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Abstract
Cylindrical roller bearings typically employ roller profile modification to equalize the load distribution, minimize the stress concentration at roller ends, and allow for a small amount of misalignment. The 1947 Lundberg-Palmgren analysis reported an inverse fourth-power relation between load and life for roller bearings with line contact. In 1952, Lundberg and Palmgren changed their load–life exponent to 10/3 for roller bearings, assuming mixed line and point contacts. The effect of the roller–crown profile was reanalyzed in this article to determine the actual load–life relation for modified roller profiles. For uncrowned rollers (line contact), the load–life exponent is p = 4, in agreement with the 1947 Lundberg-Palmgren value, but crowning reduces the value of the exponent, p. The lives of modern roller bearings made from vacuum-processed steels significantly exceed those predicted by the Lundberg-Palmgren theory. The Zaretsky rolling-element bearing life model of 1996 produces a load–life exponent of p = 5 for flat rollers, which is more consistent with test data. For the Zaretsky model with fully crowned rollers, p = 4.3. For an aerospace profile and chamfered rollers, p = 4.6. Using the 1952 Lundberg-Palmgren value p = 10/3, the value incorporated in ANSI/ABMA and ISO bearing standards, can create significant life calculation errors for roller bearings.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the technical assistance of Bob Flowers of J. V. Poplawski and Associates for developing programmed formulas for Lundberg-logarithmic crown drop.
NOMENCLATURE
| b | = | Semiwidth of Hertzian contact area in direction of rolling (mm, in.) |
| C | = | Dynamic load capacity (N, lb) |
| Ci | = | Magnitude of crown drop relief at location i (mm, in.) |
| C0 | = | Roller crown drop amplitude based on a gage point along roller (mm, in.) |
| C(x) | = | Roller crown drop at a lamina location (mm, in) |
| c | = | Stress–life exponent |
| d | = | Roller diameter (mm, in.) |
| E | = | Young's modulus of elasticity (MPa, ksi) |
| f | = | Ratio of ball bearing raceway groove radius to ball diameter (conformity) |
| h | = | Exponent in EquationEq. [1] |
| Ki | = | Inner-race stiffness for roller lamina |
| k1 | = | Constant in EquationEqs. [B1] |
| L | = | Life, millions of inner-race revolutions or hours |
| le | = | Effective roller length (mm, in.) |
| LB | = | Life of bearings in gearbox, stress cycles, millions of inner-race revolutions or hours |
| LG | = | Life of bearings in gearbox, stress cycles, millions of inner-race revolutions or hours |
| L10 | = | 10% Life: life at which 90% of a population survives, millions of inner-race revolutions or hours |
| m | = | Weibull slope |
| N | = | Number of stress cycles per inner-race revolution |
| n | = | Hertz stress–life exponent in EquationEq. [5] |
| P | = | Bearing, normal or roller load (N, lb) |
| p | = | Load–life exponent |
| pi | = | Load acting on lamina slice i (N, lb) |
| R | = | Radius (mm, in.) |
| S | = | Stress (MPa, ksi) |
| Smax | = | Maximum Hertz stress (MPa, ksi) |
| V | = | Stressed volume (mm3, in3) |
| w | = | Width of roller lamina (mm, in) |
| x | = | Dimension along roller length (mm, in) |
| Y0 | = | Roller displacement (compression) in normal direction |
| Z | = | Axial location along roller |
| z | = | Distance below surface to critical shear stress due to Hertzian load (mm, in.) |
| α | = | Exponent for deflection term, δ |
| δi | = | Deflection in roller spring element i (mm, in) |
| θ | = | Roller misalignment angle, radians |
| ν | = | Poisson's ratio |
| ξ | = | Roller lamina location, measured from roller center (mm, in) |
| σ | = | Stress (MPa, ksi) |
| τ | = | Shear stress (MPa, psi) |
| τmax | = | Maximum shear stress (MPa, psi) |
| τo | = | Maximum orthogonal shearing stress (MPa, psi) |
Subscripts
| i | = | ith Lamina on roller |
| ir, or | = | Inner or outer race of bearing |
| LP | = | Lundberg-Palmgren life model, EquationEq. [2] |
| max | = | Maximum Hertz stress or maximum shearing stress |
| o | = | Maximum orthogonal shearing stress |
| r | = | Residual stress |
| RE | = | Rolling-element set |
| S | = | Shaft and inner-ring bore |
| x | = | Tangential direction |
| Z | = | Zaretsky life model, EquationEq. [3] |
| z | = | Normal direction |