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 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.

Vibration Analysis of Clamped, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2013 ◽  
Vol 81 (1) ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
3D Analysis ◽  
Shells Of Revolution ◽  
Conical Shells ◽  
Cylindrical Coordinate ◽  
Convergence Study ◽  
3D Methods ◽  
Thin Shell Theory

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of clamped, 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. A convergence study is presented. The frequencies from the present 3D analysis are compared with those from other 3D methods and 2D thin shell theory.

Vibration Analysis of Complete Conical Shells with Variable Thickness

2014 ◽  
Vol 14 (04) ◽  
pp. 1450001 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Vibration Analysis ◽  
Present Method ◽  
Shell Theory ◽  
Ritz Method ◽  
Three Dimensional ◽  
Dynamic Equations ◽  
Algebraic Polynomials ◽  
Bottom Edge ◽  
Conical Shells ◽  
Varying Thickness

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of complete (not truncated) conical shells with linearly varying thickness. The complete conical shells free or clamped at the bottom edge with a free vertex are investigated. Unlike conventional shell theories, which are mathematically 2D, the present method is based upon the 3D dynamic equations of elasticity. Displacement components ur, uθ and uz in the radial, circumferential and axial directions, respectively, are taken to be periodic in θ and in time, and expressed by algebraic polynomials in the r- and z-directions. Potential (strain) and kinetic energies of the complete conical shell are formulated. The Ritz method is used to solve the eigenvalue problem, yielding the upper bound values of the frequencies by minimization. As the degree of the polynomials is increased, frequencies converge to the exact values, with four-digit exactitude demonstrated for the first five frequencies. The frequencies from the present 3D method are compared with those from other 3D approaches and 2D shell theory by previous researchers.

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.

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.

FREE VIBRATIONS OF COMBINED HEMISPHERICAL–CYLINDRICAL SHELLS OF REVOLUTION WITH A TOP OPENING

2013 ◽  
Vol 14 (01) ◽  
pp. 1350023 ◽  
Author(s):  
JAE-HOON KANG
Keyword(s):  
Present Method ◽  
Thin Shell ◽  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Three Dimensional ◽  
Free Vibrations ◽  
Two Dimensional ◽  
Algebraic Polynomials ◽  
Shells Of Revolution

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies of joined hemispherical–cylindrical shells of revolution with a top opening. 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, uθ and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the joined shells 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 polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from 2D thin shell theories.

Comparisons of Paraboloidal Shells and Sinusoidal-Shaped Shells in Natural Frequencies

2019 ◽  
Vol 24 (3) ◽  
pp. 451-457
Author(s):  
Yeong-Bin Yang ◽  
Jae-Hoon Kang
Keyword(s):  
Thin Shell ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thin Shells ◽  
Thick Shell ◽  
Cylindrical Coordinates ◽  
Thin Shell Theory

Natural frequencies and mode shapes are obtained for a sinusoidal-shaped shell of revolution by using the Ritz method from a three-dimensional (3-D) analysis instead of a mathematically two-dimensional (2-D) thin shell theory or high order thick shell theory. The present analysis uses circular cylindrical coordinates instead of 3-D shell coordinates, which have been used in traditional shell analyses. Convergence studies can analyze the first five frequencies to four-digit exactitude. Results are given for a variety of shallow and deep sinusoidal-shaped shells with different boundary conditions. The sinusoidal-shaped shells are very similar to paraboloidal shells in shape. The frequencies of the sinusoidal-shaped shells from the present 3-D method are compared with those from 2-D thin shell theories for paraboloidal shells. The present 3-D method is applicable to very thick as well as thin shells.

Vibration Analysis of Shallow Spherical Domes with Non-Uniform Thickness

2017 ◽  
Vol 17 (02) ◽  
pp. 1750016 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Vibration Analysis ◽  
Present Method ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Algebraic Polynomials ◽  
Uniform Thickness ◽  
3D Finite Element ◽  
3D Finite Element Method

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies of shallow spherical domes with non-uniform thickness. Unlike conventional shell theories, which are mathematically two dimensional (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 meridional, circumferential, and normal directions, respectively, are taken to be periodic in [Formula: see text] and in time, and algebraic polynomials in the [Formula: see text] and z directions. Potential (strain) and kinetic energies of the shallow spherical domes with non-uniform thickness 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 polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Natural frequencies are presented for different boundary conditions. The frequencies from the present 3D method are compared with those from a 2D exact method, a 2D thick shell theory, and a 3D finite element method by previous researchers.

Comparisons of Paraboloidal Shells and Sinusoidal-Shaped Shells in Natural Frequencies

2019 ◽  
Vol 24 (2) ◽  
pp. 451-457
Author(s):  
Yeong-Bin Yang ◽  
Jae-Hoon Kang
Keyword(s):  
Thin Shell ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Thin Shells ◽  
Thick Shell ◽  
Cylindrical Coordinates ◽  
Thin Shell Theory

Natural frequencies and mode shapes are obtained for a sinusoidal-shaped shell of revolution by using the Ritz method from a three-dimensional (3-D) analysis instead of a mathematically two-dimensional (2-D) thin shell theory or high order thick shell theory. The present analysis uses circular cylindrical coordinates instead of 3-D shell coordinates, which have been used in traditional shell analyses. Convergence studies can analyze the first five frequencies to four-digit exactitude. Results are given for a variety of shallow and deep sinusoidal-shaped shells with different boundary conditions. The sinusoidal-shaped shells are very similar to paraboloidal shells in shape. The frequencies of the sinusoidal-shaped shells from the present 3-D method are compared with those from 2-D thin shell theories for paraboloidal shells. The present 3-D method is applicable to very thick as well as thin shells.

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.

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