MODE-LOCKING AND CHAOS IN A JEFFCOTT ROTOR WITH BEARING CLEARANCES

A complex mode-locking (or entrainment) structure underlying the nonli near whirling phenomenon of a horizontal Jeffcott rotor with a discont inuous nonlinearity (bearing clearance) was identified. A winding numb er is introduced as a measure of the ratio between two frequencies inv olved in the aperiodic whirling motions of the rotor system considered . Utilizing the winding number map, it was revealed that the alternati ng periodic and quasi-periodic responses take place according to the F arey number tree. The winding number varies in the form of the so-call ed ''Devil's staircase'' as a certain system parameter varies. From th e mode-locking pattern in the parameter space of the forcing amplitude and frequency, it was observed that as the forcing amplitude increase s, the size of each locking interval increases so that its growth take place in the form of ''Arnol'd tongues,'' where the winding number re mains a rational number. Moreover, inside each locking zone, i.e., eac h ''Arnol'd tongue,'' there exist many smaller tongues similar to the main tongue, in which a sequence of period-doubling bifurcations leadi ng to chaos occurred. The boundaries of each locking zone was obtained using a fixed-point algorithm along with the Floquet theory for check ing the stability of the periodic solutions. The winding numbers were estimated utilizing a fixed-point algorithm modified to obtain quasi-p eriodic responses. A jump phenomenon was also observed by tracking mul tiple periodic solutions for several parameters of the rotor system.