Free Vibrations of Thick Hollow Circular Cylinders From Three-Dimensional Analysis

Abstract

A three-dimensional (3-D) method of analysis is developed for the free vibration frequencies of hollow circular cylinders of elastic material. The method is based upon local coordinates whose origin is attached to the center of cylindrical wall. It assumes for the three displacement components a Fourier series in the circumferential (θ) direction and algebraic polynomials in the radial (q) and axial (z) directions. Convergence studies for completely free cylinders show that the analysis can yield frequencies which are exact to five or six significant figures. These accurate frequencies are compared with those from other 3-D analyses available for free hollow circular cylinders having various length-to-outside diameter (L/Do) and inside-to-outside diameter (Di/Do) ratios. Extensive, accurate data are presented for the first 10 frequencies of each circumferential wave number 0 through 5 for hollow circular cylinders having Di/Do of 0.1, 0.5, and 0.9, with L/Do = 0.2, 1 and 5 and a Poisson’s ratio (v) = 0.3.

Two Dimensional Functionally Graded Material Finite Thick Hollow Cylinder Axisymmetric Vibration Mode Shapes Analysis Based on Exact Elasticity Theory

“…A finite element model for axisymmetric elasticity is formulated directly in the cylindrical coordinates to study the vibration of hollow, isotropic and homogeneous finite length cylinders and frequencies are computed for free-free end boundary conditions in the reference [27] and compared with the reference [13]. For solving the mentioned problem by graded finite element method developed here, we consider a thick hollow cylinder with freely supported end conditions in which the material distribution is uniform.…”

“…Studies on shells, based on three-dimensional theory of elasticity, have been presented by some researchers for infinitely long cylindrical shells [6][7][8][9]. For finite-length thick cylindrical shells, different methods, such as finite element method, series solution, the Ritz energy method are used by some researchers for both solid and hollow homogeneous cases [10][11][12][13].…”

Free Vibration Analysis of Functionally Heterogeneous Hollow Cylinder Based on Three Dimensional Elasticity Theory

“…A finite element for axisymmetric elasticity is formulated directly in the cylindrical coordinates to study the vibration of hollow, isotropic, and homogeneous finite length cylinders and frequencies are computed for free-free end boundary conditions in the reference 19 and compared with the reference. 16 For solving the aforementioned problem using the graded finite element method developed here, we considered a thick hollow cylinder with freely supported end conditions in which the material distribution is uniform. Therefore, the volume fraction exponent and property coefficients in the 2D-FGM are taken as: n z = 0, n r = 0, P c1 = P c2 = P m1 = P m2 = P , where P is a uniform material properties of the cylinder.…”

“…[9][10][11][12] For finite-length thick cylindrical shells, different methods such as the finite element method, series solution, and the Ritz energy method have been used by some researchers for both solid and hollow homogeneous cases. [13][14][15][16][17] A three-dimensional energy formulation was used by Liew et. al.…”

Three-dimensional vibration analysis of prisms with isosceles triangular cross-section

“…The ill-conditioning situation can be improved by employing the orthogonal polynomials instead of the natural ones. However, as So and Leissa [18] pointed out, this would not only complicate the analysis but may yield inaccurate results due to the truncation errors which arise in calculating orthogonal polynomials by the Gram-Schmidt process. In this paper, the Chebyshev polynomials are suggested as the alternative basic functions to study the prisms with isosceles triangular cross-section using the Ritz method.…”