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Three-Dimensional Vibration Analysis of Deep, Nonlinearly Tapered Rods and Beams with Circular Cross-Section  

심현주 (중앙대학교 대학원 건축학과)
강재훈 (중앙대학교 공과대학 건축학부)
Publication Information
Journal of the Computational Structural Engineering Institute of Korea / v.16, no.3, 2003 , pp. 251-260 More about this Journal
Abstract
A three dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of deep, tapered rods and beams with circular cross section. Unlike conventional rod and beam theories, which are mathematically one-dimensional (1-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u/sup r/, u/sub θ/ and u/sub z/, in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in , and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the rods and beams are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rods and beams. Novel numerical results are tabulated for nine different tapered rods and beams with linear, quadratic, and cubic variations of radial thickness in the axial direction using the 3D theory. Comparisons are also made with results for linearly tapered beams from 1-D classical Euler-Bernoulli beam theory.
Keywords
vibration; three dimensional analysis; deep rod and beam; tapered rod and beam; ritz method;
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