Abstract

This paper studies how bearing asymmetry affects natural frequencies and mode shapes of a rotating disk/spindle system through a numerical simulation and a perturbation analysis. Existing literature has shown that rocking motion of a rotating disk/spindle system with symmetric bearings consists of rigid body rocking of the spindle, one-nodal-diameter modes of each disk, and deformation of spindle bearings. The rocking motion, characterized by (0,1) unbalanced modes, has repeated natural frequencies when the spindle is stationary, because the disk/spindle system is axisymmetric. For a rotating spindle, (0,1) unbalanced modes evolve into forward and backward precession with circular orbits. In this paper, the numerical simulation shows that bearing asymmetry splits a pair of repeated (0,1) unbalanced modes into two modes with distinct frequencies when the spindle is stationary. Moreover, when the rotational speed increases from zero, the (0,1) unbalanced mode with lower frequency evolves into backward precession and the (0,1) unbalanced mode with higher frequency evolves into forward precession. The precession orbits are elliptical because of the bearing asymmetry. Two perturbation schemes are developed to prove the phenomena observed in the numerical simulation. For low rotational speed, a stationary disk/spindle system with symmetric bearings serves as the unperturbed system. Both the bearing asymmetry and gyroscopic effects from rotation form the perturbation. A contraction iteration predicts the effects of bearing asymmetry on natural frequencies and mode shapes. For high rotational speed, a rotating (gyroscopic) disk/spindle system with symmetric bearings serves as the unperturbed system. The bearing asymmetry forms the perturbation. To obtain a perturbation solution, the solvability condition is first derived for the unperturbed gyroscopic system. Lindsted-Poincaré approach then predicts the effects of bearing asymmetry on natural frequencies and mode shapes of the rotating disk/spindle system.

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