Boundary Characteristic Orthogonal Polynomials Research Articles

Natural frequencies of edge restrained tapered rectangular plates

Natural frequencies of tapered plates with edges elastically restrained against rotation and translation are studied by using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method and applying the Rayleigh-Schmidt method. The two methods used are quite general and can be used to study plates with any combinations of boundary conditions, thus obtaining interesting technical results. Since such combinations of boundary conditions, along with specific values for the stiffness constants for the restraints are prohibitively large in number, results are presented for only a few cases.

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.

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.

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 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.