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Nonlinear dynamic correlation between balls, cage and bearing rings of angular contact ball bearings at different number of balls and groove curvature radii

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Abstract

In this work, the dynamic model of cage and elastohydrodynamic lubrication model were merged into the existing nonlinear dynamic model of ball bearings to explore the nonlinear dynamic correlation between balls, cage and bearing rings in the bearing system. Subsequently, the number of balls was varied to study the sliding of the ball, interaction forces between bearing components, vibration of inner ring and cage whirl motion. Again, the effects of the combinations of inner and outer groove curvature radii on the dynamic stability and vibration of the bearing system were conducted. The obtained results suggest the wonderful dynamic stability of cage and the desired vibration of the bearing system can be achieved when the maximum number of balls is reduced by one, and outer groove curvature radius is close to the radius of the ball, while inner groove curvature radius is appropriately larger than the radius of the ball. These results provide the theoretical support for the structural size matching of ball bearings.

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Data availability

All data generated during this study are included in this article and the datasets are available from the corresponding author on reasonable request.

Abbreviations

a :

Major axis of the elliptical area

b :

Minor axis of the elliptical area

δ :

Displacements of bearing components

θ :

Deflection angle of the bearing ring

Q :

Contact force

α :

Contact angle

F :

Force acting on bearing components

M :

Moment

I :

Moment of inertia

ω :

Angle velocity

m :

Mass

ρ :

Density of lubricant

β :

Attitude angle of ball

η :

Viscosity of lubricant

D :

Diameter

d :

Bearing pitch diameter

T :

Temperature

h :

Oil film thickness

Δv :

Differential slipping speed

p :

Pressure in contact area

Δu :

Relative skidding speed

u :

Entrainment velocity

h 1 :

Center oil film thickness

R :

Equivalent radius of curvature

ϑ :

Elastic deformation

E′:

Equivalent modulus of elasticity

w :

External load

p H :

Maximum Hertz contact pressure

E rp :

Relative errors of pressure

E rw :

Relative errors of load

k :

Ellipticity

ϕ :

Position angle

K′:

Coefficient of contact stiffness

ξ :

Viscous damping coefficient

C :

Clearance

µ :

Friction coefficient

r :

Radius

ħ :

Eccentricity of the cage center

ħ′:

Relative eccentricity of the cage center

B :

Guide face width of the cage

ρ e :

Effective density of the oil

ζ :

Proportionality coefficient of the oil–gas mixture

A :

Acreage

ε :

Radius of vortex trajectory

L :

Acceleration level

σ :

Acceleration

Z :

Number of the ball

f :

Frequency

Г:

Thickness of cage

x/y/z :

Directions along three axes of the global coordinate system

x′/y′/z′:

Directions along three axes of the local coordinate system

x″/y″/z″:

Directions along three axes of the moving coordinate system

x c/y c/z c :

Directions along three axes of the cage coordinate system

i:

Inner ring

o:

Outer ring

n :

Represent i or o

b:

Ball

c:

Cage

j :

jth ball

τ:

Friction effect

t:

Traction effect

e:

Retardation effect of lubricant

m :

Orbital revolution direction

ς :

Centrifugal direction

q :

Gyroscopic effect

v:

Viscous effect of lubricant

s:

Spin motion of balls

0:

Initial value

p:

Cage pockets

g:

Cage guidance

χ :

Unbalanced mass effect

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Acknowledgements

The authors would like to thank the Important Science and Technology Innovation Program of Hubei province (No. 2021BAA019) and National Key Research and Development Program of China (2019YFB2004304) for the support given to this research.

Funding

This work was funded by the Important Science and Technology Innovation Program of Hubei province (No. 2021BAA019) and National Key Research and Development Program of China (2019YFB2004304).

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Authors and Affiliations

  1. Hubei Key Laboratory of Advanced Technology for Automotive Components, Wuhan University of Technology, Wuhan, 430070, China

    Song Deng, Xianlin Zhu, Feng Wang & Lin Hua

  2. Hubei Engineering Research Center for Green Precision Material Forming, Wuhan University of Technology, Wuhan, 430070, China

    Dongsheng Qian

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  1. Song Deng
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Accepted by Jeong-Bon Kim, Area Editor, 25 August 2022. This article has been with the authors for three revisions.

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Deng, S., Zhu, X., Qian, D. et al. Nonlinear dynamic correlation between balls, cage and bearing rings of angular contact ball bearings at different number of balls and groove curvature radii. Nonlinear Dyn 111, 3207–3237 (2023). https://doi.org/10.1007/s11071-022-08018-w

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