Three-Dimensional Vibrations of Thick, Circular Rings with Isosceles Trapezoidal and Triangular Cross-Sections

1999 ◽  
Vol 122 (2) ◽  
pp. 132-139 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Free Vibration ◽  
Cross Sections ◽  
Three Dimensional ◽  
Circular Ring ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Algebraic Polynomials ◽  
Free Boundaries ◽  
Circular Rings ◽  
Time Periodic

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular rings with isosceles trapezoidal and triangular cross-sections. Displacement components us,uz, and uθ in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the ϕ and z directions. Potential (strain) and kinetic energies of the circular ring are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for the circular rings with isosceles trapezoidal and equilateral triangular cross-sections having completely free boundaries. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. The method is applicable to thin rings, as well as thick and very thick ones. [S0739-3717(00)00702-9]

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.

Three-Dimensional Vibration Analysis of Solid and Hollow Hemispheres Having Varying Thicknesses With and Without Axial Conical Holes

2004 ◽  
Vol 10 (2) ◽  
pp. 199-214 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W Leissa
Keyword(s):  
Vibration Analysis ◽  
Present Method ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thickness Variation ◽  
Algebraic Polynomials ◽  
Shells Of Revolution ◽  
Time Periodic ◽  
Hemispherical Shells

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid and hollow hemispherical shells of revolution of arbitrary wall thickness having arbitrary constraints on their boundaries. Unlike conventional shell theories, which are mathematically two-dimensional, the present method is based upon the 3D dynamic equations of elasticity. Displacement components u \#966;, u z, and u \#952; in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in \#952;, and algebraic polynomials in the \#966;-direction and zdirection. Potential (strain) and kinetic energies of the hemispherical shells are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Novel numerical results are presented for solid and hollow hemispheres with linear thickness variation. The effect on frequencies of a small axial conical hole is also discussed. Comparisons are made for the frequencies of completely free, thick hemispherical shells with uniform thickness from the present 3D Ritz solutions and other 3D finite element ones.

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.

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 Hermetic Capsules by Using Ritz Method

2017 ◽  
Vol 17 (03) ◽  
pp. 1750040
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Eigenvalue Problem ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Legendre Polynomials ◽  
Present Method ◽  
Ritz Method ◽  
Three Dimensional ◽  
Dynamic Equations ◽  
Vibration Frequencies ◽  
Good Agreement

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of a hermetic capsule comprising a cylinder closed with hemi-ellipsoidal caps at both ends. Unlike conventional shell theories, which are mathematically 2D, the present method is based upon the 3D dynamic equations of elasticity. Displacement components [Formula: see text], [Formula: see text], and [Formula: see text] in the radial, circumferential, and axial directions, respectively, are taken to be periodic in [Formula: see text] and in time, and the Legendre polynomials in the r and z directions instead of ordinary ones. Potential (strain) and kinetic energies of the hermetic capsule are formulated, and the Ritz method is used to solve the eigenvalue problem, thereby yielding upper bound values of the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Typical convergence studies are carried out for the first five frequencies. The frequencies from the present 3D method are in good agreement with those obtained from other 3D approach and 2D shell theories proposed by previous researchers.

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 Coupled Dynamic Analysis for the Riser-Conductor of Deepwater Surface BOP Drilling System

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Kanhua Su ◽  
Stephen Butt ◽  
Jianming Yang ◽  
Hongyuan Qiu
Keyword(s):  
Free Vibration ◽  
Natural Frequency ◽  
Cross Sections ◽  
Lateral Displacement ◽  
Bending Moment ◽  
Equation Of Motion ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Drilling System ◽  
Natural Frequency Analysis

Deepwater surface BOP (surface blowout prevention, SBOP) drilling differs from conventional riser drilling system. To analyze the dynamic response of this system, the riser-conductor was considered as a beam with varied cross-sections subjected to loads throughout its length; then an equation of motion and free vibration of the riser-conductor string for SBOP was developed. The finite difference method was used to solve the equation of motion in time domain and a semianalytical approach based on the concept of section division and continuation was proposed to analyze free vibration. Case simulation results show that the method established for SBOP system natural frequency analysis is reasonable. The mode shapes of the riser-conductor are different between coupled and decoupled methods. The soil types surrounding the conductor under mudline have tiny effect on the natural frequency. Given that some papers have discussed the response of the SBOP riser, this work focused on the comparison of the dynamic responses on the wellhead and conductor with variable conditions. The dynamic lateral displacement, the bending moment, and the parameters’ sensitivity of the wellhead and the conductor were analyzed.

Large amplitude free vibration study of non-uniform plates with in-plane material inhomogeneity

2016 ◽  
Vol 232 (5) ◽  
pp. 371-387 ◽  
Author(s):  
Saurabh Kumar ◽  
Anirban Mitra ◽  
Haraprasad Roy
Keyword(s):  
Free Vibration ◽  
Large Amplitude ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Static Problem ◽  
Solution Procedure ◽  
Mode Shapes ◽  
Material Inhomogeneity ◽  
Contour Plots ◽  
Variational Form

Free vibration study of non-uniform plates with in-plane material inhomogeneity is carried out in the present work considering geometric nonlinearity. Inhomogeneous plates where the material properties vary along only x-axis (unidirectional) and along both x- and y-axis (bidirectional) are considered. The analysis is performed for two boundary conditions namely clamped and simply supported at all edges, under the action of a transverse uniformly distributed load. The large amplitude problem is formulated using nonlinear strain–displacement relations along with a variational form of energy method. A two-step solution procedure is utilised where, in the first part the static problem is solved and undetermined coefficients are found, subsequently the dynamic problem is taken up on the basis of previously determined coefficients. Validity of the results is successfully confirmed by comparison with the works of other researchers. The analysis reveals that the amplitude and taper parameter affect the loaded natural frequencies significantly. Three-dimensional mode shapes for linear and nonlinear cases are presented along with their respective contour plots.

Exact Vibration Solution for Exponentially Tapered Cantilever With Tip Mass

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
C.Y. Wang ◽  
C. M. Wang
Keyword(s):  
Numerical Methods ◽  
Free Vibration ◽  
Cantilever Beam ◽  
Technical Note ◽  
Mode Shapes ◽  
Constant Thickness ◽  
Vibration Frequencies ◽  
Tapered Cantilever Beam ◽  
Tip Mass ◽  
First Time

This technical note is concerned with the free vibration problem of a cantilever beam with constant thickness and exponentially decaying width. Existing analytical results for such a vibration beam problem are found to be incomplete because lower frequencies could not be obtained. Presented herein is the exact characteristic equation for generating the complete vibration frequencies for the considered vibrating beam problem. Also the note treated for the first time such a tapered cantilever beam with a tip mass. The exact solutions (frequencies and mode shapes) are important to engineers designing such tapered beams and the results serve as benchmarks for assessing the validity, convergence and accuracy of numerical methods and solutions.

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