COMPARISONS OF VIBRATION FREQUENCIES FOR RODS AND BEAMS FROM ONE-DIMENSIONAL AND 3-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 (theta) 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 live 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. (C) 1995 Acoustical Society of America.