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

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.

Download Full-text

Three-Dimensional Vibration Analysis of Thick, Complete Conical Shells

Journal of Applied Mechanics ◽

10.1115/1.1767843 ◽

2004 ◽

Vol 71 (4) ◽

pp. 502-507 ◽

Cited By ~ 15

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.

Download Full-text

Vibration of thick curved shells, with particular reference to turbine blades

Journal of Strain Analysis ◽

10.1243/03093247v053200 ◽

1970 ◽

Vol 5 (3) ◽

pp. 200-206 ◽

Cited By ~ 21

Author(s):

S Ahmad ◽

R G Anderson ◽

O C Zienkiewicz

Keyword(s):

Thin Shell ◽

Shell Theory ◽

Turbine Blades ◽

Shell Element ◽

Thin Shells ◽

Thick Shell ◽

Shear Deformations ◽

Vibration Problems ◽

Thin Shell Theory

The application of a new thick shell element is described with reference to vibration problems. The element is derived from the general isoparametric solid and therefore allows shear deformations to be included. It can take up highly distorted shapes and is useful in such studies as vibration of turbine blades for which it is superior to elements based on thin-shell theory. This element can also be used for thin shells with caution, excessive length/thickness ratios being avoided.

Download Full-text

3D Vibration Analysis of Combined Shells of Revolution

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455419500056 ◽

2019 ◽

Vol 19 (02) ◽

pp. 1950005 ◽

Cited By ~ 3

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.

Download Full-text

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

International Journal of Structural Stability and Dynamics ◽

10.1142/s0219455413500235 ◽

2013 ◽

Vol 14 (01) ◽

pp. 1350023 ◽

Cited By ~ 6

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.

Download Full-text

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

Journal of Applied Mechanics ◽

10.1115/1.4024401 ◽

2013 ◽

Vol 81 (1) ◽

Cited By ~ 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.

Download Full-text

Dynamic Actuation and Quadratic Magnetoelastic Coupling of Thin Magnetostrictive Shells

Journal of Vibration and Acoustics ◽

10.1115/1.2175089 ◽

2005 ◽

Vol 128 (3) ◽

pp. 385-391 ◽

Cited By ~ 1

Author(s):

H. S. Tzou ◽

W. K. Chai ◽

M. Hanson

Keyword(s):

Mathematical Model ◽

Thin Shell ◽

Shell Theory ◽

Thin Shells ◽

Free Energy Function ◽

Dynamic Coupling ◽

Double Curvature ◽

System Equations ◽

Gibb’S Free Energy ◽

Thin Shell Theory

Smart adaptive structures and structronic systems have been increasingly investigated and developed in the last two decades. Although smart structures made of piezoelectrics, shape-memory materials, electrostrictive materials, and electro-/magnetorheological fluids have been evaluated extensively, studies of magnetostrictive continua, especially generic mathematical model(s), are still relatively scarce. This study is to develop a generic mathematical model for adaptive and controllable magnetostrictive thin shells. Starting with fundamental constitutive magnetostrictive relations, both elastic and magnetostrictive stresses, forces, and moments of a generic double-curvature magnetostrictive shell continuum subject to small and moderate magnetic fields are defined. Dynamic magnetomechanical system equations and permissible boundary conditions are defined using Hamilton's principle, elasticity theory, Kirchhoff-Love thin shell theory and the Gibb's free energy function. Magnetomechanical behavior and dynamic characteristics of magnetostrictive shells are evaluated. Simplifications of magnetostrictive shell theory to other common geometries are demonstrated and magnetostrictive/dynamic coupling and actuation characteristics are discussed.

Download Full-text

On the Derivation of Parametric Terms in the Equations of Elastic Stability of Thin Shells

Transactions of the Canadian Society for Mechanical Engineering ◽

10.1139/tcsme-1985-0017 ◽

1985 ◽

Vol 9 (3) ◽

pp. 121-124

Author(s):

J.L. Urrutia-Galicia ◽

A.N. Sherbourne

Keyword(s):

Thin Shell ◽

Shell Theory ◽

Elastic Stability ◽

Energy Functional ◽

Thin Shells ◽

Thin Shell Theory

In this paper, the parametric terms in the equations of elastic stability of thin shells will be derived in a way that appears to be simpler than those found in the literature. The result matches the solution given by Bolotin in 1967 but has been obtained by avoiding the energy functional of thin shell theory. The parametric terms account for the influence of prebuckling loads, membrane forces and their derivatives and are therefore of some significance in formulating the comprehensive stability equations for a variety of shell geometries.

Download Full-text

Vibration Analysis of Shallow Spherical Domes with Non-Uniform Thickness

International Journal of Structural Stability and Dynamics ◽

10.1142/s021945541750016x ◽

2017 ◽

Vol 17 (02) ◽

pp. 1750016 ◽

Cited By ~ 4

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.

Download Full-text

Three-Dimensional Vibration Analysis of a Truncated Quadrangular Pyramid

Journal of Applied Mechanics ◽

10.1115/1.3172944 ◽

1987 ◽

Vol 54 (1) ◽

pp. 115-120 ◽

Cited By ~ 7

Author(s):

T. Irie ◽

G. Yamada ◽

Y. Tagawa

Keyword(s):

Vibration Analysis ◽

Natural Frequencies ◽

Ritz Method ◽

Three Dimensional ◽

Frequency Equation ◽

The Body ◽

Mode Shapes ◽

Algebraic Polynomials ◽

Transformation Of Variables ◽

Unit Edge

An analysis is presented for the three-dimensional vibration problem of determining the natural frequencies and the mode shapes of a truncated quadrangular pyramid. For this purpose, the body is transformed into a right quadrangular prism with unit edge lengths by a transformation of variables. With the displacements of the transformed prism assumed in the forms of algebraic polynomials, the dynamical energies of the prism are evaluated, and the frequency equation is derived by the Ritz method. This method is applied to quadrangular pyramids in which the base is clamped and the other sides are free, and the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically, from which the vibration characteristics arising in the pyramids are studied.

Download Full-text