A bearing ring rotating relative to the load should be mounted with an
interference fit to avoid slipping of the ring. The interference fit will then
reduce the effective clearance of the bearing. Temperature and centrifugal
forces will have an additional influence on the effective clearance and the
properties of the interference fit. An online calculation of interference fits
considering interference, temperature and centrifugal forces is provided .

Usually the interference fit is calculated using thick ring theory assuming
two cylindrical rings and plane stress. The interference between the parts is
determined by manufacturing tolerances and some embedding due to surface
roughness can be considered. DIN 7190 (2001) proposed a reduction of effective
interference by 0.8*Rz, which was reduced to 0.4*Rz in DIN 7190 (2017). The sum
of surface roughness of the two parts in contact should be considered, but as in
most cases the bearing ring will have a much smoother surface than the shaft and
the housing, it should be sufficient to consider the surface roughness of the
shaft/housing only.

For the calculation of the interference fit the ratio of inner and outer
diameter of each ring is required. For the outer diameter of the inner ring and
the inner diameter of the outer ring, the question is, how to determine this
diameter. Some bearing catalogs mention the raceway diameters others an average
diameter between raceway diameter and the shoulders.

To get an estimate about the influence of the shoulders of the bearing rings
on the change of effective clearance an axisymmetric FEA calculation was added
to the MESYS bearing calculation software. The diametral expansion on the outer
contour of the ring is shown in a diagram in comparison to the calculation of
cylindrical rings according to thick ring theory. Two options are considered for
thick ring theory. Either the pitch diameter plus/minus ball diameter
D_pw±D_w is used for the second diameter of the ring or a
diameter leading to the same cross section area as the real cross section
including shoulders.

Figure 1 shows the diametral expansion of a 71910C bearing inner ring
considering an interference of Iw = 13 mm and speed zero. The solid line shows
the expansion of the outer contour of the ring according to the FEA calculation,
the dotted line shows the expansion using thick ring theory and a diameter for
the equivalent cross section and the dashed line shows the expansion for thick
ring theory and the outside diameter D_pw-D_w.

It can be seen that the expansion in the middle of the bearing is very close
to the value not considering the shoulders and the value at the left shoulder is
closer to the value for the equivalent cross section. It should be noted, that
the difference is about 0.3 mm, so the effect of unreliable smoothing of the
surface roughness is larger than this variation.

Figure 2 shows the result of the same example, but with rotation speed of
30000 rpm. Here the expansion of the equal cross section case is larger than in
the case without considering the shoulders. The reason is the larger mass and
the larger effective diameter for the centrifugal forces.

Figure 3 shows the mesh used in these two calculations. The width of the
shaft is a little larger than the bearing width as it is in real applications.
Quadratic elements are used.

For the same example a hollow shaft with inner diameter d_si = 30mm
instead of a solid shaft is considered in figures 4 and 5. Figure 4 shows a
larger difference for the two cases based on thick ring theory. The case with
equivalent cross section is too stiff here. Including centrifugal forces the
differences are small again.

These examples show that the consideration shoulders for the calculation of
fits is not required in many cases and a calculation using a simplified approach
with D_pw±D_w can lead to more accurate results for low
speed cases. Still mostly the differences are much smaller than the effect of
surface roughness would be.

If a shaft width equal to the bearing width would be used, the expansion of
the ring based on the FEA calculation would be smaller and closer to the results
considering the equivalent cross section. But as in real applications the shaft
and housing width is larger than the bearing width in almost all cases, this was
assumed for the FEA calculation, too.