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Research Papers: Contact Mechanics

Analytical and Low-Order Numerical Modeling of Ball-to-Ball Contact Friction in Linear Ball Bearings and Ball Screws

[+] Author and Article Information
Bo Lin

Smart and Sustainable Automation Research Laboratory,
Department of Mechanical Engineering,
University of Michigan, G.G. Brown Laboratory,
2350 Hayward, Ann Arbor, MI 48109
e-mail: bolin@umich.edu

Molong Duan

Smart and Sustainable Automation Research Laboratory,
Department of Mechanical Engineering,
University of Michigan, G.G. Brown Laboratory,
2350 Hayward, Ann Arbor, MI 48109
e-mail: molong@umich.edu

Chinedum E. Okwudire

Associate Professor
Mem. ASME
Smart and Sustainable Automation Research Laboratory,
Department of Mechanical Engineering,
University of Michigan, G.G. Brown Laboratory,
2350 Hayward, Ann Arbor, MI 48109
e-mail: okwudire@umich.edu

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the Journal of Tribology. Manuscript received October 24, 2018; final manuscript received April 25, 2019; published online May 17, 2019. Assoc. Editor: Wenzhong Wang.

J. Tribol 141(7), 071401 (May 17, 2019) (17 pages) Paper No: TRIB-18-1443; doi: 10.1115/1.4043630 History: Received October 24, 2018; Accepted April 26, 2019

Analytical and low-order numerical models are very useful for studying friction behavior of rolling element machine components like ball bearings and ball screws. This is because they provide generalizable insights into friction behavior at much lower computational costs compared with high-order numerical models like finite element analysis (FEA). While analytical and low-order numerical models in the literature are mainly focused on ball-to-groove contact friction, experimental studies have shown that ball-to-ball contact friction is also very important. This is especially true for linear ball bearings/guideways and ball screws which, unlike rotary ball bearings, do not typically make use of caged balls to prevent ball-to-ball contact. Therefore, in this paper, low-order numerical models for ball-to-ball contact friction in linear ball bearings and ball screws are developed. Furthermore, an analytical model for ball-to-ball contact friction in four-point contact linear ball bearing is derived by making simplifications to its low-order numerical model. Compared with ball-to-ball friction predictions from FEA models developed in ansys, the proposed numerical models are shown in case studies to be accurate within 7%, while computing at least three orders of magnitude faster. Moreover, case studies are used to demonstrate how the developed models can be used in practice, e.g., for the mitigation of ball-to-ball contact friction in linear ball bearings and the prediction of friction variation during the operation of a ball screw.

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References

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Figures

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Fig. 3

Contact area and relative velocity field for linear ball bearing

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Fig. 2

Geometry and coordinate systems for a four-point contact linear ball bearing

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Fig. 1

Basic module of two balls in a four-point contact linear ball bearing

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Fig. 4

Flowchart for the solving process of ball motion and friction in linear ball bearing

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Fig. 7

Contact angle deviations and external loading conditions in the case study

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Fig. 8

Velocity difference of two balls as functions of side force ratio and contact angle deviation: (a) low-order numerical model and (b) analytical formula

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Fig. 9

Ball-to-ball contact forces and friction loss as functions of side force ratio and contact angle deviation

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Fig. 5

Frictional force and moment as functions of ci/ai: (a) full plot and (b) zoomed in plot near ci/ai = 0

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Fig. 6

Angular deviation of velocity center from contact center

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Fig. 13

Geometry and coordinate systems for a four-point contact ball screw

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Fig. 10

Example of ball-to-ball contact and its avoidance by optimized design

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Fig. 11

Mesh for linear ball bearing in ansys

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Fig. 12

(a) Nominal ball center pathway (helix) and coordinate systems of ball screw and (b) two-ball module in ball screw

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Fig. 14

Contact area and relative velocity field for ball screw

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Fig. 15

Cross-sectional profile of ball screw with geometric errors

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Fig. 19

Contact angle deviation with respect to azimuth angle

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Fig. 21

Total friction torque and the contribution of ball-to-groove and ball-to-ball contact

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Fig. 17

Contact angle deviation at the cross section of ball screw grooves

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Fig. 18

Mesh for ball screw in ansys

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Fig. 20

Angular distance between the balls and ball-to-ball contact status (ball-to-ball contact is marked using black dots)

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Fig. 16

Ball-to-ball contact between two balls in ball screw

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Fig. 22

The ratio of the semimajor axis ai to the ball radius RB

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