# Rolling Bearings: Modeling and Analysis in Wolfram System Modeler

## Background

Explore the contents of this article with a **free Wolfram System Modeler trial**. Rolling bearings are one of the most common machine elements today. Almost all mechanisms with a rotational part, whether electrical toothbrushes, a computer hard drive or a washing machine, have one or more rolling bearings. In bicycles and especially in cars, there are a lot of rolling bearings, typically 100–150. Bearings are crucial—and their failure can be catastrophic—in development-pushing applications such as railroad wheelsets and, lately, large wind turbine generators. The Swedish bearing manufacturer SKF estimates that the global rolling bearing market volume in 2014 reached between 330 and 340 billion bearings.

Rolling bearings are named after their shapes—for instance, cylindrical roller bearings, tapered roller bearings and spherical roller bearings. Radial deep-groove ball bearings are the most common *rolling* bearing type, accounting for almost 30% of the world bearing demand. The most common *roller* bearing type (a subtype of a rolling bearing) is the tapered roller bearing, accounting for about 20% of the world bearing market.

With so many bearings installed every year, the calculations in the design process, manufacturing quality, operation environment, etc. have improved over time. Today, bearings often last as long as the product in which they are mounted. Not that long ago, you would have needed to change the bearings in a car’s gearbox or wheel bearing several times during that car’s lifetime. You might also have needed to change the bearings in a bicycle, kitchen fan or lawn mower.

For most applications, the basic traditional bearing design concept works fine. However, for more complex multidomain systems or more advanced loads, it may be necessary to use a more advanced design software. Wolfram System Modeler has been used in advanced multidomain bearing investigations for more than 14 years. The accuracy of the rolling bearing element forces and Hertzian contact stresses are the same as the software from the largest bearing manufacturers. However, System Modeler provides the possibilities to also model the dynamics of the nonlinear and multidomain surroundings, which give the understanding necessary for solving the problems of much more complex systems. The simulation time for models developed in System Modeler is also shorter than comparable approaches.

In this blog post, I will briefly describe traditional bearing design and indicate what can be done in System Modeler. Finally, I will show two bearing examples. The first discusses bearing preload, and the second bearing monitoring with vibration analysis.

## Traditional bearing life predictions

Bearing life, i.e. how long you can expect your bearing to last, is a statistical quantity. The basic life rating is associated with 90% reliability of bearings. The basic life rating is

where *L*_{10} is the number of revolutions in millions; *C* is the basic dynamic load rating, a value supplied by the manufacturer; and *P* is the equivalent dynamic bearing load, which needs to be calculated for the bearing. Typically, the load arises from gravity, gears, belt drives, etc. Finally, *p* is the life equation exponent. For example, *p* = 3 for ball bearings, and *p* = 10/3 for roller bearings.

For bearings running at constant speed, the basic rating life in operating hours can be expressed as

where *n* is the speed in rpm. In practice, predicted life may deviate significantly from actual service life. Therefore, to adjust for lubrication type and contamination, a factor *a*_{ISO} is used:

The bearing manufacturers have guidelines for specifications for different types of machines. As an example, *L*_{10h} for household machines can be 1,500 hours, while large electrical machines can be designed for 100,000 hours or more.

## Wolfram System Modeler model

For simplicity, we will study the easiest bearing to model, the cylindrical roller bearing. It can be built up with standard Modelica MultiBody parts. The input parameters in this case are the same as those used by the world’s leading bearing manufacturer, SKF:

And the result:

To give a better understanding of how the bearing operates, the outer ring is made transparent, and the arrows indicate the forces on the rollers. In this case, the radial clearance has been chosen to give a preload:

The deflection in the rollers, the roller loads and the corresponding Hertz contact stress have been calculated by using the Hertz contact stress theory for contact between two cylinders with parallel axes. In the videos and figures below, the roller loads are shown.

If the geometry of the rollers and rings is more complex than for the cylindrical bearing described above, the bearing manufacturer can usually supply the stress and deflection constants. However, handbooks cover many of the most common shapes.

## Bearing internal clearance

As an application of the model, I will show two well-known bearing issues. The first one is the effect of bearing internal clearance on the roller loads.

Bearing internal clearance is defined at unmounted, mounted or operational speeds. The bearing manufacturer establishes the initial unmounted bearing internal clearance. Operational bearing clearance is the clearance after the bearing is fitted onto the shaft and into the housing, and when the bearing reaches the steady-state operating temperature.

Normally bearings have a certain amount of internal play when they are operating. The closer to zero, the better, but due to the uncertainties and variations of tolerances and operations, the play may vary. In some cases, it is beneficial to use negative play, i.e. preloaded rollers. For instance, precision machines get better accuracy due to the resulting stiffer bearing arrangement. This is typically used in machine tools, pinion shafts or car gearboxes. Noise may be reduced without play. Slipping for rollers may be reduced, especially for heavy rollers in the unloaded zone. The drawback is, of course, higher stress cycles for the rollers and more energy loss, i.e. higher bearing temperatures.

The figure below shows the roller forces for a loaded bearing both without preolad (left) and with preload (right):

As illustrated above, the load distribution becomes more even when there is a preload. The peak load, i.e. the force on the roller at 6 o’clock, is also lower.

The total simulation model can now be seen in the figure below. Two bearings support the shaft. The shaft is divided into two flexible beams. The bearing rings are supported in two fixed supports. In the middle of the shaft, a load is applied. A torque is applied at the left end of the shaft.

The left bearing is unloaded from the start, and the right bearing is preloaded due to negative internal clearance. If we now start to increase the load in the middle of the shaft, we can follow the roller contact load during the rotation:

The above animation shows the simulation. In the figure it is seen that at low loads the preloaded bearing has a higher roller contact force (and consequently stress), but at high loads the peak will be lower. So as with many other machine elements, a preload may reduce the stress, but the cost can be high for a bearing with increasing nominal loads. Note that for illustration purposes the applied force in the middle of the shaft has been scaled to 1/4 of its original format:

## Bearing defect analysis

One of the most common reasons machines fail is bearing failure. An entire industry has been created to monitor the condition of bearings, with the aim of predicting failures and planning for replacement in a controlled way. In many applications, there may be hundreds or even thousands of bearings, so that you cannot change all at the same time. In others you may have a few large bearings and a stop once a year—or even once every fifth year—to check and see if it is time to change a bearing. In many situations, a bearing can continue to operate many months after an initial defect is detected. The frequency of a bearing’s vibration can reveal the type of defect:

**Inner race defect frequency (BPFI):** The frequency that corresponds to the rolling elements passing a defect on the inner race; often referred to as the *ball pass frequency inner race*.

**Outer race defect frequency (BPFO):** The frequency that corresponds to the rolling elements passing a defect on the outer race; often referred to as the *ball pass frequency outer race*.

**Cage defect frequency (FTF):** The frequency that corresponds to the rotational speed of the cage. This frequency is often referred to as the *fundamental train frequency*.

**Ball spin frequency (BSF):** The frequency that corresponds to the rotational speed of the rolling elements (balls or rollers); referred to as the *ball speed frequency*.

These frequencies and multiples of these frequencies show up as spikes on a vibration analysis spectrum when bearings begin to fail.

In my experience, the most common type of defect due to peak loads are inner ring defects, but if there are contaminants in the lubrication the most common are outer ring defects. Cage and roller defects are less often a problem, at least on larger bearings. So for simplicity, let’s introduce an outer ring defect at 12 o’clock in the left bearing. Every time a roller passes this defect, an impact force will act on the bearing (see the red arrow in the figure below):

The bearing defect frequencies are tabulated by the manufacturers and can usually be found on their websites.

For the bearing in this example, FTF = 0.393 x rpm, BSF = 2.234 x rpm, BPFI = 7.281 x rpm, and BPFO = 4.719 x rpm. This means that with a rotational speed of 1,500 rpm, an outer ring defect should give a frequency response at:

The FFT (fast Fourier transform) done in the Simulation Center in System Modeler gives 11,799 Hz and some multiple of that frequency when analyzing the motion of the shaft. (To keep this example basic, I used a shaft instead of housing, but actual analyses would also evaluate bearing housing acceleration.) The peaks are very clearly seen:

The video shows the example. Note that it is run in slow motion (in this case, one-tenth of the actual speed) so we can see the vibration:

## Summary

Wolfram System Modeler is a powerful tool for studying advanced problems in the domain of rotating machinery. Combined with Mathematica, it gives tremendous opportunities to work with and analyze your models and results. I’ve shown some basic and rather simple examples of how rolling bearings can be analyzed with System Modeler. In fact, very few other software products manage the same complexity as System Modeler in dynamic bearing and surrounding analysis.

## Further Explorations

Many more questions can be answered with just small modifications to the examples in this blog—for instance:

- How does the play affect the response if the external dynamic force is less than what is needed to avoid the shaft “jumping around” in the bearing?
- What would a bearing defect on the inner ring look like?
- How will defect size affect the frequency spectrum?
- What does the vibration frequency spectrum look like if there are two defects on the inner or outer ring?
- Signal noise is an important issue during condition monitoring. Is an envelope technique better than traditional analysis with a lot of noise?
- What are the Hertzian contact stresses in the inner and outer ring during normal operation (40,000 hours for this bearing)?
- What load is needed to achieve a roller contact stress of 4,000 MPa—that is, the contact stress when the permanent deformation is 0.00001 times the roller diameter? This stress level is regarded as the max one-time peak load.

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