The Impact of Lubricant Film Thickness and Ball Bearings Failures

An effort was made to find a relationship between the lubricant thickness at the point of contact of rolling element ball bearings, and empirical equations to predict the life for bearings under constant motion. Two independent failure mechanisms were considered, fatigue failure and lubricant failure resulting in seizing of the roller bearing. A theoretical formula for both methods was established for the combined probability of failure using both failure mechanisms. Fatigue failure was modeled with the empirical equations of Lundberg and Palmgren and standardized in DIN/ISO281. The seizure failure, which this effort sought to investigate, was predicted using Greenwood and Williamson’s theories on surface roughness and asperities during lubricated contact. These two mechanisms were combined, and compared to predicted cycle lives of commercial roller bearing, and a clear correlation was demonstrated. This effort demonstrated that the Greenwood–Williams theories on the relative height of asperities versus lubricant film thickness can be used to predict the probability of a lubricant failure resulting in a roller bearing seizing during use.

empirical data on bearing failure to verify and validate it. In this aim, the L 10 empirical equations 48 provided by Svenska Kullagerfabriken (SKF) will be used as a baseline [13,14]; SKF is a Swedish company 49 founded in 1907 and is currently the world's largest manufacturer of ball bearings. They have a bearing 50 calculator that provides the L 10 life in revolutions before the bearings have a 10% chance of failure.

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The core equation for L 10 life is where C (N) is the basic dynamic load rating, P (N) is the equivalent load, and A SKF is the Life Modification 53 Factor. The value ofp was found empirically, and it is 3 for spherical bearings and 10/3 for cylindrical 54 bearings [15][16][17]. The value of A SKF is a function of the the combined influence of load and 55 contamination on fatigue β; and the viscosity ratio κ, which represents the lubrication conditions and 56 their influence on fatigue. The basis for the ( C P ) 10/3 component of equation 1 is based on empirical 57 research of Lundberg and Palmgren [15][16][17].

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The dimensionless value of κ is a ratio of the kinematic viscosity ν (m 2 /s) over the rated viscosity 59 ν 1 (m 2 /s) where ν 1 is a function of both the speed Ω rpm and the average bearing diameter d m (m) where D (m) and d (m) represent the diameter of the inner and outer bearing race.
where the mean diameter d m is in meters and the bearing speed Ω rpm is in revolutions per minute.

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Calculated values of ν 1 (m 2 /s) are plotted in units of centistokes or mm 2 /s in Figure 1. The other term necessary to determine A SKF is the dimensionless coefficient β, which is the 69 product of the cleanliness factor N c and the safety factor ratio of the fatigue load limit P u (N) over the 70 equivalent bearing load P (N) The cleanliness factor N c ranges from 0.2 to 1.0, with 0.2 representing the dirtiest possible lubricant, 72 and 1.0 representing a perfectly clean lubricant. In this analysis, the lubricant will be assumed to be 73 clean, with N c = 1. The equivalent load P (N) is a combination of radial and axial loads [18] 74 P = X a ·F a + X r ·F r , where F a (N) and F r (N) are the axial and radial loads, and X a and X r are bearing specific coefficients.

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For example, for spherical thrust bearings X a =1 and X r =1.2.

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The SKF website provides tables for the value of A SKF as a function of β and κ, as well as a 77 calculator tool, but no specific formula was given. For this reason, the least squared method was used, 78 and a close match all throughout the permissible range of β and κ yielded the empirical equation 6 79 A SKF = (C 1,1 ·κ 3 + C 2,1 ·κ 2 + C 3,1 ·κ + C 4,1 )·(C 1,2 ·β 3 + C 2,2 ·β 2 + C 3,2 ·β + C 4,2 ), where the values of C i,j is tabulated in Table 1   Friction is never constant in practice, it constantly fluctuates about a given average, therefore, this 98 failure prediction model will be normalized to a given quantity of standard deviations away from the 99 mean friction where erfc represents the complementary error function, and µ represents the Z-factor that corresponds to a given failure probability P f for a single revolution. In equation 9, x i (m) refers to 102 a given asperities height over a given duration, x i (m) refers to the mean asperities height of the roller 103 bearing race, and σ (m) refers to the standard deviation of the asperities height for the given roller race.

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The relationship between µ and the L 10 life is thus While µ is a parameter for the probability of failure, it also is a representation of the mean 106 coefficient of friction. According to Greenwood and Williamson's research [19][20][21][22][23][24], wear and friction 107 (other than from fluid stresses) occur due to random asperities exceeding the thickness of the lubricant 108 film [19][20][21][22][23][24][25][26]. Assuming the surface asperities height follows a normal distribution, the ratio of 109 metal-on-metal contact A real /A with the lubricant thickness should roughly follow where A real (m 2 ) represents the true metal-on-metal contact area, A (m 2 ) represents the apparent (but and the poisson's ratio p will be 0.3. The parametric study would calculate both the L 10 life as defined 120 in equation 1, and compare it to the predicted lubricant film thickness [10][11][12]18,[27][28][29][30][31][32][33][34][35][36][37][38], as well as the 121 relative fatigue load. The parametric study was conducted for a temperature ranging between 40 where h min (m) is the minimum film thickness, h c (m) is the central film thickness, U n is the 130 dimensionless speed parameter, G n is the dimensionless material parameter, W n is the dimensionless where R (m) is the radius of the cylindrical bearing, and E y (Pa) and p is the Young's Modulus and 137 Poisson's ratio of the bearing material.

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If there is a given friction force that will cause the bearings to seize, and the friction is affected by 139 the ratio of the height of the surface asperities (which follow a normal distribution) over the lubricant 140 film thickness, an accurate equation for µ as a function of h c (m) was realized with equation 19 where σ was predicted as 1 µm RMS for the surface asperities, and P u was defined as 4,800 N. Equation

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19 incorporated two separate failure mechanisms, where X 2 is a coefficient for the lubricant seizure coefficients for this particular design is X 1 = 8.1130, X 2 = −3.1285, and X 3 = −1.0173. By taking the

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A second parametric was conducted to see if varying the bearing size would affect the coefficients 154 for equation 19. The mean bearing radius was modeled from 30 mm to 500 mm. With a changing 155 bearing diameter, the radius of the rollers R (m) was consistently adjusted so 25 rollers in the bearings 156 would consistently fit within the roller bearing circumference where N r = 25 represents the number of cylindrical roller bearings. As observed in Figure 4, the first 158 two coefficients, X 1 and X 2 are a clear function of the inverse of the diameter 159 The following abbreviations are used in this manuscript: