A three-dimensional (3-D) method of analysis developed in a companion paper [A. W. Leissa and J. So, J. Acoust Soc. Am. 98, 2122–2135 (1995)] is used to obtain free-vibration frequencies of completely free-elastic cylinders. Extensive, accurate data are presented for the first 20 frequencies of each circumferential wave number 0–5 for free–free cylinders having length-to-diameter ratios (L/D) of 1, 1.5, 2, 3, and 5 with a Poisson’s ratio (ν) of 0.3. An additional study for L/D=1 over the full range (0≤ν<0.5) possible for isotropic materials shows that Poisson’s ratio effects can be quite important. The first known values of frequencies from 3-D analysis of cylinders having one end fixed, with all other boundaries free, are also presented in considerable detail. For free–free cylinders, comparisons are made with less accurate 3-D results published previously by other researchers.

No 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.

A method of three-dimensional (3-D) analysis is developed for the free vibration frequencies and mode shapes of solid circular cylinders of elastic material. The method assumes for the three displacement components a Fourier series in the circumferential (θ) direction and algebraic polynomials in the radial (r) and axial (z) directions. All types of boundary conditions and arbitrary length-to-diameter ratios (L/D) may be accommodated. Extensive convergence studies show that frequencies which are exact to five or six significant figures may be obtained for cylinders with free–free ends, and that three or four figure accuracy is achievable for fixed–free ends. These accurate frequencies are compared with ones determined from known elementary and improved 1-D theories for longitudinal, torsional, and bending modes for various L/D ratios, thereby establishing the ranges of accuracy of the 1-D analyses for circular cross sections.

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.