scholarly journals FREE VIBRATION OF ISOTROPIC HALF-ELLIPTIC PLATES OF LINEARLY VARYING THICKNESS WITH CLAMPED CURVED BOUNDARY

2010 ◽  
Vol 7 (2) ◽  
pp. 1-15
Author(s):  
A.P Gupta ◽  
N. Bhardwaj ◽  
K.K. Choong
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Mode Shapes ◽  
Chebyshev Collocation Method ◽  
Integral Equation Methods ◽  
Aspect Ratios ◽  
Dimensional Boundary ◽  
Title Problem ◽  
Significant Figures

Two-dimensional boundary characteristic orthonormal polynomials are used in Rayleigh-Ritz method to study the title problem. In general, it is found that this method gives better results than the other traditional method such as boundary integral equation methods, Spline methods, Chebyshev collocation method, Frobenius method etc. The thickness is taken to be linearly varying in two orthogonal directions. Comparisons in particular cases have been made with the existing results in the literature. Convergence of frequencies of at least up to five significant figures is obtained. Results showing the variation in frequencies with taper parameters and aspect ratios are presented in tabular form. Mode shapes are shown using three-dimensional graphs of plates in displaced configurations.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

Cusping, capture, and breakup of interacting drops by a curvatureless boundary-integral algorithm

1999 ◽  
Vol 391 ◽  
pp. 249-292 ◽  
Author(s):  
ALEXANDER Z. ZINCHENKO ◽  
MICHAEL A. ROTHER ◽  
ROBERT H. DAVIS
Keyword(s):  
Three Dimensional ◽  
Adaptive Mesh ◽  
Boundary Integral ◽  
Small Surface ◽  
Low Viscosity ◽  
Primary Breakup ◽  
Dimensional Boundary ◽  
Normal Vectors ◽  
Passive Technique ◽  
Line Singularities

A three-dimensional boundary-integral algorithm for interacting deformable drops in Stokes flow is developed. The algorithm is applicable to very large deformations and extreme cases, including cusped interfaces and drops closely approaching breakup. A new, curvatureless boundary-integral formulation is used, containing only the normal vectors, which are usually much less sensitive than is the curvature to discretization errors. A proper regularization makes the method applicable to small surface separations and arbitrary λ, where λ is the ratio of the viscosities of the drop and medium. The curvatureless form eliminates the difficulty with the concentrated capillary force inherent in two-dimensional cusps and allows simulation of three-dimensional drop/bubble motions with point and line singularities, while the conventional form can only handle point singularities. A combination of the curvatureless form and a special, passive technique for adaptive mesh stabilization allows three-dimensional simulations for high aspect ratio drops closely approaching breakup, using highly stretched triangulations with fixed topology. The code is applied to study relative motion of two bubbles or drops under gravity for moderately high Bond numbers [Bscr ], when cusping and breakup are typical. The deformation-induced capture efficiency of bubbles and low-viscosity drops is calculated and found to be in reasonable agreement with available experiments of Manga & Stone (1993, 1995b). Three-dimensional breakup of the smaller drop due to the interaction with a larger one for λ=O(1) is also considered, and the algorithm is shown to accurately simulate both the primary breakup moment and the volume partition by extrapolation for moderately supercritical conditions. Calculations of the breakup efficiency suggest that breakup due to interactions is significant in a sedimenting emulsion with narrow size distribution at λ=O(1) and [Bscr ][ges ]5–10. A combined capture and breakup phenomenon, when the smaller drop starts breaking without being released from the dimple formed on the larger one, is also observed in the simulations. A general classification of possible modes of two-drop interactions for λ=O(1) is made.

AXISYMMETRIC VIBRATION OF POLAR ORTHOTROPIC CIRCULAR PLATES OF QUADRATICALLY VARYING THICKNESS RESTING ON ELASTIC FOUNDATION

2005 ◽  
Vol 05 (03) ◽  
pp. 387-408 ◽  
Author(s):  
N. BHARDWAJ ◽  
A. P. GUPTA
Keyword(s):  
Elastic Foundation ◽  
Ritz Method ◽  
Three Dimensional ◽  
Normal Modes ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Circular Plates ◽  
Varying Thickness ◽  
Edge Conditions ◽  
Polar Orthotropic

This paper is concerned with the axisymmetric vibration problem of polar orthotropic circular plates of quadratically varying thickness and resting on an elastic foundation. The problem is solved by using the Rayleigh–Ritz method with boundary characteristic orthonormal polynomials for approximating the deflection function. Numerical results are computed for frequencies, nodal radii and mode shapes. Three-dimensional graphs are also plotted for the first four normal modes of axisymmetric vibration of plates with free, simply-supported and clamped edge conditions for various values of taper, orthotropy and foundation parameters.

Three-Dimensional Vibration Analysis of Thick, Complete Conical Shells

2004 ◽  
Vol 71 (4) ◽  
pp. 502-507 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Present Method ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Conical Shells ◽  
Thin Shell Theory ◽  
Time Periodic

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur,uz, and uθ in the radial, axial, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z-directions. Potential (strain) and kinetic energies of the conical shells are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the conical shells. Novel numerical results are presented for thick, complete conical shells of revolution based upon the 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and a 2D thin shell theory.

Three-Dimensional Vibrations of Truncated Hollow Cones

1995 ◽  
Vol 1 (2) ◽  
pp. 145-158 ◽  
Author(s):  
Arthur W. Leissa ◽  
Jinyoung So
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Arbitrary Boundary ◽  
Conical Shells ◽  
Hole Radius ◽  
Geometric Boundary

This work presents a three-dimensional (3-D) method of analysis for determining the free vibration frequencies and corresponding mode shapes of truncated hollow cones of arbitrary thickness and having arbitrary boundary conditions. It also supplies the first known numerical results from 3-D analysis for such problems. The analysis is based upon the Ritz method. The vibration modes are separated into their Fourier components in terms of the circumferential coordinate. For each Fourier component, displacements are expressed as algebraic polynomials in the thickness and slant length coordinates. These polynomials satisfy the geometric boundary conditions exactly. Because the displacement functions are mathematically complete, upper bound values of the vibration frequencies are obtained that are as close to the exact values as desired. This convergence is demonstrated for a representative truncated hollow cone configuration where six-digit exactitude in the frequencies is achieved. The method is then used to obtain accurate and extensive frequencies for two sets of completely free, truncated hollow cones, one set consisting of thick conical shells and the other being tori having square-generating cross sections. Frequencies are presented for combinations of two values of apex angles and two values of inner hole radius ratios for each set of problems.

3D Vibration Analysis of Combined Shells of Revolution

2019 ◽  
Vol 19 (02) ◽  
pp. 1950005 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Legendre Polynomials ◽  
Thin Shell ◽  
Ritz Method ◽  
Three Dimensional ◽  
Surface Strain ◽  
Mode Shapes ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Thick Shells ◽  
Admissible Functions

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies and the mode shapes of combined hemispherical–cylindrical shells of revolution with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2D) conventional thin shell theories or higher-order thick shell theories, the present method is based upon the 3D dynamic equations of elasticity. Mathematically, minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangent to the shell mid-surface. Strain and kinetic energies of the combined shell of revolution with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for the combined shells of revolution with or without a top opening, which are completely free and fixed at the bottom of the combined shells. The frequencies from the present 3D Ritz method are compared with those from 2D thin shell theories by previous researchers. The present analysis is applicable to very thick shells as well as very thin shells.

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.

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