Nonlinear Excitation Model of Ball Bearing Waviness in a Rigid Rotor Supported by Two or More Ball Bearings Considering Five Degrees of Freedom

Abstract

This research presents a nonlinear model to analyze the ball bearing vibration due to the waviness in a rigid rotor supported by two or more ball bearings. The waviness of a ball and each races is modeled by the superposition of sinusoidal function, and the position vectors of inner and outer groove radius center are defined with respect to the mass center of the rotor in order to consider five degrees of freedom of a general rotor-bearing system. The waviness of a ball bearing is introduced to these position vectors to use the Hertzian contact theory in order to calculate the elastic deflection and nonlinear contact force resulting from the waviness while the rotor has translational and angular motion. They can be determined by solving the nonlinear equations of motion with five degrees of freedom by using the Runge-Kutta-Fehlberg algorithm. Numerical results of this research are validated with those of prior researchers. The proposed model can calculate the translational displacement as well as the angular displacement of the rotor supported by two or more ball bearings with waviness. It also characterizes the vibration frequencies resulting from the various kinds of waviness in rolling elements, the harmonic frequencies resulting from the nonlinear load-deflection characteristics of ball bearing, and the sideband frequencies resulting from nonlinearity of the waviness interaction.

An extensive review of vibration modelling of rolling element bearings with localised and extended defects

“…The 435 former observation was also reported by Tiwari et al [105,106], and the latter by Tandon et al [46]. 436 Sopanen et al [54,55] ignored the slippage of the rolling elements (balls) [49] and neglected the centrifugal 437 forces acting on them, although it was shown that the modelled defect-related frequencies agree well with 438 the earlier results published in the literature [46,97,98,103,[105][106][107].…”

“…While Sopanen et al [54,55] did not conduct any experimental work, they 429 compared their modelling results with those of similar studies available in the literature. For example, for 430 the localised raceway defects, the predicted results were compared with the results in reference [46], and for 431 the waviness, the modelling results were compared with the results reported in references [97,98,103,107].…”

A general method for impact dynamic analysis of a planar multi-body system with a rolling ball bearing joint

“…To simplify the computational model, only the normal contact forces in bearing multi-point contacts were considered by the authors. On the other hand, the rolling friction in ball and raceway contact is usually quite small and has a smaller effect on the dynamic response of such bearing systems: many researchers [42][43][44][45][47][48][49] examining rolling element bearing dynamics neglect frictional effects.…”

“…All of these determine the complexity of the modelling and solution of the mechanical multi-body system with joint rolling element bearings. In recent years, and with regards the dynamic modelling of rolling element bearings, Jang and Jeong [42][43][44] presented a non-linear model to analyse the ball bearing vibration due to waviness in a rigid rotor supported by two or more ball bearings. Bai and Xu [45] presented a 5-DOF dynamic model to study the dynamic performance of ball bearings due to the effect of both internal clearance and waviness at high speed, where the centrifugal force and gyroscopic moment from balls were taken into account.…”

Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system

“…two or five degrees of freedom with or without ball centrifugal force, which could be applied to determine the vibration response of rotor systems supported by ball bearings. Jang and Jeong [10] proposed the fivedegree-of-freedom excitation model of ball bearing waviness to investigate the bearing vibration. Then, considering the centrifugal force and gyroscopic moment of ball, they developed an analytical method to calculate the characteristics of the ball bearing under the effect of waviness in [11].…”