Three-Dimensional Vibrations of Truncated Hollow Cones

Abstract

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.

On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

“…(17) and (18) constitute an orthogonal and mathematically complete series in the torus domain. Each of these two vibration categories (symmetric and anti-symmetric modes about the centerline plane) can be separately solved and thus, it results in two smaller sets of eigenvalue equations while maintaining the same level of accuracy.…”

“…Due to rapid increases in computer speeds and storage capacities, it is now commonplace to obtain accurate solutions for three-dimensional (3-D) elastic vibration problems, especially for bodies of revolution incorporating simple algebraic polynomials and algebraic-trigonometric polynomials as displacement trial functions in variational Ritz procedures developed in cylindrical and spherical coordinates. For example, accurate frequencies have been achieved for cylinders [15,16], hollow cones [17] and spheres [18][19][20].…”

Three-dimensional vibrations of solid cones with and without an axial circular cylindrical hole

“…The majority of the existing literature describes the vibration analysis for thin conical shells, based upon a thin shell or membrane type of shell theory (Leissa, 1973). The first contribution to the 3-D analysis of conical shells was by Leissa and So (1995) applying the Ritz method. Five years later, the conical shells were analyzed by a finite element method (Buchanan, 2000;Buchanan and Wong, 2001).…”

Three-dimensional vibration analysis of a cantilevered parallelepiped: Exact and approximate solutions

“…However, such threedimensional elastic solutions are particularly scarce. Some of the investigations are concerned with rods and beams ͑Hutchinson, 1971; Hutchinson, 1981;So, 1995a͒, parallelepiped ͑Fromme andLeissa, 1970;Hutchinson and Zillmer, 1983;Leissa and Zhang, 1983;Liew et al, 1995a͒, solid andhollow cylinders ͑Hutchinson, 1967;Hutchinson, 1980;Leissa and So, 1995b;Liew et al, 1995b;So and Leissa, 1997͒, truncated hollow cones ͑Leissa and So, 1995c͒, andopen shells ͑Lim et al, 1998c͒. To the author's knowledge, direct comparison of threedimensional elasticity solutions including and excluding transverse normal stress is only available in Hutchinson ͑1979, 1984͒. In these two papers, Hutchinson analyzed the vibration of thick, free circular plates using the Mathieu series solution and modified Pickett method to obtain exact solutions.…”