Selecting a bearing

13. Displacement

When the loads are applied to the bearings, the displacement takes place at the contact points between balls and raceways.

Radial displacement

When the loads are applied in radial directions as shown in figure13-1, 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 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 displacement, the displacement between balls and inner ring, and outer ring need to be summed because the balls are contacting both the 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

figure13-1

Axial displacement

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

Initial contact angle (α0)

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 be calculated as follows.

α0	= cos^-1		1 -	Gr
				2 (ri + re - Dw)

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 relationship between the initial contact angle and the contact angle generated by applying axial loads (Fa) is expressed as below. (Figure 13-2)

cos α0	= 1 +	c･Dw		Fa
cos α		(ri + re - Dw)		Z･Dw²･sin α

Figure 13-2

Displacement in axial direction is calculated with the formula below.

δt	= ( ri + re - Dw) ( sin α - sin α0) + c		Fa				sin α
			Z				Dw

c : Coefficient of elastic contact