Eigenfrequencies of a Three-Dimensional Arbitrarily-Curved Beam

Abstract

Research Problem The eigenvalue problem for the vibrations of an arbitrarily-curved three-dimensional beam with circular cross-section is solved by a series expansion method under various boundary conditions. Methodology The governing differential equations of motion are derived based on Euler-Bernoulli beam theory using the Hamilton’s principle. The general equations are given for any space curved beam with variable curvature and torsion, and solved for a specific example using the method of power series. Results and Conclusions The eigenfrequencies of a specific 3D beam were computed and compared with the eigenfrequencies of straight, circular, and helical beams, all having the same length. It was found that the eigenfrequencies of the 3D beam tend to increase slower compared to the other cases as the mode number increases. The main contribution of this study is the computation of the eigenfrequencies of a truly three-dimensional beam: torsion and curvature change continously along the beam length. In constrast, the most studied 3D case, helical beam, has constant curvature and torsion.