Research Papers: Applications

A Discrete Element Approach for Modeling Cage Flexibility in Ball Bearing Dynamics Simulations

[+] Author and Article Information
Nick Weinzapfel, Farshid Sadeghi

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907

J. Tribol 131(2), 021102 (Mar 03, 2009) (11 pages) doi:10.1115/1.3063817 History: Received March 25, 2008; Revised October 12, 2008; Published March 03, 2009

A model for deep-groove and angular-contact ball bearings was developed to investigate the influence of a flexible cage on bearing dynamics. The cage model introduces flexibility by representing the cage as an ensemble of discrete elements that allow deformation of the fibers connecting the elements. A finite element model of the cage was developed to establish the relationships between the nominal cage properties and those used in the flexible discrete element model. In this investigation, the raceways and balls have six degrees of freedom. The discrete elements comprising the cage each have three degrees of freedom in a cage reference frame. The cage reference frame has five degrees of freedom, enabling three-dimensional motion of the cage ensemble. Newton’s laws are used to determine the accelerations of the bearing components, and a fourth-order Runge–Kutta algorithm with constant step size is used to integrate their equations of motion. Comparing results from the dynamic bearing model with flexible and rigid cages reveals the effects of cage flexibility on bearing performance. The cage experiences nearly the same motion and angular velocity in the loading conditions investigated regardless of the cage type. However, a significant reduction in ball-cage pocket forces occurs as a result of modeling the cage as a flexible body. Inclusion of cage flexibility in the model also reduces the time required for the bearing to reach steady-state operation.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 13

Effect of cage elastic modulus on cage whirl radius

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Figure 14

Effect of ball-cage pocket traction on cage whirl radius

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Figure 15

Effect of cage type on (a) ball-inner race slip velocity and (b) ball total angular velocity

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Figure 16

Effect of cage type on ball-race pressures: (a) ball-inner race contact and (b) ball-outer race contact

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Figure 1

Ball-race traction curve

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Figure 2

Dynamic bearing model algorithm

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Figure 3

Simplified stamped cage model

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Figure 4

Rigid beams connecting pockets and flexible bridge assemblies

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Figure 5

(a) Joint model and (b) fiber model

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Figure 6

Degrees of freedom for a cage pocket or block

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Figure 7

Cage reference frame

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Figure 8

Cage motion in various bearing loading conditions: (a) X-position and (b) title angle

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Figure 9

Pocket velocity vector resolved into radial and tangential components

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Figure 10

Comparison of force-deflection relationships for finite element and discrete element cage models

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Figure 11

Effect of cage type on normal forces between all ball-pocket pairs: (a) rigid cage and (b) flexible cage

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Figure 12

Net driving force on cage pockets: (a) rigid cage and (b) flexible cage

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Figure 17

Cage tilting in combined radial and axial loaded bearings

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Figure 18

Path of the cage center of mass of the flexible cage: (a) loading types 1 and 3; (b) loading type 2

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Figure 19

Angle of ball-to-pocket vector in the pocket frame of reference

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Figure 20

Ball position and advancing force on the cage pocket of the flexible cage

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Figure 21

Conceptual illustration of force-field acting on pockets of the flexible cage

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Figure 22

Motion of a discrete mass P in a reference frame undergoing general three-dimensional motion



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