Free Vibration of a Rotating Disk–Blade Coupled System With Shrouds

1996 ◽  
Author(s):  
Yukinori Kobayashi ◽  
Gen Yamada ◽  
Takahiro Tomioka
Keyword(s):  
Free Vibration ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Coupled System ◽  
Rotating Disk ◽  
Mode Shapes ◽  
Admissible Functions

The free vibration of rotating disk–blade coupled system is investigated by the Ritz method. Centrifugal effects due to rotation are taken into account for both of the disk and blades. The boundary and continuity conditions between the disk and blades are satisfied by means of artificial springs introduced at their joints, and the orthogonal polynomials generated by using the Gram–Schmidt process are employed as admissible functions for both of the disk and blades. Frequency parameters and mode shapes of vibration are obtained to investigate the vibration of the disk–blade coupled system.

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

1991 ◽  
Author(s):  
C. Rajalingham ◽  
R. B. Bhat ◽  
G. D. Xistris
Keyword(s):  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Polar Coordinate System ◽  
Free Boundaries ◽  
Natural Modes ◽  
Polar Coordinate ◽  
Boundary Characteristic

Abstract The natural frequencies and natural modes of vibration of uniform elliptic plates with clamped, simply supported and free boundaries are investigated using Rayleigh-Ritz method. A modified polar coordinate system is used to investigate the problem. Energy expressions in 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 its symmetrical or antisymmetrical property 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.

Analysis of In-Plane Modal Characteristics of an Annular Disk With Multiple Point Supports

2009 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Finite Element ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Element Model ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Multiple Point ◽  
Annular Disk ◽  
Free Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an isotropic, elastic annular disk constrained at some points on the inner and outer boundaries are investigated. The presented study is relevant to various practical problems including disks clamped by bolts along the inner and outer edges or the railway wheel vibrations. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. The boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to realize clamped conditions at discrete points. The frequency parameters for different point constraint conditions are evaluated and compared with those computed from a finite element model to demonstrate the validity of the proposed method. The computed mode shapes are presented for a disk with different point constraints at the inner and outer boundaries to demonstrate the free in-plane vibration behavior of the disk. Results show that addition of point supports causes some of the modes to split into two different frequencies with different mode shapes. The effects of different orientations of multiple point supports on the frequency parameters and mode shapes are also discussed.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

scholarly journals In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

2011 ◽  
Vol 18 (4) ◽  
pp. 627-640 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Outer Edge ◽  
Mode Shapes ◽  
Annular Disk ◽  
Different Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries, which are applied either along the entire periphery of the disk or at a point are investigated. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

On the Use of Beam Functions for Problems of Plates Involving Free Edges

1975 ◽  
Vol 42 (4) ◽  
pp. 858-864 ◽  
Author(s):  
S. F. Bassily ◽  
S. M. Dickinson
Keyword(s):  
Free Vibration ◽  
Vibration Mode ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Static Deflection ◽  
Numerical Examples ◽  
Accurate Treatment ◽  
Beam Mode ◽  
Free Edges

The inadequacy of beam vibration mode shapes when used in the Ritz method to obtain approximate solutions for flexural problems concerning plates involving free edges is demonstrated. A new set of functions, related to beam mode shapes, is postulated which allows considerably more accurate treatment of such plates. Several numerical examples concerning static deflection and free vibration of plates involving free edges are examined and serve to illustrate the applicability and accuracy of the new functions and to further demonstrate the inadequacy of the ordinary beam functions.

Free Vibration Investigation of Submerged Thin Circular Plate

2020 ◽  
Vol 12 (03) ◽  
pp. 2050025
Author(s):  
Xi Yang ◽  
Adil El Baroudi ◽  
Jean Yves Le Pommellec
Keyword(s):  
Free Surface ◽  
Free Vibration ◽  
Circular Plate ◽  
Plate Theory ◽  
Fluid Pressure ◽  
Three Dimensional ◽  
Coupled System ◽  
Mode Shapes ◽  
Thin Circular Plate ◽  
Benchmark Solutions

Free vibration of coupled system including clamped-free thin circular plate with hole submerged in three-dimensional cylindrical container filled with inviscid, irrotational and compressible fluid is investigated in this work. Numerical approach based on the finite element method (FEM) is performed using the Comsol Multiphysics software, in order to analyze qualitatively the vibration characteristics of the plate. Plate modeling is based on Kirchhoff–Love plate theory. Velocity potential is deployed to describe the fluid motion since the small oscillations induced by the plate vibration is considered. Bernoulli’s equation together with potential theory is applied to get the fluid pressure on the free surface of the plate. To prove the reliability of the present numerical solution, a comparison is made with the results in the literature, which shows a very good agreement. Then, different parameters effect including fluid density, fluid height, free surface wave, hole radius and hole eccentricity on the natural frequencies of the coupled system is discussed in detail. Some three-dimensional mode shapes of the submerged plate are illustrated. Furthermore, the obtained results can serve as benchmark solutions for the vibration control, parameter identification and damage detection of plate.

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.

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.

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