σs = Standard deviation of shaft diameter
σI = Standard deviation of bore diameter
σf = Standard deviation of interference
σ∆0 = Standard deviation of radial clearance (before mounting)
σ∆f = Standard deviation of residual clearance (after mounting)
ms = Mean value of shaft diameter (Ø50+0.008)
mi = Mean value of bore diameter (Ø50-0.006)
m∆0 = Mean value of radial clearance (before mounting) (0.014)
m∆f = Mean value of residual clearance (after mounting)
Rs = Shaft tolerance (0.011)
Ri = Bearing bore tolerance (0.012)
R∆0 = Range in radial clearance (before mounting) (0.017)
λI = Rate of raceway expansion from apparent interference (0.75 from Figure 2)

The average amount of raceway expansion and contraction from apparent interference is calculated using:
λi (mm – mi).
The following equation is used to determine, within 99.7 percent probability, the variation in internal clearance after mounting (R∆f):

R∆f  = m∆f ± 3σ∆f  = +0.014 to -0.007

In other words, the mean value of residual clearance—(m∆f) is +0.0035—and the range are from -0.007 to 0.014 for a 6310 bearing.

Figure 2
Figure 2. Rate of inner ring raceway expansion (λi) from apparent interference

Radial Internal Clearance and Temperature
When a bearing runs under a load, the temperature of the entire bearing will rise. This includes the rolling elements. However, because this change is extremely difficult to measure or estimate, the temperature of the rolling elements is generally assumed to be the same as the inner-ring temperature.
Using a 6310 bearing again as an example, the reduction in clearance caused by a temperature difference of 5 C between the inner and outer rings can be calculated using the following equation:


δτ = Decrease in radial internal clearance caused by a temperature difference between the inner and outer rings (mm)
α = Linear thermal expansion coefficient for bearing steel, 12.5 x 10-6 (1/ C)
t = Difference in temperature between inner ring (or rolling elements) and outer ring (C)
D = Outside diameter (6310 bearing, 110 mm)
d = Bore diameter (6310 bearing, 50 mm)
De = Outer-ring raceway diameter (mm)

The following equations are used to calculate the outer-ring raceway diameter:

Formulas 3

Using the values calculated for ∆f and δt, the effective internal clearance (∆) can be determined using the following equation:

D = Df – dt = (+0.014 to -0.007) – 0.006
= +0.008 to -0.013

In Figure 3, note how the effective internal clearance influences bearing life, in this example, with a radial load of 3,350 Newtons (or approximately 5 percent of the basic load rating). The longest bearing life occurs under conditions in which the effective internal clearance is -13 micrometers. The lowest limit to the preferred effective internal clearance range is also -13 micrometers.


Figure 3
Figure 3. Relationship between the effective clearance and the bearing life for a 6310 ball bearing


In theory, targeting a slightly negative clearance is optimal for bearing life. However, in practice, end users must be careful when designing or building a pump with bearing preload. As shown in Figure 3, the life ratio peaks at -13 micrometers, but decreases dramatically with additional preload. Incorrect assumptions regarding machining tolerances or operating temperatures can easily result in a shorter life than anticipated if the bearing becomes preloaded too heavily.

On the other hand, too much clearance can result in the bearing slipping and poor pump performance. End users must evaluate the trade-offs of clearance and bearing preload based on the needs of the application. Understanding the importance of bearing internal clearance will help increase bearing life and optimize overall pump performance.  P&S