The Timoshenko beam theory includes the effects of shear deformation and rotary inertia on the vibrations of slender beams. The theory contains a shear coefficient which has been the subject of much previous research. In this paper a new formula for the shear coefficient is derived. For a circular cross section, the resulting shear coefficient that is derived is in full agreement with the value most authors have considered best. Shear coefficients for a number of different cross sections are found.

Shear coefficients for Timoshenko beam theory

Recent research advances in the dynamic behavior of shells: 1989–2000, Part 2: Homogeneous shells

Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method

Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method

Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration

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.

Comparisons of vibration frequencies for rods and beams from one‐dimensional and three‐dimensional analyses

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.

Accurate vibration frequencies of circular cylinders from three‐dimensional analysis

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 ...

Three-Dimensional Vibrations of Truncated Hollow Cones:

Response to ‘‘Comments on ‘Comparisons of vibration frequencies for rods and beams from one‐dimensional and three‐dimensional analyses’ ’’ [J. Acoust. Soc. Am. 100, 1890–1892 (1996)]

The authors respond to the comments by Hutchinson.The authors respond to the comments by Hutchinson.

Response to ‘‘Comments on ‘Accurate vibration frequencies of circular cylinders from three‐dimensional analysis’ ’’ [J. Acoust. Soc. Am. 100, 1894–1895 (1996)]

Jinyoung So

Papers

Comparisons of vibration frequencies for rods and beams from one‐dimensional and three‐dimensional analyses

Accurate vibration frequencies of circular cylinders from three‐dimensional analysis

Three-Dimensional Vibrations of Truncated Hollow Cones:

Response to ‘‘Comments on ‘Comparisons of vibration frequencies for rods and beams from one‐dimensional and three‐dimensional analyses’ ’’ [J. Acoust. Soc. Am. 100, 1890–1892 (1996)]

Response to ‘‘Comments on ‘Accurate vibration frequencies of circular cylinders from three‐dimensional analysis’ ’’ [J. Acoust. Soc. Am. 100, 1894–1895 (1996)]