Gyroscopic effects analysis in the Lagrangian formulation of rotating beams

The present paper deals with some questions related to the Lagrangian formu lation of a continuous, axisymmetric rotating Timoshenko beam. Two Lagrangi an densities, derived by other authors, are considered; they differ from ea ch other in the expression of the gyroscopic terms. It is proved that the a bove discrepancy, which originates from the different descriptions of the k inematics of the rotating beam, is not of physical nature and that both the competing formulations lead to the same differential equations of motion. In particular, by means of the theory of continuous systems, it is first sh own that the difference between the two Lagrangian densities, which turns o ut to be a special case of four-divergence, identically satisfies the Lagra nge's equations; the same result is then obtained by verifying that the Ham iltonian density of the system exhibits the same expression when both formu lations are utilized. The latter point leads also to the unambiguous defini tion of the total energy of the system.