Consider two solids having different radii of curvature in two tangential directions which are in point contact when no load is applied. Under the action of the load F, the contact area becomes elliptical. As a convention, let us assign the positive sign for the curvature of a convex surface and negative for a concave surface. The contact profile between the two solids A and B can be expressed in terms of the curvature sum, R, and curvature difference, Rd, as follows

where R1 and R2 are the effective radii of curvature of the contacting surface with the indices 1 and 2 representing the local tangent directions. The effective radii of curvature are given by

The point of contact under the action of the load expands to an ellipse with semi-major axis, ae, and semi-minor axis, be. The ellipticity parameter is defined as

This ellipticity parameter can be defined in terms of the curvature difference, Rd, and elliptic integrals of first and second kind as follows

where ξ and ζ are the elliptic integrals of first and second kind, respectively, defined as

with E' defined as

where δ is the penetration of the surfaces into each other.

When two cylindrical surfaces with parallel axes come in contact, the contact formed is a line contact. In this case, the force, F, changes the contact area into rectangular. The width of the contact area, b, can be obtained in terms of the applied load, F, equivalent radius of curvature, R, and the effective elastic modulus, E', as

where L is the length of cylindrical surfaces in contact. The normal displacement is given by

The derivative of the normal displacement, δ, with respect to the load, F, gives the compliance. The inverse of the compliance is the stiffness of the contacting surfaces.

Figure 3-2 shows the front view of a typical roller bearing. Due to the relative motion of the inner race and outer race, some of the rollers get loaded, whereas others get unloaded. This relative motion determines the contact indentation at different rollers. From Figure 3-3 the initial distance between the inner and outer race curvature centers is

The plus sign is for the second row and the minus sign is for the first row of the rollers. In case of a single row bearing ϕ0 becomes zero. eirj is the radial vector corresponding to the roller i in row j and is defined as

Let the relative center displacement between the inner and outer race be ur, and the relative tilt about local y and z directions be θ2 and θ3 respectively. Then the relative displacement between the inner and outer races at roller i in row j is

for single row bearing. Here, b is the axial distance between the roller centers and dp is the pitch diameter of the bearing. The loaded distance vector between the inner and outer race curvature centers is

Then the indentation at roller i in row j is

Only positive values of the contact forces on each roller are taken into account, and negative values are replaced by zero since such rollers are unloaded. For a single row bearing, summation over j is dropped. The contact force on the outer race acts in opposite direction to that on inner race.

In the tapered roller bearing the contact force directions at inner and outer contacts are different. In addition, there is also a contact between the bearing flange and roller. Therefore, it is not possible to directly get the equivalent stiffness of the inner and outer contacts. This requires the determination of the contact forces at inner and outer contacts individually. Geometric details of the tapered roller bearing are shown in the Figure 3-4.

There are three locations at which the roller comes into contact with the inner and outer races, namely radial contact with the outer race, radial contact with the inner race, and axial contact with the inner race. Radial contacts are line contacts, whereas the axial contact between the roller and the inner race is a point contact. In order to determine the contact forces in all these cases, contact deformation in the roller needs to be determined due to the relative motion of the inner and outer races. This can be described conveniently by using a local coordinate system (ξ, η, ζ) in which the ζ direction is along the axis of the roller, ξ is perpendicular to the ζ axis and is in the plane formed by bearing's radial and axial directions. In this coordinate system, the relative in-plane motion (ξ-ζ plane) and tilt about η axis can cause roller deformation.

Determination of the roller deformation can be done by first looking at the initial position vectors of the various points which are likely to come into contact, and then determine their current position vector due to the motion of the races. A projection of the difference between the current and initial position vectors gives an estimate to the roller deformation. Using the geometric details given in the Figure 3-4, the position vectors of the various points are as follows:

where ζcs is given by

Introduce the displacement ui and rotation vector θi at the reference point on the inner race, the displacement ue and rotation vector θe at the reference point on the outer race, and the displacement ur and rotation vector θr at the reference point on the rollers. Then, the displaced position π of the arbitrary point located at π0 from the reference point will be:

where uref and θref are pair (ui, θi) for a point on the inner race, (ue, θe) for a point on the outer race and (ur, θr) for a point on the roller.

The compression δ at the contact between roller and raceway is found by taking the projection of the current relative position vectors of the nominal contact points on their outside normal vector and removing the initial gap:

The motion of the both the races are determined from the motion of the components they are connected to. For example, the inner race motion can be determined from the rotor and the outer race motion can be determined from the motion of the bearing pedestal. The motion of the roller, however, still remains unknown. It can be determined by considering the load balance of the roller as shown in Figure 3-6.

where re and ri are the position vectors on the outer and inner contact lines, respectively, with respect to the reference point. dQe and dQi are the infinitesimal forces acting on the infinitesimal length dx on the outer and inner contact lines.

In the cylindrical roller bearing, the forces at inner and outer contact lines are aligned as in the case of the point contact roller bearings. A typical arrangement of a cylindrical roller bearing is shown in Figure 3-7. The initial gap between the outer race and inner race for roller i in row j is given by

Here, eirj is the radial vector corresponding to the roller i in row j and is defined as

Let the relative center displacement of the inner race with respect to outer race be ur and the relative tilt about local y and z directions be θ2 and θ3, respectively. Then the relative displacement of the inner with respect to outer races at roller i in row j is

for a single row bearing. b is the axial distance between the roller centers and dp is the pitch diameter of the bearing.

Then the indentation at roller i in row j is:

Only positive values of the contact forces on each roller are taken into account, and negative values are replaced by zero since the roller is unloaded. For a single row bearing, summation over j is dropped. Contact forces on the outer race acts in the opposite direction to those at the inner race.

Here, Fa is the axial preload in the bearing, e1 is the bearing axis, ur is the rotor displacement, uf is the foundation displacement and er is the contact direction between the rolling element and the races. Because races of a bearing are connected to the rotor and the foundation, respectively, the displacement of the inner race is equal to the rotor displacement, and the displacement of the outer race is equal to foundation displacement. As you can see, for single row bearings, the preload is also transmitted to the rotor on which the bearing is mounted.

Let us assume that the outer race is split and that the part corresponding to the first row moves by an axial displacement ua along the bearing axis, and that the part corresponding to the second row moves by the same axial displacement but in opposite direction. The weak contribution due to the axial preload, in this case, is

We assume that radial deformation in the bearing is ur due to the preload. Then the weak contribution due to radial preload is

Fill angle: A fill angle ψf is the angular space occupied by the rolling elements on a pitch circle.

Since each rolling element has a diameter dr and there are N number of them, the arc length occupied on the pitch circle by the rolling elements is Ndr. Furthermore, the pitch circle radius is dp/2. Thus, the angular space occupied by the rollers, that is, the fill angle is

Inner race conformity: Inner race conformity, fin, is the ratio of the inner race radius of curvature to the diameter of the rolling element. It is defined by

Outer race conformity: Outer race conformity, fout, is the ratio of the outer race radius of curvature to the diameter of the rolling element. It is defined by

For a physical bearing, the fill angle cannot be larger than 2π radians. This gives a constraint on the diameter of the roller in relation to the pitch diameter as

Usually the fill angle is much smaller than 2π due to the space occupied by the cage.

For point contact bearings, another constraint comes from the fact that the rolling elements have to fit between the inner race and the grooves of the outer race. As a result, the race radius of curvatures should be larger than the radius of the rolling elements. This means that the race conformity should be larger than 0.5 for both races.

For a tapered roller bearing another geometric constraint is that the flange contact angle, μ0, should be less than the roller cap angle, λ.