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Contact Load and Stiffness of Four-Point Contact Ball Bearings Under Loading

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Abstract

Four-point contact ball bearings (X-ACBBs) are highly effective in applications requiring support for shafts that feature a high ratio of diameter to unsupported axial length and axial loading in two directions. In an effort to predict and understand problems that have been reported in X-ACCB usage, new modeling and analysis of the contact load and stiffness of X-ACBBs has been presented. The new model features a quasi-static model for X-ACBBs subjected to five degrees-of-freedom loading and displacement. The model was validated against numerical ball contact loads available in the literature. A numerical investigation was then performed to study the effects on ball contact loads and stiffness of rotational speed, external loads, and various geometric parameters of bearings such as arching dimension and unloaded contact angle. Under pure axial load, increasing the rotational speed causes an abrupt change in X-ACBB stiffness when the ball contact changes from two-point to three-point contact. Gyroscopic moment reduces the speed at which transition from two-point to three-point contact occurs. The dependence of X-ACBB stiffness on arching dimension and unloaded contact angle reveals that those parameters are critical for the design of X-ACBBs.

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Abbreviations

\(a{ }\) :

Semi-major radius of contact ellipse, mm

\(c\) :

Load–deflection constant

\(D_{a}\) :

Ball diameter, mm

\(d_{m}\) :

Pitch diameter, mm

\(\left\{ F \right\}\) :

Load vector

\(F\) :

External load, N

\(F_{c}\) :

Centrifugal force, N

\({\text{g}}\) :

Arching dimension (shim size), mm

\(J\) :

Ball mass moment of inertia, Nmm2

\(\left[ k \right]\) :

Stiffness matrix

\(M\) :

External moment, Nmm

\(M_{g}\) :

Gyroscopic moment, Nmm

\(m\) :

Ball mass, kg

\(n\) :

Rotational speed, rpm

\(\left[ {Q^{\prime}} \right]\) :

Ball Jacobian matrix

\(\left\{ Q \right\}\) :

Inner ring contact load vector

\(Q\) :

Contact load, N

\(\left[ {R\phi } \right]\) :

Transformation matrix

\(r\) :

Raceway curvature radius, mm

\(T\) :

Contact moment, Nmm

\(\left\{ u \right\}\) :

Inner ring contact displacement vector

\(u\) :

Inner ring displacement

\(v\) :

Ball center displacement

\(Z\) :

Number of balls

\(\alpha\) :

Contact angle, rad

\(\alpha_{0}\) :

Unloaded contact angle, rad

\(\alpha_{s}\) :

Resting (shim) contact angle, rad

\(\beta\) :

Pitch angle, rad

\(\left\{ \delta \right\}\) :

Displacement vector

\(\theta\) :

Inner race cross-section rotation, rad

\(\gamma\) :

Dimensionless parameter, \(\gamma = D_{a} /d_{m}\)

\(\omega\) :

Rotational speed, rad/s

\(\omega_{m}\) :

Ball orbital speed, rad/s

\(\omega_{r}\) :

Ball pivotal speed, rad/s

\(e\) :

Outer raceway

\(i\) :

Inner raceway

\(j\) :

Rolling element index

\(l\) :

Left ring

\(r\) :

Right ring

\(x,y,z\) :

Coordinate axes

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Acknowledgements

This research was financially supported by the National Research Foundation of Korea (Grant No.: NRF-2019R1F1A1063783, NRF-2020K2A9A1A06107195).

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Authors and Affiliations

  1. Department of Mechanical System Engineering, Kumoh National Institute of Technology, 61 Daehak-ro, Gumi, Gyeongbuk, 39177, South Korea

    Gilbert Rivera & Seong-Wook Hong

  2. Mobile Display Laser Technology Group, Samsung Display Vietnam (SDV), Yen Trung, Yen Phong, Bac Ninh, 16251, Vietnam

    Van-Canh Tong

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Correspondence to Seong-Wook Hong.

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Appendix: Proper Arching Dimension

Appendix: Proper Arching Dimension

The proper arching dimension can be chosen depending on the geometry of X-ACBB. In this way, a design constraint is used to avoid overlap** the left and right outer race contacts. This implies that the arching dimension should satisfy the inequality as follows [7]:

$$a_{e,l} \cos \alpha_{e,l} + a_{e,r} \cos \alpha_{e,r} < \frac{{D_{a} }}{2}\left( {\sin \alpha_{e,l} + \sin \alpha_{e,r} } \right)$$
(28)

Here, \({\varvec{a}}_{{{\varvec{e}},{\varvec{m}}}} ,\user2{ m} = {\varvec{l}},{\varvec{r}}\) correspond to the semi-major radii of contact ellipses in ball-race contacts. The arching dimension should also satisfy that the resting angle (\({\varvec{\alpha}}_{{\varvec{s}}}\)) is greater than zero and always less than or equal to the unloaded contact angle (\({\varvec{\alpha}}_{0}\)).

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Rivera, G., Tong, VC. & Hong, SW. Contact Load and Stiffness of Four-Point Contact Ball Bearings Under Loading. Int. J. Precis. Eng. Manuf. 23, 677–687 (2022). https://doi.org/10.1007/s12541-022-00643-0

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