Abstract
The occurrence and aggravation of local defects in ball bearings are closely linked to the skidding behavior of the ball. Previous studies have given less attention to investigating the impact of localized defects on the problem of bearing skidding. To investigate the dynamic response of defective bearings due to skidding, a dynamic model of the ball bearing is developed that considers various factors, including self-rotation, revolution, and radial motion of the ball, as well as the contact forces and friction forces of ball/cage and ball/race, time-varying displacement excitation, and elastohydrodynamic lubrication (EHL). Experimental signals collected from a machinery fault simulator test rig are used to validate the accuracy of the proposed model. The impact of race defects on the vibration characteristics of the bearing is analyzed, and the patterns of variation in contact and friction forces within one cycle of inner race rotation are described. The results indicate that the presence of defects intensifies the force fluctuation of the ball and causes it to deviate from its normal rolling condition. By comparing the skidding characteristics of a healthy bearing with a defective one under slippage, local defects will increase the skidding ratio of bearings. The proposed model can investigate the impact of race defects on the vibration response of ball bearings under the skidding condition.
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Abbreviations
- A :
-
Lubricant coefficients
- B :
-
Lubricant coefficients
- c v :
-
Drag coefficient
- c h :
-
Damping coefficient
- c oil :
-
Damping of oil film between ball and races
- c r :
-
Damping of oil film in the inlet area
- c hi,c ho :
-
Structural damping between ball and races
- c in-oil, c out-oil :
-
Damping of oil film and races
- c io-oil :
-
Total damping under EHL
- c s :
-
Damping of spindle and inner race
- c p :
-
Damping of pedestal and outer race
- C r :
-
Radial clearance of bearing
- C :
-
Lubricant coefficient
- d :
-
Diameter of ball
- D :
-
Lubricant coefficient
- D m :
-
Pitch diameter of ball bearing
- D i :
-
Inner race diameter
- D o :
-
Outer race diameter
- E :
-
Elastic modulus of ball bearing material
- f c :
-
Total friction force between all balls and cage
- f inj, f outj :
-
Friction force between a single ball and races
- f io x :
-
Total oil film friction force of x-direction
- f io y :
-
Total oil film friction force of y-direction
- f s :
-
Spindle rotation frequency
- f bpi :
-
Fault characteristic frequency of inner race
- f bpo :
-
Fault characteristic frequency of outer race
- F cj :
-
Contact force between a single ball and cage
- F c :
-
Sum of contact forces of all balls and cage
- F v :
-
Viscous resistance
- F ce :
-
Centrifugal force
- F inj, F outj :
-
Contact force between a single ball and races
- F io x :
-
Total contact force applied to races of x-direction
- F io y :
-
Total contact force applied to races of y-direction
- \(\overline F \) :
-
Dimensionless material parameter
- g :
-
Acceleration of gravity
- Ḡ :
-
Dimensionless load parameter
- h min :
-
Minimum thickness of oil film
- H i, H o :
-
Time-varying displacement function
- H d :
-
Defect depth
- J b :
-
Rotational inertia of ball
- J c :
-
Rotational inertia of cage
- J m :
-
Moment of inertia of ball about ball bearing axis
- k s :
-
Stiffness of spindle and inner race
- k c :
-
Spring stiffness
- k p :
-
Stiffness of pedestal and outer race
- k oil :
-
Stiffness of oil film between ball and races
- k io-oil :
-
Total stiffness under EHL
- k in-oil, k out-oil :
-
Stiffness of oil film and races
- k i, k o :
-
Contact stiffness between ball and races
- l i, l o :
-
Distances of ball entering races defect
- m s :
-
Mass of spindle and inner race
- m p :
-
Mass of pedestal and outer race
- N b :
-
Number of balls
- r bo :
-
Position vectors from center of bearing to of ball
- r io :
-
Position vectors from center of bearing to center of inner race
- r bi :
-
Position vector from center of inner race to ball
- \(\overline R \) :
-
Equivalent radius between ball and races
- S b :
-
Ball skidding ratio
- S c :
-
Cage skidding ratio
- u :
-
Oil film extrusion speed
- Ū :
-
Dimensionless velocity parameter
- v i :
-
Velocity of inner race
- v j-i :
-
Velocity of revolution of ball relative to inner race
- v j-o :
-
Velocity of revolution of ball relative to outer race
- v bj :
-
Velocity of self-rotation of ball
- v cj :
-
Relative skidding velocity of a single ball to cage
- v iji, v ijo :
-
Relative velocity between ball and inner race
- v oji, v ojo :
-
Relative velocity between ball and outer race
- W :
-
Radial load of ball bearing
- W d :
-
Defect width
- x i, y i :
-
Horizontal and vertical displacements of inner race
- x s, y s :
-
Horizontal and vertical displacements of inner race and spindle
- x p, y p :
-
Horizontal and vertical displacements of outer race and bearing pedestal
- α :
-
Contact angle
- δ io :
-
Contact deformation between ball and races with a defect
- δ io-oil :
-
Total contact deformation under EHL
- δ ioj :
-
Total contact deformation between ball and races
- δ inj, δ out :
-
Contact deformation between ball and races
- η :
-
Viscosity of the lubricant
- θ j :
-
Angular position of the j-th ball
- θ 0 :
-
Initial position angle of ball
- θ i, θ o :
-
Defect angles of races
- θ rj :
-
Actual angular position of the j-th ball
- μ :
-
Friction coefficient
- ξ :
-
Effective fluid density of ball bearing.
- ρ i, ρ o :
-
Sum of the curvature of inner and outer race
- σ :
-
Poisson’s ratio
- φ bj :
-
Angular of self-rotation of ball
- φ j :
-
Revolution angle of the j-th ball
- φ c :
-
Rotation angle of cage
- ϕ i, ϕ o :
-
Angles of the ball entering race defects
- ω s :
-
Rotational angular velocity of inner
- ω j :
-
Angular velocity of revolution of ball
- ω bj :
-
Angular velocity of self-rotation of ball
- ω bt :
-
Theoretical rotational speed of ball
- ω ct :
-
Theoretical rotational speed of cage
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Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant nos.51765034), the Excellent Doctoral Program in Gansu Province (Project no. 22JR5RA230) and the Excellent Graduate Student “Innovation Star” project of Education Department of Gansu Province (Project no. 2023 CXZX-409).
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Yu Tian is currently a master student in the School of Mechanical and Electrical Engineering at Lanzhou University of Technology, China. His research interests include dynamic modeling of rolling bearings and fault diagnosis.
Changfeng Yan received his Ph.D. degree from Tongji University, China, in 2010. He is currently a Professor in the School of Mechanical and Electrical Engineering at Lanzhou University of Technology, doctoral supervisor, China. His research interests include fault diagnosis and signal process.
Yaofeng Liu received his M.S. degree from the School of Mechanical and Electrical Engineering, Lanzhou University of Technology, China, in 2020. He is currently pursuing a Ph.D. degree with the School of Mechanical and Electrical Engineering at Lanzhou University of Technology. His research interests include rotor-bearing dynamic models and fault diagnosis.
Wei Luo received his M.S. degree from the School of Mechanical and Electrical Engineering, Lanzhou University of Technology, China, in 2023. His research interests include dynamic modeling of rolling bearings and fault diagnosis.
Jianxiong Kang received his Ph.D. degree from the School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, China, in 2022. He is currently a lecturer in the School of Mechanical and Electrical Engineering at the Lanzhou University of Technology, master supervisor, China. His research interests include dynamic modeling and health management.
Zonggang Wang received his M.S. degree from the College of Electrical and Information Engineering, Lanzhou University of Technology, China, in 2012. He is currently an Associate Professor in the College of Physics & Electromechanical Engineering at Hexi University, China. His research interests include prognostics, condition monitoring and fault diagnosis.
Lixiao Wu received her M.S. degree in mechanical engineering from the Gansu University of Technology, China, and the degrees from the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore, in 1998 and 2006, respectively. She is currently with the School of Mechanical and Electrical Engineering at Lanzhou University of Technology, China. Her research interests are in the area of chemical-mechanical polishing and contact mechanics.
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Tian, Y., Yan, C., Liu, Y. et al. Characteristics of vibration response of ball bearing with local defect considering skidding. J Mech Sci Technol 37, 5695–5711 (2023). https://doi.org/10.1007/s12206-023-1010-8
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DOI: https://doi.org/10.1007/s12206-023-1010-8