The second spectrum in Timoshenko beam theory: a new approach for its identification

The problem of existence and identification of a second frequency spectrum in the Timoshenko beam theory is reconsidered from a novel perspective, studying a Timoshenko beam rotating with constant angular speed about its longitudinal axis. The existence of a second spectrum in the case of non–rotating beams and general boundary conditions has been much debated in the literature, since it is possible to easily identify the companion natural frequencies constituting the second spectrum only in particular cases. Its existence in a non–rotating finite–length beam has been recently demonstrated on the basis of accurate experimental results, at least for free–free boundary conditions, and also by considering free waves in beams of infinite length. In this study new evidence of existence of a second spectrum together with a novel definition for its identification are presented, possible when considering gyroscopic effects. As a secondary result, it can be stated that the whole second spectrum gives no contribution to the forward critical speeds of the rotating beam.