Introduction
Rolling bearings are one of the most critical machine components and their correct selection is an extremely important step in the design of any rotating equipment.
The processes needed to select the right bearings for a given application, to define their correct arrangement and to evaluate the right amount of preload/clearance, require deep knowledge, field experience and engineering accuracy, which has to be used in the following main calculation steps:
- Firstly, the evaluation of the loading conditions of each bearing of a rotating equipment in both static conditions as well as in operations
- Secondly, the evaluation of how the bearing load is distributed among the rolling elements inside each bearing
- Finally, the evaluation of the bearing performance based on the contact conditions between each rolling element and the contacting surfaces.
In essence: in order to evaluate bearing performance, it is fundamental to evaluate the loading conditions of each bearing of a rotating equipment. In this article a few approaches are presented with pros and cons, by focusing on limitations and possibilities.
Different approaches to evaluate bearing loading conditions
In order to tackle the above calculation steps, there are several calculation methods available: some are simplified, but they can quickly produce reliable results; others are more complicated, time demanding but more accurate. It is obvious that, as the complexity increases, a dedicated calculation software becomes necessary in order to efficiently solve the background calculation models.
Depending on the amount and the quality of the accessible information as well as the time available to extract the needed results, there are different methods that can be used to determine the load distribution in each bearing of rotating equipment:
- The traditional approach: it is useful to quickly evaluate the order of magnitude of the loads of single- or two-bearings applications
- The shaft-bearing-housing approach: it is useful to perform a more precise calculation for any arrangement of bearings by taking into account bearing, shaft a housing geometry and their mutual interaction
- The system approach: it is useful to verify and optimize bearing arrangements and the overall system, by both taking into account real geometry of different parts as well as the mutual interactions among the different components
- The system dynamics approach: it is useful to understand the dynamic behavior of any rotating equipment by means of full transient analyses.
Additionally, it has to be pointed out that each of the above approaches provides a different level of accuracy: the assumptions and simplifications made in the background numerical models need to be well-known in order to correctly interpret the results.
The traditional approach
The traditional approach allows to estimate the bearing load in applications based on a single shaft supported by one or two bearings, by means of isostatic models of simply supported beams.
This method is ideal for a quick bearing selection: with the minimum of information on shaft geometry and operating conditions, the bearing load can be assessed for a wide range of applications.
In the example shown in the picture, the traditional method can quickly provide how much of the applied load (F) is carried by each support.
However, hyper-static structures (shaft supported by more than two bearings) cannot be easily handled and more sophisticated models are needed to evaluate the distribution of the applied force into the different supports.
Additionally, one of the main assumption is that the bearings are infinitely rigid in relation to the radial displacements and that they do not offer resistance to the rotations.
Furthermore, since the internal geometry of the bearing is not required, the loaded rolling elements are assumed to be covering a load zone of ~180 degrees without any further detail calculation. 
The shaft-bearing-housing approach
The assumptions introduced by the traditional approach on the bearing stiffness (= resistance to displacements and rotations) are not always close to reality. In fact, each bearing type behaves differently: not all the bearing types are suitable for large rotation and/or large axial excursions; some other types are designed to withstand those kind of deformations. However, in order to better understand the bearing behavior, it is important to know their internal geometry. Additionally, the bearing stiffness also depends on it: it is remarkable to observe that a difference of a few micrometers in internal clearance can result into a substantially different behavior in relation to displacement and rotations.
The bearing stiffness is one of the fundamental “piece of the puzzle” that is taken into account in the shaft-bearing-housing approach. The final equilibrium depends on several combined stiffness contributions: shaft, housing and bearings. Since every micrometer counts, the accurate evaluation of the bearing behavior, depends on also on the real operating conditions, especially rotational speed and temperatures. Additionally, the fits used to fix the bearing onto the shaft and inside the housing seats are also an extremely important element to consider. In fact, fits, rotational speed and temperatures are heavily changing of the initial bearing manufacturing clearance into the operating one (see also clearance reduction).
Moreover, as shown in the picture, by considering the real internal geometry of the bearings as well as the shaft and housing geometries and mounting fits, the solution of such system is not the one that the traditional method would suggest:
- The right bearing may fill a stronger clearance reduction due to different housing geometry and possibly fits
- The right and the left bearings may present different stiffness due to a difference in operating internal clearance
- The bearings are developing bending moments due to the finite rotation stiffness offered by the ball bearing
- Because of the different internal clearance, the bearing loading conditions may not be the same even though the load is positioned in the middle of the shaft.
Additionally, for axial bearings, such as tapered roller bearings and angular ball bearings, there is the need to accurately assess the right amount of preload/clearance used in each group: the optimum value will ensure the maximum life in the given operating conditions (see more at this page). The corresponding amount of microns necessary to reach this optimum condition is one of the most difficult operation to measure on the shop floor. In this case numerical simulations are a great help to quickly tackle the preloading operation.
The system approach
As discovered above, considering bearings inside a flexible housing rather than rigidly fixed to the foundation, provides different results: the bearing behavior is impacted by the mutual interaction with what is connected to it. Moreover, in many applications housings may present shapes that are not easily represented by simple pipes. A housing with a lot of stiffening ribs may influence the bearing behavior and hence its internal loading conditions among the rolling elements. In these cases, complex housing geometry introduce a non uniformly distributed stiffness forcing the bearing to react differently along the circumference; again, the internal load distribution among the rolling elements may deviate from the expected pattern of the typical load zone. 
In case the housing 3D geometry is available, the stiffness of such components may be calculated to correctly evaluate by means of advanced numerical methods. In this way it is possible to better capture how the bearing seats may move in the space and how they deform under load. In fact, one of the powerful possibility of the system approach is the evaluation of the ovalization of both bearing rings and bearing seats. In general, with an increased level of flexibility, the load zone also increases with a correlated decreasing in the rolling element load. With a lower level of contact load on the raceway the bearing performance evaluation will automatically increase. 
The system dynamics approach
The above methods are uniquely static or quasi-static: time variation is not taken into account and inertia is not playing a role in the equilibrium of the forces acting on the system. However, the rotational speed may be taken into account to evaluate the extra centrifugal forces acting on the rotating parts.
In reality, the operating conditions are reached after the start-up phase: in such phase the amplitude of the applied forces varies in time before reaching a stable plateau.
In order to be able to simulate this start-up phase, transient analyses are needed to solve the equation of motion: in the full version of the force equilibrium the inertia effects due to the system mass (m) becomes an important element to be considered.
These kind of analyses are also used where the applied loads evolve in time: e.g. impact, high speed, rotordynamics applications, etc. In these advanced simulations, all the components of a bearing (rolling elements, rings, cage, etc.) move within certain constraints modeled by means of contact elements. As forces vary in time, displacements also vary in time and the contact elements need to be able to deal with contact conditions varying in time.
It has to be pointed out that, in order to evaluate an accurate dynamic bearing behavior, a lot of information are needed, e.g. internal geometry of rolling elements, cage geometry, raceways profiles, temperature dependent material properties and models, etc. In order to deal with all this input and to properly solve the complex numerical schemes, a highly sophisticated software is needed able to run on powerful computers, such as High Performance Cluster (HPC).
Conclusions
Bearings are critical element for all rotating equipment. A proper bearing selection starts with the correct evaluation of the loading conditions at each support: depending on the amount of information available, several approaches may be used to extract the needed results. Depending on the situation, one approach can be preferred to the another: the pros and cons have been presented to guide the reader to a conscious choice before stepping into the bearing performance evaluation.
Bearing selection is like an “art” and application engineers are mastering it for a wide variety of different applications, type of customers and an incredibly vast range of operating conditions. This is an amazing and precious knowledge: software can deliver a great support to engineers, but it cannot (yet) replace the experience and the advice from the experts in the field.
Acknowledgments
This article would not have been possible without the significant learning opportunities offered by SKF. A special thanks goes also to Christian Bacher.
author: Andrea Bachetto
