Nonlinear Harmonic Oscillations of Gyroscopic Structural Systems and the Case of a Rotating Ring

Abstract

Two related problems are investigated in order to study via a simple example the influence of gyroscopic forces on nonlinear harmonic oscillations of rotationally symmetric shell structures. First, the amplitude frequency equations are calculated for circumferentially traveling waves in a circular ring rotating about its geometrical axis. The results show that in the range of rotational speeds considered the backward traveling waves exhibit hardening type of response, whereas for the forward traveling waves there is a transition from hardening to softening type of behavior as the rotational speed increases. The second part of the paper is devoted to an analysis of interaction between the two traveling waves which is expected at low angular speeds. The results, valid for arbitrary shells of revolution, reveal the existence of secondary bifurcation points on the branches corresponding to the traveling waves, and the response on the secondary branches is found to be close to standing waves which do not appear at all as solutions of the linear free-vibration problem for the rotating shell.