Characteristic Orthogonal Polynomials Research Articles

Natural frequencies of arbitrarily shaped plates using the Rayleigh-Ritz method together with natural co-ordinate regions and normalized characteristic orthogonal polynomials

A Rayleigh-Ritz method mapping arbitrarily shaped plates on to a natural co-ordinate plane using serendipity functions is presented. The displacement function uses normalized beam characteristic orthogonal polynomials. The kinetic and strain energy equations are mapped from Cartesian co-ordinates to natural co-ordinates and the Rayleigh coefficient is minimized. Flexural free vibration analyses of various shaped plates is performed. Results are in excellent agreement with those found in the literature. This paper corrects the theory previously presented on this topic.

Vibration of mindlin plates using boundary characteristic orthogonal polynomials

Abstract The vibration analysis of shear deformable plates formulated on the basis of first order Mindlin theory is presented. The displacement and rotational functions of the plates are approximated by sets of boundary characteristic orthogonal polynomials. The ease of generation and manipulation of these polynomial functions greatly enhances the computational efficiency of the numerical method. The energy functional of the shear deformable plates derived from the Mindlin plate theory is minimized in the Ritz procedure to arrive at the governing eigenvalue equation. Corresponding natural frequencies and mode shapes can be obtained by solving the eigenvalue equation. Computed frequency results for various boundary conditions, aspect ratios a/b and thickness ratios t/b are presented to demonstrate the effects of each factor on the vibration frequencies of moderately thick plates. Vibration mode shapes in the form of contour plots are presented for a thickness ratio of t/b = 0.1. These plots are believed to be the first known in the open literature.

Boundary characteristic orthogonal polynomials in numerical approximation

AbstractThe paper describes a procedure to generate boundary characteristic orthogonal polynomials (BCOPs) over a domain in Rn. The method is based upon the Gram‐Schmidt process of orthogonalization. The orthogonal polynomial sequence (OPS) is generated from a set of linearly independent functions which satisfy the given boundary conditions. The procedure is illustrated by taking several examples in R2, the most important case from the application point of view. Extensive numerical work has been carried out for simple domains like the circle, square and isosceles right‐angled triangle. The following cases have been considered: (a) when the BCOPs vanish on the boundary; (b) when they and their normal derivatives vanish on the boundary; (c) when no conditions have been imposed on the boundary. The results for other geometries such as an ellipse, a parallelogram and an arbitrary triangle follow by some simple transformations. The procedure is also applicable to one‐dimensional problems and can be easily extended to R3.

Use of Characteristic Orthogonal Polynomials in Two Dimensions for Transverse Vibration of Elliptic and Circular Plates With Variable Thickness

Free flexural vibrations of elliptic and circular plates with variable thickness have been studied by using characteristic orthogonal polynomials satisfying the essential boundary conditions and the Rayleigh-Ritz method. Two types of variable thickness have been considered: in the first case it varies linearly or quadratically parallel to the major axis; in the second case it is taken to be the same along concentric ellipses but varies linearly or quadratically as we move from one ellipse to another. The results for a circular plate with variable thickness follow as a special case. Computations have been carried out for clamped, simply supported and free boundary. Comparison has been made with known results in special cases.

Flexural Vibration of Skew Plates Using Boundary Characteristic Orthogonal Polynomials in Two Variables

The first five frequencies have been determined for the transverse vibration of a rectangular or a skew plate under different boundary conditions by using boundary characteristic orthogonal polynomials in two variables. The given plate is first mapped into a square plate over which a set of orthonormal polynomials satisfying the essential boundary conditions is generated by using the Gram-Schmidt process. The Rayleigh-Ritz method is then used to determine the frequencies for all possible combination of the boundary conditions and with different skew angles. Comparison has been made with known results.

Natural Frequencies and Mode Shapes of Elliptic Plates With Boundary Characteristic Orthogonal Polynomials as Assumed Shape Functions

The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using the Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in the Cartesian coordinate system are transformed into the modified polar coordinate system. Boundary characteristic orthogonal polynomials in the radial direction, and trigonometric functions in the angular direction are used to express the deflection of the plate. These deflection shapes are classified into four basic categories, depending on their symmetrical or antisymmetrical properties about the major and minor axes of the ellipse. The first six natural modes in each of the above categories are presented in the form of contour plots.

Transverse Vibration Of Annular Circular And Elliptic Plates Using The Characteristic Orthogonal Polynomials In Two Dimensions

Characteristic orthogonal polynomials in two variables have been generated over the annular region occupied by a circular or elliptic plate satisfying the essential boundary conditions. These are used to express the displacement in the transverse vibration of the plate. The Rayleigh-Ritz method is then applied to study the fundamental mode of vibration. Comparison has been made with the known results for circular annular plates in some cases. The results for the elliptic annular plate are entirely new and are not available elsewhere. All the results are summarized in tables covering various combinations of the boundary conditions.

Axisymmetric vibration of circular plates and its analog in elliptical plates using characteristic orthogonal polynomials

The axisymmetric vibration of circular plates and its analog in elliptic plates with clamped, simply supported and free boundary conditions are investigated using boundary characteristic orthogonal polynomials in the Rayleigh—Ritz method. Modified polar coordinates are employed to analyze the vibration of elliptical plates, with circular plates as a special case. Axisymmetric vibration of circular plates and its analog in elliptical plates with concentric nodal elipses require only one-dimensional shape functions in the Rayleigh—Ritz method. The first six natural frequencies and the parameters associated with nodal ellipses are tabulated.

Transverse vibration of triangular plates using characteristic orthogonal polynomials in two variables

Transverse vibrations of a triangular plate were studied under various types of boundary conditions at the three sides by using orthogonal polynomials as the basis functions and applying the Rayleigh-Ritz method. The problem was formulated and studied in the most general form, and comparison of numerical results was made with known results in special cases.

Transverse vibration of simply supported elliptical and circular plates using boundary characteristic orthogonal polynomials in two variables

The first four frequencies and mode shapes have been computed for elliptical and circular plates of uniform thickness with simply supported edges by employing orthogonal polynomials in the Rayleigh-Ritz method. In general, this is found to give better results than traditional methods. Successive approximations have been worked out to ensure convergence. Comparisons have been made with existing results in some cases. Calculations have been done for several values of b a ; i.e., the ratio of the semi-major and semi-minor axes. Mode shapes have been used to obtain three-dimensional plots of plates in displaced configurations.

Vibration of elliptic plates using characteristic orthogonal polynomials in the Rayleigh-Ritz method

The vibrations of uniform elliptic plates with clamped, simply-supported and free boundary conditions are investigated using boundary-characteristic orthogonal polynomials in the Rayleigh-Ritz method. Deflection shapes in the form of products of functions in the radial coordinate and the angular coordinate, respectively, using modified polar coordinates, are assumed. Boundary-characteristic orthogonal polynomials are used in the radial direction, whereas trigonometric functions are employed along the angular coordinate. Six natural frequencies corresponding to each of the four basic categories of natural modes, such as symmetric-symmetric, symmetric-antisymmetric, antisymmetric-symmetric, antisymmetric-antisymmetric modes about the major and minor axes of the ellipse, are presented.

Transverse vibration of completely-free elliptic and circular plates using orthogonal polynomials in the Rayleigh-Ritz method

The Rayleigh-Ritz method using characteristic orthogonal polynomials has been applied to study the problem of transverse vibration of completely-free elliptic and circular plates of uniform thickness. A sequence of successive approximations has been generated to work out the frequencies and mode shapes. The process is continued until at least the first four frequencies converge to five significant digits. It has been found that the results agree completely with the known results for a circular plate. A comparison has also been made with some known experimental and theoretical results for an elliptic plate.

Numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non-uniform thickness

A comparison of natural frequencies of transverse vibration of thin, rectangular plates of non-uniform thickness with different combinations of boundary conditions is presented in this paper, as obtained by using the following methodologies: (1) the Rayleigh-Ritz method with characteristic orthogonal polynomial shape functions; (2) the Rayleigh-Ritz method with a shape function which includes two exponents that are to be determined by minimizing the fundamental frequency coefficient; (3) the optimized Kantorovich method which has been proposed rather recently; (4) the finite element method. The results are presented in tabular form and discussed.

Vibration analysis of arbitrarily shaped plates using beam characteristic orthogonal polynomials in the semi-analytical finite strip method

A semi-analytical finite strip method mapping arbitrarily shaped plates into a natural co-ordinate plane using Serendipity functions is presented. The product of normalized beam characteristic orthogonal polynomials and Hermitian polynomials form the displacement function for the strip in the natural co-ordinate domain. Flexural free vibration analysis of various shaped plates is performed. Results show that as long as the plate is modelled by high enough degree mapping functions for the specific applications requested, the flexural frequencies obtained are in excellent agreement with those found in the literature.

Transverse vibrations of a trapezoidal cantilever plate of variable thickness

N ATURAL frequencies of a trapezoidal cantilever plate of variable thickness, which model an aircraft wing structure approximately, are determined using energy techniques. Characteristic orthogonal polynomials in two variables are constructed to describe the structural deflections, which are used in the Rayleigh-Ritz method to obtain the natural frequencies. The fundamental natural frequencies for different parameter values are also obtained using a deflection function containing an optimized exponent. The first technique can be used to obtain the natural frequencies and corresponding mode shapes for the preliminary design of aircraft wing structures.

Bounds on decay constants for diffusion through inhomogeneous media

The decay constants for diffusion through inhomogeneous media are known to be proportional to the eigenvalues of the corresponding elliptic operator. A new method of obtaining a hierarchy of upper bounds on sums and products of these eigenvalues as well as the eigenvalues themselves is presented. The first member of this hierarchy is just the usual Rayleigh-Ritz quotient. The other members of the hierarchy are generalised Rayleigh-Ritz quotients which can be derived simply using properties of integrals of the solutions of the diffusion equation. Explicit bounds are presented for the first three eigenvalues, but general methods of obtaining bounds for higher-order eigenvalues are also outlined. For fixed time t, many of the bounds reduce to results given by the classical method of moments. The hierarchy of rigorous variational bounds on the eigenvalues studied may be generated using simple recursion relations based on properties of the characteristic orthogonal polynomials. The conditions on the trial functions used to obtain bounds on eigenvalues higher than the first are much simpler than those required by the traditional Rayleigh-Ritz procedure.

Flexural vibration of polygonal plates using characteristic orthogonal polynomials in two variables

A set of characteristic orthogonal polynomials in two variables is used as deflection functions to obtain the natural frequencies and mode shapes of polygonal plates by the Rayleigh-Ritz method. The set of orthogonal polynomials is constructed by employing the Gram-Schmidt orthogonalization procedure. The first six natural frequencies are numerically evaluated for triangular plates of different configurations by using these orthogonal functions and the corresponding mode shapes are plotted. Results are compared with those obtained previously by other methods.

Transverse vibrations of a rotating uniform cantilever beam with tip mass as predicted by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method

Natural frequencies and mode shapes of a rotating uniform cantilever beam with a tip mass are studied by using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The set of orthogonal polynomials which satisfy the geometrical boundary conditions are generated by using the Gram-Schmidt process. The results are compared with those obtained by the Myklestad method, the extended Galerkin method and finite element methods. The variation of natural frequencies with the speed of rotation is plotted for several parameter combinations such as setting angle, tip mass, moment of inertia of tip mass, etc. Mode shapes at different rotational speeds are also plotted. Use of orthogonal polynomials for the deflection shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any numerical difficulties, which is not the case when arbitrary polynomial expressions are used.

Plate Deflections Using Orthogonal Polynomials

Bending deflection of plates under static loading has been determined using beam characteristic orthogonal polynomials in Rayleigh‐Ritz formulation. The first member of the orthogonal polynomial set was constructed as the simplest polynomial that satisfies all the boundary conditions of the corresponding beam problems accompanying the plate problem. The rest of the set was generated using the Gram Schmidt orthogonalization process. Results are obtained for plates with all edges clamped and those with three edges clamped and one edge free. Two types of loadings, uniform and hydrostatic, are considered. The results are found to agree closely with those obtained by previous methods. Since the Gram Schmidt orthogonalization process can include a weight function also, even plates with nonuniform properties can be studied using this method.

Natural frequencies of rectangular plates using characteristic orthogonal polynomials in rayleigh-ritz method

Natural frequencies of rectangular plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generated by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. Natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods. The method yields superior results for lower modes, particularly when plates have some of the edges free.