Three-Dimensional Harmonic Vibrations of a Circular Beam

In this paper an analytical treatment for the determination of the natural frequencies of a circular uniform beam, subjected to three-dimensional harmonic loads, is presented. Each differential element of the beam has six degrees of freedom, i.e., three translations and three rotations. This problem, in the most general case of response, is associated with a partial linear differential system composed of four coupled 3 × 1 vectorial equations. The influences of transverse shear deformation and rotatory inertia are also included in the analysis. The aforementioned solution methodology is successfully demonstrated through several numerical results.