# Preload in bearing arrangements with angular contact ball or tapered roller bearings

When determining preload, the preload force required to provide an optimum combination of stiffness, bearing service life and operational reliability should be calculated first. Then calculate the preload force to be used when adjusting the bearings during mounting. When mounting, the bearings should be at ambient temperature and should not be subjected to any other load.
The appropriate preload at normal operating temperature depends on the bearing load. An angular contact ball bearing or a tapered roller bearing can accommodate radial and axial loads simultaneously. Under radial load, these bearings produce a resultant axial load which must be accommodated by a second bearing facing the opposite direction. Purely radial displacement of one bearing ring relative to the other means that half of the rolling elements are under load. The resultant axial load produced in the bearing can be determined by:
• for single row angular contact ball bearings
Fa = R Fr
• for single row tapered roller bearings
Fa = 0,5 Fr / Y

where
 Fa = axial bearing load (fig. 1) Fr = radial bearing load (fig. 1) R = variable for inside contact conditions (→ Determination of axial forces) Y = calculation factor (→ product table)

When a single bearing is subjected to a radial load Fr, an axial load Fa (external) of the same magnitude as the resultant load must be applied to the bearing if the basic load rating is to be fully exploited. If the applied external load is lighter, there are fewer rolling elements supporting the load and the load carrying capacity of the bearing is correspondingly reduced.
In a bearing arrangement consisting of two single row angular contact ball bearings or two tapered roller bearings arranged back-to-back or face-to-face, each bearing must accommodate the axial load in one direction. When these bearing arrangements are adjusted to near-zero clearance, the radial load is shared equally between the two bearings and half the rolling elements in each bearing are loaded.
In other cases, where there is an external axial load, it may be necessary to preload the bearings to compensate for the clearance that can occur when the axially loaded bearing deforms elastically. Preload also distributes the loads more favourably in an axially unloaded bearing.
Preload also increases the stiffness of a bearing arrangement. However, keep in mind that stiffness is also influenced by the elasticity of the shaft and housing, the shaft and housing fits, as well as the elastic deformation of all other components adjacent to the bearings, including the abutments. Each of these has a considerable impact on the resilience of the total bearing-shaft-housing system. The axial and radial resiliences of a bearing depend on its internal design, contact conditions (point or line contact), the number and diameter of rolling elements and the contact angle. The greater the contact angle, the higher the degree of stiffness in the axial direction.
If, as a first approximation, a linear dependence of the resilience on the load is assumed, such as a constant spring ratio, a comparison shows that the axial displacement in a bearing arrangement under preload is smaller than for a bearing arrangement without preload for the same external axial force Ka (diagram 1). A pinion arrangement design (fig. 2 and fig. 3) typically consists of two different size tapered roller bearings, A and B, with different spring constants cA and cB. Both are subjected to a preload force F0. If an axial force Ka acts on bearing A, bearing B becomes unloaded, and the additional load acting on bearing A results in an axial displacement δa that is smaller than it would be if the bearings had not been preloaded. However, B is relieved of the axial preload force and the axial displacement under additional load is the same as it is for a bearing arrangement without preload, that means determined solely by the spring constant cA, if the external axial force exceeds the value

To prevent bearing B from becoming unloaded when bearing A is subjected to an axial force Ka, the following preload force is required

Loads and elastic displacements in a preloaded bearing arrangement, as well as the effects of a change in preload, are easily understood from a preload force / axial displacement diagram (diagram 2). This consists of the spring curves of the components that are adjusted against each other to apply preload and enables the following:
• the relationship of the preload force and axial displacement within the preloaded bearing arrangement
• the relationship between an externally applied axial force Ka and the bearing load for a preloaded bearing arrangement, as well as the elastic deformation produced by an external load
In diagram 2, all the components subjected to external loads in operation are represented by the curves that increase from left to right, and all the unloaded components by the curves that increase from right to left. Curves 1, 2 and 3 are for different preload forces (F01, F02 < F01 and F03 = 0). The broken lines represent individual bearings, while the solid lines represent the total bearing arrangements (bearing(s) and associated components) for different preload forces.
From diagram 2, it is also possible to explain the relationship between components, for example in a pinion arrangement design (fig 2.), where bearing A is located adjacent to a gear and is adjusted against bearing B to provide preload. The external axial force Ka (axial component of tooth forces) is superimposed on the preload force F01 (curve 1) in such a way that bearing A is subjected to additional load while bearing B is unloaded. The load on bearing A is designated FaA and on bearing B it is designated FaB. Under the influence of the axial force Ka, the pinion shaft is axially displaced by the amount δa1.
The smaller preload force F02 (curve 2) has been chosen so that bearing B is just unloaded by the axial force Ka, that means FaB = 0 and FaA = Ka. The pinion shaft is displaced in this case by the amount δa2 > δa1.
When the arrangement is not preloaded (curve 3), the axial displacement of the pinion shaft is greatest (δa3 > δa2).