Recreating back-to-back bearing life calculation 1

[electricpete]

Attached is a calculation for a "back-to-back" arrangement of a SKF 7318 bearing (40 degree angle contact) and a SKF QJ318N2 (4-point contact / Gothic arch bearing).

I am interested in recreating the thought process of this particular author in this particular calculation. Everything seems straightforward, except one question written on 1st page and one on 2nd page.

1st question (1st page): Is there a way to estimate what fraction of axial load each bearing would carry based on the available info? (in absence of input from bearing OEM).

2nd question (2nd page): After calculating a life for each bearing (based on it's share of the load), this author combines the two lives using
Life = 1 / (1/Life1 + 1/Life2)
I'm trying to figure out the logic of this equation. I can imagine that the statistical life of the combination is slightly less than the shorter of the two lives (which is what this equation gives), but I'm not sure how to arrive at this exact equation. Has anyone seen it derived or stated in a reference?

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(2B)+(2B)' ?

 

[electricpete]

Fugeguy - I have been playing around with smath and it seems pretty handy. I have a question: is it possible to resize graphics that have been inserted into the worksheet?

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[Fugeguy]

electricpete,

I have only used SMATH a few times and there does not seem to a good way to re-size graphics.

I usually pre-size and take a screen shot with something that lets me choose the window size.

Also, it is not correct to split the load since the x and y values in the equations take that into account. That is why the y value for btb is .93 and it is .57 for tandem. Because in one arrangement (tandem) the bearings share the load and the other (BTB) they don't. The reason the tandem exponents are not half the btb exponents is governed by the equation for combined load rated i^.7* load rating where i = number of bearings sharing the load.

Here 2^.7 * .93 = .57.

- fuge

 

[electricpete]

Thanks. My last comments were off-base since I hadn't noticed that the Y values you selected were any different than the typical ones, these ones selected specifically for tandem application. Good observation that the 2^0.7 is built into this tandem Y value (2^.7 * .93 = .57), that helps tie it together.

I am hoping to tie it together a little more. I would propose that the 2^0.7 representation of the rating of the combined tandem unit is an attempt to express the same thing as the life combination of individual bearings using L = 1/(1/L1+1/L2) under the assumption that identical tandem bearings will share the load equally... each sees half of the total load.

Attached is an attempt to explore this. The two approaches differ by approx 7%... pretty close imo.

What do you think?

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[electricpete]

Attached is the smath file if anyone wants it.

Note, if you try to use the life combination formula with exponent 1.1 instead of 1.0, results were more complicated.... that's why I stuck with 1.0.

But we can qualitatively see that if we used 1.1, it is as if the inidividual lives increased by exponent 1.1, which pushes the combined life of part 1 higher, and the ratio of L10_Part1/L10_Part2 higher and closer to 1.0.

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(2B)+(2B)' ?

 

[electricpete]

under the assumption that identical tandem bearings will share the load equally

This is an important assumption to talk about. Even for identical bearings, they must be a matched set as stated above to share load equally.

The bearings of original post were not equal. However if the bearings are supplied as a matched set of unequal bearings, I think as a first guess (in absence of OEM info), one would guess that the OEM would match them so that they shared load in proportion to their rating.... or if load sharing is non-linear (load-sharing fraction varies with load level) that they reached their rating load level at the same time. This would be logical way for an OEM to optimize the combination, wouldn't it?

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(2B)+(2B)' ?

 

[electricpete]

equation for combined load rated i^.7* load rating where i = number of bearings sharing the load.

Does anyone have a reference for this equation? It is vaguely familiar to me, but I can't find it in any refernces.



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[electricpete]

I found that reference:

5.1.1.3 The basic radial load rating for two or more similar single row angular contact ball bearings mounted side-by-side on
the same shaft such that they operate as a unit (paired or stack mounting) in “tandem” arrangement, properly manufactured and
mounted for equal load distribution, is the number of bearings to the power of 0.7, times the rating of one single row bearing.

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[electricpete]

Interesting that the references which include the statistical life combination equation L = 1/(1/L1+1/L2) (such as Koyo and NTN) do not include the revised rating equation C=C1*n^0.7 for n identical bearings, and the reference that includes the rating equation C=C1*n^0.7 (AFBMA) does not include the statistical life combination equaion L = 1/(1/L1+1/L2). So each of these references lists one or the other but not both. I think the reason is that they represent the same thing, so you can only use one or the other in your calc. If unequal bearings, you are forced to use the life combination equation.

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[electricpete]

In my smath file, I left out the factor Y. In my analysis it is assumed constant, so does not affect the conclusion regarding the ratio.

As an interesting side note, the factor Y is used to capture the statistical life combination in the catalogue that Fugeguy used (which bearing oem was that?).

So in summary, there are actually three different ways that exact the same effect (statistical combination of identical bearings sharing load equally) can be captured:
1 – By combining L10 lives of individual beairngs using L=1/(1/L1+1/L2) as suggested in Koyo and NTN catalogues
2 – By using a revised rating of the combined unit Ccomb = Cindividual * n^0.7 as suggested in AFBMA standard above.
3 – By adjustment of the Y factor as in the bearing catalogue referenced by Fugeguy

With three different possible ways to capture the same effect, it is important that we use one and only one. If you mixed and matched coefficients or equations from different sources you could get in trouble. The value of Y needs to be coordinated with the value of C and also with the treatment as single unit or multiple units whose life is combined at the end.

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(2B)+(2B)' ?

 

[dinjin]

If the bearings are a matched set and you knew the approximate life calcs, would it not make sense to interchange the two bearings at half life to get longer life from them?

 

[electricpete]

The angle contact 7318 has to always be on bottom and so there nothing you can swap to extend it's life.

The split inner race 4-point contact QJ318 is bi-directional, so assuming (*) it is ground exactly the same in both direction, you could flip it around to swap the intermittent up-thrust surfaces into the continuous downthrust duty, which might almost double the L10 life of the bearing. (* Not a good assumption. The MRC pairs come with arrows on both the angle-contact and the split inner inner inner ring bearing which are supposed to be installed in directino of continuous thrust, so they have a preferred direction and swapping would be outside OEM's recommendation).

In this particular case, according to the original calc attached at beginning, the 7318 has lower life and is limiting, so flipping the QJ318 wouldn't buy you much even if it was ground identically.

More importantly, it is our general practice to replace a bearing every time it is removed. This is based on
1 - concern about the same bearing being originally installed then removed/reinstalled onto inner race, which brings some trauma to the bearing.
2 - a lot of the cost is manpower anyway. So if you're spending the manpower to disassemble and swap, you might as well use a new bearing.

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[electricpete]

Correction in bold:

The angle contact 7318 has to always be on bottom and so there nothing you can swap to extend it's life.

should have been:

The angle contact 7318 has to always be on top and so there nothing you can swap to extend it's life.



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(2B)+(2B)' ?

 

[electricpete]

2nd question (2nd page): After calculating a life for each bearing (based on it's share of the load), this author combines the two lives using
Life = 1 / (1/Life1 + 1/Life2)
I'm trying to figure out the logic of this equation. I can imagine that the statistical life of the combination is slightly less than the shorter of the two lives (which is what this equation gives), but I'm not sure how to arrive at this exact equation. Has anyone seen it derived or stated in a reference?

I did figure out this appears to be a very basic textbook statistics principle, sometimes referred to as "series" combination of components.

It is discussed a little here.

http://www.theriac.org/DeskReference/viewDocument.php?id=219#section3

I have also attached some basic proofs. If the probability of surviving to time t for bearing 1 is exp(-lambda1*t) and for bearing 2 is exp(-lambda2*t), then I proved the MTTF for bearing 1 alone is MTTF1 = 1/lambda1 and for bearing 2 is MTTF2 = 1/lambda2 and for the series combination has MTTFtot = 1/(lambda1+lambda2) = 1/(1/MTTF1 + 1/MTTF2). If we substitute L10 life for MTTF, we see this is identical to the equation posted in original post. I'll have to think a little bit about whether there is basis for this substitution, but it looks pretty close to me.


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