The dynamic behavior of a high speed unbalanced rotor supported on roller bearings with damping has been studied, focusing particular attention on its nonlinear aspects. The non-linearity in the rotor bearing system has been considered mainly due to Hertzian contact, unbalanced rotor effect and radial internal clearance. This is modeled as an oscillating spring-mass-damper system. Various techniques like Time Response curves, Poincaré maps, Orbits plots, fast Fourier transformation, Hopf bifurcation and Phase Trajectory are used to study the nature of response. The motion of an unbalanced rotor is categorized with respect to the ratio of the Forcing/Natural frequency of the system as Harmonic, Sub-harmonic, Quasi periodic and Chaotic. The results show the appearance of instability and chaos in the dynamic response as the speed of the rotor-bearing system is changed. Period doubling and mechanism of intermittency have been observed that lead to chaos. The outcomes illustrate the appearance of instability and chaos in the dynamic response as the speed of the rotor-bearing system is changed. This work differs from the previous studies in the way that the complex model simulates nonlinear vibrations, considering that both the lubricated nonlinear contact stiffness and damping correspond to the conservative and dissipative energies, respectively. The comprehensive model developed in this investigation is a useful tool to predict the system behavior and for performance evaluation of a rotor bearing system.

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In this article, a mathematical model has been developed to predict the non-linear dynamic behaviour of a high-speed unbalanced rotating shaft due to ball size variations and varying number of balls. In the mathematical formulation, the contact between balls and inner/outer races is considered as a non-linear spring, whose stiffness is obtained using Hertzian elastic deformation theory. A detailed contact-damping model reflecting the influences of the surface profiles and the speeds of both contacting elements is developed and applied in the rolling element bearing formulation. This formulation also accounts for sources of non-linearity such as Hertzian contact force, varying number of balls, and off-sized balls resulting transition from noncontact to contact state between balls and races. The equations of motion of a rolling element bearing are formulated using Lagrange's equation considering the vibration characteristics of individual constituents such as inner race, outer race, rolling elements, and shaft. To overcome the strong non-linearity of the governing equations of motion, a modified Newmark-β time integration technique has been used to solve the equations of motion numerically. All results have been presented in the form of fast Fourier transformations and Poincaré maps. The highest radial vibrations due to ball size variation are at a speed of the number of balls times the cage speed (ω = kω cage Hz). The other vibrations due to ball size variation also occurr at (X ± kω cage ), where k is a constant. When the number of balls is increased, the centre of oscillations approaches zero, implying a stiffer system. From this it can be predicted that increasing the number of balls will reduce the effect of the BPF. Hence it is implied from the analysis that the number of balls and the size variation of balls are important parameters and should be considered at the design stage.

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Purpose -The non-recessed hybrid journal bearings of cylindrical type, when operating at higher rotational speeds can suffer self-exited vibrations(oilwhirl Instability), which can cause excessive rotor motion causing bearing and sometimes total machine failure. The multi-lobe journal bearing exhibits better stability as well as a superior capability to suppress oil-whirl. The paper aims to present a theoretical study pertaining to a two-lobe hole-entry hybrid journal bearing by considering the combined influence of surface roughness and journal misalignment on the performance of the bearing. Design/methodology/approach -The average Reynolds equation governing the flow of lubricant in the clearance space between the rough bearing surfaces together with the equation of flow through a capillary restrictor has been solved using FEM. The bearing performance characteristics have been simulated for a two-lobe hole-entry hybrid journal bearing for the various values of offset factor, restrictor design parameter, surface roughness parameter, surface pattern parameter and journal misalignment parameters. Findings -The two-lobe hole-entry hybrid journal bearing system with an offset factor greater than one indicates significant improvement of the order of 15-25 percent in the values of direct stiffness and direct damping coefficients compared to a circular hole-entry hybrid journal bearing system. Also the lubricant flow of a two-lobe hole-entry hybrid journal bearing is reduced by 25 percent vis-à -vis circular bearing. Originality/value -The present work is original of its kind, in case of two-lobe hole-entry hybrid journal bearing. The results are quite useful for the bearing designer.

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