Chaotic motions of a rigid rotor in short journal bearings

In the present paper the conditions that give rise to chaotic motions in a rigid rotor on short journal bearings are investigated and determined. A suitable symmetry was given to the rotor, to the supporting system, to the acting system of forces and to the system of initial conditions, in order to restrict the motions of the rotor to translatory whirl. For an assigned distance between the supports, the ratio between the transverse and the polar mass moments of the rotor was selected conveniently small, with the aim of avoiding conical instability. Since the theoretical analysis of a system's chaotic motions can only be carried out by means of numerical investigation, the procedure here adopted by the authors consists of numerical integration of the rotor's equations of motion, with trial and error regarding the three parameters that characterise the theoretical model of the system: m, the half non-dimensional mass of the rotor, , the modified Sommerfeld number relating to the lubricated bearings, and , the dimensionless value of rotor unbalance. In the rotor's equations of motion, the forces due to the lubricating film are written under the assumption of isothermal and laminar flow in short bearings. The number of numerical trials needed to find the system's chaotic responses has been greatly reduced by recognition of the fact that chaotic motions become possible when the value of the dimensionless static eccentricity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabew7aLnaaBaaaleaacaWGZbaabeaaaaa!4046!\[\varepsilon _s \] is greater than 0.4. In these conditions, non-periodic motions can be obtained even when rotor unbalance values are not particularly high (=0.05), whereas higher values (>0.4) make the rotor motion periodic and synchronous with the driving rotation. The present investigation has also identified the route that leads an assigned rotor to chaos when its angular speed is varied with prefixed values of the dimensionless unbalance . The theoretical results obtained have then been compared with experimental data. Both the theoretical and the experimental data have pointed out that in the circumstances investigated chaotic motions deserve more attention, from a technical point of view, than is normally ascribed to behaviours of this sort. This is mainly because such behaviours are usually considered of scarce practical significance owing to the typically bounded nature of chaotic evolution. The present analysis has shown that when the rotor exhibits chaotic motions, the centres of the journals describe orbits that alternate between small and large in an unpredictable and disordered manner. In these conditions the thickness of the lubricating film can assume values that are extremely low and such as to compromise the efficiency of the bearings, whereas the rotor is affected by inertia forces that are so high as to determine severe vibrations of the supports.Nomenclature C radial clearance of bearing (m) - D diameter of bearing (m) - e dimensional eccentricity of journal (m) - e _s value of e corresponding to the static position of the journal - E dimensional static unbalance of rotor (m) - f _x, f _y =F _x/(P), F _y/(P): non-dimensional components of fluid film force - F _x, F _y dimensional components of fluid film force (N) - g acceleration of gravity (m/s²) - L axial length of bearing (m) - m % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabg2da9maalaaabaGaeqyYdC3aaWbaaSqa% beaacaaIYaaaaaGcbaGaeqyYdC3aa0baaSqaaGabciaa-bdaaeaaca% WFYaaaaaaakiabg2da9maalaaabaGaeqyYdC3aaWbaaSqabeaacaaI% YaaaaOGaam4qaaqaaiabeo8aZjaadEgaaaaaaa!4C14!\[ = \frac{{\omega ^2 }}{{\omega _0^2 }} = \frac{{\omega ^2 C}}{{\sigma g}}\]: half non-dimensional mass of rotor - M half mass of rotor (kg) - n angular speed of rotor (in r.p.m.=60/2) - t time     

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