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

Abstract

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.

A hierarchy of dynamic equations for solid isotropic circular cylinders

“…Dirichlet or Neumann), various approximate methods of solutions have been developed. There exists on one hand analytical solutions based on expansion in terms of Bessel functions [8,9] where part of the boundary conditions are satisfied approximately, and on the other hand numerical solutions such as the Ritz method [10,11,12,13] or the finite element method [14]. These works using approximate methods concentrate mainly on eigenfrequency analyzes.…”

Study of pure transverse motion in free cylinders and plates in flexural vibration by Ritz’s method

“…B) Each of the frequencies shown in Table 1, third row, is smaller than those shown in Table II of Leissa and So (1995a) for the same number of analogous terms. Hence the introduction of the added terms improves the calculation of the natural frequencies of vibration of a short cylinder.…”

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

“…Hutchinson [6,7] studied the vibration of a completely free prism with circular cross-section and compared his 3-D solutions with the approximate 1-D solutions [8]. Such a prism was also studied by Leissa and So [9] using the Ritz method for the free-free and the fixed-free end conditions. Fromme and Leissa [10] used the so-called Fourier associated-periodicity extension method to study the 3-D vibration of a completely free prism with rectangular cross-section.…”

“…Lim [16] investigated the effect of hypothetical assumption of neglecting flexural normal stress in vibration analysis for a cantilevered prism with rectangular cross-section. Up to now, most of the research work on 3-D vibration analysis of structural components is performed by the Ritz method because of its simplicity in analysis [9,[12][13][14][15][16], in which two types of polynomial functions are commonly used as the basis of admissible functions: the simple algebraic polynomials [9,[12][13][14]16] and the orthogonal polynomials [15]. It is well known that the simple algebraic polynomials have the straightforward and convenient characteristics in programming and computing.…”