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Nonlinear dynamic model of ball bearings with elastohydrodynamic lubrication and cage whirl motion, influences of structural sizes, and materials of cage

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Abstract

This paper proposes a nonlinear dynamic model of angular contact ball bearings with elastohydrodynamic lubrication and cage whirl motion. On this basis, the correlation between the dynamic stability of cage, interaction forces of bearing components, sliding of the ball, and vibration of inner ring is elaborated. Next, the effects of pocket diameter, small diameter and large diameter of cage on the dynamic behaviors of the bearing system are revealed. Also, the influence of material density of cage on the dynamic behaviors is investigated. The results show that in the design stage of ball bearings, the wonderful dynamic stability of cage should firstly be guaranteed by the optimal ratio of pocket clearance to guiding clearance; on this basis, guiding clearance should be adjusted to obtain the acceptable vibration of inner ring. Moreover, copper cage is disadvantageous for attenuating the vibration of the bearing system, while light material cage is beneficial for improving the dynamic stability of cage and attenuating the vibration of the bearing system, especially, the best is polytetrafluoroethylene, next is phenolic laminate and third is polyamide.

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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

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

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

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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

The study 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, Hongyang Chang, Feng Wang & Lin Hua

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

    Song Deng, Dongsheng Qian & Lin Hua

  3. Luoyang Bearing Research Institute Co., Ltd., Luoyang, 471039, China

    Song Deng & Shaofeng Jiang

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  1. Song Deng
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Correspondence to Dongsheng Qian or Feng Wang.

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Deng, S., Chang, H., Qian, D. et al. Nonlinear dynamic model of ball bearings with elastohydrodynamic lubrication and cage whirl motion, influences of structural sizes, and materials of cage. Nonlinear Dyn 110, 2129–2163 (2022). https://doi.org/10.1007/s11071-022-07683-1

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