NONPERIODIC MOTIONS OF A JEFFCOTT ROTOR WITH NONLINEAR ELASTIC RESTORING FORCES

The conditions that give rise to non-periodic motions of a Jeffcott ro tor in the presence of non-linear elastic restoring forces are examine d. It is well known that non-periodic behaviours that characterise the dynamics of a rotor are fundamentally a consequence of two aspects: t he non-linearity of the hydrodynamic forces in the lubricated bearings of the supports and the non-linearity that affects the elastic restor ing forces in the shaft of the rotor. In the present research the anal ysis was restricted to the influence of the non-linearity that charact erises the elastic restoring forces in the shaft, adopting a system th at was selected the simplest as possible. This system was represented by a Jeffcott rotor with a shaft of mass that was negligible respect t o the one of the disk, and supported with ball bearings. In order to c heck in a straightforward manner the non-linearity of the system and t o confirm the results obtained through theoretical analysis, an invest igation was carried out using an experimental model consisting of a ro tating disk fitted in the middle of a piano wire pulled taut at its en ds but leaving the tension adjustable. The adopted length/diameter rat io was high enough to assume the wire itself was perfectly flexible wh ile its mass was negligible compared to that of the disk. Under such h ypotheses the motion of the disk centre can be expressed by means of t wo ordinary, non-linear and coupled differential equations. The condit ions that make the above motion non-periodic or chaotic were found thr ough numerical integration of the equations of motion. A number of num erical trials were carried out using a 4th order Runge-Kutta routine w ith adaptive stepsize control. This procedure made it possible to plot the trajectories of the disk centre and the phase diagrams of the com ponent motions, taken along two orthogonal coordinate axes, with their projections of the Poincare sections. On the basis of the theoretical results obtained, the conditions that give rise to non-periodic motio ns of the experimental rotor were identified.