Modeling three-dimensional vibration of elliptic bars

This paper presents a three-dimensional vibration analysis of elliptic bars with various end constraints. The theoretical derivation is established based on the linear, small-strain, three-dimensional elasticity principle. With the expansion of the displacements in a three-dimensional spatial coordinate, the integral expressions for strain and kinetic energies are developed. Using the orthogonal polynomials to represent the three-dimensional displacement field, a governing eigenvalue equation is obtained by minimizing the energy functional according to the Ritz principle. To demonstrate the applicability of this method, several numerical examples are solved. The accuracy of the frequency solutions is established by checking with the existing refined beam theories. First known results in terms of frequency parameters and displacement mode shape plots of the elliptic bar with different end support conditions are presented. The new results may serve as a useful reference to future research into the refined beam theories. © 1995, Acoustical Society of America. All rights reserved.