Nonlinear Vibration Analysis of an Elastic Rotor Supported on Angular Contact Ball Bearings Considering Six Degrees of Freedom and Waviness on Balls and Races

Abstract

Nonlinear vibration analysis of an elastically deformable shaft supported on two lubricated angular contact ball bearings is reported herein considering six-degrees of freedom (6-DOF) and waviness on races and balls. This is an extension work of the investigation published by the authors Babu, C. K., Tandon, N., and Pandey, R. K., 2012, “Vibration Modeling of a Rigid Rotor Supported on the Lubricated Angular Contact Ball Bearings Considering Six Degree of Freedom and Waviness on Balls and Races,” ASME J. Vib. Acoust., 134, p. 011006. Elastic deformation of shaft, frictional moment, and waviness on races and balls have been incorporated in the model for the vibration investigations of rotor's CG. Two noded 3D Timoshenko beam element having 6-DOF has been employed in the computation of the shaft's deformation. Governing equations with appropriate boundary conditions have been solved using 4th order Runge–Kutta method. It is observed that vibration amplitude enhances considerably after incorporating the elastic deformation in comparison to the amplitude achieved using rigid rotor model approach. Moreover, the influence of outer race's radial waviness is large on the amplitudes of vibrations in comparison to radial waviness of inner race. However, it is worth noting here that in case of rigid rotor model the presence of radial waviness on inner race yields high amplitudes of vibrations.

The influence of elastohydrodynamic lubrication on the stiffness of deep groove ball bearing

“…The schematic diagram for the relations between bearing loads and displacements for dry contact ball bearing are shown in Fig. 3 [21,29]. As seen from Fig.…”

“…The bearing load-displacement relations for dry contact have been elaborated in [21,28] and can be derived by considering the relations of (i) the displacements of inner ring and the deformation of outer raceway-ball element-inner raceway, (ii) the load and the deformation for outer raceway-ball element-inner raceway contact, and (iii) the normal loads on all ball elements and bearing forces and moments. The lubricated ball bearing load-displacement relations can be also derived on the basis of above steps, but the load and deformation relation for outer raceway-ball element-inner raceway for dry contact (denoted as (ii)) needs to be replaced with the lubricated contact force and mutual approach relation as shown in Eq.…”

“…(11). The lubricated ball bearing load-displacement relations is given directly as follow and more detailed derivation process can be found in [21,28]…”

“…Liu [20] calculated the time-varying contact stiffness between the roller and races due to the nonuniform surface waviness and investigated its effect on the vibration response of the lubricated roller bearing. Babu [21] analyzed the nonlinear vibration of an elastically deformable shaft at high rotational speeds supported on two lubricated angular contact ball bearings with surface waviness on races and balls, however, the detailed calculation process for the stiffness and damping of the lubricated contacts were not given, and the influence of the EHL on the bearing stiffness coefficients were also not investigated.…”

“…These relationships are used on the establishment of the lubricated bearing equilibrium model describing the relation of bearing forces and displacements. An enhanced five-dimensional stiffness matrix of the lubricated bearing is given based on the comprehensive dry contact bearing stiffness matrix proposed by Lim and Singh [21]. Furthermore, the stiffness coefficients from both dry and lubricated bearings under different combinations of loads are compared, and the effects of EHL on stiffness coefficients are discussed.…”