Gyroscopic Effects Analysis in the Lagrangian Formulation of Rotating Beams

Abstract

The present paper deals with some questions related to the Lagrangian formulation of a continuous, axisymmetric rotating Timoshenko beam. Two Lagrangian densities, derived by other authors, are considered; they differ from each other in the expression of the gyroscopic terms. It is proved that the above discrepancy, which originates from the different descriptions of the kinematics 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 shown that the difference between the two Lagrangian densities, which turns out to be a special case of four-divergence, identically satisfies the Lagrange's equations; the same result is then obtained by verifying that the Hamiltonian density of the system exhibits the same expression when both formulations are utilized. The latter point leads also to the unambiguous definition of the total energy of the system.