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)