Ball Bearing Displacement

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When the loads are applied to the bearings, the displacement takes place at the contact points between balls and raceways. When the ball bearing is running under a load, the balls drive the force between the two rings of the ball bearing. As the contact area between the rings and the balls is negligible, the amount of stress that is created can have an impact on the life and performance of the ball bearings. For maximum performance, the internal working of bearings must be considered carefully; this includes the contact angle, raceways, and radial and axial play. The radial and axial play refers to the slight amount of looseness between the balls and the raceways, which allows the balls to rotate smoothly but affects the bearing’s performance.
The radial play is the maximum distance that one ring can be displaced from the other ring in a perpendicular direction to the axis of rotation occurring along the circumference. Axial play is the maximum relative displacement of the rings in a parallel direction to the axis of rotation or the shaft axis where the disc is attached. Though both the radial and axial play are mutually dependent due to the looseness between the components, they differ in values.
Contact angle refers to the point where the ball in the bearing makes contact with the inner raceways and outer raceways. Radial play and the initial contact angle are directly related. If the radial play is higher, the contact angle of the ball and the raceways is also high. For pure radial loads, a low contact angle is preferred. On the other hand, where the thrust load is more demanding, a higher contact angle is required. Thrust loads also give the position of the ball inside the raceways. When thrust loads are applied to a ball bearing, the balls in the bearing will roll away from the median planes of the raceways and take up a position between the raceway edges and the deepest part of the raceway.

Radial Displacement

When loads are applied in radial directions in a bearing as shown in Figure 1, Q is expressed as
Q = 5 / Z * Fr.
(Fr, Q, and Z represent a radial load, the maximum load applied to the balls, and the number of balls, respectively.)
Radial displacement at the contact points between balls and raceways of a bearing is expressed as below.
: Coefficient based on the relationship between balls and raceways
Σρ : Total major curvature of contact point
In order to determine the total bearing displacement, the displacement between balls and inner ring, and outer ring ring of a ball bearing need to be summed because the balls are contacting both the bearing inner ring and outer ring.
δr   : Total radial displacement
δi   : Radial displacement between balls and inner ring raceway
δe   : Radial displacement between balls and outer ring raceway
Total displacement is represented as follows:
δr = δi + δe
Radial Bearing DisplacementFigure 1: Radial Displacement

Axial Displacement

Axial bearing displacement (Fa) with axial loads applied is calculated as follows:

Initial contact angle (α0)

To calculate the initial contact angle of the bearing, which had an initial clearance (Gr) that was eliminated by moving the raceway rings in the axial directions, can employ the following formula.

α0 = cos-1 1 - Gr
2 (ri + reDw)
Gr : Radial clearance
ri : Inner ring groove radius
re : Outer ring groove radius
Dw : Ball diameter

Relationship between Initial contact angle (α0) and contact angle (α)

The formula below expresses the relationship between the initial contact angle and the contact angle generated by applying axial loads (Fa) (Figure 2)

cos α0 = 1 + c•Dw Fa
cos α (ri + reDw) ZDw2•sin α
Axial bearing displacement Contact Angle RelationshipFigure 2: Contact Angle Relationship

The formula below calculates bearing displacement in axial direction.

δt = ( ri + reDw) ( sin α – sin α0) + c Fa sin α
Z Dw

c : Coefficient of elastic contact