Abstract
This paper established geometric coordination equations of deep groove ball bearings (DGBB) and 4-point ball bearings (4PBB) in which the contact sliding state is defined specifically. Combined with the classic Hertzian contact theory, a set of five nonlinear equations was established. The discrete Newton method algorithm was used to solve this nonlinear equation. According to the calculation results, the load-displacement curve of the bearing was obtained. Theoretically, the load-displacement relationship of different parameters (such as DGBB or 4PBB, clearance, raceway curvature, etc.) was compared. Then a test bench for measuring the load-displacement relationship was established, the axial load-displacement relationship and tilting load-displacement relationship were measured. Afterward, the axial load-displacement and the tilting load-displacement curves of the sample bearings with five different design parameters were measured and compared with the theory. Finally, three ways to decrease the bearing’s displacement under load were concluded from the study.
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Abbreviations
- A 0 :
-
Inner and outer raceway curvature center distance
- \(A_0^\prime \) :
-
Inner and outer raceway curvature center distance after deformation
- D w :
-
Ball diameter
- d 1, \({d_1},\,d_1^\prime \) :
-
Inner and outer raceway curvature center radial distance (“’” means after deformation)
- d 2, \({d_2},\,d_2^\prime \) :
-
Inner and outer raceway curvature center axial distance (“’” means after deformation)
- d f1, \({d_{f1}},\,d_{f1}^\prime \) :
-
4PBB first row inner and outer raceway curvature center radial distance (“’” means after deformation)
- d f2, \({d_{f2}},\,d_{f2}^\prime \) :
-
4PBB first row inner and outer raceway curvature center axial distance (“’” means after deformation)
- d s2, \({d_{s2}},\,d_{s2}^\prime \) :
-
4PBB second-row inner and outer raceway curvature center radial distance (“’” means after deformation)
- d s2, \({d_{s2}},\,d_{s2}^\prime \) :
-
4PBB second-row inner and outer raceway curvature center axial distance (“’” means after deformation)
- d m :
-
Ball group pitch circle diameter
- d mi :
-
Tilt rotating diameter at i-th steel ball
- E′ :
-
Comprehensive elastic modulus, 1.132×105 for steel bearings
- F x, F y, F z, M y, M z :
-
External load in 5 directions
- K n :
-
The coefficient that characterizes Hertz’s contact stiffness
- O i :
-
Curvature center of inner raceway
- O e, \({O_e},\,O_e^\prime \) :
-
Curvature center of outer raceway (“’” means after deformation)
- O fi :
-
4PBB first row inner raceway curvature center
- O fe, \({O_{fe}},\,O_{fe}^\prime \) :
-
4PBB first row outer raceway curvature center (“’” means after deformation)
- O si :
-
4PBB second-row inner raceway curvature center
- O se, \({O_{se}},\,O_{se}^\prime \) :
-
4PBB second-row outer raceway curvature center (“’” means after deformation)
- Q :
-
Normal contact load
- Q Ni :
-
Normal contact load of i-th steel ball (DGBB)
- Q xi, Q yi, Q zi :
-
Force component of QNi in x, y, z directions
- Q fNi :
-
Normal contact load of i-th steel ball on first row (4PBB)
- Q fxi, Q fyi, Q fzi :
-
Force component of QfNi in x, y, z directions
- Q sNi :
-
Normal contact load of i-th steel ball on the second row (4PBB)
- Q sxi, Q syi, Q szi :
-
Force component of QsNi in x, y, z directions
- r i :
-
Curvature radius of the inner raceway
- r e :
-
Curvature radius of the outer raceway
- Z :
-
Number of steel balls
- α, α′ :
-
Contact angle (“’” means after deformation)
- \(\alpha _{fi}^\prime \) :
-
First-row contact angle after deformation (4PBB)
- \(\alpha _{si}^\prime \) :
-
Second-row contact angle after deformation (4PBB)
- δ :
-
Normal elastic deformation
- δ i :
-
Normal elastic deformation of the steel ball in contact with inner raceways
- δ e :
-
Normal elastic deformation of the steel ball in contact with outer raceways
- δ n :
-
Total normal elastic deformation of the steel ball in contact with inner and outer raceways
- δ x, δ y, δ z, δ y, δ z :
-
Displacement of the ring in 5 directions
- δ ni :
-
Total normal elastic deformation of i-th steel ball in contact with inner and outer raceways
- δ xi :
-
Displacement of raceway in x direction at i-th steel ball
- δ yi :
-
Displacement of raceway in y direction at i-th steel ball
- δ θzxi :
-
Displacement of raceway in x direction at i-th steel ball caused by θz
- δ θzyi :
-
Displacement of raceway in y direction at i-th steel ball caused by θz
- δ θyxi :
-
Displacement of raceway in x direction at i-th steel ball caused by θy
- δ θyzi :
-
Displacement of raceway in y direction at i-th steel ball caused by θy
- δ* :
-
Intermediate parameters in Hertz contact calculation
- \(\delta _i^ \ast \) :
-
Intermediate parameters in Hertz contact calculation (inner ring)
- \(\delta _e^ \ast \) :
-
Intermediate parameters in Hertz contact calculation (outer ring)
- Σρ :
-
Curvature parameter
- Σρ i :
-
Curvature parameter (inner ring)
- Σρ e :
-
Curvature parameter (outer ring)
- ϕ i :
-
Azimuth angle of i-th steel ball
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Kaiyuan Li is currently a Ph.D. candidate in Mechanical Engineering at the Southeast University. His main research interests are contact mechanics and ball bearing optimal design.
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Li, K., Tang, W. Load-displacement relationship model and measurement of deep groove ball bearing and 4-point contact ball bearing. J Mech Sci Technol 35, 3045–3058 (2021). https://doi.org/10.1007/s12206-021-0627-8
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DOI: https://doi.org/10.1007/s12206-021-0627-8