration analysis of railway bridges. A rectangular Kirchhoff plate is considered, simply supported on two opposite edges and free on the other two edges, loaded by a partially distributed mass travelling in the parallel direction with respect to the free edges. The formulation includes damping, and it is accomplished by the Rayleigh–Ritz method, expressing the solution in semi-analytical form. The shape functions for describing the transverse displacement field of the plate are selected as tensor products of linearly independent eigenfunctions of homogeneous uniform beams in flexural vibration, yielding a low-order model with time-dependent coefficients. Numerical examples are then presented and discussed, aimed at investigating the effects of each of the model governing parameters.

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.

Flexural Vibration of In-Plane Loaded Plates with Straight Line/Curved Internal Supports

1993 ◽  
Vol 115 (4) ◽  
pp. 441-447 ◽  
Author(s):  
K. M. Liew ◽  
C. M. Wang
Keyword(s):  
Ritz Method ◽  
Flexural Vibration ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Internal Support ◽  
Straight Line ◽  
Contour Plots ◽  
Complete Set ◽  
Clamped Edge ◽  
Internal Line Supports

An investigation into the vibration analysis of a class of in-plane loaded rectangular plates with internal supports of arbitrary contour is conducted. Solutions to this vibration problem are obtained based on the pb-2 Rayleigh-Ritz method. The Ritz function for this method is defined as the product of (1) a two-dimensional polynomial function expanded in a new manner, (2) equations of the internal support and (3) equations of the boundary supports each raised to the power of either 0, 1, or 2 corresponding to a free, simply supported or clamped edge, respectively. A comparison study on the convergence between the proposed set of polynomials and mathematically complete set of polynomials is conducted. The simplicity and accuracy of the method are demonstrated by analyzing square plates with either two intersecting internal line supports or a central ring support. The influence of the in-plane loads on the natural frequencies will be studied. Note that this paper presents some first known solutions to in-plane loaded rectangular plates with internal supports of arbitrary contour. The mode shapes for these plates are also presented in contour plots.

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