Mode-locking in nonlinear rotordynamics

G.H.M. van der Heijden

We present a computer-assisted study of the dynamics of two nonlinearly coupled driven oscillators with rotational symmetry which arise in rotordynamics (the nonlinearity coming from bearing clearance). The nonlinearity causes a splitting of the twofold degenerate natural frequency of the associated linear model, leading to three interacting frequencies in the system. Partial mode-locking, then, yields a bi-infinite series of attracting invariant 2-tori carrying (quasi-)periodic motion. Due to the resonance nature, the (quasi-)periodic solutions become periodic in a co-rotating co-ordinate system. They can be viewed as entrainments of periodic solutions of the associated linear problem; one presumably infinite family is generated by (scaled) driving frequencies omega=1+2/n, n=1,2,3,..., another one by frequencies omega=m, m=4,5,6,.... Both integers n and m can be related to discrete symmetry properties of the particular periodic solutions. Under a perturbation that breaks the rotational symmetry, more complicated behaviour is possible. In particular, a second rational relation between the frequencies can be established, resulting in fully mode-locked periodic motion.

keywords: rotor dynamics, bearing clearance, mode-locking, resonance

J. Nonlinear Sci. 5, 257-283 (1995)