# Determining preload force

Preload may be expressed as a force or as a path (distance), although the preload force is the primary specification factor. Depending on the adjustment method, preload is also indirectly related to the frictional moment in the bearing.
Empirical values for the optimum preload can be obtained from proven designs and can be applied to similar designs. For new designs SKF recommends calculating the preload force and checking its accuracy by testing. As generally not all influencing factors of the actual operation are accurately known, corrections may be necessary in practice. The reliability of the calculation depends above all on how well the assumptions made regarding the temperature conditions in operation and the elastic behaviour of the associated components - most importantly the housing - coincide with the actual conditions.
When determining the preload, the preload force required to give an optimum combination of stiffness, bearing 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 not subjected to an operating 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, a force acting in the axial direction will be produced in the bearing, and this must generally be accommodated by a second bearing, which faces in the opposite direction to the first one. Purely radial displacement of one bearing ring in relation to the other will mean that half of the bearing circumference (i.e. half of the rolling elements) is under load and the axial force produced in the bearing will be

Fa = 0,5 Fr/Y

where Fr is the radial bearing load (fig 1)

The values of the axial factor Y can be found in the product tables.
When a single bearing is subjected to a radial load Fr, an external axial force Fa of the above magnitude must be applied if the prerequisite for the basic load ratings (half of the bearing circumference under load) is to be fulfilled. If the applied external force is smaller, the number of rolling elements supporting the load will be smaller and the load carrying capacity of the bearing will be correspondingly reduced.
Preloading also increases the stiffness of the bearing arrangement. When considering stiffness it should be remembered that it is not only influenced by the resilience of the bearings but also by the elasticity of the shaft and housing, the fits with which the rings are mounted and the elastic deformation of all other components in the force field including the abutments. These all have a considerable impact on the resilience of the total shaft system. The axial and radial resilience of a bearing depend on its internal design, i.e. on the contact conditions (point or line contact), the number and diameter of the rolling elements and the contact angle; the greater the contact angle, the greater the stiffness of the bearing in the axial direction.
If, as a first approximation, a linear dependence of the resilience on the load is assumed, i.e. a constant spring ratio, then 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 bearing arrangement, for example, consists of two tapered roller bearings A and B of different size having the spring constants cA and cB and is subjected to a preload force F0. If the axial force Ka acts on bearing A, bearing B will be unloaded, and the additional load acting on bearing A and the axial displacement δa will be smaller than for a bearing without preload. However, if the external axial force
exceeds the value

Ka = F0 [1 + (cA/cB)]

then bearing B will be relieved of the axial preload force and the axial displacement under additional load will be the same as it is for a bearing arrangement without preload, i.e. determined solely by the spring constant of bearing A. To prevent complete unloading of bearing B when bearing A is subjected to load Ka, the following preload force will thus be required

F0= Ka cB/(cA + cB)
The forces and elastic displacements in a preloaded bearing arrangement as well as the effects of a change in preload force are most easily recognized from a preload force/preload path diagram (diagram 2). This consists of the spring curves of the components that are adjusted against each other to preload and enables the following
• the relationship of the preload force and preload path 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 the external force.
In the diagram 3, all the components subjected to additional loads by the operational forces 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 refer to the bearings themselves whereas the unbroken lines are for the bearing position in total (bearing with associated components).
Using diagram 4 it is possible to explain the relationships, for example, for a pinion bearing arrangement (fig 2) where bearing A is adjusted against bearing B via the shaft and housing to give 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 at bearing position A is designated FaA, that at bearing position B, FaB.
Under the influence of the 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, i.e. 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 at its greatest (δa3 > δa2).