Archive > 2015 > February 2015 > Desktop Engineering - How to Calculate Dynamic and Static Load Ratings

# Desktop Engineering – How to Calculate Dynamic and Static Load Ratings

Introduction When comparing bearing suppliers, engineers are often left with few options other than to compare dynamic load ratings and corresponding life calculations. Of course, we can look at steel and manufacturing quality; but if we are comparing sources of similar quality, those items may not provide a large contrast. It often surprises people to learn that bearing capacities are calculated values, not tested values. Lately, however, a trend is emerging for bearing suppliers to increase their ratings for higher performance bearings that have premium features such as higher quality steel and specilaized heat treatment. Bearing companies are under intense competitive pressure to make every feature add to the dynamic capacity of their bearings because it is very well understood that an increase in capacity adds to the bottom line. As a result, it is important that the end user develop a keen understanding of how capacity ratings and subsequent life calculations are generated in order to make a true comparison, or be left to comparing the claims of well-heeled marketing departments.

Dynamic Capacity

Nearly every calculation surrounding
bearing life begins with the dynamic capacity,
*C _{r}. C_{r}* is the equivalent load that
would result in an average service life of
one million revolutions. The formula is
imperfect and standard bearings aren’t
designed to handle 100% of

*C*; but, those are annoying nuances we have to understand and live with. In the past, most bearing companies would follow the dynamic capacity formula to ISO or ABMA standards and then increase the resulting life calculation by some factor based on heat treatment and other premium features in addition to the increase that would come through ISO 281 or 16281 using a-iso factors. The issues that end users have with this is that enhanced life calculation factors are not always shown on the bearing print and the print is the primary legal document that exists between the end user and the bearing manufacturer. Some end users interpret this as being an escape route for the bearing companies if something goes wrong – which it is not. Consequently, the competition and end users are pushing bearing companies to increase capacity ratings on the print. This is where the math starts getting fuzzy. We will walk through the formula with a couple of real bearings and determine where these numbers are coming from. The ISO/ABMA

_{r}*C*formula:

_{r}For this example, let’s look at two high-quality competitors, each producing their own design of the HM804846/10, a popular inch-series tapered roller bearing. We’ll refer to these as Company A and Company B.

Right off, we see *b _{m}* is defined by ISO
as 1.1. For

*i*, both bearings have 1 row;

*i*= 1. The bearing half-angle, a, will be provided by the manufacturer, so we can skip that measurement. Both of these bearings are around 20° (though a side-by-side comparison clearly shows they are not identical angles).

*Z*, the number of rollers, is easy enough to count — both have 18 rollers. The remaining values —

*f*,

_{c}*L*,

_{we}*D*and

_{we}*D*— are often not provided, but we can physically measure these features. Customer models will typically leave off just enough features to prevent an accurate measurement. We could get fancy and have these set up on a CMM and measure to 3 decimal places, but if you glance at the load ratings in the catalog you will see everything is rounded to the nearest 500 N. None of these factors will change your results greater than the rounding error if you are within 0.5 mm of accuracy. This sounds like a job for calipers.

_{pw}We will skip fc for now because that is
a tabulated value which we need two of
our other unknowns for. Let’s start with
the effective roller length *L _{we}*. ABMA defines

*L*as:

_{we}*The theoretical maximum length of
contact between a roller and that raceway
where the contact is shortest. NOTE:
This is normally taken to be either the
distance between the theoretically sharp
corners of the roller minus the roller
chamfers, or the raceway width excluding
the grinding undercuts — whichever
is the smaller.*

The roller chamfer can be be hard to identify with the naked eye, and will usually involve a little guesswork.

Usually, the *L _{we}* will be 1-1.5 mm
shorter than the entire length of the
roller. We can check ourselves before
we are done, so don’t worry too much
about your estimate for now. For a
21 mm roller, an

*L*of 19.5 mm is a good guess.

_{we}Now on to *D _{we}* — the mean roller diameter.
This is very straightforward;
measure the large diameter at the bottom
and the small diameter at the top
and average the values for

*D*.

_{we}The final measurement, *D _{pw}*, is also
fairly simple.

*D*, the pitch diameter of the roller set, is the theoretical centerline that the rollers run on. This is measured in similar fashion as were the rollers; measure the large and small diameters of the inner ring raceway; take the average to find the diameter in the center, and then add 1

_{pw}*D*to get the pitch diameter at the center of the rollers, at the center of the raceway.

_{we}With those values measured, we can
now find *f _{c}*, which is a tabulated value
based on the quotient.

For example, Company A

*D _{we}* = 10.2

*D*= 71.2

_{pw}a = 20

The quotient calculates to:

Let’s compare our values and results:

Plugging these values back into the
formula:

*C _{r}* = 1.1 · 87.4 (1 · 21.3 ·

*cos*20)

^{7/9}18

^{3/4}10.2

^{29/27}

*C*Company A: 104,675 N

_{r}*C*Company B: 106,144 N

_{r}If your calculated value is more than 1% different than the published value, adjust the

*L*until the calculated

_{we}*C*matches the book value.

_{r}Static Capacity

By definition, the static capacity *C _{or}*
is the calculated maximum-recommended
static load value which loosely
represents the yield point of the bearing
steel. Ideally, this value should
represent peak stress levels around
4,000 MPa — the ISO-recommended
stress limit. Just due to geometry, the
highest stress will occur on the inner
ring/roller interface. The ball-ball contact
between the inner ring and roller
has a smaller contact area than the
ball-socket contact pattern on the outer
ring.

*C*is a useful maximum load value if you don’t have bearing software to calculate actual stress values. The benefit with using stress values is that the effects of crowning can be taken into account, and if the bearing has premium heat treatment features that produce a harder surface, stress values up to 4,200 MPa or higher may be permissible. Comparing catalog values of Cor can be very useful because there are no places to add non-standard factors; the formula is completely based on geometry. If you need a quick comparison for the physical amount of steel contact between two different bearings, forget

_{or}*C*—

_{r}*C*is what you want to compare.

_{or}The other good news is, if you collected
your *C _{r}* values, you already
have everything you need to calculate
Cor .

*C _{or}* Company A: 139,926 N

*C*Company B: 142,337 N

_{or}If a bearing company wanted to increase the static rating on paper for a premium bearing, they could easily justify using a 4,200 MPa as a baseline for their rating, though it is not standard ISO/ABMA practice and not a fair comparison to another company that is strictly following ISO standards.

Let’s compare all of our calculated values next to the published catalog values for both companies.

The calculated values
for Company A
came within 1% of the published values.
However, something is quite different
with Company B; the published *C _{r}*
is 38% higher than our calculated value
and the published

*C*is 10% higher than our calculated value. Company A claims to have similar quality and performance as Company B, but we certainly cannot ignore the fact that Company B has a 41% higher

_{or}*C*and a 12% higher

_{r}*C*. This is a significant difference between two relatively similar bearings. What is going on here?

_{or}Company B claims that they have
lab-tested proof to show that their increased
*C _{r}* is legitimate and they do not
want to be held to an artificially low ISO
or ABMA formula, and therefore do not
adhere to the standards. On the other
hand, Company A claims that they are
able to add a performance factor to the
calculated

*L*life that will give them nearly the same calculated life as Company B. Let’s revisit the basic

_{10}*L*formula so that we can play along:

_{10}Where *L _{10}* is measured in millions of
revolutions and

*P*is the applied load. Mathematically, an increase of

*X*in

*C*does this:

_{r}While a performance factor does this:

Because Cr is raised to the exponent
of 10/3, a small increase nets large increases
in calculated *L _{10}*. Let’s see what
type of performance factor a 38% increase
in

*C*would yield:

_{r}This means that company A could
multiply their calculated *L _{10}* by 3.2
times and effectively match the results
of Company B. Company A states they
are comfortable going with a performance
factor of 2.6, but not 3.2 (Note:
Until recently, Company B had a

*C*of 141 kN that was exactly equivalent to a 2.6 performance factor. Two completely separate companies

_{r}*coincidentally*had performance factors of 2.6). What the end users want the bearing companies to do is take the 2 or 2.6 performance factor and increase

*C*by that amount on the print rather than just increasing the calculated

_{r}*L*. For example, a performance factor of 2 would mean:

_{10}Let Cr/P = 1, then X = 1.23. This means
that every 23% increase in dynamic
capacity doubles the calculated life.
End users want to see 1.23 × Cr, rather
than *2 × L _{10}*. The perceived benefit is
that the increased Cr is shown on the
print — which is a legal document. The
risk in doing this for the bearing companies
is that, right or wrong, some engineers
are accustomed to designing to
rules of thumb based on the published

*C*. If

_{r}*C*is artificially increased on the print, these practices may very easily result in a bearing that is under-designed for the application in terms of operating load and peak static stress.

_{r}The increased rating for *C _{or}* is easier
to explain. As mentioned earlier,
if you calculate the load required
to reach a higher-than-ISO-recom-mended peak stress value of 4,000 MPa,
you can easily justify the higher rating
on the print. Though again, this is not
standard practice.

From here it becomes difficult to make a rational decision, because there seems to be a lot of subjectivity going on with the calculations. We have tested vs. calculated dynamic load ratings, performance factors that have questionable origins, and less-than-obvious methods of increasing static load ratings. Recall the earlier statement that the static load rating calculations can be valuable for comparison. If we only compare our calculated static capacities (recall, true steel on steel contact area) we see a marginal difference of only 1%.

With that, we absolutely know that we have similar amounts of surface contact area. Armed with the knowledge that we have comparable geometry between the two bearings, the only real performance difference should be in the rolling fatigue performance of the steel. Again, we are assuming these are both top-shelf companies, so bearing design, manufacturing quality, surface finishes, etc., should be comparable. All of the fancy calculation methods beyond this point are useless for comparing these two bearings; only dyno or field performance tests over the entire loading range will conclusively separate the two. These formulas are easy to set up in a spreadsheet format that will facilitate future comparisons and provide real insight when dealing with your bearing suppliers.

Conclusion

There is an undeniable level of comfort when you see a huge capacity rating on a print that puts your safety factors well into “good night’s sleep” territory. It can be argued that both Companies A and B have valid points in the way they handle the premium features. One does not want to be held to capacity ratings that they can outperform by 50%, and the other does not want to deviate from the standards.

The main point of this article is to
show that load ratings are based on simple
formulas that you can calculate on
your own. You should ask a prospective
supplier if their capacity ratings and life
calculations are based on ISO 281:2007
and ISO 76:2006. If not, you need to
completely understand how and why
they are using their value. Likewise for
any performance factors added to the
calculated *L _{10}* life; double-check their
work and ask questions. Secondly, a
supplier is not off the hook just because
they don’t put their performance factor
on the print. If their calculations are
well-documented with all of the latest
information you gave them, their analysis
is a legal form of communication
(though be forewarned — contamination
levels, temperatures, alignment,
roundness of shaft and bores…all of the
factors that go into ISO 281 are subject
to review). Finally, capacity ratings are
pushed from an engineering and marketing
perspective. Companies are expected
to live up to their ratings, but
with the wide scatter of failure points in
any type of fatigue test, it can be difficult
to pinpoint a true 20% difference during
bench or field testing with a limited
number of parts. We need to account
for genuine high-performance features
on our bearings because we use those
factors in our designs. Just be sure that
you know how to compare the different
methods being used to account for
those features.

References

- Brandlein, Eschmann, Hasbargen and Weigand. Ball and Roller Bearings: Theory, Design and Application, Third Edition, Wiley & Sons 1999.
- International Organization for Standardization. Rolling Bearings: Static Load Ratings,
- ISO 76:2006(E), ISO 2006, Third Edition, 2006-05-01.
- International Organization for Standardization. Rolling Bearings: Dynamic Load Ratings and Rating Life, ISO 281:2007(E), ISO 2007, Second Edition 2007- 02-15.
- American Bearing Manufacturers Association. Load Ratings and Fatigue Life for Roller Bearings, ANSI/ABMA 11:2014 (Revision of ANSI/ABMA 11:1990).

The article "Desktop Engineering - How to Calculate Dynamic and Static Load Ratings" appeared in the February 2015 issue of Power Transmission Engineering.