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

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.